# New look at Milnor Spheres

Leonardo F. Cavenaghi  College of Arts and Sciences, Department of
Mathematics, University of Miami, Ungar Bldg, 1365 Memorial Dr 515, Coral
Gables, FL 33146. & Instituto de Matemática, Estatística e Computação
Científica da Universidade Estadual de Campinas (Unicamp), Cidade
Universitária, Campinas - SP, 13083-856<EMAIL_ADDRESS>,  Lino
Grama  Instituto de Matemática, Estatística e Computação Científica (IMECC) da
Universidade Estadual de Campinas (Unicamp), Cidade Universitária, Campinas -
SP, 13083-856<EMAIL_ADDRESS>and  Ludmil Katzarkov  College of Arts and
Sciences, Department of Mathematics, University of Miami, & Institute of the
Mathematical Sciences of the Americas (IMSA). Ungar Bldg, 1365 Memorial Dr
515, Coral Gables, FL 33146. & Higher School of Economics (HSE University),
Moscow-Russia. & International Center for Mathematical Sciences (ICMS), Sofia-
Bulgaria<EMAIL_ADDRESS>
###### Abstract.

In [32], S. Donaldson disproved the smooth $\mathrm{h}$-cobordism conjecture
in dimension $4$ by studying invariants coming from the moduli space of
connections on $\mathrm{SO}(3)$-bundles over smooth $4$-dimension manifolds
$X$. In this paper, we reverse his point of view in dimension $8$. Namely, we
consider fibrations coming from diffeomorphisms classes of homotopy Hopf
manifolds $\Sigma^{7}\times\mathrm{S}^{1}$, i.e., $\Sigma^{7}$ is a homotopy
sphere, seeing the bases as moduli spaces for additional structures. For
instance, recall that in [21], it was shown that these manifolds admit complex
structures.

Our overarching objective is to establish a correspondence between the smooth
invariants of Kervaire–Milnor and categorical invariants. One interprets
pairwise non-diffeomorphic smooth structures on
$\Sigma^{7}\times\mathrm{S}^{1}$ in terms of non-rationality. This paper’s
contributions are diverse, each carrying its significance. We believe profound
connections exist between the themes addressed, which connect hypersurface
singularity theory, Homological Mirror Symmetry, Topological Modular forms
($\mathrm{tmf}$), and Spherical T-duality.

###### Key words and phrases:

###### Contents

  1. 1 Introduction
  2. 2 $\star$-diagrams: A concise account and new results
  3. 3 Spherical T-duality and $\star$-diagrams
  4. 4 Higher-Dimension Logarithmic Transformation
  5. 5 Fano stacks and their cohomologies
  6. 6 $\mathrm{tmf}$, K3-surfaces, and stably framed bordisms
  7. 7 Gluing induced complex structures on some homotopy Hopf manifolds
  8. A Straightening angles: the smooth and the holomorphic cases
  9. B Chen-Ruan orbifold cohomology

##  1\. Introduction

This paper focuses on 8-dimension _homotopy Hopf manifolds_ within the rich
theoretical framework of category theory, hypersurface singularity theory, and
Homological Mirror Symmetry (HMS).

By a $8$-dimension homotopy Hopf manifold, we mean a non-Kähler complex
manifold $\Sigma^{7}\times\mathrm{S}^{1}$ where $\Sigma^{7}$ is a homotopy
sphere in dimension $7$ (the existence of complex structures on these
manifolds was first obtained in [21]). The present is the inaugural paper of a
long research program we briefly outline below. A detailed program and
additional results shall appear elsewhere, [25, 24].

In [32], S. Donaldson introduces new invariants for certain $4$-dimension
smooth manifolds. His approach starts with rationality to disprove the smooth
_$\mathrm{h}$ -cobordism_ conjecture. In this paper, we reverse his point of
view. We interpret the existence of pairwise non-diffeomorphic complex
manifolds $\Sigma^{7}\times\mathrm{S}^{1}$ in terms of non-rationality.

Recall that, in the point-set topology realm, the _$\mathrm{h}$ -cobordism_
theorem states that any $\mathrm{h}$-cobordism $W^{5}$ between
$M^{4},\leavevmode\nobreak\ N^{4}$ is trivial if $M,\leavevmode\nobreak\ N$
are simply connected, meaning that

$W\stackrel{{\scriptstyle\text{homeomorphic}}}{{\cong}}M\times[0,1]\,\,\mbox{
and }\,\,M\stackrel{{\scriptstyle\text{homeomorphic}}}{{\cong}}N.$

Donaldson invariants were first used to show that this is no longer true if
one changes the adjective “homeomorphic” to “diffeomorphic”, i.e.,
$M,\leavevmode\nobreak\ N$ need not be diffeomorphic. Such achievement led to
establishing irreducible $4$-manifolds that are not complex, [47].

Let $X$ be a smooth $4$-dimension manifold with a cyclic fundamental group and
$b_{+}^{2}(X)\geq 3,\leavevmode\nobreak\ b_{-}^{2}$ odd. Let
$\mathcal{P}\rightarrow X$ be a principal $\mathrm{SO}(3)$-bundle with
$\langle p_{1}(\mathcal{P}),[X]\rangle=l$ and
$w_{2}(\mathcal{P})=\eta\in\mathrm{H}^{2}(X,\mathbb{Z}_{2})$. The quantities
$l,\leavevmode\nobreak\ \eta$ fully determine the bundle type. Let
$\mathcal{A}_{\mathcal{P}}$ stand to the space of connections on
$\mathcal{P}$. Let $\mathcal{G}_{\mathcal{P}}$ be the group of automorphisms
$s$ of $\mathcal{P}$ which, when seen as equivariant maps
$s:\mathcal{P}\rightarrow\mathrm{SO}(3)$, lift to $\mathrm{SU}(2)$. Let
$B(l,\eta):=\mathcal{A}_{\mathcal{P}}/\mathcal{G}_{\mathcal{P}}$. For
$l=-\frac{3}{2}(1+b_{+}^{2}(X))$ the dimension of $B(l,\eta)$ is zero. With
this, the moduli space $B(l,\eta)$ is discrete, and the Donaldson invariants
can be read from this more easily, Theorem 0.1 in [47]. Taking advantage of
that and constructions based upon the Kummer surface, authors disprove the
_Decomposition Conjecture_.

In higher dimensions, i.e., for $\mathrm{h}$-cobordisms between $n$-dimension
manifolds with $n\geq 5$, one can establish the $\mathrm{h}$-cobordism theorem
in the category of smooth manifolds. The set of equivalence classes of the
$\mathrm{h}$-cobordism relation in dimension $n$ is a commutative monoid
$\Theta^{n}$ under the operation of the connected sum. The identity element is
represented by the class of manifolds bounding a contractible manifold. The
group $\theta^{n}$ of invertible elements consists of homology spheres, [69,
Theorem 5.5, p. 158]. If $n=7$, then $\theta^{7}$ has as its elements
diffeomorphisms classes of smooth structures on homology $7$-dimension
spheres.

We consider fibrations from $\Sigma^{7}\times\mathrm{S}^{1}$, for
$\Sigma^{7}\in\theta^{7}$, and understand the possible quotient spaces as
moduli spaces. When looking for fibrations from $\mathrm{T}^{2}$-actions, the
orbit spaces are Fano orbifolds and stacks from Fano orbifolds. Given this
algebraic nature, trying to obtain its algebraic-geometric properties is
natural. For instance, determine if different choices of smooth structures in
homology spheres lead to non-isomorphic Fano orbifolds.

We also consider fibrations from $\mathrm{S}^{3}\times\mathrm{S}^{1}$-group
actions. To that, we perceive $\Sigma^{7}$ as the total space of
$\mathrm{SO}(4)$-bundles with typical fiber $\mathrm{S}^{3}$, known in the
literature as _Milnor bundles_. For these fibrations, the quotient spaces are
$\mathrm{S}^{4}$, carrying no complex structures. Here, these fibrations are
explored in the light of Spherical T-duality ([8, 9]).

A remarkable tool used both in [32] and [47] is the concept of _logarithmic
transformations_ , [48]. Although it is usually employed in the presence of
elliptic surfaces, we extend it to the set of diffeomorphisms classes of Hopf
homotopy manifolds $\sigma^{8}$, proving it defines a group structure on it.
This paper’s techniques are natural generalizations of already studied
techniques in the interplay of differential topology, complex geometry, and
algebraic geometry. A huge difference, however, relies on the exploration of
$\star$-diagrams, as in [26], to test the proposed connections between the
treated subjects.

For each $k\in\\{1,\ldots,28\\}$, let $\Sigma^{7}_{k}$ be a homology sphere
admitting a realization as the link of a certain hypersurface singularity,
[17]. They fiber over Fano orbifolds $V_{k}$ or over stacks of Fano orbifolds
$[V_{k}/G]$, where $G$ is a finite group, depending on logarithmic
transformations (Theorem H and Corollary I). In [68], Kontsevich, Pestun, and
Tschinkel introduce an equivariant birational invariant. They consider finite
group $G$ actions on certain projective varieties $X$ (of dimension $d$) and
associate to them classes $\beta(X)\in\mathcal{B}_{d}(G)$. We are tempted to
conjecture that different logarithmic transformations may produce examples of
$G$-actions on $V_{k}$ with different $\beta(V_{k})$. Thus providing non
$G$-birational $V_{k}$. Given the orbifold nature of $V_{k}$, understanding
the invariants given in [73] is interesting. Further details on the relation
of non $G$-birational Fano quotients shall appear elsewhere, [25, 24]. Here,
we only perceive this through Chen-Ruan cohomology of $V_{k}/G$, see Theorem
K.

In summary, this paper inaugurates the following program:

  * •

We aim to relate the birational invariants appearing in [71, 73, 68] with
smooth ones. Here, we explore $\mathrm{T}^{2}$-fibrations from homotopy Hopf
manifolds $\Sigma^{7}\times\mathrm{S}^{1}$ to some Fano orbifolds $V$ embedded
in weighted projective spaces. These Fano orbifolds have the same homotopy
groups as $\mathbb{P}^{3}$ and the same cohomology, as we can check from the
long homotopy sequence for fibrations and spectral sequences, respectively. It
raises whether we can relate smooth invariants to some equivariant birational
invariants.

  * •

Let $[[V_{k}]]$ denotes a class of $G$-birational Fano orbifolds to $V_{k}$.
Understanding the possible diffeomorphism type of homotopy Hopf manifolds that
fibers over $[[V_{k}]]$ is interesting. We believe that to determine this, we
may rely upon $\mathrm{tmf}$, [62, 79]. When considering
$\mathrm{T}^{2}$-bundles from $\Sigma^{7}\times\mathrm{S}^{1}$, we use
hypersurface singularity to describe the diffeomorphism types of $\Sigma^{7}$
(and hence of $\Sigma^{7}\times\mathrm{S}^{1}$). As one knows from [85],
$\mathrm{S}^{3}$ can also be described as the link for a certain (family of)
hypersurface singularities. Computing the orbifold elliptic genus (as in [74])
for the orbifolds coming from the quotients of $\mathrm{T}^{2}$-actions from
$\mathrm{S}^{3}\times\mathrm{S}^{1}$ one observes, however, that the
hypersurface singularity information used to define
$\mathrm{S}^{3}\times\mathrm{S}^{1}$ is missing. This does not hold for the
corresponding $\Sigma^{7}\times\mathrm{S}^{1}$-example.111We did not include
this computation here, it shall appear in [25].

  * •

Recall that the elliptic genus is a ring homomorphism
$E:\Omega^{*}_{\mathrm{SU}}\rightarrow R$ where $R$ is some commutative ring,
[62, 74, 79]. Based on the work of M. Hopkins ([61, 62]), one observes that by
picking $R=\mathrm{tmf}$ we may connect the invariants on the quotients for
fibrations from $\Sigma^{7}\times\mathrm{S}^{1}$ to (invariants of)
diffeomorphisms classes of the former.

  * •

The analysis of the described problems passes to understand the relations
between $G$-birational invariants in [68, 73] with the Chen-Ruan cohomology,
[23]. Results regarding this shall be presented in [25].

  * •

Rationality can be indirectly tested by studying mirror symmetry, [65]. We
believe that the relationship between rationality and smooth structures also
can. On the one hand, one learns from [100] how to study Homological Mirror
Symmetry on the homotopy Hopf manifold $\mathrm{S}^{3}\times\mathrm{S}^{1}$
and its logarithmic transformations. On the other hand, we point out that HMS
for $\Sigma^{7}\times\mathrm{S}^{1}$ can reveal more interesting phenomena, as
logarithmic transformation (in our context) can lead to different smooth
structures of $\Sigma^{7}\times\mathrm{S}^{1}$, even keeping the point-set-
topology. This does not hold for $\mathrm{S}^{3}\times\mathrm{S}^{1}$, as
proved in [101]. We aim to approach HMF for $\Sigma^{7}\times\mathrm{S}^{1}$
elsewhere; however, we present preliminary results and conjectures here.

Below, we discuss our results in more detail.

Homotopy Hopf manifolds represent a fascinating intersection of classical
topological invariants and modern categorical approaches in topology. Let
$\theta^{7}$ be the set of diffeomorphisms classes of seven-dimension smooth
manifolds $M^{7}$ that are homeomorphic to $\mathrm{S}^{7}$. It has a group
structure from connecting sums and is isomorphic to $\mathbb{Z}_{28}$, as
explained in [66]. We define an 8-dimension homotopy Hopf manifold as
$\Sigma^{7}\times\mathrm{S}^{1}$, where $\Sigma^{7}\in\theta^{7}$. Our long-
term broader objective revolves around establishing a correspondence between
the smooth invariants of Kervaire–Milnor and categorical invariants. Here, we
begin paving the road to accomplish such a goal.

Our approach is based on Homological Mirror Symmetry (HMS), [67]. We pursue
elaboration on the ideas of A. Ward in [100]. HMS for
$\mathrm{S}^{1}\times\mathrm{S}^{3}$ can be decoupled from HMS for
$E_{\tau}\times\mathbb{P}^{1}$ where $E_{\tau}$ is the elliptic curve
$\mathbb{C}\setminus 0/z\sim\lambda z,\leavevmode\nobreak\
\lambda\in\mathbb{C}^{*}$ where $\tau$ is the parameter in
$\lambda=e^{2\pi\mathbf{i}\tau}$ combined with logarithmic transformations.
Here, we take the first step on extending this idea for any Hopf fibration
$\mathrm{S}^{1}\times\mathrm{S}^{2n+1}$ and more generally
$\mathrm{S}^{1}\times\mathrm{S}^{2n+1}$ for any homotopy sphere
$\Sigma^{2n+1}$.

Our contributions vary significantly in nature, each possessing its
significance. The authors believe that there are deeper connections between
the themes addressed. In the next diagram, the arrows mean that the source
subject leads to information on the target subject.

Hypersurface sing.H. Mirror Symm.tmf Sph. T-duality Fano orb.Homotopy
HopfHomology spheresLog. transformMilnor spheres

What the following diagram encodes can be further elaborated accordingly

  * •

In Section 3, we explore the concept of Spherical T-duality, as introduced in
[8, 9], with a focus on oriented $\mathrm{S}^{3}$-bundles over
$\mathrm{S}^{4}$. These bundles are characterized by two integers, usually
labeled as $m$ and $n$, which parameterize their total spaces, denoted as
$M_{m,n}$. The numbers $m+n$ and $2(m-n)$ are interpreted via characteristic
classes.

In Section 2, we recall the concept of $\star$-diagrams, as systematically
described in [26]. They provide geometric realizations to Spherical T-duality
over $\mathrm{S}^{4}$. Applications include interchanging complex structures
and deriving analogous Buscher rules [20] for Spherical T-duality.

  * •

Some particularly noteworthy examples of manifolds within the class of total
spaces represented as $M_{m,n}$ are those where $m+n$ equals $\pm 1$. These
examples encompass 16 of the 28 possible diffeomorphism types of $7$-spheres.
We particularly examine the products $M_{m,1-m}\times\mathrm{S}^{1}$ as
primary instances of _$8$ -dimension homotopy Hopf manifolds_. One of our
objectives in this paper is to highlight the significance of
$M_{m,1-m}\times\mathrm{S}^{1}$ as playing a role analogous to minimal
elliptic fibrations with no multiple fibers in dimension $4$. In pursuit of
this goal, we extend the concept of logarithmic transformation (Section 4),
commonly used in elliptic fibrations, to fibrations such as

$\mathrm{S}^{3}\times\mathrm{S}^{1}\rightarrow
M_{m,1-m}\times\mathrm{S}^{1}\rightarrow\mathrm{S}^{4}.$

  * •

Not every $7$-dimension sphere admits a fibration with fibers $\mathrm{S}^{3}$
over $\mathrm{S}^{4}$. Moreover, in cases where such fibrations exist, further
analysis reveals limitations in utilizing Spherical T-duality to distinguish
complex structures of other _admissible geometries_ , as they realize Morita
equivalences between $M_{m,1-m}\times\mathrm{S}^{1}$ and
$M_{m^{\prime},1-m^{\prime}}\times\mathrm{S}^{1}$ for every $m$ and
$m^{\prime}$ (Theorem 2.15). In light of this, we focus on
$\mathrm{T}^{2}$-actions on $\Sigma^{7}\times\mathrm{S}^{1}$. Following [16],
this yields fibrations of the form:

$\mathrm{T}^{2}\rightarrow\Sigma^{7}\times\mathrm{S}^{1}\rightarrow V$

where $V$ represents certain _Fano_ orbifolds on weighted projective spaces
(Section 5).

  * •

Interestingly enough, the Fano orbifolds $V$ appearing above are all _Fake
Complex Projective spaces_ ([43, 82]), that is, they are homotopy and have the
same cohomology of $\mathbb{P}^{3}$. We wonder whether $V$ shall be
homeomorphic to $\mathbb{P}^{3}$. We prove it is when $\Sigma^{7}$ is the
standard sphere (Theorem N).

  * •

Fano orbifolds carry plenty of K3-surfaces. We wonder whether one can _lift_
such K3-surfaces in $V$ to $\Sigma^{7}\times\mathrm{S}^{1}$. We show this is
possible when $\Sigma^{7}$ is diffeomorphic to $\mathrm{S}^{7}$ and ask
whether such lifting is possible if, and only if, $V$ is homeomorphic to a
projective space (Theorem N).

  * •

Let $h_{\mathbb{H}}:\mathrm{S}^{7}\rightarrow\mathrm{S}^{4}$ be the standard
Hopf fibration ($M_{1,0}=\mathrm{S}^{7}$), and recall that
$[h_{\mathbb{H}}]\in\pi_{3}^{S}$, i.e., the Hopf fibration defines a generator
class to the third stable homotopy group of spheres. The Thom–Pontryagin
isomorphism ensures that the relation $24[h_{\mathbb{H}}]=0$ can be perceived
by the K3-surface (Proposition 6.7). Hopkins showed in [61] that we can
perceive the class $[h_{\mathbb{H}}]$ in some expansion of the Weierstrass
$\wp$-function via its identification with the number $\frac{1}{12}$, which is
the only non-integer coefficient for that expansion. We extend Hopkins’ result
by observing that, using our proposed logarithmic transformation, one can go
from any two of $M_{m,1-m},\leavevmode\nobreak\
M_{m^{\prime},1-m^{\prime}},\leavevmode\nobreak\ m\neq m^{\prime}$ (since they
fit manifolds appearing in certain Spherical T-duality) and thus associate any
of $M_{m,1-m}$ with the same $\frac{1}{12}$ (Section 6.1).

  * •

We utilize the bundle description of $M_{m,1-m}$ to understand _special_
complex structures on $M_{m,1-m}\times\mathrm{S}^{1}$ and indicate that they
can be determined from $\mathrm{S}^{3}\times\mathrm{S}^{1}$ (Section 7).

  * •

We begin (Section 5.2) to discuss some homological mirror symmetric to
homotopy Hopf manifolds via their description as a link of Brieskorn-Pham
singularities ([16]) using the results presented in [42]. We aim to develop
this in light of [100].

