# Equivariant formality in complex-oriented theories

Shaoyun Bai  Department of Mathematics, Columbia University, New York, NY
10027, USA<EMAIL_ADDRESS>and  Daniel Pomerleano  University of
Massachusetts, Boston, 100 William T, Morrissey Blvd, Boston, MA 02125, USA
<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a product of unitary groups and let $(M,\omega)$ be a compact
symplectic manifold with Hamiltonian $G$-action. We prove an equivariant
formality result for any complex-oriented cohomology theory $\mathbb{E}^{*}$
(in particular, integral cohomology). This generalizes the celebrated result
of Atiyah–Bott–Kirwan for rational cohomology from the 1980s. The proof does
not use classical ideas but instead relies on a recent cohomological splitting
result of Abouzaid–McLean–Smith [AMS21] for Hamiltonian fibrations over
$\mathbb{CP}^{1}.$ Moreover, we establish analogues of the “localization” and
“injectivity to fixed points” theorems for certain cohomology theories studied
by Hopkins–Kuhn–Ravenel in [HKR00]. As an application of these results, we
establish a Goresky–Kottwitz–MacPherson theorem with Morava $K$-theory
coefficients for Hamiltonian $T$-manifolds.

##  1\. Introduction

In [GKM98], Goresky, Kottwitz, and MacPherson introduced the notion of an
equivariantly formal space. Let $G$ be a compact, connected Lie group,
$T\subset G$ a maximal torus and $R$ a commutative ring. A $G$-space $M$ is
called equivariantly formal over $R$ if the Leray spectral sequence

$\displaystyle H^{p}(BG;H^{q}(M;R))=>H_{G}^{p+q}(M;R)$ (1.1)

degenerates. The main examples of equivariantly formal-spaces (c.f. [GKM98,
§14.1]) arise from the following celebrated result of Atiyah–Bott–Kirwan
([AB84, §7], [Kir84, §5]):

######  Theorem 1.1.

Let $(M,\omega)$ be a compact symplectic manifold equipped with a Hamiltonian
$G$-action. Then $M$ is equivariantly formal over any $\mathbb{Q}$-algebra
$R$.

The proof of Theorem 1.1 easily reduces to the case $G=T$ via taking Weyl
invariants. In the torus-equivariant case, the idea is to argue that the
moment map for a generic one-parameter subgroup of $T$ is equivariantly
perfect. Theorem 1.1 naturally leads one to ask what happens with integral
coefficients. The proof of this claim in these classical references relies on
the so-called Atiyah–Bott lemma ([AB83, Proposition 13.4],[Kir84, Lemma
2.18]), whose proof may fail when $R=\mathbb{Z}$ (or even $R=\mathbb{F}_{p})$
and the cohomology of the critical sets have torsion (see e.g. [AB83, §13],
[HL07, Remark 2.4], [TW99, §4]). The main result of this note is the following
somewhat surprising generalization of Theorem 1.1 to any complex-oriented
cohomology theory $\mathbb{E}^{*}$, including integral cohomology as a special
case:

######  Theorem 1.2.

Let $(M,\omega)$ be a compact symplectic manifold with Hamiltonian $G$-action.

  1. (1)

Suppose $G$ is a product of unitary groups. For any complex oriented
cohomology theory $\mathbb{E}^{*}$, the Leray spectral sequence

$\displaystyle H^{p}(BG;\mathbb{E}^{q}(M))=>\mathbb{E}_{G}^{p+q}(M)$ (1.2)

degenerates. Moreover, there is a splitting:

$\displaystyle\mathbb{E}^{*}_{G}(M)\cong\mathbb{E}^{*}(M\times BG)$ (1.3)

of $\mathbb{E}^{*}(BG)$ modules.111Contrary to the usual convention in
equivariant homotopy theory, we use the notation $\mathbb{E}_{G}^{*}(M)$ to
denote the Borel-equivariant cohomology.

  2. (2)

For general $G$, suppose that $\mathbb{E}^{*}(pt)$ is an algebra over a field
$\mathbf{k}$ for which the natural map

$\displaystyle H^{*}(BG;\mathbf{k})\to H^{*}(BT;\mathbf{k})$ (1.4)

is injective. Then the Leray spectral sequence (1.2) degenerates.

Specializing to the case of classical cohomology implies the following result,
which already appears to be new:

######  Theorem 1.3.

Let $(M,\omega)$ be a compact symplectic manifold with Hamiltonian $G$-action.

  1. (1)

If $G$ is a product of unitary groups, then $M$ is equivariantly formal over
any commutative ring $R$. In fact, there is a (non-canonical) isomorphism of
$H^{*}(BG;R)$-modules

$\displaystyle H_{G}^{*}(M;R)\cong H^{*}(M;R)\otimes_{R}H_{G}^{*}(pt;R).$
(1.5)

  2. (2)

For general $G$, let $\mathbf{k}$ be a field such that the natural map

$\displaystyle H^{*}(BG;\mathbf{k})\to H^{*}(BT;\mathbf{k})$ (1.6)

is injective and let $R$ be any $\mathbf{k}$-algebra. Then $M$ is
equivariantly formal over $R.$

######  Remark 1.4.

As far as the authors know, for $R=\mathbb{Z}$, the sharpest known results in
the literature along the lines of Theorem 1.3 (1) for a general Hamiltonian
$G$-manifold $M$ were:

  * •

the short exact sequence (not proven to be split) for cohomology groups in
degree $2$ (cf. [HL07, Theorem A.3])

$0\longrightarrow
H^{2}_{G}(pt;\mathbb{Z})\otimes_{\mathbb{Z}}H^{0}(M;\mathbb{Z})\longrightarrow
H^{2}_{G}(M;\mathbb{Z})\longrightarrow H^{2}(M;\mathbb{Z})\longrightarrow 0.$
(1.7)

  * •

for $G$ a torus, equivariant formality over $\mathbb{Z}$ was known when all
the isotropy groups are _connected_ (cf. [FP07, Theorem 5.1]).

######  Remark 1.5.

When $R$ is not a field, the splitting from (1.5) does not follow from
equivariant formality; see [FP07, Example 5.2] for a simple example of a space
which is equivariantly formal but for which no splitting of this form exists.

######  Remark 1.6.

For semi-simple Lie groups, Theorem 1.3 (2) is sharp in the following sense.
The cohomology $H^{*}(BT,\mathbf{k})$ is isomorphic to
$H_{G}^{*}(G/T,\mathbf{k})$, where $G$ acts by left-translation. This action
is Hamiltonian for an appropriate symplectic form on the flag manifold $G/T$,
and so if (1.4) is not injective, $G/T$ is not equivariantly formal over
$\mathbf{k}.$

######  Example 1.7.

We illustrate Theorem 1.3 (2) for some classical groups:

  * •

If $G=U(n),SU(n),$ or $Sp(2n)$, then the result holds over any field
$\mathbf{k}$; see e.g. [Tod87, Theorem 2.1] for the calculation of
$H^{*}(BG;\mathbf{k})$ for classical groups $G$.

  * •

If $G=SO(n)$, then the result holds for any field $\mathbf{k}$ with
$\operatorname{char}(\mathbf{k})\neq 2.$

  * •

If $G=G_{2}$, then the result holds provided
$\operatorname{char}(\mathbf{k})\neq 2$ ([AK16, Theorem 2.19]).

Theorem 1.3 may be unexpected as many of the other well-known consequences of
the Atiyah–Bott lemma are known to fail with integral coefficients (or finite
field coefficients). For example, it is easy to construct examples of
Hamiltonian $T$-manifolds $M$ for which the restriction to the fixed point
locus $j^{*}:H_{T}^{*}(M,\mathbb{Z})\to H_{T}^{*}(M^{T},\mathbb{Z})$ fails to
be injective. Indeed, we do not know how to prove Theorem 1.2 (or even Theorem
1.3) using this circle of ideas.

Instead, we deduce Theorem 1.2 from a more general result in symplectic
topology. Recall that a locally trivial fiber bundle $M\to P\to B$ with fiber
a closed symplectic manifold $(M,\omega)$ is said to be Hamiltonian if its
structure group can be reduced to the group of Hamiltonian symplectomorphisms
$\operatorname{Ham}(M,\omega)$. We prove the following cohomological splitting
result for Hamiltonian fibrations:

######  Proposition 1.8.

Let $M\to P\to B$ be Hamiltonian fibration over a base $B$ which is a product
of complex Grassmannians. Then there is a non-canonical isomorphism of
$\mathbb{E}^{*}(B)$ modules

$\displaystyle\mathbb{E}^{*}(P)\cong\mathbb{E}^{*}(M)\otimes_{\mathbb{E}^{*}(pt)}\mathbb{E}^{*}(B)$
(1.8)

The case where $B=\mathbb{CP}^{1}$ is an important recent result of
Abouzaid–McLean–Smith [AMS21] (see also [BX22] for a more direct argument in
the case of integral cohomology). Their argument is based on “hard” pseudo-
holomorphic curve methods first introduced by [LMP99]. We are able to formally
“bootstrap” this result to prove Proposition 1.8 by sharpening the “soft”
topological approach to cohomological splitting of Hamiltonian fibrations by
Lalonde–McDuff [LM03].

######  Remark 1.9.

In the introduction to their paper [AMS21], the authors suggest that one could
use the arguments of [LM03] to prove Proposition 1.8 for $B$ a product of
projective spaces and cohomology with coefficients in any _field_. The goal of
this note is to carry out and considerably generalize this idea (by
establishing cohomological splitting for Hamiltonian fibrations over more
general $B$ and with coefficients in any complex-oriented cohomology theory)
as well as to make the connection to classical concepts in equivariant
symplectic geometry.

To finish this note, we include two applications of our equivariant formality
result. Our applications both concern a class of cohomology theories studied
by Hopkins–Kuhn–Ravenel in [HKR00] (cf. Definition 3.1). For the first
application, we establish an Atiyah–Bott style localization theorem (Lemma
3.6) for HKR theories which are futhermore integral and of finite Krull
dimension in the sense of Definition 3.3. Combining this localization theorem
with our equivariant formality result, we deduce the following:

######  Theorem 1.10.

Let $(M,\omega)$ be a compact symplectic manifold with Hamiltonian $T$-action.
Let $\mathbb{E}^{*}$ be an HKR cohomology theory which is integral and of
finite Krull dimension. Suppose that $\mathbb{E}^{*}(M)$ is free as a module
over the ground ring $\mathbb{E}_{*}$. Then the restriction map on equivariant
cohomology

$\displaystyle j^{*}:\mathbb{E}_{T}^{*}(M)\to\mathbb{E}_{T}^{*}(M^{T})$ (1.9)

is injective. In particular, if $\mathbb{E}_{*}$ is a graded field, $j^{*}$ is
injective for any compact symplectic manifold $M$ with Hamiltonian $T$-action.

This is in contrast to the aforementioned failure of injectivity of $j^{*}$
for ordinary cohomology with finite field coefficients. The ingredients in the
proof can further be used to prove the Atiyah–Bott “integration formula” with
$\mathbb{E}^{*}$-coefficients, which is completely analogous to the classical
statement for the rational cohomology; see Theorem 3.7.

The second application concerns one of the most interesting classes of HKR
theories, the Morava $K$-theories $K_{p}(n)$. For $\mathbb{E}=K_{p}(n)$, we
can prove the following Goresky–Kottwitz–MacPherson type theorem which
calculates the (equivariant) Morava $K$-theories of GKM spaces.

Let $(M,\omega)$ be a compact symplectic manifold with Hamiltonian torus
action by $T=(S^{1})^{m}$. It is said to satisfy the _GKM conditions_ if $T$
acts with only finitely many fixed points $x_{1},\dots,x_{k}$, and the union
of $T$-orbits of corank at most $1$ is a union of invariant two-spheres
$\overline{E}_{1},\dots,\overline{E}_{l}$. Each $\overline{E}_{j}$ is
isomorphic to
$S^{2}=\mathbb{C}^{\times}\cup\\{x_{j_{0}}\\}\cup\\{x_{j_{\infty}}\\}=E_{j}\cup\\{x_{j_{0}}\\}\cup\\{x_{j_{\infty}}\\}$
for some $x_{j_{0}},x_{j_{\infty}}\in\\{x_{1},\dots,x_{k}\\}$, and the action
of $T$ on $E_{j}$ factors through a group homomorphism $\Theta_{j}:T\to S^{1}$
where $S^{1}$ rotates the sphere $\overline{E}_{j}$ with fixed points
$x_{j_{0}}$ and $x_{j_{\infty}}$.

######  Theorem 1.11.

Let $(M,\omega)$ be a compact symplectic manifold with Hamiltonian torus
action satisfying the GKM conditions and let $\mathbb{E}^{*}=K_{p}(n)$. The
restriction to fixed points

$\mathbb{E}^{*}_{T}(M)\to\bigoplus_{x_{i},1\leq i\leq
k}\mathbb{E}^{*}(pt)[\\![u_{1},\dots,u_{m}]\\!]$ (1.10)

is injective. Furthermore, the image is given by

$\big{\\{}(f_{1},\dots,f_{k})\quad|\quad\forall 1\leq j\leq l,\quad
f_{j_{0}}|_{\mathbb{E}^{*}_{\ker(\Theta_{j})}}=f_{j_{\infty}}|_{\mathbb{E}^{*}_{\ker(\Theta_{j})}}\big{\\}},$
(1.11)

where the restriction $|_{\mathbb{E}^{*}_{\ker(\Theta_{j})}}$ denotes the
homomorphism $\mathbb{E}^{*}_{T}(pt)\to\mathbb{E}^{*}_{\ker(\Theta_{j})}(pt)$
induced by the inclusion $\ker(\Theta_{j})\hookrightarrow T$ of the kernel of
$\Theta_{j}$.

This statement reduces the calculation of the cohomology rings
$\mathbb{E}^{*}(M)$ and $\mathbb{E}^{*}_{T}(M)$ to the combinatorics of the
$T$-invariant $2$-spheres. We leave the applications of this circle of ideas
to topology and enumerative geometry for future investigations.

### Acknowledgements

We thank David Anderson, Yael Karshon, Allen Knuston, and Che Shen for sharing
their expertise on Bott–Samelson resolutions. D.P. was partially funded by NSF
grant DMS-2306204 while working on this note.

##  2\. Complex-oriented splitting

###  2.1. Complex-oriented theories

Suppose $\mathbb{E}^{*}$ is a complex-oriented cohomology theory with
coefficient ring $\mathbb{E}_{*}$. Recall that this means that there exists a
class $u\in\mathbb{E}^{2}(BS^{1})$ that restricts to a generator of the
reduced cohomology, $\tilde{\mathbb{E}}^{*}(\mathbb{CP}^{1})$ (a free rank one
$\mathbb{E}_{*}$–module). Such a class $u$ is called a complex orientation of
$\mathbb{E}^{*}$. The universal example of such a cohomology theory is complex
cobordism $MU^{*}$. For a multiplicative cohomology theory $\mathbb{E}^{*}$,
complex orientations are in bijective correspondence with multiplicative
natural transformations to $\mathbb{E}^{*}$ from complex cobordism $MU^{*}$. A
choice of orientation $u\in\mathbb{E}^{2}(BS^{1})$ gives rise to isomorphisms

$\displaystyle\mathbb{E}^{*}(BS^{1})=\mathbb{E}_{*}[\\![u]\\!],\quad\mathbb{E}^{*}(B(S^{1})^{m})=\mathbb{E}_{*}[\\![u_{1},u_{2},\cdots,u_{m}]\\!].$
(2.1)

It follows from this that the complex orientation of $\mathbb{E}^{*}$ has an
associated formal group law. Let $\mu:BS^{1}\times BS^{1}\to BS^{1}$ denote
the universal classifying map for tensor product of line bundles. The formal
group law is by definition the pull-back of $u$ under this classifying map:

$\displaystyle
u_{1}+_{L_{\mathbb{E}}}u_{2}=L_{\mathbb{E}}(u_{1},u_{2})=\mu^{*}(u)\in\mathbb{E}_{*}[\\![u_{1},u_{2}]\\!].$
(2.2)

From the formal group law, one can extract the $\ell$-series, which is given
by:

$\displaystyle[\ell]\cdot_{\mathbb{E}}u=\underbrace{u+_{L_{\mathbb{E}}}u+_{L_{\mathbb{E}}}\cdots+_{L_{\mathbb{E}}}u}_{l\text{
times}}\in\mathbb{E}_{*}[\\![u]\\!].$ (2.3)

We also record the following technical fact which will be used in the sequel:

######  Lemma 2.1.

Let $\mathbb{E}^{*}$ be a complex-oriented cohomology theory and let $B$ be a
finite-type CW complex (meaning $B$ has finitely many cells in each degree)
with only even cells. The Atiyah–Hirzebruch spectral sequence

$\displaystyle H^{p}(B,\mathbb{E}^{q}(pt))=>\mathbb{E}^{p+q}(B)$ (2.4)

degenerates.