Below, we summarize the paper’s content and suggest questions and conjectures.
At the end of this section, we present the paper’s overall organization.

###  1.1. Spherical T-duality

The current study originated from the idea of _Spherical T-duality_ , explored
in [8, 9]. We noticed its potential when combined with $\star$-diagrams, as
discussed in [92, 26].

T-duality is essentially about swapping geometric information for _2-dimension
conformal field theories_ , explained in [6]. T. Buscher is credited with its
discovery [20]. In mathematical terms, T-duality arises in the context of
torus bundles, with significant developments in papers like [10, 11, 12, 13].
Later, in [8, 9], T-duality was expanded to include Spherical T-duality.

######  Definition A ([8]).

Given a pair $(\pi:\mathcal{P}\rightarrow M,H)$ consisting of a
$\mathrm{S}^{3}$-principal bundle $\pi$ and an element
$H\in\mathrm{H}^{7}(\mathcal{P})$, its _spherical T-dual_
$\widehat{\pi}:\widehat{\mathcal{P}}\rightarrow M$ is any pair
$(\hat{\pi}:\widehat{\mathcal{P}}\rightarrow M,\widehat{H})$ satisfying the
following conditions:

  * •

$\widehat{\mathcal{P}}$ is a principal $\mathrm{S}^{3}$-bundle.

  * •

$\mathrm{c}_{2}(\widehat{\mathcal{P}})=\pi_{*}H$.

  * •

$\widehat{H}$ is defined by
$\widehat{\pi}_{*}\widehat{H}=\mathrm{c}_{2}(\mathcal{P})$.

  * •

$\widehat{p}^{*}H=p^{*}\widehat{H}$ in
$\mathcal{P}\times_{M}\widehat{\mathcal{P}}$, fitting the diagram:

(1.1)
$\textstyle{\mathcal{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathcal{P}\times_{M}\widehat{\mathcal{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p=\pi\otimes
1}$$\scriptstyle{\widehat{p}=1\otimes\widehat{\pi}}$$\textstyle{M}$$\textstyle{\widehat{\mathcal{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\pi}}$

To the elements $H\in\mathrm{H}^{7}(\mathcal{P}),\leavevmode\nobreak\
\widehat{H}\in\mathrm{H}^{7}(\widehat{\mathcal{P}})$ we term _H-fluxes_.

The former definition was expanded in [9] to encompass _oriented_
$\mathrm{S}^{3}$-bundles associated to principal $\mathrm{SO}(4)$-bundles.
Subsequently, in [76], a topological approach to Spherical T-Duality was
introduced, and it was generalized to include non-oriented sphere bundles.

######  Remark 1.1.

In this paper, when referring to spherical T-dual pairs, we always mean the
underlined manifolds are the total space of oriented $\mathrm{S}^{3}$-bundles,
not necessarily principal bundles. When we gain information for stressing
differences, we explicitly say if we are assuming the principal bundle case. ∎

Recall that within $\theta^{7}$, 16 out of 28 elements are realized as
oriented $\mathrm{S}^{3}$-bundles over $\mathrm{S}^{4}$ associated with
$\mathrm{SO}(4)$-principal bundles $\mathcal{P}$ [39]. These can be
represented as $M_{m,1-m}$ for $m\in\mathbb{Z}$, where $2(1-2m)$ denotes the
second Chern class of $E:=\mathcal{P}\times_{\mathrm{S}^{3}}\mathbb{H}$ (here
and throughout $\mathbb{H}$ stands for the quaternions), and the Euler class
is $1$. In full generality, we name an oriented $\mathrm{S}^{3}$-bundle over
$\mathrm{S}^{4}$ a _Milnor bundle_ if its structure group is $\mathrm{SO}(4)$
acting linearly on $\mathrm{S}^{3}$. These bundles are fully characterized by
two parameters: its Euler class $m+n:=k$ and first Pontryagin class $m-n$
(Theorem 3.10). We denote them by $M_{m,n}$.

######  Theorem A (=Theorem 3.11).

Let $M_{m,k-m}$ be a Milnor bundle with $H$-flux
$H=[j]\in\mathrm{H}^{7}(M_{m,k-m},\mathbb{Z})\cong\mathrm{H}^{4}(\mathrm{S}^{4},\mathbb{Z})\cong\mathbb{Z}$.
Then $(M_{m,k-m},[j])$ and $(M_{j,k-j},[m])$ are spherical T-dual, i.e., they
fit the following diagram

(1.2)
$\textstyle{(M_{m,k-m}\times_{\mathrm{S}^{4}}M_{j,k-j},\widehat{p}^{*}([j])-p^{*}([m])=0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{p}=1\otimes\widehat{\pi}}$$\scriptstyle{p=\pi\otimes
1}$$\textstyle{(M_{m,k-m},[j])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{(M_{j,k-j},[m])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\pi}}$$\textstyle{\mathrm{S}^{4}}$

If $k=1$ then both $M_{m,1-m}$ and $M_{j,1-j}$ are homeomorphic to
$\mathrm{S}^{7}$ but not diffeomorphic for

(1.3) ${(-1+2m)^{2}\not\equiv 1\pmod{7}}\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ {(-1+2j)^{2}\not\equiv 1\pmod{7}}.$

Furthermore, for $m=1$, the bundle $M_{1,0}$ is isomorphic to the Hopf
fibration. Consequently, spherical T-duality may occur for two homeomorphic
but not diffeomorphic manifolds.

Theorem A is obtained from the geometric realization of the manifolds involved
in Spherical T-duality. More precisely,

######  Theorem B (=Theorems 3.13, 2.15, 2.18, 2.20+Proposition 2.8).

Let $(E,H),\leavevmode\nobreak\ (\widehat{E},\widehat{H})$ be a spherical
T-dual pair over $\mathrm{S}^{4}$. Then there exists a
$\mathrm{S}^{3}$-$\mathrm{S}^{3}$ manifold $\mathcal{Q}$ fitting a
$\star$-diagram

(1.4)
$\textstyle{\mathrm{S}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{\mathrm{S}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\star}$$\textstyle{\mathcal{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$$\textstyle{\widehat{E}}$$\textstyle{E}$

i.e.,

  1. (a)

$\pi,\leavevmode\nobreak\ \pi^{\prime}$ are principal $\mathrm{S}^{3}$-bundles
over $E,\leavevmode\nobreak\ \widehat{E}$, respectively, with a common total
space $\mathcal{Q}$ carrying commuting $\mathrm{S}^{3}$-actions, denoted as
$\bullet,\leavevmode\nobreak\ \star$

  2. (b)

the manifolds $E,\leavevmode\nobreak\ \widehat{E}$ admit
$\mathrm{S}^{3}$-actions having equivariantly diffeomorphic orbit types,
consisting in a _Morita equivalence_ for the respective action groupoids

  3. (c)

    1. (i)

if the induced $\mathrm{S}^{3}$-action on $E$ is almost free, so it is the
corresponding $\mathrm{S}^{3}$-action on $\widehat{E}$, and there exists an
almost-injective (i.e., injective on an open dense subset) bundle morphism
$\Theta$ fitting the diagram below for appropriate $\rho,\leavevmode\nobreak\
\rho^{\prime}$

(1.5)
${E\times\mathfrak{g}}$${TE}$${{\widehat{E}}\times\mathfrak{g}}$${T{\widehat{E}}}$$\scriptstyle{\rho}$$\scriptstyle{(f,\mathrm{Id})}$$\scriptstyle{\Theta}$$\scriptstyle{\rho^{\prime}}$

where $f:E\rightarrow\widehat{E}$ is a bijection that is a diffeomorphism
between an open and dense subset $E^{*}\subset E$ and its image

    2. (ii)

there exists an isomorphism between the following equivariant cohomologies

$\mathrm{H}^{*}_{G}(E;\mathbb{R})\cong\mathrm{H}^{*}_{G}(\widehat{E};\mathbb{R})\cong\mathrm{H}^{*}_{T}(E;\mathbb{R})^{W},$

where $W=N(T)/T$ is the Weyl group of any maximal torus $T<G$.

Moreover, the characteristic classes of the vector bundles

$\displaystyle EG\times_{G}(E\times\mathfrak{g})$ $\displaystyle\rightarrow
BG\times_{G}E,$ $\displaystyle EG\times_{G}(\widehat{E}\times\mathfrak{g})$
$\displaystyle\rightarrow BG\times_{G}\widehat{E},$

coincide after proper identification and are determined by the characteristic
classes of the vector bundle

$ET\times_{T}(E\times\mathfrak{t})\rightarrow BT\times_{T}E,$

where $\mathfrak{t}$ is the Lie algebra of $T$.

Conversely, given a $\star$-diagram such as (1.4) for which
$E,\leavevmode\nobreak\ \widehat{E}$ are $\mathrm{S}^{3}$-bundles over
$\mathrm{S}^{4}$, there exist
$H\in\mathrm{H}^{4}(\mathrm{S}^{4}),\leavevmode\nobreak\
\widehat{H}\in\mathrm{H}^{4}(\mathrm{S}^{4})$ such that
$(E,H),\leavevmode\nobreak\ (\widehat{E},\widehat{H})$ are spherical T-duals.

As explained in [8], the relationship between duality (to our interests,
Spherical T-duality) and physical theories may appear as symmetries for the
former. The geometric picture of dualities should determine a set of
transformation rules analogous to the Buscher rules [20] for ordinary
T-duality. To each principal $G$-bundle $\pi:\mathcal{P}\to M$ is associated a
2nd Chern class $\mathrm{c}_{2}(\mathcal{P})\in\mathrm{H}^{4}(M,\mathbb{Z})$.
In the case of $G=\mathrm{S}^{3}$, this class plays a crucial role in the
Gysin sequence relating the cohomology of $\mathcal{P}$ to the cohomology of
the base space $M$ (Theorem 3.3). For any $\mathrm{S}^{3}$-principal bundle
$\pi$ over $\mathrm{S}^{4}$ we can identify the Euler class $\mathrm{e}$ of
$\pi$ with the second Chern class $\mathrm{c}_{2}$ of
$\mathcal{P}\times_{\mathrm{S}^{3}}\mathbb{H}$

(1.6)
$\mathrm{e}(\mathcal{P}\times_{\mathrm{S}^{3}}\mathbb{H})=\mathrm{c}_{2}(\mathcal{P}\times_{\mathrm{S}^{3}}\mathbb{H}).$

Given (principal) Spherical T-duals $(\mathcal{P},[j]),\leavevmode\nobreak\
(\widehat{\mathcal{P}},[m])$, the pull-back bundle
$\pi^{*}(\mathcal{P})=\mathcal{P}\times_{M}\mathcal{P}$ under
$\pi:\mathcal{P}\to M$ is trivial, so $\pi^{*}\mathrm{c}_{2}$ is exact on
$\mathcal{P}$. In [8] it is shown that

$\pi^{*}\mathrm{c}_{2}=\pi^{*}([m])=\mathrm{d}\,\mathrm{CS}(\omega)\,,$

where $\mathrm{CS}(\omega)\in\Omega^{3}(\mathcal{P})$ is the Chern-Simons
3-form where $\omega:T\mathcal{P}\rightarrow\mathfrak{su}(2)$ is a principal
connection on $\pi:\mathcal{P}\rightarrow\mathrm{S}^{4}$. Our next result
encompasses the construction appearing in Section 8 in [8] for (principal)
Spherical T-duality over $\mathrm{S}^{4}$, and can be seen as analogous
(coordinate-free) Buscher rules constraints in the context of principal
Spherical T-duality.

######  Theorem C (=Theorem 3.18).

For every (principal) spherical T-duals $(\mathcal{P},H),\leavevmode\nobreak\
(\widehat{\mathcal{P}},\widehat{H})$ over $\mathrm{S}^{4}$ there exist
principal connections
$\omega:T\mathcal{P}\rightarrow\mathfrak{su}(2),\leavevmode\nobreak\
\widehat{\omega}:T{\widehat{\mathcal{P}}}\rightarrow\mathfrak{su}(2)$ and
invariant metrics $\operatorname{\mathsf{g}}_{\mathcal{P}}$ on $\mathcal{P}$
and $\operatorname{\mathsf{g}}_{\widehat{\mathcal{P}}}$ on
$\widehat{\mathcal{P}}$ such that

  1. (a)

$(\mathcal{H},\operatorname{\mathsf{g}}_{\mathcal{P}}|_{\mathcal{H}})$ and
$(\widehat{\mathcal{H}},\operatorname{\mathsf{g}}_{\widehat{\mathcal{P}}}|_{\widehat{\mathcal{H}}})$
are isometric, where

$\mathcal{H}=(\ker\omega)^{\operatorname{\mathsf{g}}_{\mathcal{P}}},\leavevmode\nobreak\
{\widehat{\mathcal{H}}}=(\ker\widehat{\omega})^{\operatorname{\mathsf{g}}_{\widehat{\mathcal{P}}}}$

and the fibers $\mathrm{S}^{3}$ on $\mathcal{P}$ and $\widehat{\mathcal{P}}$
are totally geodesic

  2. (b)

$8\pi^{2}\mathrm{c}_{2}(\mathcal{P})=\mathrm{tr}_{\operatorname{\mathsf{g}}_{\mathcal{P}}}(F_{\omega^{\mathcal{P}}}\wedge
F_{\omega^{\mathcal{P}}})=\widehat{H}$
$8\pi^{2}\mathrm{c}_{2}({\widehat{\mathcal{P}}})=\mathrm{tr}_{\operatorname{\mathsf{g}}_{\widehat{\mathcal{P}}}}(F_{\omega^{\widehat{\mathcal{P}}}}\wedge
F_{\omega^{\widehat{\mathcal{P}}}})=H$

with $\pi^{*}(\widehat{H})=\mathrm{d}\mathrm{CS}(\omega^{\mathcal{P}})$ and
${\widehat{\pi}}^{*}(H)=\mathrm{d}\mathrm{CS}(\omega^{\widehat{\mathcal{P}}})$,
where $\pi:\mathcal{P}\rightarrow\mathrm{S}^{4}$,
$\widehat{\pi}:\widehat{\mathcal{P}}\rightarrow\mathrm{S}^{4}$ are the bundle
projections and $\mathrm{CS}(\omega^{\mathcal{P}}),\leavevmode\nobreak\
\mathrm{CS}(\omega^{\widehat{\mathcal{P}}})$ the corresponding Chern–Simons
form

  3. (c)

it holds

$\displaystyle{\widehat{p}}^{*}\pi^{*}(\widehat{H})-{p}^{*}{\widehat{\pi}^{*}}(H)$
$\displaystyle={\widehat{p}}^{*}(\mathrm{d}\mathrm{CS}(\omega^{\mathcal{P}}))-{p}^{*}(\mathrm{d}\mathrm{CS}(\omega^{\widehat{\mathcal{P}}}))$
$\displaystyle=0\in\mathrm{H}^{7}(\mathcal{P}\times_{\mathrm{S}^{4}}\widehat{\mathcal{P}}).$

An application from Theorem B is that of _transport_ “$G$-invariant complex
structures” between spherical T-dual manifolds.

######  Definition B.

We say that a complex manifold $M$ with an effective action by a compact
connected Lie group $G<\mathrm{Aut}(M)$ has a $G$-invariant complex structure
if, for each point $x$, there exists a $G$-invariant tubular neighborhood
$\mathrm{Tub}(Gx)$ centered at the orbit $Gx$ and an equivariant
biholomorphism

$\mathrm{Tub}(Gx)\cong(\mathbb{C}^{n-k}/\Lambda)\times_{H_{x}}S_{x}$

where $\mathbb{C}^{n-k}/\Lambda$ is a complex torus,
$H_{x}\subset\mathrm{Gl}(n-k;\mathbb{C})$ is a Lie subgroup, $k=\dim Gx$, and
$S_{x}$ is a _slice_ at $x$ ([3, Definition 3.47, p.64]).

We have

######  Theorem D (=Theorem2.12).

Let
$M\stackrel{{\scriptstyle\pi}}{{\leftarrow}}\mathcal{P}\stackrel{{\scriptstyle\pi^{\prime}}}{{\to}}M^{\prime}$
be a $\star$-diagram. $M$ admits a $G$-invariant complex structure if and only
if $M^{\prime}$ also admits.

###  1.2. Logarithmic transformations

Recall that a _cobordism_ in dimension $n$ is a triple of manifolds
$\\{V_{0},W,V_{1}\\}$ with $\dim V_{0}=\dim V_{1}=n-1$ such that $\partial
W=V_{0}\cup V_{1}$ and $V_{0},\leavevmode\nobreak\ V_{1}$ are disjoint open
subsets of $\partial W$. An $\mathrm{h}$-cobordism is a cobordism in which the
inclusions $V_{0}\hookrightarrow W,\leavevmode\nobreak\ V_{1}\hookrightarrow
W$ are homotopy equivalences.

In classical elliptic fibration theory, _simply connected minimal elliptic
surfaces_ form families of $4$-dimension closed manifolds for each positive
integer $n\in\mathbb{N}$. Within a family, all surfaces share the same
intersection form, defining an $\mathrm{h}$-cobordism (see [48]). Although one
family may contain multiple diffeomorphism types, simply connected minimal
elliptic surfaces can be uniquely parameterized by a single positive integer
representing the self-intersection number of a section. These surfaces are
part of Kodaira fibrations, per Kodaira’s classification of elliptic
fibrations on surfaces.

######  Theorem 1.2 (Theorem 5.5, p. 158 in [69]).

Fix a positive integer $n$. The set of equivalence classes of the
$\mathrm{h}$-cobordism relation in dimension $n$ is a commutative monoid,
usually denoted by $\Theta^{n}$, under the operation of the connected sum. The
identity element is represented by the class of manifolds bounding a
contractible manifold, and the group of invertible elements consists of
homology spheres.

Let $\theta^{n}$ be the group of homology spheres. For $n\geq 5$, $\theta^{n}$
can be identified with the group of smooth structures on the topological
$n$-spheres. We learn from [66] that when $n=7$ it holds
$\theta^{n}\cong\mathbb{Z}_{28}$. Moreover, all homology spheres in dimension
$7$ bound a parallelizable manifold, p.512 in [66]. When $n=4$, the group
$\theta^{4}$ is trivial.

The classical significance of Kervaire–Milnor’s theory, addressing distinct
smooth structures on $7$-dimension spheres, remains a cornerstone in classical
topology. To comprehend these invariants and establish connections with new
constructions from diverse fields, we shift the focus away from exotic
$7$-spheres $\Sigma^{7}$. Instead, we focus on homotopy Hopf manifolds
$\Sigma\times\mathrm{S}^{1}$. We pass from surgery on $\Sigma^{7}$ to
_generalized logarithmic transformations_ on $\Sigma^{7}\times\mathrm{S}^{1}$.
In essence, we propose that, for various applications, the $8$-dimension
analog to simply connected minimal elliptic surfaces should be represented by
families of homotopy Hopf manifolds $\Sigma^{7}\times\mathrm{S}^{1}$ for
$\Sigma^{7}\in\theta^{7}$.

######  Definition C (Higher-dimension logarithmic transformation for
$\mathrm{S}^{n+k+1}\times\mathrm{S}^{1}$).

Let
$\mathrm{D}^{k+1}\times\mathrm{S}^{n}\times\mathrm{S}^{1}\cup_{\Phi^{-1}\circ\Psi}\mathrm{S}^{k}\times\mathrm{D}^{n+1}\times\mathrm{S}^{1}$
be a straightening angles description (Appendix A) of the homotopy Hopf
manifold $\mathrm{S}^{k+n+1}\times\mathrm{S}^{1}$. Given diffeomorphisms
$\phi,\leavevmode\nobreak\
\psi:\mathrm{S}^{k}\times\mathrm{S}^{n}\times\mathrm{S}^{1}\rightarrow\mathrm{S}^{k}\times\mathrm{S}^{n}\times\mathrm{S}^{1}$,
we say that a higher-dimension logarithmic transformation from
$\mathrm{S}^{n+k+1}\times\mathrm{S}^{1}$ is the smooth manifold
$\mathrm{S}^{n+k+1}\times\mathrm{S}^{1}_{\phi,\psi}$ obtained from
straightening angles of

$\mathrm{D}^{k+1}\times\mathrm{S}^{n}\times\mathrm{S}^{1}\cup_{\Phi^{-1}\circ\phi^{-1}\circ\psi\circ\Psi}\mathrm{S}^{k}\times\mathrm{D}^{n+1}\times\mathrm{S}^{1}$

Let $\sigma^{8}$ denote the set of diffeomorphism classes of homotopy Hopf
manifolds. Theorem E below ensures that it has a natural group structure given
by logarithmic transformations. The logarithmic transformations in this class
are in bijective correspondence (and have group structure from that induced)
with
$\pi_{3}(\mathrm{SO}(3))\otimes\pi_{3}(\mathrm{SO}(3))\otimes\pi_{1}(\mathrm{SO}(2))$.
This construction is based upon Milnor bilinear pairs [84].