###### Proof.

Because the cohomology groups $H^{p}(B,\mathbb{Z})$ are free, the $E_{2}$ page
of the spectral sequence can be identified with

$\displaystyle E_{2}^{p,q}=H^{p}(B)\otimes\mathbb{E}^{q}(pt).$ (2.5)

The spectral sequence is linear over $\mathbb{E}^{q}(pt)$ and it suffices to
prove that

$\displaystyle d^{r}(\alpha\otimes 1)=0\text{ for all }\alpha\in H^{*}(B).$
(2.6)

If $\mathbb{E}^{*}=MU^{*}$ is complex cobordism the spectral sequence
collapses because the $E_{2}$ page is concentrated in even degree. In general,
the complex orientation of $\mathbb{E}^{*}$ is induced by a natural
transformation (of multiplicative cohomology theories)
$MU^{*}=>\mathbb{E}^{*}$ and the Atiyah-Hirzebruch sequence is natural in the
cohomology theory. The result therefore follows from the case of complex
cobordism. ∎

###  2.2. Splittings

Next, we discuss cohomological splittings with $\mathbb{E}^{*}$ coefficients,
where $\mathbb{E}^{*}$ is a complex-oriented cohomology theory.

######  Definition 2.2.

Suppose that we have a fibration $M\xrightarrow{i}P\xrightarrow{\pi}B$. A
fibration is _c-split_ if there exists a map of $\mathbb{E}_{*}$ modules
$s:\mathbb{E}^{*}(M)\to\mathbb{E}^{*}(P)$ so that $i^{*}\circ
s=\operatorname{id}.$ We will refer to $s$ as a _c-splitting_.

######  Definition 2.3.

A smooth manifold $B$ is called _Hamiltonian c-split_ for $\mathbb{E}^{*}$ if
every Hamiltonian fibration $M\xrightarrow{i}P\xrightarrow{\pi}B$ is c-split.

We suppress the cohomology theory $\mathbb{E}$ when it is clear from the
context. The key result for us is the following result of
Abouzaid–McLean–Smith:

######  Theorem 2.4 ([AMS21]).

The one-dimensional projective space, $\mathbb{CP}^{1}$, is Hamiltonian
c-split.

######  Remark 2.5.

When $\mathbb{E}^{*}$ is ordinary cohomology with coefficients in a
commutative ring $R$, the first author and Xu ([BX22]) have given a more
direct proof of Theorem 2.4 which avoids the chromatic homotopy theory
arguments from [AMS21].

The following two Lemmas are refinements of arguments in [LM03] (cf. Corollary
4.2 and Lemma 4.3 of _loc. cit._).

######  Lemma 2.6.

Let $\tau^{B}:\tilde{B}\to B$ be a birational map of smooth projective
varieties. If $\tilde{B}$ is Hamiltonian c-split, then so is $B$.

###### Proof.

Suppose that we have a Hamiltonian fibration
$M\xrightarrow{i}P\xrightarrow{\pi}B$ over $B$ and let
$M\xrightarrow{\tilde{i}}\tilde{P}\xrightarrow{\tilde{\pi}}\tilde{B}$ be the
pull-back Hamiltonian fibration, which admits a map $\tau^{P}:\tilde{P}\to P.$
We can factor $\tau^{B}$ as an inclusion followed by a projection
$\tilde{B}\to\mathbb{CP}^{r}\times B\to B$ and similarly for $\tau^{P}$. This
shows that the map $\tau^{P}$ is compatible with complex-orientations. Set
$U\subset B$ to be the locus over which $\tau^{B}$ is an isomorphism and let
$P_{U}$ denote the restriction of the bundle to this locus. Note that we can
assume that our reference fiber $M_{b}$ for both $\tilde{P}$ and $P$ lies over
a point $b\in U$. We have a Cartesian diagram of complex-oriented
manifolds/maps:

$\textstyle{P_{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{j}}$$\textstyle{\tilde{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau^{P}}$$\textstyle{P_{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{P.}$

For such Cartesian squares, the Umkëhr maps on cohomology are compatible with
pullback/restriction maps; see [Qui71, 1.7] for complex cobordism, [BE90,
E(iii)] in general, and [CJ98, Proposition 12.9] for a statement at the level
of spectra, which implies the statement for cohomology by taking homotopy
groups. In the present case this means that:

$\displaystyle j^{*}\circ\tau^{P}_{*}=\tilde{j^{*}}\text{ and hence
}i^{*}\circ\tau^{P}_{*}=\tilde{i}^{*}.$ (2.7)

Suppose we are given a splitting
$\tilde{s}:\mathbb{E}^{*}(M)\to\mathbb{E}^{*}(\tilde{P})$ such that
$(\tilde{i})^{*}\circ\tilde{s}=\operatorname{id}$. Then it follows from (2.7)
that

$\displaystyle
s=\tau^{P}_{*}\circ\tilde{s}:\mathbb{E}^{*}(M)\to\mathbb{E}^{*}(P)$ (2.8)

satisfies $i^{*}\circ s=\operatorname{id}.$ ∎

######  Lemma 2.7.

Suppose that we have a Hamiltonian fibration
$M\xrightarrow{i}P\xrightarrow{\pi}B$ with $B$ simply connected. Suppose that
$M$ and $B$ are Hamiltonian c-split. Then $P$ is also Hamiltonian c-split.

###### Proof.

Let $F\to E\to P$ be a Hamiltonian bundle over $P$. After restricting to $M$
we get a Hamiltonian bundle $F\to W\to M$ which by assumption gives rise to a
c-splitting:

$\displaystyle s_{M}:\mathbb{E}^{*}(F)\to\mathbb{E}^{*}(W).$ (2.9)

The composite bundle $W\to E\to B$ is also Hamiltonian [LM03, Lemma 3.9] and
thus gives rise to a c-splitting:

$\displaystyle s_{B}:\mathbb{E}^{*}(W)\to\mathbb{E}^{*}(E).$ (2.10)

The desired c-splitting for $F\to E\to P$ is the map $s=s_{B}\circ s_{M}$. ∎

######  Lemma 2.8.

Let $B$ be a product of complex projective spaces. Then $B$ is Hamiltonian
c-split.

###### Proof.

In view of Lemma 2.7, it suffices to consider the case where
$B=\mathbb{CP}^{k}$ is a single projective space. We proceed by induction on
$k$. The base case where $k=1$ is Theorem 2.4. Consider the blow-up
$\tilde{B}$ of $\mathbb{CP}^{k}$ at a point. Then $\tilde{B}$ can be
identified with the $\mathbb{CP}^{1}$-bundle:

$\displaystyle\mathbb{P}(\mathcal{O}_{\mathbb{CP}^{k-1}}\oplus\mathcal{O}_{\mathbb{CP}^{k-1}}(-1))\to\mathbb{CP}^{k-1}.$
(2.11)

The inductive hypothesis together with Lemma 2.7 shows that $\tilde{B}$ is
Hamiltonian c-split. It then follows from Lemma 2.6 that $\mathbb{CP}^{k}$ is
c-splitting. ∎

More generally, we have the following result:

######  Lemma 2.9.

Let $B$ be a product of Grassmannians. Then $B$ is Hamiltonian c-split.

###### Proof.

Again, it suffices to consider a single Grassmannian. Recall that the Schubert
varieties of the complete flag variety $GL(n,{\mathbb{C}})/B$ are indexed by
elements $w$ of the symmetric group $S_{n}$. If we consider the Schubert
variety $Z_{w}$ corresponding to the permutation
$w(i)=i+k(\operatorname{mod}n)$, this maps birationally onto the complex
Grassmannian $\mathrm{Gr}(k,n)$ via the projection
$GL(n,{\mathbb{C}})/B\to\mathrm{Gr}(k,n)$ (see discussion above Example 1.2.3
in [Bri05]). Now take the Bott–Samelson variety $Z_{\underline{w}}$ associated
to a reduced decomposition $\underline{w}$ of $w$. By e.g. [Bri05, Proposition
2.2.1 (iii), (iv)], this is an iterated fibration of Kähler manifolds

$\displaystyle M_{k}\to M_{k-1}\to\cdots\to M_{1}=\mathbb{CP}^{1}$ (2.12)

with fiber $\mathbb{CP}^{1}$, which maps birationally onto $Z_{w}$ and hence
onto $\mathrm{Gr}(k,n)$. The statement again follows from Lemma 2.6 and Lemma
2.7. ∎

###  2.3. Proof of Theorems

We now turn to giving our main arguments. As in previous sections,
$\mathbb{E}^{*}$ will denote a complex-oriented cohomology theory. We will
make use of the following technical variant of the Leray–Hirsch theorem:

######  Lemma 2.10.

Suppose that we have a c-split fibration
$M\xrightarrow{i}P\xrightarrow{\pi}B$, where $B$ is a finite type CW-complex
with only even dimensional cells. Then the following hold:

  1. (1)

If $B$ is finite, the natural map

$\displaystyle\phi:\mathbb{E}^{*}(M)\otimes_{\mathbb{E}_{*}}\mathbb{E}^{*}(B)\to\mathbb{E}^{*}(P)$
(2.13)

is an isomorphism.