######  Theorem E (=Theorem 4.11).

The set

$\sigma^{8}:=\\{\text{$8$-dimension smooth manifolds $X$ that are homotopic
equivalent to}\leavevmode\nobreak\
\mathrm{S}^{7}\times\mathrm{S}^{1}\\}/\langle\text{diffeomorphism}\rangle$

carries a group structure for the operation of logarithmic transformations.
Moreover,

  1. (a)

there exists a bilinear map

$\sigma_{3,3}:\pi_{3}(\mathrm{SO}(3))\otimes\pi_{3}(\mathrm{SO}(3))\rightarrow\theta^{7}$

such that Gromoll–Meyer exotic sphere $M_{2,-1}$ is the image of $(2,1)$ under
the epimorphism $\sigma_{3,3}$. Moreover, $\sigma_{3,3}(1,1)$ coincides with
the Hopf fibration and can be identified with the image of the map
$t_{10}:\mathrm{S}^{3}\rightarrow\mathrm{SO}(4)$

$t_{10}(x)v:=xv,\leavevmode\nobreak\ v\in\mathbb{H},$

under the Whitehead’s map

$J:\pi_{3}\mathrm{SO}\rightarrow\pi_{3}^{S},$

where $\mathrm{SO}$ is the infinite special orthogonal group, and
$\pi_{3}^{S}$ is the third stable homotopy group of spheres, isomorphic to
$\mathbb{Z}_{24}$

  2. (b)

similarly, there exists a trilinear surjective map

$\widetilde{\sigma}:\pi_{3}(\mathrm{SO}(3))\otimes\pi_{3}(\mathrm{SO}(3))\otimes\pi_{1}(\mathrm{SO}(2))\rightarrow\sigma^{8}$

such that the Homotopy Hopf Manifolds $M_{2,-1}\times\mathrm{S}^{1}$ is the
image of $(2,1,1)$ under the epimorphism $\widetilde{\sigma}$.

In [14, Section 4.6], it is observed that there is a group structure
associated with spherical T-duality over 4-dimension manifolds. We have
expanded upon this by recognizing these groups as the ones with operation
given by logarithmic transformations. Theorem F below establishes the relation
between our introduced logarithmic transformations for
$\Sigma^{7}\times\mathrm{S}^{1}$ and spherical T-duality.

######  Theorem F (=Proposition4.12).

Let $M_{m,1-m}$ be a homotopy seven sphere manifold with $H$-flux
$H=[j]\in\mathrm{H}^{7}(M_{m,1-m},\mathbb{Z})\cong\mathrm{H}^{4}(\mathrm{S}^{4},\mathbb{Z})\cong\mathbb{Z}$.
We have the following spherical T-duality diagram

${{M_{m,1-m}\times\mathrm{S}^{1}}}$${{M_{m,1-m}\times_{\mathrm{S}^{4}}M_{j,1-j}}}$${{M_{j,1-j}\times\mathrm{S}^{1}}}$${{(M_{m,1-m},[j])}}$${{(M_{j,1-j},[m])}}$${\mathrm{S}^{4}}$$\scriptstyle{\mathrm{p}_{1}}$log.
transform$\scriptstyle{\widehat{p}=1\otimes\widehat{\pi}}$$\scriptstyle{p=\pi\otimes
1}$$\scriptstyle{\mathrm{p}_{1}}$

###  1.3. Homological Mirror Symmetry

It is noteworthy that any $7$-dimension homotopy sphere can be described as
the link of the Brieskorn-Pham singularity singularity
$f(u,v,z_{0},z_{1},z_{2})=u^{6k-1}+v^{3}+z_{0}^{2}+z_{1}^{2}+z_{2}^{2}$ for
$(u,v,z_{0},z_{1},z_{2})\in\mathbb{C}^{5}$, where $k\in\\{1,\ldots,28\\}$,
[17]. According to [16], it is possible to consider
$\mathrm{S}^{1}$-fibrations from the link of the singularity $f$. We show that
these quotients are Fano orbifolds $V_{k}$.

Algebraically, let $A$ denote the coordinate ring of $f^{-1}(0)$. Then
$X:=\mathrm{Proj}\leavevmode\nobreak\ A$ defines a degree $6(6k-1)$
hypersurface in $\mathbb{P}(6,2(6k-1),3(6k-1),3(6k-1),3(6k-1))$ that coincides
with $V_{k}$. In [42], some orbifold Fano stacks $Y=[X/G]$ are considered,
where $G$ is a finite group acting on $X$. The results in [42] prove that for
certain Briskorn-Pham singularities $g$, the corresponding construction —
specifically, $X_{g}$ and $Y_{g}$ — establish homological mirror symmetry:

$\mathrm{D}^{b}\mathfrak{Fuk}\leavevmode\nobreak\
{\widetilde{g}}\cong\mathrm{D}^{b}\mathrm{coh}^{G}\leavevmode\nobreak\ Y_{g},$

where $\widetilde{g}$ represents a Morsification of $g$. Here, we prove

######  Theorem G (=Theorem 5.6).

For each $k\in\\{1,\ldots,28\\}$ it holds the following equivalence of
triangulated categories

$\mathrm{D}^{b}{\mathfrak{Fuk}\leavevmode\nobreak\
\widetilde{f}}\cong\mathrm{D}^{b}({\mathfrak{A}}_{6k-2}\otimes{\mathfrak{A}}_{2}\otimes{\mathfrak{A}}_{1}\otimes{\mathfrak{A}}_{1}\otimes\mathfrak{A}_{1})\cong\mathrm{D}^{\mathrm{gr}}_{\mathrm{Sg}}(A),$

where $\widetilde{f}$ is a Morsification of
$f=u^{6k-1}+v^{3}+z_{0}^{2}+z_{1}^{2}+z_{2}^{2}$, $\mathfrak{A}_{i}$ are
certain differential graded categories and
$\mathrm{D}_{\mathrm{Sg}}^{\mathrm{gr}}(A)=\mathrm{D}^{b}(\mathrm{gr}A)/\mathrm{D}^{\mathrm{perf}}(\mathrm{gr}A)$,
for $\mathrm{D}^{\mathrm{perf}}(A)$ the subcategory of perfect complexes,
$\mathrm{D}^{b}(\mathrm{gr}A)$ the bounded derived category of finitely
generated graded $A$-modules and $\mathrm{D}^{\mathrm{perf}}(\mathrm{gr}A)$
its full subcategory consisting of bounded complexes of projective modules.

We proceed in the direction of understanding the orbifold stacks $[V_{k}/G]$
for $V_{k}$ appearing as the base orbifold in the fibrations

${\mathrm{T}^{2}}$${{W^{7}_{6k-1,3}\times\mathrm{S}^{1}}}$${{W^{7}_{6k-1,3}}}$${V_{k}}$$\scriptstyle{\mathrm{p}_{1}}$$\scriptstyle{\pi\circ\mathrm{p}_{1}}$$\scriptstyle{\pi}$

Consider the product $W^{7}_{6k-1,3}\times\mathrm{S}^{1}$. We can project it
into the smash product $W^{7}_{6k-1,3}\wedge\mathrm{S}^{1}$, which is
homeomorphic to a homotopy $\mathrm{S}^{8}$. Denote
$\Sigma^{8}_{k}:=W^{7}_{6k-1,3}\wedge\mathrm{S}^{1}$. We look for descending
$\mathrm{T}^{2}$-actions from $W^{7}_{6k-1,3}\times\mathrm{S}^{1}$ to
$\Sigma^{8}_{k}$ and let $V_{k,\rho}$ be the corresponding orbit space.
Summarily:

${\mathrm{T}^{2}}$${{W^{7}_{6k-1,3}\times\mathrm{S}^{1}}}$${\Sigma^{8}_{k}}$${\mathrm{T}^{2}}$${V_{k}}$${{V_{k,\rho}}}$$\scriptstyle{\pi_{\wedge}^{k}}$$\scriptstyle{\pi_{k,\rho}\circ\pi^{k}_{\wedge}}$$\scriptstyle{\pi_{k,\rho}}$

Our first result provides conditions to obtain orbifold stacks from $V_{k}$:

######  Theorem H (=Theorem5.8).

For each $k\in\\{1,\ldots,28\\}$ it holds that

  1. (1)

let $\rho:\mathrm{S}^{1}\rightarrow\mathrm{SO}(2)$ be such that for _no_
$q\in\mathrm{S}^{1}$ we have

$\\{\lambda\in\mathrm{S}^{1}:\rho(\lambda)q=q\\}=\mathrm{S}^{1}.$

Then $V_{k,\rho}$ is an orbifold

  2. (2)

for $\rho$ equivalent (as representation) to the left-multiplication
representation, it holds that $V_{k,\rho}$ is isomorphic as orbifolds to
$V_{k}$

  3. (3)

there is $\rho$ not equivalent (as representation) to the left-multiplication
representation such that $V_{k,\rho}$ is a Fano orbifold given by $V_{k}/G$
where $G$ is a finite group acting on $V_{k}$.

Theorems E and H show that the orbifold stacks $[V_{k}/G]$ are in direct
relation with logarithmic transformations.

######  Corollary I (=Corollary 5.9).

We can obtain different Fano orbifolds from fibrations from
$W^{7}_{6k-1,3}\times\mathrm{S}^{1}$ via logarithmic transformations according
to choices of generators of $\pi_{1}\mathrm{SO}(2)$. These are given by the
image of the map $\widetilde{\sigma}$ appearing in Theorem E.

Due to the relation of orbistructures with elements in
$\pi_{1}(\mathrm{SO}(2))$, we can provide a group structure in the Fano
orbifolds. These orbifolds appear as bases for fibrations for homotopy Hopf
manifolds:

######  Theorem J (=Theorem5.11).

Fix a fibration $W^{7}_{6k-1,3}\times\mathrm{S}^{1}\rightarrow
V_{k},\leavevmode\nobreak\ k\in\\{1,\ldots,28\\}$. Consider the moduli space
$\mathcal{F}:=\\{Y=[X_{k}/G]:G<\mathrm{S}^{1}\leavevmode\nobreak\ \text{is
discrete}\\}/\langle\text{orbifold isomorphism}\rangle$. Then $\mathcal{F}$
admits a group structure with neutral element $[V_{k}]$ and is isomorphic to
the Abelian group
$\\{\rho:\mathrm{S}^{1}\rightarrow\mathrm{SO}(2)\\}/\langle\text{equivalent
representations}\rangle$.

The set of diffeomorphism classes of fake $\mathbb{P}^{3}$ admits a group
structure given by the wedge sum [82, p.484-485] in which $[\mathbb{P}^{3}]$
is the unit element. Theorem J provides an extension in the sense that the
diffeomorphism type of each $V_{k}$ (a fake $\mathbb{P}^{3}$) is fixed, and we
provide a group structure to its orbifold stacks.

Recall that the Chen-Ruan cohomology [4] extends conventional cohomology to
orbifolds encapsulating equivariant information. We can use them to
distinguish the geometric information from the Fano orbifolds $Y$:

######  Theorem K (=Theorem5.12).

Let $X=\mathrm{Proj}\leavevmode\nobreak\ A$ where $A$ is the coordinate ring
of $f=u^{6k-1}+v^{3}+z_{0}^{2}+z_{1}^{2}+z_{2}^{2}$. Let $G$ be any finite
group acting effectively on $X$ and denote its corresponding stack by
$Y=[X/G]$. Let

  1. (a)

$(a_{0},\ldots,a_{4})=(6,2(6k-1),3(6k-1),3(6k-1),3(6k-1))$

  2. (b)

$\\{I\\}$ be the collection of all non-empty multi-indexes of
$\\{0,\ldots,4\\}$

  3. (c)

for each multi-index $I$,
$V_{I}=\\{(x_{0},x_{1},x_{2},x_{3},x_{4})\in\mathbb{C}^{5}:x_{i}\neq
0\leavevmode\nobreak\ \text{for}\leavevmode\nobreak\ i\in I\\}$

Then the Chen-Ruan cohomology $H^{*}_{\mathrm{orb}}(Y)$ is completely
determined in terms of twisted forms222see Appendix B.1, Definition B.8. by
the following data

$\bigcup_{I\leavevmode\nobreak\ \text{multi-index of}\leavevmode\nobreak\
\\{0,\ldots,4\\},\leavevmode\nobreak\
V_{I}\cap\mathrm{S}^{9}\neq\emptyset}\left\\{\prod_{i\not\in
I}\left(\frac{\widetilde{\xi_{i}}}{a_{i}}\right)\prod_{i\in
I}\left(\frac{\widetilde{\xi_{i}}}{a_{i}}\right)^{\frac{1}{2\pi}\mathrm{Arg}\lambda^{a_{i}}}:\lambda\in\mathrm{S}^{1}\right\\}$

where $\widetilde{\xi_{i}}$ is $G$-equivariant Chern class of the line bundle
defined by $V_{I}\cap\mathrm{S}^{9}/G$.

In Section 1.6, we elaborate on how to approach Mirror Symmetry for
$\Sigma^{7}\times\mathrm{S}^{1}$ following the steps initiated in [100], to be
obtained in subsequential works. Additional questions relating to the obtained
group structures and some topological invariants are left at this section’s
end.

###  1.4. Homotopy Hopf manifolds and $\mathrm{tmf}$

In [61], Hopkins draws a connection between the Hopf fibration
$\mathrm{S}^{7}\rightarrow\mathrm{S}^{4}$ and the constant term $\frac{1}{12}$
in the Fourier expansion of the Weierstrass $\wp$-function:

$\wp(z,\tau)dz^{2}=\left(\sum_{s\in\mathbb{Z}}\frac{q^{s}u}{(1-q^{s}u)^{2}}+\frac{1}{12}-2\sum_{s\geq
1}\frac{q^{s}}{(1-q^{s})^{2}}\right)\left(\frac{du}{u}\right)^{2},$

where $u=e^{2\pi\mathbf{i}z}$, $q=e^{2\pi\mathbf{i}\tau}$. We further
elaborate on Hopkins’ argumentation to establish a similar relation applicable
to homotopy Hopf manifolds. His observation stems from the following: if
$S^{0}$ represents the _sphere spectrum_ – [61, Section 3.2], then the cyclic
group $\pi_{3}^{S}\cong\mathbb{Z}_{24}$ is generated by the Hopf fibration, as
noted in [1].

######  Theorem L (=Theorem 6.2).

For each Milnor sphere $M_{m,1-m}$ there exist a $\mathrm{S}^{3}$-action
$\star$ in $\mathrm{S}^{4}$ and homomorphisms classes
$\beta,\leavevmode\nobreak\ \gamma\in\pi_{3}(\mathrm{SO}(4))$ yielding a
principal bundle $J\gamma^{*}\mathrm{S}^{7}$ such that

$J\gamma^{*}\mathrm{S}^{7}/\star\cong M_{m,1-m}.$

Consequently, after a logarithmic transformation, we can associate (via
$\mathrm{tmf}$) $M_{m,1-m}$ to the constant $\tfrac{1}{12}$ in the Fourier
expansion of the Weierstrass $\wp$-function

$\wp(z,\tau)dz^{2}=\left(\sum_{s\in\mathbb{Z}}\frac{q^{s}u}{(1-q^{s}u)^{2}}+\frac{1}{12}-2\sum_{s\geq
1}\frac{q^{s}}{(1-q^{s})^{2}}\right)\left(\frac{du}{u}\right)^{2},$

where $u=e^{2\pi\mathbf{i}z},q=e^{2\pi\mathbf{i}\tau}$. I.e., there exists a
map

$\mathcal{L}:M_{m,1-m}\stackrel{{\scriptstyle\textrm{log.
transform}}}{{\longmapsto}}M_{1,0}\cong\mathrm{S}^{7}\mapsto[h:\mathrm{S}^{7}\rightarrow\mathrm{S}^{4}]\in\pi_{3}^{S}$

######  Theorem M (=Theorem 6.3).

Consider the standard fibration

$\mathrm{T}^{2}\hookrightarrow W^{7}_{6k-1,3}\times\mathrm{S}^{1}\rightarrow
V_{k}\subset\mathbb{P}(6,2(6k-1),3(6k-1),3(6k-1),3(6k-1))$

given by diagram (5.5). Then, for $16$ of the $28$ homotopy Hopf manifolds in
dimension $8$, the $\mathrm{tmf}$ theory identifies
$W^{7}_{6k-1,3}\times\mathrm{S}^{1}$ with $\tfrac{1}{12}$. Moreover,

  1. (i)

there is a constant map from the subset of
$\sigma^{8}_{\mathrm{Milnor}}\subset\sigma^{8}$ consisting in homotopy Hopf
manifolds with $7$-dimension factor a Milnor sphere

$\sigma^{8}_{\mathrm{Milnor}}\rightarrow\pi_{3}^{S}$

obtained from the composing

$\sigma^{8}_{\mathrm{Milnor}}\stackrel{{\scriptstyle\mathrm{p}_{1}}}{{\rightarrow}}\theta^{7}\stackrel{{\scriptstyle\mathcal{L}}}{{\rightarrow}}\pi_{3}^{S}$
$M_{m,1-m}\times\mathrm{S}^{1}\mapsto
M_{m,1-m}\mapsto\mathrm{S}^{7}\mapsto[\mathrm{S}^{7}\rightarrow\mathrm{S}^{4}].$

Since $\mathrm{tmf}$ identifies its image with $\frac{1}{12}$, we can
attribute to it the numerical value $\frac{1}{12}$.

  2. (ii)

every element $M_{m,1-m}\times\mathrm{S}^{1}\in\sigma^{8}$ defines a fibration
onto $\mathbb{P}^{3}$ up to a logarithmic transformation.

Conjectures B, D, ask whether the former results provide a numerical map from
$\sigma^{8}\rightarrow\mathbb{R}$. Studying its level sets could distinguish
topological and geometric properties of homotopy Hopf manifolds.

###  1.5. Fano orbifolds

Following [16], from the link $\Sigma^{7}$ of the singularity
$u^{6k-1}+v^{3}+z_{0}^{2}+z_{1}^{2}+z_{2}^{2}=0$ one obtain fibrations as

$\mathrm{T}^{2}\rightarrow\Sigma^{7}\times\mathrm{S}^{1}\rightarrow V$

where $V$ are _Fano_ orbifolds on some weighted projective spaces. More
concretely, the Fano orbifolds $V$ appearing above are all _Fake Complex
Projective spaces_ , that is, they are homotopy and have the same cohomology
of $\mathbb{P}^{3}$. Fano orbifolds carry plenty of K3-surfaces. We wonder
whether one can _lift_ such K3-surfaces in $V$ to
$\Sigma^{7}\times\mathrm{S}^{1}$. We show this is possible when $\Sigma^{7}$
is diffeomorphic to $\mathrm{S}^{7}$ (Theorem 6.8), and ask whether such
lifting is possible if, and only if, $V$ is homeomorphic to a projective
space.

######  Theorem N (=Theorem 6.8).

Consider the fibration $\mathrm{T}^{2}\hookrightarrow
W^{7}_{5,3}\times\mathrm{S}^{1}\cong\mathrm{S}^{7}\times\mathrm{S}^{1}\rightarrow
V_{1}\subset\mathbb{P}(6,10,15,15,15)$. Then
$V_{1}\cong\mathbb{P}(6,10,15,15)$ and there exists a family of K3-surfaces
embedded in $\mathrm{S}^{7}\times\mathrm{S}^{1}$.