  2. (2)

For general $B$ as above, the c-splitting induces an isomorphism

$\displaystyle\mathbb{E}^{*}(M\times B)\cong\mathbb{E}^{*}(P).$ (2.14)

of $\mathbb{E}^{*}(B)$-modules.

###### Proof.

_Part (1):_ We follow the proof of the Leray–Hirsch theorem in [Hat02, Theorem
4D.1]. We assume that $B$ has dimension $m$ and proceed by induction on $m$.
We let $B^{\prime}$ be the subspace obtained by deleting a point $x_{\alpha}$
from each $m$ cell $U_{\alpha}$ and let $P^{\prime}$ denote the restriction of
the fibration to $B^{\prime}$. Then $M\to P^{\prime}\to B^{\prime}$ is also a
c-split fibration. We let $V_{\alpha}\subset U_{\alpha}$ be small disc about
$x_{\alpha}$ over which the fibration is trivial.

We note that the cohomologies $\mathbb{E}^{*}(B,B^{\prime})$,
$\mathbb{E}^{*}(B)$, $\mathbb{E}^{*}(B^{\prime})$ are all free over
$\mathbb{E}_{*}$. For $\mathbb{E}^{*}(B)$, $\mathbb{E}^{*}(B^{\prime})$ this
follows from Lemma 2.1, together with the fact that $B^{\prime}$ retracts onto
the $m-2$ skeleton. For $\mathbb{E}^{*}(B,B^{\prime})$, this follows form the
excision isomorphism
$\mathbb{E}^{*}(B,B^{\prime})\cong\mathbb{E}^{*}(V_{\alpha},V_{\alpha}-x_{\alpha}).$
Lemma 2.1 implies that the maps
$\mathbb{E}^{*}(B)\to\mathbb{E}^{*}(B^{\prime})$ are surjections. It follows
from the long exact sequence for the pair $(B,B^{\prime})$ that the connecting
maps

$\displaystyle\delta:\mathbb{E}^{*}(B^{\prime})\to\mathbb{E}^{*+1}(B,B^{\prime})\text{
are zero.}$ (2.15)

This means that the long exact sequence gives rise to a short exact sequence:

$\displaystyle
0\to\mathbb{E}^{*}(B,B^{\prime})\to\mathbb{E}^{*}(B)\to\mathbb{E}^{*}(B^{\prime})\to
0$ (2.16)

Because all of these groups are finite free over $\mathbb{E}_{*}$, we learn
that

$\displaystyle
0\to\mathbb{E}^{*}(B,B^{\prime})\otimes_{\mathbb{E}_{*}}\mathbb{E}^{*}(M)\to\mathbb{E}^{*}(B)\otimes_{\mathbb{E}_{*}}\mathbb{E}^{*}(M)\to\mathbb{E}^{*}(B^{\prime})\otimes_{\mathbb{E}_{*}}\mathbb{E}^{*}(M)\to
0$ (2.17)

is also exact. Let
$(\mathbb{E}^{*}(B,B^{\prime})\otimes_{\mathbb{E}_{*}}\mathbb{E}^{*}(M))^{q}$
denote the degree $q$ piece of the tensor product and similarly for the other
groups. Then there is a commutative diagram:

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{E}^{*}(B,B^{\prime})\otimes_{\mathbb{E}_{*}}\mathbb{E}^{*}(M))^{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{(\mathbb{E}^{*}(B)\otimes_{\mathbb{E}_{*}}\mathbb{E}^{*}(M))^{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{(\mathbb{E}^{*}(B^{\prime})\otimes_{\mathbb{E}_{*}}\mathbb{E}^{*}(M))^{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{E}^{q}(P,P^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{E}^{q}(P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{E}^{q}(P^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

The top row is exact by (2.17) and (2.15) and the commutativity of the diagram
follows as in the proof of [Hat02, Theorem 4D.1]. It is then easy to establish
that
$\phi:\mathbb{E}^{*}(B,B^{\prime})\otimes_{\mathbb{E}_{*}}\mathbb{E}^{*}(M)\to\mathbb{E}^{*}(P,P^{\prime})$
is an isomorphism by retracting everything on to $V_{\alpha}.$ Because
$B^{\prime}$ retracts onto the $m-2$ skeleton, the induction hypothesis
implies that the map
$\phi:\mathbb{E}^{*}(B^{\prime})\otimes_{\mathbb{E}_{*}}\mathbb{E}^{*}(M)\to\mathbb{E}^{*}(P^{\prime})$
is also an isomorphism. The claim now follows from the five-lemma.

_Part (2):_ We let $B^{m}$ denote the $m$-skeleton of $B$ and let $P^{m}\to
B^{m}$ denote the restriction of $P$ to $B^{m}$. Using the splitting $s$, we
can identify

$\displaystyle\phi^{m}:\mathbb{E}^{*}(M)\otimes_{\mathbb{E}_{*}}\mathbb{E}^{*}(B^{m})\to\mathbb{E}^{*}(P^{m})$
(2.18)

In particular, all of the restriction maps
$\mathbb{E}^{*}(P^{m+1})\to\mathbb{E}^{*}(P^{m})$ are surjective. Applying the
Milnor exact sequence together with the Mittag-Leffler property, we obtain
isomorphisms:

$\displaystyle\varprojlim_{m}\mathbb{E}^{*}(M)\otimes_{\mathbb{E}_{*}}\mathbb{E}^{*}(B^{m})\cong\varprojlim_{m}\mathbb{E}^{*}(P^{m})\cong\mathbb{E}^{*}(P).$
(2.19)

Finally, we note that the same argument shows that the inverse system
$\varprojlim_{m}\mathbb{E}^{*}(M)\otimes_{\mathbb{E}_{*}}\mathbb{E}^{*}(B^{m})$
is canonically isomorphic to $\mathbb{E}^{*}(M\times B)$, proving the claim.

∎

######  Remark 2.11.

One can also use standard spectral sequence arguments to give a proof of Lemma
2.10.

######  Corollary 2.12.

Suppose that we have a Hamiltonian fibration
$M\xrightarrow{i}P\xrightarrow{\pi}B$, where $B$ is a product of
Grassmannians. Then there is a non-canonical isomorphism of
$\mathbb{E}^{*}(B)$ modules:

$\displaystyle\mathbb{E}^{*}(M)\otimes_{\mathbb{E}_{*}}\mathbb{E}^{*}(B)\to\mathbb{E}^{*}(P)$
(2.20)

###### Proof.

This follows immediately by combining Lemma 2.8 and Lemma 2.10. ∎

######  Lemma 2.13.

Suppose that we have a c-split fibration
$M\xrightarrow{i}P\xrightarrow{\pi}B$, where $B$ is a finite type CW-complex
with only even dimensional cells. Let $i:B_{0}\subset B$ be a sub-complex,
$P_{0}\to B_{0}$ denote the restriction of the fibration to $B_{0}$, and
$\iota:P_{0}\to P$ the inclusion. Given a fixed c-splitting
$s_{0}:\mathbb{E}^{*}(M)\to\mathbb{E}^{*}(P_{0})$, we can find a c-splitting
$s:\mathbb{E}^{*}(M)\to\mathbb{E}^{*}(P)$ such that

$s_{0}=\iota^{*}\circ s.$ (2.21)

###### Proof.

Let $\tilde{s}$ be the c-splitting over $B$. This gives rise to a commutative
diagram:

$\textstyle{\phi:\mathbb{E}^{*}(M)\otimes_{\mathbb{E}_{*}}\mathbb{E}^{*}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\otimes
i^{*}}$$\textstyle{\mathbb{E}^{*}(P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{*}}$$\textstyle{\phi:\mathbb{E}^{*}(M)\otimes_{\mathbb{E}_{*}}\mathbb{E}^{*}(B_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{E}^{*}(P_{0})}$

where the horizontal arrows are isomorphisms. Thus, because
$\mathbb{E}^{*}(B)$, $\mathbb{E}^{*}(B_{0})$ are free, we may choose a
splitting of $\mathbb{E}_{*}$ modules
$\mathbb{E}^{*}(B_{0})\to\mathbb{E}^{*}(B)$ for $i^{*}$. This defines a
splitting $\tau:\mathbb{E}^{*}(P_{0})\hookrightarrow\mathbb{E}^{*}(P)$ of
$\mathbb{E}_{*}$\- modules for $\iota^{*}.$ We can thus split the cohomology
$\mathbb{E}^{*}(P)$ as

$\displaystyle\mathbb{E}^{*}(P)\cong\tau(\mathbb{E}^{*}(P_{0}))\oplus\operatorname{ker}(\iota^{*})$
(2.22)

We then define a new splitting $s:\mathbb{E}^{*}(M)\to\mathbb{E}^{*}(P)$ using
$s_{0}$ by letting $s=(\tau\circ s_{0},0)$ in this direct sum decomposition.
By construction $s$ satisfies (2.21).

∎

_Proof of Theorem 1.2:_ _Part (1):_ Recall that there is a convenient model
for the total space of the universal $U(k)$-bundle, $EU(k)$, given by taking
the set of orthonormal $k$-frames in a complex Hilbert space $\mathcal{H}.$
Topologically, this is the colimit of spaces of $k$-frames in $\mathbb{C}^{n}$
as $n\to\infty$:

$EU(k)_{1}\subset\cdots\subset EU(k)_{n}\subset\cdots.$

The group $U(k)$ acts freely on this space and the quotient is the infinite
Grassmannian of $k$-planes $Gr_{k}(\mathcal{H}).$ Given a product of unitary
groups $G=\prod U(k_{i})$, we can similarly take $EG:=\prod EU(k_{i}).$ The
classifying space for $G$, $BG$, therefore admits finite dimensional
approximations by spaces $BG_{n}$ which are products of Grassmannians. We set
$M_{G}:=M\times_{G}EG$ and $M_{G,n}:=M\times_{G}EG_{n}$. In view of Lemma 2.9,
we have c-splittings $s_{n}:\mathbb{E}^{*}(M)\to\mathbb{E}^{*}(M_{G,n})$ which
by Corollary 2.12 give rise to identifications:

$\displaystyle\mathbb{E}^{*}(M_{G,n})\cong\mathbb{E}^{*}(M)\otimes\mathbb{E}^{*}(BG_{n}).$
(2.23)

The identifications from (2.23) imply that the restrictions
$\mathbb{E}^{*}(M_{G,n+1})\to\mathbb{E}^{*}(M_{G,n})$ are surjective and we
have that

$\displaystyle\mathbb{E}^{*}(M_{G})\cong\varprojlim_{n}\mathbb{E}^{*}(M_{G,n}).$
(2.24)

By Lemma 2.13, we can arrange that the c-splittings $s_{n}$ are compatible
with restriction and thus give rise to a splitting:

$\displaystyle s:\mathbb{E}^{*}(M)\to\mathbb{E}^{*}(M_{G}).$ (2.25)

Using Lemma 2.10 (ii) (by construction the model for $BG$ has only even-
dimensional cells), we obtain that

$\displaystyle\mathbb{E}^{*}(M_{G})\cong\mathbb{E}^{*}(M\times BG).$ (2.26)

It remains to prove that the Leray-spectral sequence
$\\{E_{r}^{p,q}(M_{G})\\}$ for the fibration $M_{G}\to BG$ degenerates. It
follows easily from (2.23) that the Leray spectral sequences
$\\{E_{r}^{p,q}(M_{G,n})\\}$ associated to the fibrations $M_{G,n}\to BG_{n}$
degenerate. The $E_{2}$ page $E_{2}^{p,q}(M_{G})$ is an inverse limit of the
$E_{2}$ pages $E_{2}^{p,q}(M_{G,n})$, which implies that the differential
vanishes on $E_{2}^{p,q}(M_{G})$. The same argument then implies that the
differentials on the subsequent pages $E_{r}^{p,q}(M_{G})$ are zero, which
proves degeneration. _Part (2):_ We let $EG$ be any CW-complex on which $G$
(and hence $T\subset G$) acts freely. We again set $M_{T}=M\times_{T}EG$ and
$M_{G}:=M\times_{G}EG$ and consider the diagram

$\textstyle{M_{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\textstyle{M_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BT\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BG}$

We let $\\{E_{r}^{p,q}(M_{T})\\}$ and $\\{E_{r}^{p,q}(M_{G})\\}$ be the Leray
spectral sequences associated to the fibrations over the respective Borel
spaces $BT$ and $BG$. By functoriality of the Leray spectral sequence, $r$
induces maps $r^{*}:E_{r}(M_{G})\to E_{r}(M_{T}).$ We can identify the $E_{2}$
pages of the spectral sequence with
$E_{2}(M_{G})=\mathbb{E}^{*}(M)\otimes_{\mathbf{k}}H^{*}(BG,\mathbf{k})$ and
$E_{2}(M_{T})=\mathbb{E}^{*}(M)\otimes_{\mathbf{k}}H^{*}(BT,\mathbf{k})$
respectively. By hypothesis $r^{*}$ is injective on the $E_{2}$ pages and
hence there are no differentials on $E_{2}(M_{G}).$ The same argument shows
there are therefore no differentials on subsequent pages and thus the spectral
sequence $E_{r}^{p,q}(M_{G})=>\mathbb{E}^{*}(M_{G})$ degenerates at the second
page as required. ∎

######  Remark 2.14.

We note that in case of classical cohomology, the argument for Part (1)
actually simplifies a little bit. This is because we can choose $BG_{n}$ large
enough so that the restriction map

$\displaystyle j^{*}:H^{d}(M_{G})\to H^{d}(M_{G,n})\text{ is an isomorphism in
all degrees }d\leq\operatorname{dim}(M).$ (2.27)

By Lemma 2.9, we can produce a c-splitting $\bar{s}:H^{*}(M)\to
H^{*}(M_{G,n})$ for the fibration $M_{G,n}\to BG_{n}.$ Then by composing with
the inverse maps to (2.27), we get a c-splitting $s:H^{*}(M)\to H^{*}(M_{G})$.

We have the following Corollary.

######  Corollary 2.15.

Let $T\cong(S^{1})^{m}$ be a torus and $M$ a symplectic manifold with
Hamiltonian $T$-action. There is an isomorphism of
$\mathbb{E}^{*}(BT)\cong\mathbb{E}_{*}[\\![u_{1},\cdots,u_{m}]\\!]$ modules:

$\displaystyle\mathbb{E}^{*}_{T}(M)\cong\mathbb{E}^{*}(M)[\\![u_{1},\cdots,u_{m}]\\!].$
(2.28)

###### Proof.

By equivariant formality, $\mathbb{E}^{*}_{T}(M)\cong\mathbb{E}^{*}(M\times
BT).$ We also have an identification $\mathbb{E}^{*}(M\times
BT)\cong\mathbb{E}^{*}(M)[\\![u_{1},\cdots,u_{m}]\\!]$ from which the result
follows. ∎

##  3\. Localization and GKM

###  3.1. HKR theories

We next turn to discussing a particular class of complex-oriented cohomology
theories. Throughout the discussion, we fix a prime number $p$. We let
$\mathbb{Z}_{p}$ denote the $p$-adic integers and let
$\mathbb{Z}_{(p)}=\mathbb{Z}_{p}\cap\mathbb{Q}.$ Suppose we have complex-
oriented cohomology theory $\mathbb{E}^{*}$ where $\mathbb{E}_{*}$ is a local
ring with graded maximal ideal $\mathfrak{m}$ such that $p\in\mathfrak{m}$.
For such $\mathbb{E}^{*}$, the mod $\mathfrak{m}$ reduction, $L_{0}$, of
$L_{\mathbb{E}}$ is a formal group over a graded field of characteristic
$p>0$. The height of such a cohomology theory is the height of the formal
group law $L_{0}$, which we denote by $n$.