Throughout the manuscript, we leave some questions and conjectures to be
approached in subsequential works. We chose to re-state that below for
readers’ convenience.

###  1.6. Questions, conjectures, and future developments

Here, we pose our conjectures and questions to be further explored. We divide
it into smaller sections and provide some context to them.

####  1.6.1. Homological Mirror Symmetry in $\Sigma^{7}\times\mathrm{S}^{1}$

In [100] it is obtained the analogous homological mirror symmetry for the Hopf
surface $\mathrm{S}^{3}\times\mathrm{S}^{1}$. This is the $4$-dimension case
of our considered homotopy Hopf manifolds $\Sigma^{7}\times\mathrm{S}^{1}$.
$\mathrm{S}^{3}\times\mathrm{S}^{1}$ is obtained as the quotient space

$\mathrm{S}^{3}\times\mathrm{S}^{1}\cong(\mathbb{C}^{2}\setminus\\{0\\})/(z_{0},z_{1})\sim\lambda(z_{0},z_{1}),\leavevmode\nobreak\
\lambda\in\mathbb{C}^{*},\leavevmode\nobreak\ |\lambda|<1.$

As one can check,
$\mathrm{H}^{2}(\mathrm{S}^{3}\times\mathrm{S}^{1};\mathbb{Z})=0$ thus
$\mathrm{S}^{3}\times\mathrm{S}^{1}$ is neither algebraic nor symplectic [21,
80]. The main scope of the paper [100] is to show that despite that,
$\mathrm{S}^{3}\times\mathrm{S}^{1}$ still exhibits homological symmetry
behavior, namely, there is a mirror space $Y$ and a Fukaya category
$\mathfrak{F}$ associated to $Y$ such that there is a correspondence between
coherent analytic sheaves in $Y$ and objects in $\mathfrak{F}$.

There are two options to obtain analogous homological mirror symmetry to
$\Sigma^{7}\times\mathrm{S}^{1}$. The first amounts consider the $\Sigma^{7}$
admiting fibration description as
$\mathrm{S}^{3}\hookrightarrow\Sigma^{7}\rightarrow\mathrm{S}^{4}$. A huge
downstep is because $\mathrm{S}^{4}$ does not admit complex structure.
Nevertheless, we could explore the disc decomposition of
$\Sigma^{7}\times\mathrm{S}^{1}$ in two pieces
$\mathrm{D}^{4}\times\mathrm{S}^{3}\times\mathrm{S}^{1}$, to which we can
furnish appropriate complex structures (Proposition 7.11). Appropriate gluing
of these complex structures extends to complex structures in
$\Sigma^{7}\times\mathrm{S}^{1}$ (Theorem 7.1). The main thing here is that
each piece contains a copy of $\mathrm{S}^{3}\times\mathrm{S}^{1}$, indicating
that we can explore it to derive the notion of homological mirror symmetry to
$\Sigma^{7}\times\mathrm{S}^{1}$.

The second approach is more algebraic. As we can consider the fibrations
$\mathrm{T}^{2}\hookrightarrow W^{7}_{6k-1,3}\times\mathrm{S}^{1}\rightarrow
V_{k}$, intersecting
$W^{7}_{6(1)-1,3}\times\mathrm{S}^{1}\cap\\{z_{1}=z_{2}=0\\}$, one recovers
$(\mathrm{S}^{3}/\Gamma)\times\mathrm{S}^{1}$ where $\Gamma$ is a subgroup of
$\mathrm{SU}(2)$ of order $120$, see [85]. So for $k=1$, one has a very
natural copy of (a quotient of) $\mathrm{S}^{3}\times\mathrm{S}^{1}$ sitting
inside $\mathrm{S}^{7}\times\mathrm{S}^{1}$. As
$\mathrm{S}^{7}\times\mathrm{S}^{1}$ can be obtained as the quotient

$(\mathbb{C}^{4}\setminus\\{0\\})/(z_{0},\ldots,z_{3})\sim\lambda(z_{0},\ldots,z_{3}),\leavevmode\nobreak\
\lambda\in\mathbb{C}^{*},\leavevmode\nobreak\ |\lambda|<1$

it promptly suggests that the methods in [100] should adapt to
$\mathrm{S}^{7}\times\mathrm{S}^{1}$. A crucial observation, however, is that
logarithmic transformations do not change the diffeomorphism type of
$\mathrm{S}^{3}\times\mathrm{S}^{1}$ ([101]), but our proposed logarithmic
transformations may change that of $\mathrm{S}^{7}\times\mathrm{S}^{1}$
(Theorem E). It is left to understand the meaning of this (possible) smooth
structure change once adapting the content of [100].

It is worth pointing out that the general spirit in [100] is based on the
Strominger-Yau-Zaslow (SYZ) philosophy. Corollary I establishes that distinct
generalized logarithmic transformations correspond to different Fano
orbifolds.

######  Question A.

Do logarithmic transformations encompass all possible parameterizations for
lagrangians and SYZ structures of homotopy Hopf manifolds?

Considering the results in [100], it is very natural to pose:

######  Conjecture A.

The mirror of $W^{7}_{6k-1,3}\times\mathrm{S}^{1}$ is the mirror of
$\mathrm{T}^{2}\times V_{k}$.

We expect that the mirror of
$\mathrm{S}^{7}\times\mathrm{S}^{1}\rightarrow\mathbb{P}(6,10,15,15)$ is a
$\mathrm{T}^{2}$-bundle over another weighted projective space. In full
generality, recall that for each $k$ the homotopy Hopf manifold
$W^{7}_{6k-1,3}\times\mathrm{S}^{1}$ fibers over (with $\mathrm{T}^{2}$ as
fibers) Fano orbifolds $V_{k}$ of degree $6(6k-1)$ in
$\mathbb{P}(6,2(6k-1),3(6k-1),3(6k-1),3(6k-1))$. For instance,

  1. (1)

for $k=2$ we have a $\mathrm{T}^{2}$-bundle from
$W^{7}_{11,3}\times\mathrm{S}^{1}\rightarrow V_{2}$ where $V_{2}$ is a Fano
orbifold of degree $66$ in $\mathbb{P}(6,22,33,33,33)$

  2. (2)

for $k=3$ we have a $\mathrm{T}^{2}$-bundle from
$W^{7}_{17,3}\times\mathrm{S}^{1}\rightarrow V_{3}$ where $V_{3}$ is a Fano
orbifold of degree $102$ in $\mathbb{P}(6,34,51,51,51)$.

We expect that for each $k$ the mirrors of
$W^{7}_{6k-1,3}\times\mathrm{S}^{1}$ are $\mathrm{T}^{2}$-bundles over Fano
orbifolds.

Once we have the Shimada Spheres [90] (the fifteen-dimension analogous to the
Milnor spheres in dimension seven), we believe our analysis should be
generalized to this case.

Recall that in the realm of algebraic geometry, when dealing with Fano
threefolds, a fundamental tool for analyzing their derived categories is the
notion of semi-orthogonal decomposition (SOD). As for the singularity
$f=u^{6k-1}+v^{3}+z_{0}^{2}+z_{1}^{2}+z_{2}^{2}$ we can recover every homotopy
sphere in dimension $7$, considering $\mathrm{D}^{b}(V_{k})$ may lead to
further connections of algebraic geometry invariant and (smooth) topological
ones.

An exceptional collection in the derived category $\mathrm{D}^{b}(X)$ of a
Fano orbifold $X$ is a sequence of objects $E_{1},E_{2},...,E_{n}$ that are
mutually exceptional. This means they have no higher Ext-groups between them,
i.e., $\text{Ext}^{i}_{X}(E_{j},E_{k})=0$ for $i\neq 0$ and all $j,k$, and
$\text{Hom}_{X}(E_{j},E_{k})=0$ for $j\neq k$. For any Fano orbifold $X$, it
is known that

$\mathrm{D}^{b}(X)=\langle\mathcal{A},E_{1},\ldots,E_{n}\rangle.$

Here $\mathcal{A}$ is the so-called Griffiths component,
$\langle\mathcal{A},E_{1},\ldots,E_{n}\rangle$ a semi-orthogonal
decomposition, and $\mathcal{A}$ is a fractional Calabi-Yau category, see
[64].

######  Question B.

Are the Griffiths components smooth invariants?

######  Question C.

Are the Griffiths components distinguished by $\mathrm{tmf}$ and the elliptic
genus?

As it follows from the work of Katzarkov, Kontsevich, Pantev, Yu, [65], we can
associate to the $\mathcal{A}$ a noncommutative Hodge structure - an Atom
$\mathrm{Atom}(\mathcal{A})$ depending on $\mathcal{A}$. To it, we associate a
noncommutative spectra.

######  Question D.

Is the noncommutative spectra of $\mathrm{Atom}(\mathcal{A})$ related to
spectra of the singularity
$f(u,v,z_{0},z_{1},z_{2})=u^{6k-1}+v^{3}+z_{0}^{2}+z_{1}^{2}+z_{2}^{2}$ and to
the elliptic genus (as in [74])? Is it related to the $\mathrm{tmf}$?

######  Question E.

Can the similar $\mathrm{tmf}$, elliptic genus invariants be defined for every
Fano and its Griffiths componnent $\mathcal{A}$?

######  Question F.

What are the mirror counterparts of $\mathrm{tmf}$, elliptic genus invariants?
What captures the geometry of the logarithmic transformation for the mirror?

####  1.6.2. Topological Modular forms

As one also learns from [100], the quotient space

$(\mathbb{C}\setminus\\{0\\})/z\sim\lambda z,\leavevmode\nobreak\
\lambda\in\mathbb{C}^{*},\leavevmode\nobreak\ |\lambda|<1$

is an elliptic curve $E_{\tau}$ with modualr parameter $\tau$ satisfying
$q=e^{2\pi\mathbf{i}\tau}$. This elliptic curve plays a major role in the
homological mirror symmetry approach in [100]. Consider the Fourier expansion
of the Weierstrass $\wp$-function

$\wp(z,\tau)dz^{2}=\left(\sum_{s\in\mathbb{Z}}\frac{q^{s}u}{(1-q^{s}u)^{2}}+\frac{1}{12}-2\sum_{s\geq
1}\frac{q^{s}}{(1-q^{s})^{2}}\right)\left(\frac{du}{u}\right)^{2},$

where $u=e^{2\pi\mathbf{i}z}$, $q=e^{2\pi\mathbf{i}\tau}$. As we explained,
Hopkins showed that the term $\frac{1}{12}$ is identified via
$\mathrm{tmf}$-theory with the Hopf fibration
$\mathrm{S}^{7}\rightarrow\mathrm{S}^{1}$. Since
$\mathbb{C}\setminus\\{0\\}\subset\mathbb{C}^{2}\setminus\\{0\\}\subset\mathbb{C}^{3}\setminus\\{0\\}$,
we have that $\mathrm{S}^{3}\times\mathrm{S}^{1}$ sits very naturally inside
of $\mathrm{S}^{7}\times\mathrm{S}^{1}$, and $E_{\tau}$ embeds in
$\mathrm{S}^{3}\times\mathrm{S}^{1}$. We picture

${\mathrm{S}^{7}\times\mathrm{S}^{1}}$${\mathrm{S}^{4}}$${\mathrm{S}^{3}\times\mathrm{S}^{1}}$${E_{\tau}}$

The following are natural questions

######  Question G.

Do $E_{\tau}$ play the analogous role to obtained homological mirror symmetry
in $\mathrm{S}^{7}\times\mathrm{S}^{1}$ to that it plays in the case of
$\mathrm{S}^{3}\times\mathrm{S}^{1}$ in [100]? Is it true that $E_{\tau}$ also
embedds in every homotopy Hopf manifold of the form
$M_{m,1-m}\times\mathrm{S}^{1},\leavevmode\nobreak\ m\neq 1$?

As Theorem M shows, for the subset
$\sigma^{8}_{\textrm{Milnor}}\subset\sigma^{8}$ we can associate the number
$\frac{1}{12}$ extending Hopkins’ observation. We wonder if the existence of a
numerical function $N:\sigma^{8}\rightarrow\mathbb{R}$ whose level sets
distinguish topological (or geometric) among homotopy Hopf manifolds is true.

######  Conjecture B.

There is a numerical map

$N:\sigma^{8}\rightarrow\mathbb{R}$

which is the constant $\frac{1}{12}$ in $\sigma^{8}_{\textrm{Milnor}}$.

Let $BO\langle 8\rangle$ be the seven connected cover of $B\mathrm{Spin}$ and
$MO\langle 8\rangle$ be its Thom spectrum [61, Section 3.2]. According to [61,
Theorem 6.25], for any multiplicative map $MO\langle 8\rangle\to\mathrm{tmf}$
whose underlying genus is the Witten genus, the induced map of homotopy groups
$\pi_{\ast}MO\langle 8\rangle\to\pi_{\ast}\mathrm{tmf}$ is surjective. Since
$\pi_{\ast}\mathrm{tmf}$ is isomorphic to $\pi_{\ast}S^{0}$, the numerical map
$N$ could be decoupled from the following

######  Conjecture C.

  1. (1)

The map

$\mathcal{L}\circ\mathrm{p}_{1}:\sigma^{8}\rightarrow\pi^{S}_{3}$

in Theorem M induces a map

$\mathfrak{L}_{3}:\pi_{3}MO\langle 8\rangle\to\pi_{3}^{S}$

that can be seen as a factor of the ring isomorphism
$\pi_{\ast}\mathrm{tmf}\cong\pi_{\ast}S^{0}$.

  2. (2)

Restricting the domain of $\mathfrak{L}_{3}$ properly, it can be identified
(as a constant function) to the term $\frac{1}{12}$ in the Fourier expansion
of the $\wp$-Weierstrass function.

  3. (3)

For each positive integer $n$ there exists a map
$\mathfrak{L}_{n}:\pi_{n}MO\langle 8\rangle\to\pi_{n}^{S}$ which we can
associate to some coefficient in the Fourier expansion of Weierstrass
$\wp$-function.

We justify the importance of the conjectural numerical map $N$ as:

######  Conjecture D.

Given a homotopy Hopf manifold $X\in\sigma^{8}$, if we _can not_ associate $X$
with the number $\frac{1}{12}$ in the Fourier expansion of the
$\wp$-Weierstrass function, then

  1. (i)

either $X$ is the image under $\widetilde{\sigma}$ of an element in
$(m,1-m,l)\in\pi_{3}\mathrm{SO}(3)\otimes\pi_{3}\mathrm{SO}(3)\otimes\pi_{1}\mathrm{SO}(2)$
with $l\neq 1$

  2. (ii)

or $X=\Sigma^{7}\times\mathrm{S}^{1}$ for $\Sigma^{7}\in\theta^{7}$ a homotopy
seven sphere that admits _no_ realization as the total space of a
$\mathrm{S}^{3}$-bundle over $\mathrm{S}^{4}$, i.e., $\Sigma^{7}$ is not the
total space of a Milnor bundle.

A more profound conjecture relating $\mathrm{tmf}$, orbifold Elliptic genus,
and rationality is posed next (Conjecture G).

####  1.6.3. Fano orbifolds, Kähler-Einstein metrics and the K3

The third stable homotopy group of spheres (the third stable stem) is the
cyclic group of order $24$:

$\pi_{3}^{S}\cong\mathbb{Z}_{24}$ $[h_{\mathbb{H}}]\mapsto[1]$

where the generator $[1]\in\mathbb{Z}_{24}$ is represented by the quaternionic
Hopf fibration $\mathrm{S}^{7}\stackrel{{\scriptstyle
h_{\mathbb{H}}}}{{\longrightarrow}}\mathrm{S}^{4}$.

Under the Pontrjagin-Thom isomorphism, we can identify the stable homotopy
groups of spheres with the bordism ring $\Omega^{\mathrm{fr}}_{\bullet}$ of
stably framed manifolds with the generator $[h_{\mathbb{H}}]$ being
represented by the $3$-sphere (with its left-invariant framing induced from
the identification with the Lie group $\mathrm{SU}(2)\cong\mathrm{S}^{3}$)

$\pi_{3}^{S}\cong\Omega_{3}^{\mathrm{fr}}$
$[h_{\mathbb{H}}]\leftrightarrow[\mathrm{S}^{3}].$

The relation $24[\mathrm{S}^{3}]\cong 0$ is represented by the bordism given
by the complement of $24$ open balls inside the K3-manifold (Proposition 6.7).

Consider the fibrations

$\mathrm{T}^{2}\hookrightarrow
W^{7}_{6k-1,3}\times\mathrm{S}^{1}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}V_{k}\subset\mathbb{P}(6,2(6k-1),3(6k-1),3(6k-1),3(6k-1)),$

where $V_{k}$ is a Fano orbifold, introduced in Section 5. Fano orbifolds have
many K3 surfaces embedded in them, which we will describe next. Let $-K$ be
the anticanonical bundle over $V_{k}$. The zero sets for the bundle sections
$\pi_{K}:-K\rightarrow V_{k}$ give us plenty of K3 surfaces. Now let us pull
back $-K$ to $W^{7}_{6k-1,3}\times\mathrm{S}^{1}$, realizing the diagram

${\pi^{*}(-K)}$${-K}$${{W^{7}_{6k-1,3}\times\mathrm{S}^{1}}}$${V_{k}}$$\scriptstyle{\pi}$$\scriptstyle{\pi_{K}}$$\scriptstyle{\pi^{*}}$$\scriptstyle{\pi}$

where

$\pi^{*}(-K)=\\{(x,v)\in
W^{7}_{6k-1,3}\times\mathrm{S}^{1}\times{-K}:\pi(x)=\pi_{K}(v)\\}.$

Each section $s\in C^{\infty}(-K)$ is a map $s:V_{k}\rightarrow-K$ so that we
can define the pulling back for these sections by
$s\circ\pi:W^{7}_{6k-1,3}\times\mathrm{S}^{1}\rightarrow\pi^{*}(-K)$. Once in
hand Theorem N, we ask

######  Question H.

Is it true that the zeroes of the pulled-back sections
$\\{s\circ\pi:W^{7}_{6k-1,3}\times\mathrm{S}^{1}\rightarrow\pi^{*}(-K)\\}$ are
embedded K3-surfaces in $W^{7}_{6k-1,3}\times\mathrm{S}^{1}$?

######  Question I.

Is it true that Question H has a positive answer only for $16$ values of $k$,
the ones in correspondence with the diffeomorphism type of $W^{7}_{6k-1,3}$
compatible with an $\mathrm{S}^{3}$-bundle over $\mathrm{S}^{4}$ realization?
Being this true, is it also true that the corresponding $V_{k}$ is homomorphic
to $\mathbb{P}^{3}$?

The main goal in the paper [16] is to provide the existence of Einstein
metrics on homotopy spheres. They do this by searching for the existence of
Kähler-Einstein metrics on the Fano orbifolds $V$ obtained as orbit spaces for
the weighted $\mathrm{S}^{1}$-action in $W^{7}_{6k-1,3}$ described above. As
we can readily check, their results do not ensure the existence of Kähler-
Einstein metrics on our Fano orbifolds described above (see Theorem 34 in [16,
p.572]). Nevertheless, in the case $k=1$, we have that
$W^{7}_{6(1)-1,3}\stackrel{{\scriptstyle\text{diffeomorphic}}}{{\cong}}\mathrm{S}^{7}$.
Theorem 6.8 ensures that the Fano orbifold, in this case, is a weighted
projective $\mathbb{P}^{3}$.

######  Question J.

Is it true that the Fano orbifold $V_{k}$ admits a Kähler-Einstein metric if,
and only if, $V_{k}$ is homeomorphic to $\mathbb{P}^{3}$?

####  1.6.4. How logarithmic transformations patches everything together

Our introduced logarithmic transformations are showcased in spherical
T-duality (Theorem F), offering distinct smooth structures on homotopy Hopf
manifolds (Theorem E). Moreover, they ensure diverse orbifold stacks as the
base for fibrations from the former (Theorem H). The orbifold structures, as
elucidated in Theorem K, can be discerned through Chen-Ruan cohomology.
Consequently, certain choices of logarithmic transformations can be evaluated
using Chen-Ruan cohomology. This prompts the inquiry: How many orbifold stacks
and logarithmic transformations can be amalgamated to characterize a
diffeomorphism type for a homotopy Hopf manifold uniquely? We believe the
following is true

######  Conjecture E.

Let $[X/G]$ be a Fano stack where $G$ is some finite subgroup of
$\mathrm{S}^{1}$ and $X$ is a fake $\mathbb{P}^{3}$. Then there is an element
in
$\pi_{3}(\mathrm{SO}(3))\otimes\pi_{3}(\mathrm{SO}(3))\otimes\pi_{1}(\mathrm{SO}(2))$
whose image under $\widetilde{\sigma}$ (given in Theorem E) is the total space
of a $\mathrm{T}^{2}$-fibration over $X$.

A very strong support for Conjecture E is Lemma 6 in [82]. It ensures a
correspondence between (smooth) fake $\mathbb{P}^{3}$ and circle actions on
homotopy spheres. Our conjecture is further elaborated to establish a more
concrete relation between the orbifold stacks from fake $\mathbb{P}^{3}$ and
logarithmic transformations.

A more speculating question is based on Conjecture D. Putting conjecture E in
perspective with that, it may be the case that partial data associated with
logarithmic transformations can be inferred from $\mathrm{tmf}$. We pose

######  Question K.

Is there a connection between $\mathrm{tmf}$ and logarithmic transformations?
How are conjectures E and D related?

Observe that, fixed a Fano orbifold $V_{k}$, logarithmic transformations
induce new quotient stacks $[V_{k}/G]$ for certain finite groups $G$, these
are Theorem J and Corollary I. In [25], we give further evidence that these
logarithmic transformations can produce examples of non-equivariantly
birational $V_{k}$, combining its Chen-Ruan description (Theorem K) with the
$\beta$-classes introduced in [68].

######  Conjecture F.

For each Fano orbifold $V_{k}$ coming from a $\mathrm{T}^{2}$-fibration
$\pi_{k}$ with $\Sigma^{7}_{1}\times\mathrm{S}^{1}$ as total space, there
exist logarithmic transformations of $\pi_{k}$ leading to non $G$-birational
pairs $(V_{k},G,\rho),\leavevmode\nobreak\ (V_{k},G,\rho^{\prime})$, where
$\rho,\leavevmode\nobreak\ \rho^{\prime}$ are $G$-representations on
$\mathrm{Iso}(V_{k})$, the isomorphism classes of $V_{k}$, and
$G<\pi_{1}(\mathrm{SO}(2))$.

The more profound connections relating $\mathrm{tmf}$, orbifold Elliptic
Genus, and birational invariants shall be explored in [24]. To give further
details, _gerbes with connections_ ([56, 77]) correspond with twisted bundles
$\mathcal{L}$ over $V_{k}$. The Grothendieck group generated by the
isomorphism classes of $\mathcal{L}$ is the $\mathcal{L}$ twisted K-theory
${}^{\mathcal{L}}K_{\mathrm{grp}}(V_{k})$. Under mild hypothesis on $V_{k}$,
it holds that (apply for instance the argumentation in [77])

${}^{\mathcal{L}}K_{\mathrm{grp}}(V_{k})\otimes{\mathbb{C}}\cong\mathrm{H}^{*}_{CR}(V_{k};\mathbb{C}).$

We are tempted to conjecture the following.

######  Conjecture G.

The ring $\mathrm{tmf}$ can recover “global information” for orbifolds
$V_{k}$. This means that we can use it combined with the theory of Chen-Ruan
cohomology to classify all the possible $\mathrm{T}^{2}$-fibratios whose base
is $V_{k}$.

### This paper is organized as follows

In Section 2, we revisit the concept of $\star$-diagrams, closely following
[26]. We cover basic definitions and recall known results. Additionally, we
introduce new results that extend beyond the scope of this work. While not
directly related to our current focus, they complement our discussion on
spherical T-duality. The only $\star$-diagrams referring claims are proved in
Theorems B, D.

We introduce Spherical T-duality in Section 3. The section begins with the
recapitulation on Milnor bundles. Next, we recall spherical T-duality
following [8, 9] and conclude the proof of Theorems A, B. This section
finishes with the proof of Theorem C.

We discuss logarithmic transformations in Section 4. After recalling the
basics of logarithmic transformations in elliptic fibrations, we provide a
straightening angle description of every homotopy Hopf manifold and prove
Theorem E.

In Section 5, we start discussing homological mirror symmetry for homotopy
Hopf manifolds and accomplish with the proof of Theorems G-K.

Section 6 presents the discussion on $\mathrm{tmf}$, K3-surfaces and Kähler-
Einstein metrics, culminating in the proof of Theorems M-N. Lastly, in Section
7, we discuss complex structures on homotopy Hopf manifolds induced by gluing.

The present paper contains two appendixes. The first regarding a recap on
straightening angles, which extrapolates to its analogous in the holomorphic
case (Appendix A), and the second on the basics of orbifolds and Chen-Ruan
cohomology (Appendix B).

##  2\. $\star$-diagrams: A concise account and new results

The notion of an _exotic sphere_ originates from John Milnor’s groundbreaking
work in the 1950s [83]. Milnor introduced a family of $7$-dimension manifolds,
now referred to as exotic spheres, which are homeomorphic to, but not
diffeomorphic with, $7$-dimension spheres. The terminology extends so that an
_exotic manifold_ refers to a smooth manifold $M^{\prime}$ homeomorphic to
another smooth manifold $M$, yet not diffeomorphic.

In [92], following [37, 87], Sperança introduced a general procedure to
construct exotic manifolds $M^{\prime}$ from a _classical_ , or _standard_ ,
realization $M$, establishing a dictionary between their _invariant_
geometries. Let $\mathcal{P}$ denote a principal $G$-manifold with principal
action $\bullet$, where $G$ is a compact Lie group, and assume there exists
another $G$-commuting action on $\mathcal{P}$, denoted by $\star$. Let $M$ and
$M^{\prime}$ be the corresponding orbit spaces for the $\bullet$ and
$\star$-actions, respectively. We patch these data in the following diagram

(2.1)
$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\star}$$\textstyle{\mathcal{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$$\textstyle{M^{\prime}}$$\textstyle{M}$

The starting example is:

######  Example 2.1 (The Gromoll–Meyer exotic sphere).

Further details can be found in [45, 37, 92].

Consider the compact Lie group

(2.2) $\mathrm{Sp}(2)=\left\\{\begin{pmatrix}a&c\\\
b&\mathrm{d}\end{pmatrix}\in\mathrm{S}^{7}\times\mathrm{S}^{7}\leavevmode\nobreak\
\Big{|}\leavevmode\nobreak\ a\bar{b}+c\bar{d}=0\right\\},$

where $a,b,c,\mathrm{d}\in\mathbb{H}$ are quaternions with their usual
conjugation, multiplication, and norm. The projection
$\pi:\mathrm{Sp}(2)\to\mathrm{S}^{7}$ of an element to its first column
defines a principal $\mathrm{S}^{3}$-bundle with principal action:

(2.3) $\begin{pmatrix}a&c\\\
b&d\end{pmatrix}\bar{q}=\begin{pmatrix}a&c\overline{q}\\\
b&d\overline{q}\end{pmatrix}.$

In [45] it is also considered the $\star$-action

(2.4) $q\begin{pmatrix}a&c\\\
b&d\end{pmatrix}=\begin{pmatrix}qa\overline{q}&qc\\\
qb\overline{q}&qd\end{pmatrix},$

whose quotient for the map $\pi^{\prime}:\begin{pmatrix}a&c\\\
b&d\end{pmatrix}\mapsto(2\bar{c}d,|c|^{2}-|d|^{2})$ is an exotic 7-sphere
named _the Gromoll–Meyer exotic sphere_ , therein denoted by
$\Sigma_{GM}^{7}$. We encompasse actions (2.3), (2.4) to the principal bundle
diagram

(2.5)
$\textstyle{\mathrm{S}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{\mathrm{S}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\star}$$\textstyle{\mathrm{Sp}(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$$\textstyle{\Sigma^{7}_{GM}}$$\textstyle{\mathrm{S}^{7}}$

$\blacksquare$

A construction based on $\star$-diagrams, as in (2.1), does not necessarily
yield exotic smooth structures. The choice of the $\star$-action, encoded in a
cocycle condition (see Definition 2.3), determines whether exotic structures
emerge.

######  Example 2.2 (Pairs of diffeomorphic manifolds via $\star$-diagrams).

Let $M$ be a smooth manifold with an effective smooth action by a compact Lie
group $G$, which we denote by $\cdot$. Consider the product manifold $M\times
G$ with the following $\star$-action

$g\star(x,g^{\prime}):=(g\cdot x,gg^{\prime}),\leavevmode\nobreak\ x\in
M,\leavevmode\nobreak\ g,g^{\prime}\in G.$

Let $\bullet$ be the following $G$-action on $M\times G$:

$g\bullet(x,g^{\prime}):=(x,(g^{\prime})g^{-1}),\leavevmode\nobreak\ x\in
M,\leavevmode\nobreak\ g,g^{\prime}\in G.$

Both $\bullet,\leavevmode\nobreak\ \star$ are free and commuting actions on
$M\times G$. Orbit maps in which actions are, respectively, $\pi:M\times
G\rightarrow M,\leavevmode\nobreak\ (x,g^{\prime})\mapsto
x,\leavevmode\nobreak\ \pi^{\prime}:M\times G\rightarrow
M,\leavevmode\nobreak\ (x,g^{\prime})\mapsto(g^{\prime})^{-1}x$. We can build
the corresponding $\star$-diagram

(2.6)
$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\star}$$\textstyle{M\times
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$$\textstyle{M}$$\textstyle{M}$

$\blacksquare$

In contrast, Examples 2.1, 2.2 reinforce that to build an exotic smooth
structure out of $M$, we need to consider more “twisted constructions”. We
revisit the procedure detailed in [26], providing a recipe for constructing
diagrams akin to (2.1). Theorem 2.5 substantiates this approach, demonstrating
that the actions in Example 2.2 consistently serve as local descriptions for
any $\bullet$ and $\star$-action within a $\star$-diagram.

###  2.1. Manufacturing exotic manifolds: the recipe for $\star$-diagrams

Below, we outline the general theory for constructing $\star$-bundles,
systematically described in [26].

Let $M$ be a $G$-manifold (with action on the left) and $\\{U_{i}\\}$ be a
collection of $G$-invariant open sets of $M$. Reducing $U_{i}$ if necessary,
we can assume that $GU_{i}=U_{i}.$ Given two open $G$-invariant sets
$U_{i},\leavevmode\nobreak\ U_{j},$ let $U_{ij}:=U_{i}\cap U_{j}.$

######  Definition 2.3.

A collection $\phi_{ij}:U_{ij}\to G$ is said to be a _$\star$ -collection_ if
it satisfies the cocycle condition

(2.7) $\phi_{ij}(x)\phi_{jk}(x)\phi_{ki}(x)=\phi_{ii}(x),\leavevmode\nobreak\
\forall x\in U_{ij},$

and also,

(2.8) $\phi_{ij}(gx)=g\phi_{ij}(x)g^{-1},\leavevmode\nobreak\ \forall
i,j,\leavevmode\nobreak\ \forall g\in G,\leavevmode\nobreak\ \forall x\in
U_{ij}.$

The condition (2.8) guarantees that the adjoint map

$\widehat{\phi}_{ij}:U_{ij}\to U_{ij}$ (2.9)
$\widehat{\phi}_{ij}:x\mapsto\phi_{ij}(x)x.$

is equivariant. In this way, $\widehat{\phi}_{ij}$ defines an equivariant
diffeomorphism over $U_{ij}$. We register it in the lemma below.

######  Lemma 2.4 (Lemma 2.3 in [26]).

Let $U$ be a $G$-manifold and $\theta,\theta^{\prime}:U\to G$ smooth maps
satisfying (2.8). Then $\widehat{\theta}$ and $\widehat{\theta^{\prime}}$ are
equivariant diffeomorphisms such that

$\widehat{\theta\theta^{\prime}}=\widehat{\theta^{\prime}}\widehat{\theta},$

where
$\theta\theta^{\prime}(x):=\theta(x)\theta^{\prime}(x),\leavevmode\nobreak\
\forall x\in U.$

Let $\\{\phi_{ij}:U_{ij}\to G\\}$ be a $\star$-collection. We define

$\bigcup_{\widehat{\phi}_{ij}}U_{i}$

as the quotient space under the equivalence relation $x\in
U_{ij}\sim\widehat{\phi}_{ij}(x)\in U_{ij}$. Theorem 2.5, proved [26],
establishes the recipe for constructing $\star$-diagrams.

######  Theorem 2.5 (Theorem 2.2 in [26]).

Let $\pi:\mathcal{P}\to M$ be the principal bundle associated with a
$\star$-collection given by $\\{\phi_{ij}:U_{ij}\to G\\}.$ Then $\mathcal{P}$
admits a new action, denoted by $\star$, such that

  1. $(i)$

the $\star$-action in $\mathcal{P}$ is free

  2. $(ii)$

the quotient $\mathcal{P}/\star$ is a $G$-manifold equivariantly diffeomorphic
to

$M^{\prime}:=\bigcup_{\widehat{\phi}_{ij}}U_{i},$

  3. $(iii)$

the $\star$-diagram (2.1) obtained from this construction is such that if
$(G\times G)_{p}$ denotes the isotropy group on $p$ with respect to the action
by juxtaposition

$(r,s)p:=rps^{-1},$

where $G\times\\{e\\}$ represents the $\star$-action, $\\{e\\}\times G$
represents the principal action, and “$e$” denotes the unity element of $G$,
then there exists $g\in G$ such that

$(G\times G)_{p}=\\{(h,ghg^{-1}):h\in G_{\pi(p)}\\}.$

###### Proof.

  1. $(i)$

Let $\\{\phi_{ij}:U_{ij}\to G\\}$ be the $\star$-collection associated with
$\pi:\mathcal{P}\to M$ provided by the hypothesis. We can trivialize
$\mathcal{P}$ so that $\mathcal{P}|_{\pi^{-1}(U_{i})}\cong U_{i}\times G$.
Thus, we define the free $\star$-action by the local expression

$r(x,g):=(r\cdot x,gr^{-1}),\leavevmode\nobreak\ \forall r\in G.$

To verify the good global definition of $\star$, recall that $\mathcal{P}$ can
be retrieved via the quotient space

$\bigcup_{\phi_{ij}}U_{i}\times G=\mathcal{P},$

where the elements $(x,g),\leavevmode\nobreak\ (y,h)\in\bigcup_{i}U_{i}\times
G$ are equivalent if and only if there exist $i,j$ such that

$x=y\in U_{ij},\leavevmode\nobreak\ g\phi_{ij}(x)=h.$

Since for any $x\in U_{ij}$ we have

$(q\cdot x,gq^{-1}\phi_{ij}(qx))=(q\cdot
x,g\phi_{ij}(x)q^{-1}),\leavevmode\nobreak\ x\in U_{ij},\leavevmode\nobreak\
q,g\in G$

one gets the desired.

  2. $(ii)$

Lemma 2.4 ensures that the adjoint maps $\widehat{\phi}_{ij}$ are equivariant
diffeomorphisms, so we can define the projection

$\pi^{\prime}:\mathcal{P}\to M^{\prime}$

from a local trivialization

$\pi^{\prime}:(x,g)\in U_{i}\times G\mapsto g\cdot x.$

Note that $\pi^{\prime}$ is well defined since

$\displaystyle\pi^{\prime}(x,g\phi_{ij}(x))$
$\displaystyle=(g\phi_{ij}(x))\cdot x$ $\displaystyle=g\widehat{\phi}_{ij}(x)$
$\displaystyle=\widehat{\phi}_{ij}(g\cdot x)$ $\displaystyle=\phi_{ij}(g\cdot
x)g\cdot x$ $\displaystyle=\phi_{ij}(\pi^{\prime}(x,g))\pi^{\prime}(x,g)$
$\displaystyle=\widehat{\phi}_{ij}(\pi^{\prime}(x,g)).$

That is, $\pi^{\prime}(x,g\phi_{ij}(x))$ and $\pi^{\prime}(x,g)$ define the
same equivalence class in $M^{\prime}.$

Furthermore, $\pi^{\prime}(rp)=\pi^{\prime}(p)$, and hence $\pi^{\prime}$,
defines a submersion whose fibers are the orbits of the action $\star.$ In
particular, the map

$\iota_{i}(x):=(x,e),\leavevmode\nobreak\ \text{where $e$ denotes the unity
element of $G$}.$

defines a local section for both $\pi$ and $\pi^{\prime}.$ Therefore,
$U_{i}\times G/\star$ and $U_{i}$ are equivariantly diffeomorphic open sets.

  3. $(iii)$

This follows immediately from the fact that in a local trivialization for
$\mathcal{P}$

$(r,s)(x,g):=(r\cdot x,sgr^{-1}),\leavevmode\nobreak\ x\in
U_{i},\leavevmode\nobreak\ r,s,g\in G$

since $(r,s)(x,g)=(x,g)$ if and only if $r\in G_{x}$ and $s=grg^{-1}.$ So,

$(G\times G)_{p}=\left\\{(r,grg^{-1}):r\in G_{x}\right\\}.\qed$

###  2.2. Pulling-back $\star$-diagrams

A prolific method for building examples is to take pullbacks of $\mathcal{P}$
on a $\star$-diagram by appropriate smooth functions. Next, we elucidate this
fact.

Let $M,N$ be $G$-manifolds and $f:N\to M$ be smooth equivariant maps. Let
$\\{\phi_{ij}:U_{ij}\to G\\}$ be a $\star$-collection associated with the
bundle $\pi:\mathcal{P}\to M.$ Recall that the pullback bundle by $f$ over $N$
has as total space

$f^{*}(\mathcal{P}):=\\{N\times\mathcal{P}:f(n)=\pi(p),\leavevmode\nobreak\
\forall n\in N,\leavevmode\nobreak\ \forall p\in\mathcal{P}\\},$

and projection $\pi_{f}(n,p):=n.$ Thus, the principal action induced in
$f^{*}\left(\mathcal{P}\right)$ is defined by

$(n,p)s^{-1}:=(n,sp)$

while $\star$-action induced on $f^{*}(\mathcal{P})$ is defined by

$r(n,p):=(rn,pr^{-1}).$

Proposition 2.6 gives conditions to the pullback of $\star$-bundle to produce
another $\star$-bundle.

######  Proposition 2.6 (Proposition 2.4 in [26]).

Let $f:N\to M$ be a smooth and $G$-equivariant function and
$M^{\prime}\leftarrow\mathcal{P}\rightarrow M$ be a $\star$-bundle obtained
from a $\star$-collection given by $\\{\phi_{ij}:U_{ij}\to G\\}$. Then,

  1. $(i)$

$\pi_{f}$ consists of a bundle that is equivariantly diffeomorphic to the
$\star$-bundle obtained by the $\star$-collection given by $\\{\phi_{ij}\circ
f:f^{-1}(U_{ij})\to G\\}$

  2. $(ii)$

the quotient $f^{*}(\mathcal{P})/\star$ is equivariantly diffeomorphic to

$N^{\prime}=\bigcup_{\widehat{\phi_{ij}\circ f}}f^{-1}(U_{i})$

  3. $(iii)$

a map $f^{\prime}:N^{\prime}\to M^{\prime}$ is well defined and satisfies
$f^{\prime}\big{|}_{f^{-1}(U_{i})}=f\big{|}_{f^{-1}(U_{i})}.$

######  Remark 2.7.

Therein, we sometimes employ the short notation
$M\stackrel{{\scriptstyle\pi}}{{\leftarrow}}\mathcal{P}\stackrel{{\scriptstyle\pi^{\prime}}}{{\to}}M^{\prime}$
for a $\star$-diagram, possibly omitting the projections
$\pi,\leavevmode\nobreak\ \pi^{\prime}$. ∎

Next, we show how _invariant_ complex structures can be “transported” from $M$
in a $\star$-diagram to $M^{\prime}$, and vice-versa.