######  Definition 3.1.

We say that a complex-oriented cohomology theory $\mathbb{E}^{*}$ is of _HKR-
type_ if $\mathbb{E}_{*}$ is a Noetherian, complete, local ring with (graded-)
maximal ideal $\mathfrak{m}$ such that $p\in\mathfrak{m}$ and $\mathbb{E}^{*}$
is of finite height $n<\infty$.

Cohomology theories of this kind were considered by Hopkins–Kuhn–Ravenel
[HKR00, §5]. Many interesting examples of such theories arise from
Johnson–Wilson cohomology $E(n)^{*}$, which is a cohomology theory with
coefficients:

$\displaystyle
E(n)_{*}=\mathbb{Z}_{(p)}[v_{1},\cdots,v_{n-1},v_{n},v_{n}^{-1}],$ (3.1)

where the periodic variable $v_{n}$ plays a special role and has degree
$|v_{n}|=2(p^{n}-1)$. From $E(n)^{*}$, one can construct the following
examples of HKR-type cohomology theories: :

  1. (1)

The $I_{n}$-adically complete version of $E(n)^{*}$, $\hat{E}(n)^{*}$,
([HKR00, §1.3]) is a cohomology theory with coefficients the completion of
$E(n)^{*}$ at the ideal

$I_{n}=(p,v_{1},\cdots,v_{n-1}).$

  2. (2)

The integral lifts of Morava $K$-theory $\hat{K}(n)$ arise by taking the
$p$-completion of the quotient $E(n)/(v_{1},\cdots,v_{n-1}).$ See [GS96, §1.3]
for more details. These theories have coefficients given by:

$\displaystyle\hat{K}(n)_{*}=\mathbb{Z}_{p}[v_{n},v_{n}^{-1}].$ (3.2)

  3. (3)

Finally the Morava $K$-theories $K(n)$ are the cohomology theories which arise
by further quotienting by $p$. These cohomology theories have coefficients
given by

$\displaystyle K(n)_{*}=\mathbb{F}_{p}[v_{n},v_{n}^{-1}].$ (3.3)

The key to the localization result is the following computation of the
$\mathbb{E}^{*}$-theory for classifying spaces of finite abelian groups from
[HKR00, §5.2].

######  Lemma 3.2.

Let $\mathbb{E}^{*}$ be an HKR theory of height $n<\infty$. Let $A$ be a
finite cyclic group of order $\ell$. Write $\ell=p^{r}s$ with $(s,p^{r})=1$.
Then

  1. (1)

$\mathbb{E}^{*}(BA)$ is free over $\mathbb{E}_{*}$ of rank $p^{rn}$.

  2. (2)

In the case $\mathbb{E}^{*}=K(n)$ is the Morava $K$-theory, we have:

$\displaystyle\mathbb{E}^{*}(BA)\cong\mathbb{E}_{*}[[u]]/(u^{p^{rn}}).$ (3.4)

###### Proof.

Part (1) is established in [HKR00, Cor. 5.8]. To establish Part (2), write
$\ell=p^{r}s$ with $(s,p^{r})=1$. Note that by by Part (1), we have that the
pull-back map

$\displaystyle\mathbb{E}^{*}(BA)\to\mathbb{E}^{*}(B\mathbb{Z}/p^{r}\mathbb{Z})$
(3.5)

is an isomorphism. Thus, it suffices to consider the case where $\ell=p^{r}.$
View $A$ as a finite subgroup $A\subset S^{1}$. Regard $\mathbb{E}^{*}(BA)$ as
a module over $\mathbb{E}_{*}[\\![u]\\!]$ via the map $BA\to BS^{1}$. By
[HKR00, Lemma 5.7] we have that

$\displaystyle\mathbb{E}^{*}(BA)\cong\mathbb{E}_{*}[\\![u]\\!]/([p^{r}]\cdot_{\mathbb{E}}u).$
(3.6)

The formal group law $L_{\mathbb{E}}$ for Morava $K$-theory is essentially
characterized by the fact that its $p$-series satisfies:

$\displaystyle[p]\cdot_{\mathbb{E}}u=v_{n}u^{p^{n}}.$ (3.7)

Iterating the formal group law, we obtain

$\displaystyle[p^{r}]\cdot_{\mathbb{E}}u=v_{n}^{d(r)}u^{p^{rn}}.$ (3.8)

for some integer $d(r)>0$. The result follows immediately from this. ∎

###  3.2. Localization

######  Definition 3.3.

An HKR theory $\mathbb{E}^{*}$ is _integral of finite Krull dimension_ if the
coefficient ring $\mathbb{E}_{*}$ is an integral domain of finite Krull
dimension.

######  Remark 3.4.

All of the examples $\hat{E}(n)^{*},\hat{K}(n)^{*},K(n)^{*}$ are integral of
finite Krull dimension.

Fix an HKR cohomology theory which integral and of finite Krull dimension. The
goal of this section is to extend the Atiyah–Bott localization theorem ([AB84,
Theorem 3.5]) to $\mathbb{E}^{*}$ coefficients. Throughout this section we
take our group to be a torus $T\cong(S^{1})^{m}$. We have by [Sta24, Tag 0BNH]
that $\mathbb{E}^{*}(BT)$ is Noetherian222Strictly speaking [Sta24, Tag 0BNH]
considers standard completions as opposed to graded completions. However the
result and proof apply without change to graded completions.  and it is also
an integral domain. We may therefore set
$R=\operatorname{Frac}(\mathbb{E}^{*}(BT))$ to be the fraction field of
$\mathbb{E}^{*}(BT).$ We have the following version of the Atiyah-Bott support
lemma (cf. [AB84, Lemma 3.4]):

######  Lemma 3.5.

Let $X=T/K$ be an orbit of $T$, where $K$ is a closed subgroup of $T$. Let
$\operatorname{supp}(\mathbb{E}^{*}_{T}(X))\subset\operatorname{Spec}(\mathbb{E}^{*}(BT))$
denote the support of $\mathbb{E}^{*}_{T}(X)$ as a module over
$\mathbb{E}^{*}(BT)).$ Then:

  1. (1)

$\operatorname{supp}(\mathbb{E}^{*}_{T}(X))$ is a closed subspace of
codimension $\operatorname{dim}(T)-\operatorname{dim}(K)$.

  2. (2)

In the case where $\mathbb{E}^{*}=K(n)^{*}$ is Morava K-theory,
$\operatorname{supp}(\mathbb{E}^{*}_{T}(T/K))=\operatorname{supp}(\mathbb{E}^{*}_{T}(T/K^{0}))$,
where $K^{0}\subset K$ is the connected component of $K$ at the identity.

###### Proof.

We have that $\mathbb{E}_{T}^{*}(X)\cong\mathbb{E}^{*}(BK)$ and the induced
$\mathbb{E}^{*}(BT)$ module structure on this group is induced by the
inclusion $K\subset T$. We first argue that the map

$\displaystyle\mathbb{E}^{*}(BT)\to\mathbb{E}^{*}(BK)$ (3.9)

is surjective so that $\operatorname{Spec}(\mathbb{E}^{*}(BK))$ is a closed
subscheme of $\operatorname{Spec}(\mathbb{E}^{*}(BT))$. As above, we let
$K^{0}$ denote the connected component of $K$. After choosing a complementary
torus $H$ to $K^{0}$, we may write $K=A\times K^{0}$, where $A$ is a finite
abelian subgroup of $H$. The by the Kunneth isomorphism [HKR00, Cor. 5.11], we
have

$\displaystyle\mathbb{E}^{*}(BA)\otimes\mathbb{E}^{*}(BK^{0})\cong\mathbb{E}^{*}(BK).$
(3.10)

Moreover, the Kunneth map fits into a diagram:

$\textstyle{\mathbb{E}^{*}(BH)\otimes\mathbb{E}^{*}(BK^{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{E}^{*}(BT)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{E}^{*}(BA)\otimes\mathbb{E}^{*}(BK^{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{E}^{*}(BK)}$

It therefore suffices to assume that $K=A$ is a finite abelian group.
Moreover, by Part (1) of Lemma 3.2 we can assume that
$A\cong\mathbb{Z}/p^{r_{1}}\times\mathbb{Z}/p^{r_{2}}\times\cdots\times\mathbb{Z}/p^{r_{i}}$
is a product of $p$-groups. It is easy to see that we may find:

  * •

a rank one subgroup $H^{\prime}\cong S^{1}\subset T$ so that
$\mathbb{Z}/p^{r_{1}}\subset H^{\prime}$

  * •

a complementary torus $H^{\prime\prime}$ (for $H^{\prime}$) together with a
subgroup $A^{\prime}\subset H^{\prime\prime}$ so that
$A\cong\mathbb{Z}/p^{r_{1}}\times A^{\prime}$ (the group
$A^{\prime}\cong\mathbb{Z}/p^{r_{2}}\times\cdots\times\mathbb{Z}/p^{r_{i}}$).