###  2.3. Interchangeability of complex structures via $\star$-diagrams

As discussed in [26], the geometries of $M$ and $M^{\prime}$ can be compared
by considering a $G\times G$-invariant Riemannian metric
$\operatorname{\mathsf{g}}$ on $\mathcal{P}$. Let $\mathcal{H}^{\prime\prime}$
be the _bihorizontal space_ defined as

$\mathcal{H}^{\prime\prime}=\\{X\in T\mathcal{P}\leavevmode\nobreak\
|\leavevmode\nobreak\ X\perp T(G{\times}G)\\}.$

Following the approach in Section 5 of [26], we can construct Riemannian
metrics $\operatorname{\mathsf{g}}_{M}$ and
$\operatorname{\mathsf{g}}_{M^{\prime}}$ on $M$ and $M^{\prime}$,
respectively, making $\pi$ and $\pi^{\prime}$ Riemannian submersions.
Throughout, denote the pointwise collection of vector spaces whose elements
are vectors orthogonal to the $G$-orbits on $M$ and $M^{\prime}$,
respectively, by $\mathcal{H}\subseteq TM$ and $\mathcal{H}^{\prime}\subseteq
TM^{\prime}$. For such chosen metrics it also holds that the restrictions
$\mathrm{d}\pi\Big{|}_{\mathcal{H}^{\prime\prime}}:(\mathcal{H}^{\prime\prime},\operatorname{\mathsf{g}})\rightarrow(\mathcal{H},\operatorname{\mathsf{g}}_{M})$,
$\mathrm{d}\pi^{\prime}\Big{|}_{\mathcal{H}^{\prime\prime}}:(\mathcal{H}^{\prime\prime},\operatorname{\mathsf{g}})\rightarrow(\mathcal{H}^{\prime},\operatorname{\mathsf{g}}_{M^{\prime}})$
are isometries. We present a sketch of such construction next.

To construct $\operatorname{\mathsf{g}}_{M^{\prime}}$, we recall that a
connection $1$-form on a $G$-principal bundle is a differential $1$-form
$\omega_{0}\colon T\mathcal{P}\to\mathfrak{g}$ satisfying, for every
$\xi\in\mathfrak{g}$, $X\in T\mathcal{P}$, and $g\in G$:

  * •

$(\omega_{0})_{p}(\xi^{*})=\xi$,

  * •

$(\omega_{0})_{pg}(Xg)=\operatorname{Ad}_{g^{-1}}(\omega_{0})_{p}(X)$.

Here, $\xi^{*}$ is the action field defined by $\xi$. Averaging $\omega_{0}$
along the $G\times G$-orbits gives an invariant form $\omega$. Thus, if
$\operatorname{\mathsf{g}}_{M}$ is a $G$-invariant metric in $M$, and $Q$ is a
bi-invariant metric in $G$, we define a $G{\times}G$-invariant Kaluza–Klein
metric in $\mathcal{P}$ as

$\operatorname{\mathsf{g}}:=\pi^{\ast}\operatorname{\mathsf{g}}_{M}+Q\circ\omega\otimes\omega.$

The desired metric in $\operatorname{\mathsf{g}}_{M^{\prime}}$ is obtained as
$\operatorname{\mathsf{g}}_{M^{\prime}}:=\pi^{\prime}_{\ast}(\operatorname{\mathsf{g}})$.
For the remaining section, we assume that $M,\leavevmode\nobreak\
M^{\prime},\mathcal{P}$ are equipped with those mentioned above invariant
Riemannian metrics.

Define $\bar{\mathcal{H}}:=\ker\omega$ and choose
$x^{\prime}\in\pi^{\prime}(\pi^{-1}(x))$ for some fixed $x\in M$. Let $\nu Gx$
and $\nu Gx^{\prime}$ represent the normal bundles of the $G$-orbits through
$x$ and $x^{\prime}$, respectively. In preparation for the rest of this
section and to set the tone for the proof of Theorem 2.12, we revisit
Proposition 5.3 in [26].

######  Proposition 2.8 (Proposition 5.3, p.28 in [26]).

Given $x\in M$, there exist $x^{\prime}\in\pi^{\prime}(\pi^{-1}(x))$ and an
isomorphism $\Phi\colon\nu Gx\to\nu Gx^{\prime}$ such that, given
$\mathcal{O}\subseteq\nu Gx$, if $\exp|_{\mathcal{O}}\colon\mathcal{O}\to M$
is a diffeomorphism onto its image, then
$\exp|_{\Phi(\mathcal{O})}\colon\Phi(\mathcal{O})\to M^{\prime}$ is a
diffeomorphism onto its image.

###### Proof.

According to item (iii) of Theorem 2.5, for any $x$ there is $p\in\pi^{-1}(x)$
whose isotropy subgroup $(G{\times}G)_{p}$ is the diagonal $\Delta
G_{x}=\\{(q,q)\leavevmode\nobreak\ |\leavevmode\nobreak\ q\in G_{x}\\}$. Set
$x^{\prime}=\pi^{\prime}(p)$.

Let us consider the map $\Psi\colon Gx\to\mathcal{P}$ as $\Psi(rx)=rpr^{-1}$,
which is is well defined since $(G{\times}G)_{p}=\Delta G_{x}$, thus
$rpr^{-1}=p$ when $r\in G_{x}$. Given $X\in TM$ and $q\in\pi^{-1}(x)$, denote
by $\mathcal{L}_{q}(X)\in\bar{\mathcal{H}}_{q}$ the unique element of
$\bar{\mathcal{H}}$ such that $\mathrm{d}\pi(\mathcal{L}_{q}(X))=X$. It can be
straightforwardly verified that both $\Psi$ and $\mathcal{L}_{q}$ are
equivariant:

(2.10)
$\Psi(ry)=r\Psi(y)r^{-1},\qquad\mathcal{L}_{rqs^{-1}}(rX)=r\mathcal{L}_{q}(X)s^{-1}.$

Now define $\Phi\colon\nu Gx\to\nu Gx^{\prime}$ as

(2.11) $\Phi_{rx}(X)=\mathrm{d}\pi^{\prime}(\mathcal{L}_{\Psi(rx)}(X)).$

We claim that $\Phi_{rx}$ defines an isometry between $\mathcal{H}_{rx}$ and
$\mathcal{H}^{\prime}_{rx^{\prime}}$. Since both $\mathcal{L}_{q}$ and
$\mathrm{d}\pi^{\prime}|_{\mathcal{H}^{\prime\prime}}$ are isometries it
suffices to show that for every $q$ it holds that
$\mathcal{L}_{q}(\mathcal{H}_{x})=\mathcal{H}^{\prime\prime}_{q}$ .

Using that $\pi(rqs^{-1})=rx$ we conclude that the image of vectors tangent to
$(G{\times}G)$-orbits by $\mathrm{d}\pi$ are vectors tangent to $G$-orbits on
$M$. Once $\mathrm{d}\pi|_{\bar{\mathcal{H}}}$ is an isometry, then
$\mathcal{L}_{q}$ maps vectors orthogonal to $Gx$ to vectors orthogonal to
$(G{\times}G)q$. Now dimension comparison ensures that
$\mathcal{L}_{q}(\mathcal{H}_{x})=\mathcal{H}^{\prime\prime}_{q}$.

Let $\mathcal{O}\subseteq\nu Gx$ be such that $\exp|_{\mathcal{O}}$ is a
diffeomorphism. Thus $\tilde{\Psi}\colon\exp(\mathcal{O})\times
G\to\mathcal{P}$,

$\tilde{\Psi}(\exp_{rx}(v),g)=\exp_{\Psi(rx)}(\mathcal{L}_{\Psi(rx)}(v))g^{-1},$

is a trivialization for $\pi$ along $\exp(\mathcal{O})$. Moreover, for every
$y\in Gx$,

$\displaystyle\tilde{\Psi}(\exp_{ry}(rv),sgr^{-1})$
$\displaystyle=\exp_{\Psi(ry)}(\mathcal{L}_{\Psi(ry)}(rv))r(sg)^{-1}=\exp_{\Psi(ry)}(\mathcal{L}_{r\Psi(y)r^{-1}}(rv))r(sg)^{-1}$
(2.12)
$\displaystyle=\exp_{\Psi(ry)}(r\mathcal{L}_{\Psi(y)}(v)r^{-1})r(sg)^{-1}=r\tilde{\Psi}(\exp_{y}(v),g)s^{-1}.$

Hence, $\iota(\exp_{y}(v))=\tilde{\Psi}(\exp_{y}(v),e)$ intersects each
$\\{e\\}{\times}G$\- and $G{\times}\\{e\\}$-orbit at most once. Thus,

$\exp_{y}(v)\mapsto\pi^{\prime}(\tilde{\Psi}(\exp_{y}(v),e))$

defines a diffeomorphism between $\exp(\mathcal{O})$ and
$\pi^{\prime}(\tilde{\Psi}(\mathcal{O}))$. On the other hand, since
$\pi^{\prime}$ is a Riemannian submersion and
$v\in(\ker\mathrm{d}\pi^{\prime})^{\bot}$,

$\displaystyle\pi^{\prime}(\tilde{\Psi}(\exp_{rx}(v),e))$
$\displaystyle=\pi^{\prime}\exp_{rxr^{-1}}(\mathcal{L}_{rxr^{-1}}(v))$
$\displaystyle=\exp_{\pi^{\prime}(rxr^{-1})}(\mathrm{d}\pi^{\prime}\mathcal{L}_{rxr^{-1}}(v))=\exp_{rx^{\prime}}(\Phi(v)).$

We conclude that $\pi^{\prime}\leavevmode\nobreak\ \circ\leavevmode\nobreak\
\tilde{\Psi}=\exp\leavevmode\nobreak\ \circ\leavevmode\nobreak\
\Phi\leavevmode\nobreak\ \circ\leavevmode\nobreak\ \exp^{-1}|_{\mathcal{O}}$
is a diffeomorphism. ∎

A useful consequence of Proposition 2.8 is in hand:

######  Theorem 2.9.

Let $\mathcal{P}\to M$ be a $G$-principal bundle with another commuting action
by $G$ for which the isotropy of the juxtaposition action satisfies item (iii)
of Theorem 2.5. Then $M$ admits a $\star$-collection and so, yields a
$\star$-diagram.

###### Proof.

Let us construct $\tilde{\Psi}$ (equation (2.12)). The proof relies on its
$G$-equivariance.

The only prerequisite for $\widetilde{\psi}$ construction is that the
juxtaposition action from another $G$-action should commute with the
$G$-principal action in $\mathcal{P}\rightarrow M$. This aligns with Item
(iii) of Theorem 2.5.

By mimicking the construction of $\tilde{\Psi}$ in the proof of Proposition
2.8, we find open tubular neighborhoods $U_{x}$ for every orbit $Gx\subset M$
of $Gx$ and $\mathcal{O}_{x}\subseteq\nu Gx$ for the zero section such that
$\exp|_{\mathcal{O}_{x}}\colon\mathcal{O}_{x}\to U_{x}$ is a diffeomorphism.
The trivialization $\tilde{\Psi}_{x}\colon U_{x}\times G\to\mathcal{P}$ and
equivariance of $\tilde{\Psi}$ guarantees that the transition functions
related to the open cover $\\{U_{x}\\}_{x\in M}$ satisfy the condition given
by the equation (2.8), hence consisting in a $\star$-collection. ∎

Theorem 2.12 needs the concept of _$G$ -invariant complex structures_.
Consider a $n$-dimension manifold $M^{n}$ with an open cover
$\\{U_{x}\\}_{x\in M}$ such that the coordinate changes have Jacobians that
are elements of a certain Lie group. Complex analytic manifolds possess this
property with the related Lie group being $\mathrm{Gl}(n,\mathbb{C})$. We
pursue connecting existing complex structures on $M$ (so $n\equiv 0\pmod{2}$)
with that of a given $G$-action on $M$ by a compact Lie group $G$.

When a Lie group $G$ effectively acts on $M$, it induces a homogeneous
singular foliation on $M$ with leaves being the orbits $\\{Gx\\}_{x\in M}$.
There exists an open dense subset $M^{reg}\subset M$ where every two points
$x,y\in M^{reg}$ have conjugate isotropy subgroups ([3, Theorem 3.82, p. 76]).
However, for a point $x\in M^{reg}$, there may be a $y\in M\setminus M^{reg}$
such that the isotropy subgroup $G_{x}$ is conjugate to a subgroup of $G_{y}$,
justifying the term _singular_. In light of this, expecting a complex
structure with transition function Jacobians taking values on the entire $G$
is unreasonable.

Assume that $M$ is a complex manifold and let $\mathrm{Aut}(M)$ be the group
of biholomorphisms acting on $M$ (see [2]). In Definition 2.10 below, we term
by a $G$-invariant complex structure any effective holomorphic action by a
compact connected Lie group $G<\mathrm{Aut}(M)$ with additional property on
the orbit types. Compare it with [70] and Section 1 in [46]. Also, see [53,
Section 9].

######  Definition 2.10.

We say that a complex manifold $M$ with an effective action by a compact
connected Lie group $G<\mathrm{Aut}(M)$ has a $G$-invariant complex structure
if, for each point $x$, there exist a $G$-invariant tubular neighborhood
$\mathrm{Tub}(Gx)$ centered at the orbit $Gx$ and an equivariant
biholomorphism

$\mathrm{Tub}(Gx)\cong(\mathbb{C}^{n-k}/\Lambda)\times_{H_{x}}S_{x}$

where $\mathbb{C}^{n-k}/\Lambda$ is a complex torus,
$H_{x}\subset\mathrm{Gl}(n-k;\mathbb{C})$ is a Lie subgroup, $k=\dim Gx$, and
$S_{x}$ is a _slice_ at $x$ ([3, Definition 3.47, p.64]).

######  Example 2.11 (The Hopf manifold $\mathrm{S}^{3}\times\mathrm{S}^{1}$).

Consider the smooth manifold $\mathrm{S}^{3}\times\mathrm{S}^{1}$. Here,
$\mathrm{S}^{3}$ is defined as
$\\{(z,w)\in\mathbb{C}\oplus\mathbb{C}:|z|^{2}+|w|^{2}=1\\}$, and
$\mathrm{S}^{1}$ as $\\{r\in\mathbb{C}:|r|^{2}=1\\}$.

The action of
$\mathrm{T}^{2}=\\{(r_{0},r_{1}):r_{i}\in\mathrm{S}^{1};\leavevmode\nobreak\
i=0,1\\}$ on $\mathrm{S}^{3}\times\mathrm{S}^{1}$ is free. It’s defined as
$(r_{0},r_{1})\cdot(z,w,r):=(r_{0}z,r_{1}w,r_{0}rr_{1}^{-1})$.

Utilizing the Tubular Neighborhood Theorem in conjunction with the Slice
Theorem (refer to [3, Theorems 3.49 and 3.57]), we find that the tubular
neighborhood of any orbit of $\mathrm{T}^{2}$ in
$\mathrm{S}^{3}\times\mathrm{S}^{1}$ is equivariantly diffeomorphic to
$\mathrm{T}^{2}\times\mathrm{D}^{2}$. Here,
$\mathrm{D}^{2}\subset\mathrm{S}^{3}\times\mathrm{S}^{1}$ denotes an open
disc.  $\blacksquare$

Next, we show that $G$-invariant complex structures can be interchanged on $M$
and $M^{\prime}$ in a $\star$-diagram.

######  Theorem 2.12.

Let
$M\stackrel{{\scriptstyle\pi}}{{\leftarrow}}\mathcal{P}\stackrel{{\scriptstyle\pi^{\prime}}}{{\to}}M^{\prime}$
be a $\star$-diagram. Assume that $M$ is a complex manifold that admits a
$G$-invariant complex structure. Then $M^{\prime}$ is a complex manifold and
admits a $G$-invariant complex structure. Moreover, the roles of
$M,\leavevmode\nobreak\ M^{\prime}$ can be interchanged in the statement.

###### Proof.

According to Item (ii) in Theorem 2.9, the open sets $\\{U_{i}\\}$ in a
$\star$-collection $\\{\phi_{ij}:U_{ij}\rightarrow G\\}$ that generates the
$\star$-diagram furnish a common open cover for $M$ and $M^{\prime}$. Pick an
atlas $\mathcal{A}$ for $M$ with coordinate open sets $\\{V_{\mathcal{A}}\\}$
and coordinate maps $\\{\psi_{\mathcal{A}}\\}$. Consider the collection
$\\{U_{i}\cap V_{\mathcal{A}}:=U_{i,\mathcal{A}}\\}$ and maps
$\psi_{\mathcal{A},i}:=\psi_{\mathcal{A}}\Big{|}_{U_{i,\mathcal{A}}}$. From
Lemma 2.4 for each $i,\mathcal{A}$ the maps
$\phi_{ij}^{\mathcal{A}}:U_{ij,\mathcal{A}}\rightarrow G$ define equivariant
diffeormophism
$\widehat{\phi_{ij}^{\mathcal{A}}}:U_{ij,\mathcal{A}}\rightarrow
U_{ij,\mathcal{A}}$ where $U_{ij,\mathcal{A}}:=U_{i}\cap U_{j}\cap
V_{\mathcal{A}}$. Applying Item (ii) in Theorem 2.9 once more yields that
$M^{\prime}=\cup_{\widehat{\phi_{ij}^{\mathcal{A}}}}U_{i,\mathcal{A}}$.

By assumption, $GU_{i}\subset U_{i}$ for each $i$. Thus, for each
$i,\mathcal{A}$ it holds that $GU_{i,\mathcal{A}}\subset U_{i,\mathcal{A}}$.
For each $i,\mathcal{A}$ pose
$\widehat{\psi}_{i,\mathcal{A}}(x):=\int_{G}\psi_{i,\mathcal{A}}(gx)\mathrm{d}\mu(g),\leavevmode\nobreak\
x\in U_{i,\mathcal{A}}$, where $\mathrm{d}\mu$ is a Haar measure in $G$. We
claim that the map $\widehat{\psi_{\mathcal{A},i}}$ descends to a map in
$M^{\prime}$ and that $\\{U_{i,\mathcal{A}},\widehat{\psi_{\mathcal{A},i}}\\}$
is a holomorphic atlas for $M^{\prime}$. Indeed, take
$y=\phi_{ij,\mathcal{A}}(x)(x)$. Using that $\widehat{\psi_{i,\mathcal{A}}}$
is constant along the $G$-orbits one has

$\displaystyle\widehat{\psi_{\mathcal{A},i}}(y)$
$\displaystyle=\widehat{\psi_{\mathcal{A},i}}(\phi_{ij,\mathcal{A}}(x)(x))$
$\displaystyle=\widehat{\psi_{\mathcal{A},i}}(x)$

Now consider the composition
$\widehat{\psi_{i,\mathcal{A}}}\circ[\widehat{\psi_{j,\beta}}]^{-1}$ defined
in $\widehat{\psi_{j,\beta}}(U_{ij,\mathcal{A}}\cap U_{ij,\beta})$. Using that
$\mathrm{Tub}(Gx)$ is equivariantly diffeomorphic with
$\mathbb{C}^{n-k}/\Lambda$ and $\widehat{\psi_{i,\mathcal{A}}}$ is constant on
each tubular neighborhood, it is clear that
$\widehat{\psi_{i,\mathcal{A}}}\circ[\widehat{\psi_{j,\beta}}]^{-1}$ is a
biholomorphism from $\mathbb{C}^{n-k}/\Lambda$ to itself, concluding desired.