The surjectivity of the restriction $\mathbb{E}^{*}(BT)\to\mathbb{E}^{*}(BA)$
follows from the Kunneth isomorphism together with induction on the number of
factors in the product decomposition of $A$. To prove the claim on dimensions,
we note that the claim is obvious for connected groups where $K=K^{0}$. In
view of the Kunneth isomorphism and Part (1) of Lemma 3.2, we have that
$\mathbb{E}^{*}(BK)$ is finite over $\mathbb{E}^{*}(BK^{0})$, which implies
that they have equal dimensions [Sta24, Tag 01WG]. To prove Part (2) of the
Lemma, we again write our group $K$ as $A\times K^{0}$. Without loss of
generality, we can assume
$A\cong\mathbb{Z}/p^{r_{1}}\times\mathbb{Z}/p^{r_{2}}\times\cdots\times\mathbb{Z}/p^{r_{i}}$.
Then Part (2) of Lemma 3.2 shows that $K(n)^{*}(BA)$ is a tensor product of
the algebras in (3.4). In particular, $K(n)^{*}(BA)$ is a nilpotent extension
of $K(n)_{*}$ for any finite abelian group. The claim now follows from the
Kunneth isomorphism (3.10).∎

For the rest of this section, unless otherwise specified, let $M$ be a smooth
compact manifold with $T$-action and denote by $M^{T}$ the set of fixed points
of the $T$-action.

######  Lemma 3.6.

Let $j:M^{T}\to M$ denote the inclusion of $M^{T}$ into $M$. Then the maps:

$\displaystyle
j^{*}:\mathbb{E}_{T}^{*}(M)\otimes_{\mathbb{E}^{*}(BT)}R\to\mathbb{E}_{T}^{*}(M^{T})\otimes_{\mathbb{E}^{*}(BT)}R$
(3.11)

is an isomorphism.

###### Proof.

We stratify the complement of an equivariant tubular neighborhood $U$ of
$M^{T}$ by finitely many open sets which equivariantly retract onto non-
constant orbits. Note that for a non-constant orbit $X$, Lemma 3.5 implies in
particular that

$\displaystyle\mathbb{E}_{T}^{*}(X)\otimes_{\mathbb{E}^{*}(BT)}R=0.$ (3.12)

The result then follows from Lemma 3.2, Lemma 3.5, and the Mayer-Vietoris
sequence. ∎

###### Proof of Theorem 1.10.

The assumption that $\mathbb{E}_{*}$ is free implies by Corollary 2.15 that

$\displaystyle\mathbb{E}_{T}^{*}(M)\cong\mathbb{E}^{*}(M)[[u_{1},\cdots,u_{m}]]\cong\mathbb{E}^{*}(M)\otimes_{\mathbb{E}_{*}}\mathbb{E}^{*}(BT).$
(3.13)

Hence, $\mathbb{E}_{T}^{*}(M)$ is free as a module over $\mathbb{E}^{*}(BT).$
By Lemma 3.6, the kernel of $j^{*}$ is a torsion module over
$\mathbb{E}^{*}_{T}(pt)$. Because $\mathbb{E}^{*}_{T}(M)$ is free, it follows
that the kernel of $j^{*}$ is trivial. ∎

To end this subsection, we include an Atiyah–Bott localization formula for HKR
theories.

######  Theorem 3.7.

Suppose that $M$ has an almost complex structure which is preserved under the
$T$-action. Index the connected components of $M^{T}$ by the set $I$, and for
each $i\in I$, write the normal bundle of the corresponding connected
component $\nu_{i}$. Then the pushforward on the localized cohomology

$j_{*}:\mathbb{E}_{T}^{*}(M^{T})\otimes_{\mathbb{E}^{*}(BT)}R\to\mathbb{E}_{T}^{*}(M)\otimes_{\mathbb{E}^{*}(BT)}R$
(3.14)

is an isomorphism with inverse

$\sum_{i\in I}\frac{1}{\mathit{eu}^{T}_{\mathbb{E}}}(\nu_{i})j_{i}^{*}.$
(3.15)

###### Proof.

By the complex-orientation assumption and its compatibility with the
$T$-action, we see that the pushforward in equivariant cohomology $j_{*}$ and
the Euler class of the normal bundle $\nu_{i}$ are well-defined. Because for
each $i\in I$, we have
$j_{i}^{*}j_{*}(1)=\mathit{eu}^{T}_{\mathbb{E}}(\nu_{i})$. Using the
Mayer–Vietoris long exact sequence for the pair $(M,M\setminus M^{T})$, as in
Lemma 3.7, one can show that for the pushforward $j_{*}$, both $\ker(j_{*})$
and $\mathrm{coker}(j_{*})$ are both torsion modules over
$\mathbb{E}^{*}(BT)$. Combining with Lemma 3.7, it follows that $j^{*}j_{*}$
is an isomorphism. This proves the invertibility of the Euler classes, so the
statement follows accordingly. ∎

######  Remark 3.8.

Following [AB84, Section 3], we can prove Theorem 3.7 in a slightly different
way. For the connected component $M^{T}_{i}$, we can decompose its normal
bundle into a direct sum of irreducible $T$-eigenbundles

$\nu_{i}=\nu_{i}^{(1)}\oplus\cdots\oplus\nu_{i}^{(k)}$ (3.16)

such that for each $1\leq\alpha\leq k$, the corresponding character
$\varphi_{\alpha}:T\to U(1)$ is nontrivial. If we write
$u_{\alpha}\in\mathbb{E}_{*}[\\![u_{1},\dots,u_{m}]\\!]$ the image of the
generator of $\mathbb{E}^{*}(BS^{1})$ in $\mathbb{E}^{*}(BT)$ induced by
$\varphi_{\alpha}$, then

$\mathit{eu}^{T}_{\mathbb{E}}(\nu_{i})=\prod_{1\leq\alpha\leq
k}(\mathit{eu}_{\mathbb{E}}(\nu_{i}^{(\alpha)})+_{\mathbb{E}}u_{\alpha}),$
(3.17)

where we abuse the notation using $+_{\mathbb{E}}$ to denote the element in
$\mathbb{E}_{T}^{*}(M^{T})$ obtained by plugging in the given two elements in
the formal group law associated with $\mathbb{E}^{*}$. The Euler class of line
bundles, which coincides with the first Chern class in $\mathbb{E}^{*}$, is
nilpotent. Indeed, for any finite-dimensional CW complex $X$, the set of
isomorphism classes of complex bundles on $X$ have a one-to-one correspondence
with the set of homotopy classes of maps from $X$ to $\mathbb{CP}^{\infty}$.
If a line bundle $L\to X$ is classified by $f:X\to\mathbb{CP}^{\infty}$, then
$\mathit{eu}_{\mathbb{E}}(L)=f^{*}(u)$, where $u$ is the generator of
$\mathbb{E}^{*}(\mathbb{CP}^{\infty})$. By naturality, we know that
$\mathit{eu}_{\mathbb{E}}(L)^{N}=f^{*}(u^{N})$ for any $N\in\mathbb{Z}$. On
the other hand, $u^{N}$ is the Poincaré dual of a complex codimension $N$
subspace in $\mathbb{CP}^{\infty}$. So, for $N\gg 1$, we can choose a homotopy
of $f$ to show that $f^{*}(u^{N})=0$, which proves the nilpotency.
Accordingly, we see that $\mathit{eu}^{T}_{\mathbb{E}}(\nu_{i})$ is invertible
after tensor with the fraction field $R$.

###  3.3. GKM theorem

In this subsection, we specialize our discussion to the case where
$\mathbb{E}^{*}$ is a Morava $K$-theory $K(n)$. We show that the so-called
Chang–Skjelbred lemma works verbatim for $K(n)$ coefficients, and deduce the
GKM calculation of equivariant cohomology as a corollary.

Let $M$ be a compact smooth $T$-manifold. Denote by $\mathcal{P}$ the set of
closed subschemes
$\mathcal{Z}_{K}:=\operatorname{supp}(\mathbb{E}^{*}_{T}(T/K))\subset\mathrm{Spec}(\mathbb{E}_{*}[\\![u_{1},\dots,u_{m}]\\!])$
where $K$ is a stabilizer of some point $x\in M\setminus M^{T}$ (we give
$\mathcal{Z}_{K}$ the reduced-induced scheme structure). This is a partially
ordered set, with partial order given by inclusion. Note that by Lemma 3.5
(2), the scheme $\mathcal{Z}_{K}$ only depends on the connected component of
the identity element $K^{0}$.

Let $\delta^{\prime}$ denote the connecting homomorphism of the
$\mathbb{E}^{*}_{T}$-cohomology of the pair $(M,M^{T})$. Given
$\xi\in\mathbb{E}^{*}_{T}(M^{T})$, write $I(\xi)$ the ideal of annihilators of
of $\delta^{\prime}(\xi)$. On the other hand, for each
$\mathfrak{m}\in\mathcal{P}$, we write $M^{\mathfrak{m}}$ the subset of $M$
consisting of points $x\in M$ whose stabilizer group $K_{x}$ satisfies
$\mathfrak{m}\subset\mathcal{Z}_{K_{x}}$. Denote the connecting homomorphism
of $\mathbb{E}^{*}_{T}$-cohomology of the pair $(M,M^{\mathfrak{m}})$ by
$\delta^{\mathfrak{m}}$. The following technical statement is proved in the
same way as [CS74, Lemma 2.3] and [GKM98, Lemma 15.8].

######  Lemma 3.9.

For $\xi\in\mathbb{E}^{*}_{T}(M^{T})$, if $\delta^{\prime}(\xi)\neq 0$, then
the variety defined by $I(\xi)$ satisfies

$V(I(\xi))\subset\bigcup_{\begin{subarray}{c}\mathfrak{m}\in\mathcal{P}\\\
\delta^{\mathfrak{m}}(\xi)\neq 0\end{subarray}}\mathfrak{m}.$ (3.18)

in the scheme $\mathrm{Spec}(\mathbb{E}_{*}[\\![u_{1},\dots,u_{m}]\\!])$. ∎

Denote by $M_{1}\subset M$ the set of points consisting of the union of $0$
and $1$-dimensional orbits of $T$. In particular, $M^{T}\subset M_{1}$.

######  Proposition 3.10.

Let $\mathbb{E}^{*}$ be a Morava $K$-theory. If the equivariant cohomology
$\mathbb{E}^{*}_{T}(M)$ is a free module over $\mathbb{E}^{*}(BT)$, then the
sequence

$0\longrightarrow\mathbb{E}^{*}_{T}(M)\xrightarrow{j^{*}}\mathbb{E}^{*}_{T}(M^{T})\xrightarrow{\delta}\mathbb{E}^{*+1}_{T}(M_{1},M^{T})$
(3.19)

is exact, where $\mathbb{E}^{*+1}_{T}(M_{1},M^{T})$ denotes the relative
equivariant cohomology of the pair $(M_{1},M^{T})$ and $\delta$ is the
connecting homomorphism.

###### Proof.

The fact that $\ker(j)^{*}$ is trivial follows immediately from Theorem 1.10
because the coefficient ring is a graded field. Because the connecting
homomorphism $\delta$ factors as the composition

$\mathbb{E}^{*}_{T}(M^{T})\xrightarrow{\delta^{\prime}}\mathbb{E}^{*+1}_{T}(M,M^{T})\rightarrow\mathbb{E}^{*+1}_{T}(M_{1},M^{T})$
(3.20)

where recall that $\delta^{\prime}$ is the connecting homomorphism for the
pair $(M,M^{T})$, the long exact sequence implies that $\delta\circ j^{*}=0$.
To finish the proof, suppose $\xi\in\mathbb{E}^{*}_{T}(M^{T})$ satisfies
$\delta(\xi)=0$. To show it lies in the image of $j^{*}$, we need to show that
$\delta^{\prime}(\xi)=0$.

Suppose this is not the case. We claim that the ideal $I(\xi)$ is a principal.
We follow the proof of [CS74, Lemma 2.2]. Indeed, let $0\neq a\in I(\xi)$ be
an element which is not divisible by any other element and suppose $b\in
I(\xi)$. Choosing a basis $\\{x_{i}\\}\subset\mathbb{E}^{*}_{T}(M)$, we can
find $a_{i},b_{i}\in\mathbb{E}^{*}_{T}(pt)$ such that $a\xi=j^{*}(\sum
a_{i}x_{i})$ and $b\xi=j^{*}(\sum b_{i}x_{i})$ using the long exact sequence
for the pair $(M,M^{T})$. Then we know that $j^{*}(\sum
ba_{i}x_{i})=j^{*}(\sum ab_{i}x_{i})$, so $\sum ba_{i}x_{i}=\sum ab_{i}x_{i}$
by the injectivity of $j^{*}$. This shows that for each $i$, we have
$ba_{i}=ab_{i}$. As the $a_{i}$ must have no common factor, we see that $b$ is
divisible by $a$, which proves the claim.

By Lemma 3.9, as $I(\xi)$ is principal, there must be one $\mathfrak{m}$ which
has codimension $1$ in $T$. Because of the inclusion $M^{\mathfrak{m}}\subset
M_{1}$, the connecting homomorphism $\delta^{K}$ factors as

$\mathbb{E}^{*}_{T}(M^{T})\xrightarrow{\delta}\mathbb{E}^{*+1}_{T}(M_{1},M^{T})\to\mathbb{E}^{*+1}_{T}(M^{\mathfrak{m}},M^{T}),$
(3.21)

which tells us that $\delta(\xi)\neq 0$, leading to a contradiction. ∎

###### Proof of Theorem 1.11:.

This follows from the proof of [GKM98, Theorem 7.2], which we reproduce for
completeness. Because $\mathbb{E}^{*}_{T}(M)$ is a free module over
$\mathbb{E}^{*}(BT)$, using the exact sequence (3.19), it suffices to
determine the kernel of
$\delta:\mathbb{E}^{*}_{T}(M^{T})\to\mathbb{E}^{*+1}_{T}(M_{1},M^{T})$. Given
the complement of poles $E_{j}$, by definition, its stabilizer group is
$\ker(\Theta_{j})$. Therefore,

$\mathbb{E}^{*}_{T}(E_{j})\cong\mathbb{E}^{*}_{T}(T/\ker(\Theta_{j}))\cong\mathbb{E}^{*}_{\ker(\Theta_{j})}(pt).$
(3.22)

Choosing a $T$-invariant open cover of $\overline{E}_{j}=U_{j}\cup V_{j}$
where $U_{j}=\overline{E}_{j}\setminus\\{x_{j_{0}}\\}$ and
$V_{j}=\overline{E}_{j}\setminus\\{x_{j_{\infty}}\\}$. Then the corresponding
Mayer–Vietoris long exact sequence splits into short exact sequences because
all the cohomology groups concentrate in even degrees. Furthermore, we have a
commutative diagram

${0}$${\mathbb{E}^{*}_{T}(\overline{E}_{j})}$${\mathbb{E}^{*}_{T}(U_{j})\oplus\mathbb{E}^{*}_{T}(V_{j})}$${\mathbb{E}^{*}_{T}(E_{j})}$${0}$${0}$${\mathbb{E}^{*}_{T}(\overline{E}_{j})}$${\mathbb{E}^{*}_{T}(x_{j_{\infty}})\oplus\mathbb{E}^{*}_{T}(x_{j_{0}})}$${{\mathbb{E}_{T}^{*+1}(\overline{E}_{j},\\{x_{j_{0}},x_{j_{\infty}}\\})}}$${0,}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\delta}$
(3.23)

which tells us that $\delta$ agrees with the map

$(f_{j_{0}},f_{j_{\infty}})\mapsto
f_{j_{0}}|_{\mathbb{E}^{*}_{\ker(\Theta_{j})}}-f_{j_{\infty}}|_{\mathbb{E}^{*}_{\ker(\Theta_{j})}}.$
(3.24)

Ranging over all $j$, the theorem is proved. ∎

######  Remark 3.11.

In view of the calculation Lemma 3.2 (2), one can see that for an invariant
$2$-sphere with orbit type $T/K$ such that in the decomposition $K=A\times
K^{0}$, the finite group component $A$ is has order greater than $1$, the
formula in Theorem 1.11 depends on the height for the cohomology theory
$\mathbb{E}^{*}$. We thank Ivan Smith for a related question.

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