As for the last part, observe that one can find for every orbit $Gx\subset M$
open tubular neighborhoods $U_{x}$ of $Gx$ and $\mathcal{O}_{x}\subseteq\nu
Gx$ of the zero section such that
$\exp|_{\mathcal{O}_{x}}\colon\mathcal{O}_{x}\to U_{x}$ is a diffeomorphism.
If $M$ satisfies Definition 2.10, up to reducing $U_{x}$, we can assume that
there exist equivariant biholomorphisms $\zeta_{x}$ between $U_{x}$ and
$(\mathbb{C}^{n-k}/\Lambda)\times_{H_{x}}\mathrm{S}_{x}$, where $k=\dim Gx$.
Let $\Phi$ be as in Proposition 2.8, and collect the open sets
$\\{\Phi(U_{x})\\}$. Since $U_{x}$ and $\Phi(U_{x})$ are equivariantly
diffeomorphic and $U_{x}$ is biholomorphic with
$(\mathbb{C}^{n-k}/\Lambda)\times_{H_{x}}\mathrm{S}_{x}$ one concludes the
proof. ∎

We finish this section presenting an application of Theorem 2.12.

######  Example 2.13 (Some homotopy $8$-dimension Hopf manifolds).

From the point of view of differential topology, a Hopf manifold is
diffeomorphic to $\mathrm{S}^{1}\times\mathrm{S}^{2n-1}$. As complex analytic
manifolds, these are homogeneous fiber bundles over the complex projective
space $\mathbb{P}^{n-1}$ with fiber an elliptic curve. An interesting aspect
of these manifolds is that for $n\geq 2$, they never admit a Kähler structure.

From Example 2.1, we can build the following $\star$-diagram whose $M$ is
precisely the $8$-dimension Hopf manifold
$\mathrm{S}^{7}\times\mathrm{S}^{1}$, and $M^{\prime}$ an _exotic Hopf
manifold_ , such as in [21]:

(2.13)
$\textstyle{\mathrm{S}^{3}\times\mathrm{S}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{\mathrm{S}^{3}\times\mathrm{S}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\star}$$\textstyle{\mathrm{Sp}(2)\times\mathrm{S}^{1}\times\mathrm{S}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi\times\mathrm{p}}$$\scriptstyle{\pi^{\prime}\times\mathrm{p}^{\prime}}$$\textstyle{\Sigma^{7}_{GM}\times\mathrm{S}^{1}}$$\textstyle{\mathrm{S}^{7}\times\mathrm{S}^{1}}$

where
$\mathrm{S}^{1}=\left\\{A(\theta):=\begin{pmatrix}\cos(\theta)&-\sin(\theta)\\\
\sin(\theta)&\cos(\theta)\end{pmatrix}:\theta\in[0,2\pi[\right\\}\subset\mathrm{Sp}(2)$
and $\mathrm{S}^{3}\times\mathrm{S}^{1}$ acts on
$\mathrm{Sp}(2)\times\mathrm{S}^{1}\times\mathrm{S}^{1}$ as

  1. (a)

the $\bullet$-action

$\left(\begin{pmatrix}a&c\\\
b&d\end{pmatrix},A(\theta),A(\theta^{\prime})\right)\left(\bar{q},A(\theta^{\prime\prime})\right):=\left(\begin{pmatrix}a&c\bar{q}\\\
b&d\bar{q}\end{pmatrix},A(\theta),A(\theta^{\prime})A(\theta^{\prime\prime})\right)$

  2. (b)

the $\star$-action

$(q,A(\theta^{\prime}))\left(\begin{pmatrix}a&c\\\
b&d\end{pmatrix}\right):=\left(\begin{pmatrix}qa\overline{q}&qc\\\
qb\overline{q}&qd\end{pmatrix},A(\theta^{\prime\prime})A(\theta),A(\theta^{\prime})[A(\theta^{\prime\prime})]^{-1}\right)$

The respective projections are

  1. (a)

$\pi\times\mathrm{p}\left(\left(\begin{pmatrix}a&c\bar{q}\\\
b&d\bar{q}\end{pmatrix},A(\theta),A(\theta^{\prime\prime})A(\theta^{\prime})\right)\right)=\left(\begin{pmatrix}a\\\
b\end{pmatrix},A(\theta)\right)$

  2. (b)

$\pi^{\prime}\times\bar{\mathrm{p}}\left(\left(\begin{pmatrix}qa\bar{q}&qc\\\
qb\bar{q}&qd\end{pmatrix},A(\theta^{\prime\prime})A(\theta),A(\theta^{\prime})[A(\theta^{\prime\prime})]^{-1}\right)\right)=\left(\begin{pmatrix}2\bar{c}d,|c|^{2}-|d|^{2}\end{pmatrix},A(\theta^{\prime})A(\theta)\right).$

Providing a complex structure for $\mathrm{S}^{7}\times\mathrm{S}^{1}$ is
straightforward due to the following realization. Consider
$\mathbb{C}^{4}\setminus\\{0\\}$, and let
$\mathrm{Aut}^{c}(\mathbb{C}^{4}\setminus\\{0\\})$ be the subgroup of
automorphisms of $\mathbb{C}^{4}\setminus\\{0\\}$ generated by the
transformation $z\mapsto cz$ where $|c|\neq 1$. The quotient space for this
action is precisely $\mathrm{S}^{7}\times\mathrm{S}^{1}$. We present an
equivalent construction based on diagram (2.13).

Observe that $\mathrm{S}^{7}$ is obtained from the quotient map

$\begin{pmatrix}a&c\\\ b&d\end{pmatrix}\mapsto\begin{pmatrix}a\\\
b\end{pmatrix}$

where $a,b\in\mathbb{H}\oplus\mathbb{H}\cong\mathbb{C}^{4}$. Therefore,
$\mathrm{S}^{7}\subset\mathbb{H}\oplus\mathbb{H}$, and indeed,
$\mathrm{S}^{7}\subset\mathbb{H}\oplus\mathbb{H}\setminus\\{0\\}$ is a strong
deformation retract of $\mathbb{H}\oplus\mathbb{H}\setminus\\{0\\}$.
Identifying $\mathbb{H}\oplus\mathbb{H}\cong\mathbb{C}^{4}$, we can
appropriately identify
$\mathrm{Aut}^{c}(\mathbb{C}^{4}\setminus\\{0\\})\cong\mathrm{Aut}^{c}(\mathbb{H}\oplus\mathbb{H}\setminus\\{0\\})$.
The action of $\mathrm{Aut}^{c}(\mathbb{H}\oplus\mathbb{H}\setminus\\{0\\})$
in $\mathbb{H}\oplus\mathbb{H}\setminus\\{0\\}$ is properly discontinuous, so
$\mathrm{S}^{7}\times\mathrm{S}^{1}$ inherits holomorphic coordinate charts
induced from the covering map

$\mathrm{pr}:\mathbb{H}\oplus\mathbb{H}\setminus\\{0\\}\rightarrow\mathrm{S}^{7}\times\mathrm{S}^{1}.$

Since we can find the $\mathrm{S}^{3}\times\mathrm{S}^{1}$-invariant tubular
neighborhoods given by Theorem 2.12, it suffices to determine $\Phi$ (as
appearing in the proof of Theorem 2.12) to explicitly provide a complex
structure in $\Sigma^{7}\times\mathrm{S}^{1}$.

On the one hand, to obtain $\Phi$, foremost, we need a $\star$-collection.
Fortunately, the analogous $\Phi$ for diagram (2.5) was already explicitly
determined and used in different contexts [37, 34, 92]. Explicit local
expressions for $\tilde{\Psi}$ are already computed. Let $\mathrm{D}^{7}$
stand either for $\mathrm{S}^{7}\setminus\\{N\\}$ or
$\mathrm{S}^{7}\setminus\mathrm{S}$, where $N$ is the North-pole of
$\mathrm{S}^{7}$ and $S$ its South-pole. We have

$\displaystyle\tilde{\Psi}_{\pm}:\mathrm{D}^{7}\times\mathrm{S}^{3}\rightarrow\mathrm{Sp}(2)$
$\displaystyle\Psi_{\pm}(x,g):=g\bullet\widetilde{\exp}_{\pm}(x)$

where $\widetilde{\exp}_{+}:\mathrm{Im}\leavevmode\nobreak\
\mathbb{H}\times\mathbb{H}\rightarrow\mathrm{Sp}(2)$ is the Riemannian
exponential obtained in [37, Section 4] and
$\widetilde{\exp}_{-}(\cdot):=-\widetilde{\exp}(-\cdot)$.

For $x=(\frac{1}{2}\uppi v,\frac{1}{2}\uppi
w)\in\mathrm{S}^{6}\subset\mathrm{Im}\leavevmode\nobreak\
\mathbb{H}\times\mathbb{H}$, it holds

$\tilde{\Psi}_{-}(x,1)^{-1}\circ\tilde{\Psi}_{+}(x,1)=\begin{pmatrix}1&0\\\
0&b(x)\end{pmatrix}$

where

$b(x)=\frac{w}{|w|}e^{\uppi v}\frac{\bar{w}}{|w|}.$

A global trivialization for (2.13) is given by

(2.14)
$\\{(\tilde{\Psi}_{\pm},\mathrm{Id},\mathrm{Id}):\mathrm{D}^{7}\times\mathrm{S}^{3}\times\mathrm{S}^{1}\times\mathrm{S}^{1}\rightarrow\mathrm{Sp}(2)\times\mathrm{S}^{1}\times\mathrm{S}^{1}\\}$

and the single-element $\star$-collection is explicitly given by

(2.15) $\left\\{\left(\begin{pmatrix}1&0\\\
0&b(x)\end{pmatrix},\begin{pmatrix}\mathrm{Id}&0\\\
0&\mathrm{Id}\end{pmatrix}\right)\right\\}$

The desired map $\Phi$ is locally decomposed as
$\Phi_{\pm}:=(\pi^{\prime}\circ\Psi_{\pm},\mathrm{Id},\mathrm{Id})$. The open
sets
$\\{\mathrm{D}^{7}\times\mathrm{S}^{3}\times\mathrm{S}^{1}\\}=\\{\mathrm{S}^{7}\setminus\\{N\\}\times\mathrm{S}^{3}\times\mathrm{S}^{1},\leavevmode\nobreak\
\mathrm{S}^{7}\setminus\\{S\\}\times\mathrm{S}^{3}\times\mathrm{S}^{1}\\}$ can
be seen as the open sets associated with the local holomorphic coordinates
charts of the manifold $\mathrm{S}^{7}\times\mathrm{S}^{1}$. At the same time,
$\left\\{\pi^{\prime}\circ\Psi_{\pm}\left(\mathrm{D}^{7}\times\mathrm{S}^{3}\times\mathrm{S}^{1}\times\mathrm{S}^{1}\right)\right\\}$
are the corresponding open sets for the holomorphic coordinates in
$\Sigma^{7}\times\mathrm{S}^{1}$.  $\blacksquare$

Later, in Section 3.4, we find a partial result on the relation between the
Hodge numbers on two manifolds $M,M^{\prime}$ with complex structures –
Theorem 3.15.

###  2.4. $\star$-diagrams constitute an example of Morita equivalence

A more categorical description of $\star$-diagrams can be recognized when
studied from the perspective of Lie groupoids. A crucial aspect of this
approach is that two manifolds $M,M^{\prime}$ fitting as the base manifolds on
a $\star$-diagram, shortly written as $M\leftarrow\mathcal{P}\rightarrow
M^{\prime}$, are _Morita equivalent_ 333This fact was pointed out by O. Brahic
and C. Ortiz in 2015 to L. Sperança, who communicated it to the authors. in a
precise sense. This is what we discuss next, further showing how, from the
point of view of equivariant cohomology, these two manifolds $M,M^{\prime}$
with induced $G$-actions are indistinguishable.

In what follows, we will consider two action groupoids, employing the notation
$X//G$ to mean that $X$ is a smooth manifold with a smooth action by a Lie
group $G$ and $X//G$ is thought as the Lie action groupoid.

Let $M\leftarrow\mathcal{P}\rightarrow M^{\prime}$ be a $\star$-diagram with a
structure group $G$. Recall that $M$ can be seen as a $G$-manifold with the
induced $\star$-action on it while $M^{\prime}$ is a $G$-manifold with the
restriction of the $\bullet$-action. We establish an isomorphism of action
groupoids $M//G,\leavevmode\nobreak\ M^{\prime}//G$. Such an isomorphism is
given by a _Morita equivalence_.

######  Definition 2.14 (Morita equivalence of Lie groupoids. Definition 2.25
in [50]).

Two Lie groupoids $M//G$ and $N//H$ are Morita equivalent if there exist a
smooth manifold $\mathcal{P}$, and two surjective submersions
$\pi_{M}:\mathcal{P}\rightarrow M$ and $\pi_{N}:\mathcal{P}\rightarrow N$ such
that $\pi_{M}^{*}(G)\cong\pi_{N}^{*}(H)$. More precisely, the pullback
groupoids – [50, Definition 1.12] are equivalent in $\mathcal{P}$.

To avoid introducing further details, we remark that any $\star$-diagram
fulfills Definition 2.1 in [50] considering the singular foliation induced by
the $\star$-orbits on $M$ and $\bullet$-orbits on $M^{\prime}$. Note that
$\star$-diagrams also encode what is known as a _Hausdorff–Morita equivalence_
– Corollary 2.17 in [50]. We register this in the following

######  Theorem 2.15.

Any $\star$-bundle $M\leftarrow\mathcal{P}\rightarrow M^{\prime}$ is an
example of a Hausdorff–Morita equivalence. More precisely, the singular
foliation induced by the $G$-orbits on $M$ and $M^{\prime}$ can be pulled back
to isomorphic singular foliations in $\mathcal{P}$ via the corresponding
projections.

###### Proof.

As Corollary 2.17 in [50] states, if two connected Lie groups
$G_{1},\leavevmode\nobreak\ G_{2}$ act freely and properly on a manifold
$\mathcal{P}$ with commuting actions, the singular foliation on
$\mathcal{P}/G_{1}$ given by the induced $G_{2}$-action in $\mathcal{P}/G_{1}$
and the singular foliation in $\mathcal{P}/G_{2}$ given by the induced
$G_{1}$-action on it are Morita equivalent. Our considered $\star$-diagrams
are precisely of this form with the observation that $G_{1}=G_{2}$. Then
$P/G_{1}=M$ and $P/G_{2}=M^{\prime}$. ∎

As a next observation, we shall check (Example 2.17) that if one of $\bullet$,
$\star$-actions is almost-free, i.e., every isotropy subgroup is discrete,
then the foliation induced by the connected components of the orbits of the
$\star$-action on $M$ and the $\bullet$-action on $M^{\prime}$ are examples of
a _projective foliation_ – Definition 2.16.

######  Definition 2.16 (Projective foliations).

A Riemannian manifold $(M,\operatorname{\mathsf{g}})$ with a singular
Riemannian foliation $\mathcal{F}$, for instance, the one consisting of the
collection of connected components of the orbits for an isometric action, is
termed _projective_ if there is a vector bundle $E\to M$ such that the local
sheaf of vector fields associated with $\mathcal{F}$ is recovered by
$C^{\infty}_{c}(E)$ as $C^{\infty}(M)$-modules, where $C^{\infty}_{c}(E)$
denotes the smooth sections on $E$ with compact support. In this case, $E$
acquires the structure of an _almost injective Lie algebroid_ – [28, p. 484],
[72, Chapter 3].

Example 2.17 below provides a natural class of projective foliations. This
class appears in our Theorems 2.18, 2.20. More emphatically, the to-be-
described concepts already appear in the statement of Theorem 2.18.

######  Example 2.17 (The foliation $\mathcal{F}$ induced by the connected
components of the orbit of an almost free isometric action by a Lie group is
an example of a projective foliation).

Let $X$ be a smooth manifold with an effective almost-free action by a compact
Lie group $G$, denoted by $\mu:G\times X\rightarrow X$. Denote by
$\mathfrak{g}$ the Lie algebra of $G$.

Take $E=X\times\mathfrak{g}$ and define the _anchor map_ $\rho:E\rightarrow
TX$ by

$\rho(x,v):=\mathrm{d}\mu(e,x)((v,0))=v^{*}(x),$

where $0\in T_{x}X$, $v\in\mathfrak{g}$, and $e$ is the unity element in $G$.
In other words, $\rho$ essentially computes _action (fundamental) fields_. If
the $G$-action is almost free, then one can check that $\rho$ is almost
injective (injective on an open dense set) and is an _anchor_ map between $E$
and $TM$, see Example 2, p. 496 in [28] for additional details.
$\blacksquare$

Proposition 2.7 in [50] tells us that an existing Hausdorff–Morita equivalence
between two manifolds with singular foliations ensures that if one of such
foliations is projective, the other one is also projective. Thus, we have the
following theorem:

######  Theorem 2.18.

Let $M\leftarrow\mathcal{P}\rightarrow M^{\prime}$ be a $\star$-diagram with
structure group $G$. Assume the induced $G$-action on $M$ is almost free.
Then, the induced $G$-action on $M^{\prime}$ is almost free. If $M$ is compact
and connected, there exists an almost-injective (i.e., injective on an open
dense subset) bundle morphism $\Theta$ fitting the diagram below

(2.16)
${M\times\mathfrak{g}}$${TM}$${M^{\prime}\times\mathfrak{g}}$${TM^{\prime}}$$\scriptstyle{\rho}$$\scriptstyle{(f,\mathrm{Id})}$$\scriptstyle{\Theta}$$\scriptstyle{\rho^{\prime}}$

where $f:M\rightarrow M^{\prime}$ is a bijection that is a diffeomorphism
between an open and dense subset $M^{*}\subset M$ and its image,
$\mathrm{Id}:\mathfrak{g}\rightarrow\mathfrak{g}$ is the identity; and
$\rho,\rho^{\prime}$ are as in Example 2.17.

###### Sketch of the proof.

To the existence of the map $f$, first notice that on a manifold $M$ with
effective smooth action, we can find an open and dense subset, denoted as
$M^{reg}$ enjoying the property that any two points $x,\leavevmode\nobreak\
y\in M^{reg}$ have the same _local orbit type_ (see [3, Theorem 3.82, p.75]).
On the other hand, Proposition 2.8 teaches us how to define $f$ in $M^{reg}.$
For a generic orbit $Gx$ in $M^{reg}$ we simply define $f|_{M^{reg}}(x)$ as
the image of $\Phi(x)$ (for $\Phi$ as in Proposition 2.8) with domain a
sufficiently small tubular neighborhood $\nu Gx$. Using the fact that $M$ is
compact, one can cover it with finitely many invariant tubular neighborhoods
$\\{U_{i}\\}_{i=1}^{N}$ encoding different orbit types ([3, Theorem 3.9.1,
p.78]). We pose

$f(x):=\Phi_{i}(x),\leavevmode\nobreak\ x\in U_{i}$

for $\Phi_{i}$ as in Proposition 2.8.

Lastly, our hypotheses ensure that $M$ with singular foliation induced by the
$G$-orbits is an example of a projective foliation. Since $\star$-diagrams are
an example of Hausdorff–Morita equivalence, we use Proposition 2.7 in [50] to
guarantee that the same holds for $M^{\prime}$ with the respective singular
foliation induced by the $G$-action on it. The corresponding
$\rho,\leavevmode\nobreak\ \rho^{\prime}$ “computing action fields” maps are
as in Example 2.17.

Finally, since $f$ is bijection and $\rho,\leavevmode\nobreak\ \rho^{\prime}$
are almost injective, we can define $\Theta$ via the relation

$\Theta\circ\rho=\rho^{\prime}\circ(f,\mathrm{Id}).$

A routine argument guarantees that $\Theta$ is an almost injective bundle
morphism. ∎

The equivariant cohomologies of $M$ and $M^{\prime}$ in a coefficient ring
$\Lambda$, denoted by $\mathrm{H}^{*}_{G}(M;\Lambda)$ and
$\mathrm{H}^{*}_{G}(M^{\prime};\Lambda)$, are defined as

$\displaystyle\mathrm{H}^{*}_{G}(M;\Lambda)$
$\displaystyle:=\mathrm{H}^{*}(M\times EG/\Delta(G\times G);\Lambda),$
$\displaystyle\mathrm{H}^{*}_{G}(M^{\prime};\Lambda)$
$\displaystyle:=\mathrm{H}^{*}(M^{\prime}\times EG/\Delta(G\times
G);\Lambda).$

Here, $EG$ represents any contractible space where $G$ acts freely, and
$\Delta(G\times G)$ denotes the diagonal action on $M\times EG$, akin to the
construction in the associated bundle (Section 2.7 in [49]). Alternatively, we
may use the notation $EG\times_{G}M\equiv M\times EG/\Delta(G\times G)$. The
quotient space $BG=EG/G$ is commonly called the _classifying space of $G$_.

Consider $M$ and $M^{\prime}$ equipped with $G$-invariant Riemannian metrics
$\operatorname{\mathsf{g}}_{M}$ and $\operatorname{\mathsf{g}}_{M^{\prime}}$
respectively. Let us examine the vector bundles
$\pi_{TM}\circ\rho:M\times\mathfrak{g}\rightarrow M$ and
$\pi_{TM^{\prime}}\circ\rho^{\prime}:M^{\prime}\times\mathfrak{g}\rightarrow
M^{\prime}$, each equipped with a unique (up to scale) connection metric, as
described in [49, Proposition 2.7.1, p. 97]. These connections are compatible
with $\operatorname{\mathsf{g}}_{M}$ and
$\operatorname{\mathsf{g}}_{M^{\prime}}$, ensuring that the fibers (which are
copies of $\mathfrak{g}$) are totally geodesic and flat. Let $\nabla$ and
$\nabla^{\prime}$ be the corresponding metric connections on
$\pi_{TM}\circ\rho$ and $\pi_{TM^{\prime}}\circ\rho^{\prime}$ respectively. We
can define equivariant characteristic classes for the vector bundles
$\pi_{TM}\circ\rho$ and $\pi_{TM^{\prime}}\circ\rho^{\prime}$ (using $\nabla$
and $\nabla^{\prime}$) by considering them as the characteristic classes of
the vector bundles

$\displaystyle EG\times_{G}(M\times\mathfrak{g})$ $\displaystyle\rightarrow
BG\times_{G}M,$ $\displaystyle EG\times_{G}(M^{\prime}\times\mathfrak{g})$
$\displaystyle\rightarrow BG\times_{G}M^{\prime}.$

We recall the following.

######  Lemma 2.19 (Theorem 6.8.2 in [52]).

Let $T<G$ be a maximal torus. Denote by $N(T)$ the normalizer of $T$ in $G$.
Then
$\mathrm{H}^{*}(G/N(T);\mathbb{R})\cong\mathrm{H}^{*}(\\{\mathrm{pt}\\};\mathbb{R})$.
Moreover, the Weyl group $W\equiv W(T):=N(T)/T$ acts on $M_{T}:=ET\times_{T}M$
and $M^{\prime}_{T}:=ET\times_{T}M^{\prime}$, and

$\displaystyle\mathrm{H}^{*}_{G}(M;\mathbb{R})$
$\displaystyle\cong\mathrm{H}^{*}_{T}(M;\mathbb{R})^{W},$
$\displaystyle\mathrm{H}^{*}_{G}(M^{\prime};\mathbb{R})$
$\displaystyle\cong\mathrm{H}^{*}_{T}(M^{\prime};\mathbb{R})^{W}.$

We are ready to prove the following in the possession of Lemma 2.19.

######  Theorem 2.20.

Let $M\leftarrow\mathcal{P}\rightarrow M^{\prime}$ be a $\star$-diagram with a
compact connected structure group $G$. Assume the induced $G$-action on $M$ is
effective and almost free. Then the induced $G$-action on $M^{\prime}$ is
effective and almost free, and there exists an isomorphism between the
following equivariant cohomologies

$\mathrm{H}^{*}_{G}(M;\mathbb{R})\cong\mathrm{H}^{*}_{G}(M^{\prime};\mathbb{R})\cong\mathrm{H}^{*}_{T}(M;\mathbb{R})^{W},$

where $W=N(T)/T$ is the Weyl group of any maximal torus $T<G$.

Moreover, the characteristic classes of the vector bundles

$\displaystyle EG\times_{G}(M\times\mathfrak{g})$ $\displaystyle\rightarrow
BG\times_{G}M,$ $\displaystyle EG\times_{G}(M^{\prime}\times\mathfrak{g})$
$\displaystyle\rightarrow BG\times_{G}M^{\prime},$

coincide after proper identification and are determined by the characteristic
classes of the vector bundle

$ET\times_{T}(M\times\mathfrak{t})\rightarrow BT\times_{T}M,$

where $\mathfrak{t}$ is the Lie algebra of $T$.

###### Proof.

Following Lemma 2.19, it suffices to establish the isomorphism between
$\mathrm{H}^{*}_{T}(M;\mathbb{R})^{W}$ and
$\mathrm{H}^{*}_{T}(M^{\prime};\mathbb{R})^{W}$.

Recall that
$\mathrm{H}^{*}_{T}(M;\mathbb{R}):=\mathrm{H}^{*}(ET\times_{T}M;\mathbb{R})$,
$\mathrm{H}^{*}_{T}(M^{\prime};\mathbb{R}):=\mathrm{H}^{*}(ET\times_{T}M^{\prime};\mathbb{R})$.
According to Proposition 2.8, the $T$-action on $M$ is locally equivalent with
the $T$-action on $M^{\prime}$ via equivariant local diffeomorphisms, thus
$M/T\cong M^{\prime}/T$. Similarly, if the $G$-action on $M$ is effective, the
$G$-action on $M^{\prime}$ is also effective, the same for the restriction of
the actions to $T<G$. Hence, the principal isotropy subgroup is trivial, i.e.,
the $T$-actions on $M$ and $M^{\prime}$ are principal on open and dense
subsets. Any other isotropy subgroup belongs to different components of the
singular strata.

Now, recall that $ET\times_{T}M:=(ET\times M)/T$, that is, $ET\times_{T}M$
coincides with the total space associated bundle with fiber $M$ to the
principal $T$-bundle $ET\rightarrow BT$, the quotient space is determined by
the $T$-action on $M$. Once for each of these $T$-actions it holds that
different isotropy is in correspondence with varying types of orbit, we can
use the maps in Proposition 2.8 to establish the required isomorphism between
the cohomologies, finishing the proof.

For the second part of the statement, it suffices to observe that the
equivariant characteristic classes for the vector bundles
$\pi_{TM}\circ\rho,\leavevmode\nobreak\ \pi_{TM^{\prime}}\circ\rho^{\prime}$
(defined via $\nabla,\leavevmode\nobreak\ \nabla^{\prime}$) are obtained as
the characteristic classes of the vector bundle

$\displaystyle EG\times_{G}(M\times\mathfrak{g})$ $\displaystyle\rightarrow
BG\times_{G}M,$ $\displaystyle EG\times_{G}(M^{\prime}\times\mathfrak{g})$
$\displaystyle\rightarrow BG\times_{G}M^{\prime}.$

Using the first part of the statement joint with Theorem 2.18, we have that

$\mathrm{H}^{*}(EG\times_{G}(M\times\mathfrak{g});\mathbb{R})\cong\mathrm{H}^{*}(EG\times_{G}(M^{\prime}\times\mathfrak{g});\mathbb{R})\cong\mathrm{H}^{*}(ET\times_{T}(M\times\mathfrak{t});\mathbb{R})^{W},$

finishing the proof. ∎

In the next section, we shall discuss the concepts of spherical T-duality.
Later (Theorem 2.9), we shall establish that in some cases, spherical
T-duality is nothing but a realization of a $\star$-diagram. Consequently, an
important consequence of Theorem 2.20 is that two spherical T-dual manifolds
can not be distinguished via equivariant cohomology unless we manage to
reduce, or break spherical symmetry.

##  3\. Spherical T-duality and $\star$-diagrams

In the series of works [10, 11, 12, 13], it is shown that for each pair
$(\mathcal{P},H)$ comprising a manifold $\mathcal{P}$ equipped with a free
circle action and an integral 3-cocycle $H$ on $\mathcal{P}$, there exists a
unique association with a T-dual pair $(\widehat{\mathcal{P}},\widehat{H})$.
Here, $\widehat{\mathcal{P}}$ represents a manifold with a free circle action
and a cocycle $\widehat{H}$, while the space of orbits for both circle actions
remains the same. Despite $\mathcal{P}$ and $\widehat{\mathcal{P}}$ being
generally not homeomorphic, it is proved that T-duality induces various
degree-shifting isomorphisms between different structures, such as twisted
cohomology and twisted K-theory on $\mathcal{P}$ and $\widehat{\mathcal{P}}$.
Subsequent works [51] and [27] established that T-duality also leads to
isomorphisms on Dirac structures, Courant algebroids, generalized complex
structures, and generalized Kähler structures.

Later, in [8, 9], the authors inaugurate a higher-dimension version of
T-duality, focusing on sphere bundles. Its generalization aspect can be
justified by observing the following. The presence of a free circle action on
$\mathcal{P}$ endows it with the structure of a principal
$\mathrm{U}(1)$-bundle $\mathcal{P}\rightarrow M$. To it, we can associate a
complex line bundle so that the $\mathrm{U}(1)$-bundle represents the sphere
$\mathrm{S}^{1}$-subbundle. Now consider a $\mathrm{S}^{3}$-bundle with
structure group $\mathrm{SO}(4)$. We can decouple from it a principal
$\mathrm{SO}(4)$-bundle and associate a vector bundle with quaternionic fibers
$\mathbb{H}$. A justification for studying the to-be-recalled spherical
T-duality is that of understanding _how much of this structure carries over to
the case of $\mathrm{S}^{3}\subset\mathbb{H}$ subbundles of quaternionic line
bundles?_

In this section, we delve deeper into the work presented in [8] and its
extension in [9], focusing on a comprehensive approach to geometrically
constructing spherical T-duality. Our primary objective is to establish
topological relationships between two spherical T-dual pairs and to realize
them geometrically. To achieve this, we utilize $\star$-diagrams as our
framework.

Specifically, for $\mathrm{S}^{3}$-bundles over $\mathrm{S}^{4}$, we prove
that a $\star$-diagram can realize the manifolds emerging in every spherical
T-dual pair, and vice versa (refer to Theorems 2.9 and 3.13). Consequently,
many results in [8, 9] can be derived from the fact that $\star$-diagrams
establish Morita equivalence (as shown in Theorem 2.15).

We also observe a potential correspondence between $\mathrm{S}^{1}$-bundles
fitting in corresponding $\star$-diagrams and T-duality for principal torus
bundles. While we do not establish this correspondence here, we acknowledge
this problem and leave it for formal proof and further investigation.
Understanding the feasibility of the works in [51, 27] within this context
seems promising. This is supported by the insights provided in Theorem 2.12,
indicating an explicit “transportation” of complex structures.

###  3.1. Milnor bundles: A recall

Let us recall the following definition:

######  Definition 3.1.

A sphere bundle $\mathrm{S}^{k{-}1}\hookrightarrow M\to B$ is called linear if
$\mathrm{O}(k)$ (acting in the usual way on $\mathrm{S}^{k{-}1}$) is a
structure group. Equivalently, if there is a set of transition functions
$\\{\phi_{ij}:U_{i}\cap U_{j}\to\mathrm{O}(k)\\}$.

It is worth noting that linear sphere bundles can always be suspended:
consider $s_{k}:\mathrm{O}(k)\to\mathrm{O}(k{+}1)$, the inclusion of
$\mathrm{O}(k)$ as the subgroup of $\mathrm{O}(k{+}1)$ with 1 in the upper-
left corner. If $\\{\phi_{ij}:U_{i}\cap U_{j}\to\mathrm{O}(k)\\}$ are
transition functions for $\mathrm{S}^{k-1}\hookrightarrow
M\stackrel{{\scriptstyle\pi}}{{\to}}B$, then $\\{s_{k}\phi_{ij}:U_{i}\cap
U_{j}\to\mathrm{O}(k{+}1)\\}$ are transition functions for a linear
$\mathrm{S}^{k}$-bundle over $B$.

The usual boundary map in the long homotopy sequence of the fibration $EG\to
BG$, $G=\mathrm{SO}(k)$, provides a bijection between the set of linear
$\mathrm{S}^{k-1}$-bundles over $\mathrm{S}^{l}$ and
$\pi_{n-1}\mathrm{SO}(k)$. Let $k=l=4$. For topological computations aiming to
determine $\pi_{n-1}\mathrm{SO}(4)$ for $n\geq 2$, it suffices to obtain
$\pi_{n-1}(\mathrm{S}^{3}\times\mathrm{S}^{3})$ since we have a covering map

$\Psi:\mathrm{S}^{3}\times\mathrm{S}^{3}\rightarrow\mathrm{SO}(4)$
$\mathrm{S}^{3}\times\mathrm{S}^{3}\ni(x,y)\mapsto\psi(x,y):v\mapsto
xvy^{-1},\leavevmode\nobreak\ v\in\mathbb{H}.$

Specializing for $n=4$, we obtain
$\pi_{3}(\mathrm{SO}(4))\cong\pi_{3}(\mathrm{S}^{3}\times\mathrm{S}^{3})\cong\mathbb{Z}\oplus\mathbb{Z}$.
The linear $\mathrm{S}^{3}$-bundles over $\mathrm{S}^{4}$ are usually called
_Milnor bundles_.

As in [83], for pairs of integers $(m,n)\in\pi_{3}(\mathrm{SO}(4))$, we can
build the maps $t_{mn}:\mathrm{S}^{3}\to\mathrm{SO}(4)$, which describe an
explicit isomorphism between $\pi_{3}(\mathrm{SO}(4))$ and Milnor bundles. Let

$t_{mn}(x)v=x^{m}vx^{n},\quad v\in\mathbb{H}$

be representatives of $\pi_{3}\mathrm{SO}(4)\cong\mathbb{Z}\oplus\mathbb{Z}$,
and $f_{t_{mn}}$ be defined as the diffeomorphism

$f_{t_{mn}}(x,g):=(x,gt_{mn}(x))\in\mathrm{S}^{3}\times\mathrm{S}^{3},\leavevmode\nobreak\
(x,g)\in\mathrm{S}^{3}\times\mathrm{S}^{3}.$

The manifold
$M_{m,n}=\mathrm{D}^{4}\times\mathrm{S}^{3}\cup_{f_{t_{mn}}}\mathrm{D}^{4}\times\mathrm{S}^{3}$
obtained from the above gluing along a common boundary
$\mathrm{S}^{3}\times\mathrm{S}^{3}$ of $\mathrm{D}^{4}\times\mathrm{S}^{3}$
and $\mathrm{S}^{3}\times\mathrm{D}^{4}$ gives rise to a Milnor bundle. In
[83], Milnor observed that $M_{m,n}$ is homeomorphic to $\mathrm{S}^{7}$ if,
and only if, $m+n=\pm 1$, but not diffeomorphic when $m=2$, what can be
observed due to the construction of the _$\lambda$ -invariant_

(3.1) $\lambda(M_{m,1-m})=(-1+2m)^{2}-1\pmod{7}$

It holds that $\lambda(M_{m,1-m})\neq 0$ if, and only if, $M_{m,1-m}$ is
homeomorphic but not diffeomorphic to the standard seven-dimension sphere
$\mathrm{S}^{7}$. Next, we recall that every Milnor bundle can be obtained
from some $\mathcal{P}_{n}$ in a $\star$-diagram. This was already known from
[26], we provide a different proof adapted to our purposes, see Proposition
3.2.

From [39], a Milnor bundle $M_{m,n}$ is principal if, and only if, $m=0$ or
$n=0$. Consider the $\mathrm{S}^{3}$-principal Milnor bundles
$\pi_{-n}\colon\mathcal{P}_{-n}\to\mathrm{S}^{4}$,
$\pi_{m}\colon\mathcal{P}_{m}\to\mathrm{S}^{4}$, that is,
$\mathcal{P}_{m}=M_{m,0}$ and $\mathcal{P}_{-n}=M_{-n,0}$. It can be checked
that $\pi_{m}$ can be seen as the $\star$-bundle with the $\star$-action
locally described as

(3.2) $r(x,q)=(rx\bar{r},q\bar{r}),\leavevmode\nobreak\
x\in\mathrm{S}^{4},\leavevmode\nobreak\ q,r\in\mathrm{S}^{3}.$

Now consider $\mathcal{P}_{-n}$ as the $\mathrm{S}^{3}$-manifold with a
$\mathrm{S}^{3}$-action locally described as

(3.3) $r\cdot(x,q)=(rx\bar{r},rq\bar{r}),\leavevmode\nobreak\
x\in\mathrm{S}^{4},\leavevmode\nobreak\ q,r\in\mathrm{S}^{3}.$

We prove (for another proof, check [26, Section 3.3]):

######  Proposition 3.2.

The total space $M_{m,n}$ of any given Milnor bundle can be realized as a base
on a $\star$-diagram
$\mathcal{P}_{-n}\leftarrow\mathcal{\pi}^{\ast}_{-n}(M_{m,0})\to M_{m,n}$.

###### Proof.

We apply Proposition 2.6 to the proof. Let us take the
$\mathrm{S}^{3}$-manifold $\mathcal{P}_{-n}$ with $\star$-action (3.3). It is
straightforward from equations (3.3) and (3.2) that the bundle projection map
$\pi_{-n}:\mathcal{P}_{-n}\rightarrow\mathrm{S}^{4}$ is
$\mathrm{S}^{3}$-equivariant for the action (3.3) on $\mathcal{P}_{-n}$ and
the induced $\star$-action on $\mathrm{S}^{4}$, locally described as the
first-factor-projection of the right-hand-side of equation (3.2). We can apply
Proposition 2.6 to get the desired. Last, we claim the other manifold fitting
the base on the pulledback $\star$-diagram is $M_{m,n}$.

Following Proposition 2.6, it suffices to determine the $\star$-collection
$\\{\phi_{ij}:U_{ij}\rightarrow\mathrm{S}^{3}\\}$ associated with the
$\star$-diagram
$\mathrm{S}^{4}\leftarrow\mathcal{P}_{m}\rightarrow\mathrm{S}^{4}$ to conclude
$M_{m,n}={\bigcup}_{\widehat{\phi_{ij}\circ\pi_{-n}}}\pi_{-n}^{-1}(U_{i}).$

From Section 3.1, we learn that taking $U_{0}=\mathrm{S}^{4}\setminus\\{N\\}$,
$U_{1}=\mathrm{S}^{4}\setminus\\{S\\}$ where $N$ and $S$ are the North and
South poles on $\mathrm{S}^{4}$, respectively, results in $U_{01}=U_{0}\cap
U_{1}\cong\mathrm{S}^{3}$ with
$\phi_{01}=t_{m0}:\mathrm{S}^{3}\rightarrow\mathrm{S}^{3}\subset\mathrm{SO}(4)$
given by $t_{m0}(x)v:=x^{m}v,\leavevmode\nobreak\ v\in\mathbb{H}$.
Furthermore, $\pi_{-n}^{-1}(U_{0}\cap
U_{1}\cong\mathrm{S}^{3})=\mathrm{S}^{3}\times\mathrm{S}^{3}$ and
$t_{m0}\circ\pi_{-n}|_{\pi_{-n}^{-1}(U_{01}\times\mathrm{S}^{3})}=f_{t_{mn}}$
with $f_{t_{mn}}(x,g)=(x,gt_{mn}(x))$. Therefore, we have the diagram

$M_{0,n}\leftarrow\pi^{*}_{-n}(M_{m,0})\rightarrow
U_{0}\times\mathrm{S}^{3}\bigcup_{f_{t_{mn}}}\mathrm{S}^{3}\times
U_{1}=M_{m,n},$

what finishes the proof. ∎

###  3.2. Principal Spherical T-duality

Here, we recall the definition of Spherical T-duality in the principal bundle
case, first introduced in [8]. Following the authors, we shall motivate such a
concept according to the following:

######  Theorem 3.3 (Theorem 1.1 in [8]).

Let $\pi:\mathcal{P}\to M$ be a principal $\mathrm{S}^{3}$-bundle. We have the
following exact sequence, known as the Gysin sequence of Čech cohomology
groups over the integers

(3.4)