# Circle actions and Suspension operations on Smooth manifolds

Haibao Duan Supported by National Science foundation of China: 11771427;
11961131004;

###### Abstract

Let $M$ be a smooth manifold with $\dim M\geq 3$ and a base point $x_{0}$.
Surgeries along the oriented circle $S^{1}\times\\{x_{0}\\}$ on the product
$S^{1}\times M$ yields two manifolds $\Sigma_{0}M$ and $\Sigma_{1}M$, called
the suspensions of $M$.

The suspension operations $\Sigma_{i}$ play a basic role in the construction
and classification of the smooth manifolds which admit free $S^{1}$-actions.
We illustrate this by a number of applications.

##  1 Introduction and main results

Let $\mathcal{T}_{n}$ be the set of diffeomorphism classes of closed, oriented
and smooth manifolds of dimension $n$. A free circle action $S^{1}\times
M\rightarrow M$ on a manifold $M\in\mathcal{T}_{n}$ is called regular if the
quotient space $M/S^{1}$, with the induced topology and orientation, belongs
to $\mathcal{T}_{n-1}$. For a given $M\in\mathcal{T}_{n}$ we would like to
known whether there is a regular circle action on $M$. If the answer is
affirmative, we would like to decide furthermore the diffeomorphism types of
the quotients $M/S^{1}$ resulted by such actions. If $M$ is a homotopy sphere,
these problems have been studied by Montgemory-Yang, Edmonds, Schultz, Madsen-
Milgram and Wall [22, 9, 23, 21, 26]. In the case $M$ is an $1$-connected
$5$-manifolds, the problems have been solved by Duan and Liang [8].

This paper introduces a method to construct smooth manifolds $M$ that admit
regular circle actions, while the diffeomorphism types of the quotients
$M/S^{1}$ can be made explicitly. The central ingredient of the construction
is the suspension operations
$\Sigma_{i}:\mathcal{T}_{n}\rightarrow\mathcal{T}_{n+1}$ (with $i=0,1$ and
$n\geq 3$) we are about to describe. Recall that, for the special orthogonal
group $SO(n)$ of order $n\geq 3$, the fundamental group $\pi_{1}(SO(n))$ is
isomorphic to $\mathbb{Z}_{2}$ with generator

> $\alpha(t)=\left(\begin{array}[]{cc}\cos 2\pi t&-\sin 2\pi t\\\ \sin 2\pi
> t&\cos 2\pi t\end{array}\right)\oplus I_{n-2}$, $t\in[0,1]$,

where $I_{r}$ denotes the identity matrix of rank $r$. It gives rise to the
diffeomorphism of $S^{1}\times D^{n}$

  1. (1.1)

$\widetilde{\alpha}:S^{1}\times D^{n}\rightarrow S^{1}\times D^{n}$ by
$\widetilde{\alpha}(t,x)=(t,\alpha(t)x)$,

whose restriction on the boundary $S^{1}\times S^{n-1}$ is denoted by $\tau$.
Recall that any orientation preserving diffeomorphism of $S^{1}\times S^{n-1}$
is isotopic to either $id$ or $\tau$ (e.g. [26, p.241]).

For a manifold $N\in\mathcal{T}_{n}$ with a base point $x_{0}\in N$, and an
orientation preserving smooth embedding $(D^{n},0)\subset(N,x_{0})$ centered
at $x_{0}$, define two adjoint manifolds $\Sigma_{i}N$, $i=0,1$, by

  1. (1.2)

$\Sigma_{0}N:=D^{2}\times
S^{n-1}\cup_{id}(S^{1}\times(N-\overset{\circ}{D^{n}}))$,

$\Sigma_{1}N:=D^{2}\times
S^{n-1}\cup_{\tau}(S^{1}\times(N-\overset{\circ}{D^{n}}))$,

respectively, where $\cup_{f}$ means gluing the boundaries together by the
diffeomorphism $f$. Alternatively, the manifolds $\Sigma_{0}N$ and
$\Sigma_{1}N$ are obtained by surgeries along the oriented circle
$S^{1}\times\\{x_{0}\\}$ on $S^{1}\times N$ with the framings $id$ and
$\widetilde{\alpha}$, respectively. By the homogeneity lemma [19, §4] the
diffeomorphism types of both $\Sigma_{i}N$ are irrelevant to the choice of a
smooth embedding $(D^{n},0)\subset(N,x_{0})$, which justifies the following
notion.

Definition 1.1. For $n\geq 2$ the correspondences
$\Sigma_{i}:\mathcal{T}_{n}\rightarrow\mathcal{T}_{n+1}$ by $N\rightarrow$
$\Sigma_{i}N$, $i=0,1$, are called the suspension operations.$\square$

Let $Diff(N,x_{0})$ be the group of diffeomorphisms of $N\in\mathcal{T}_{n}$
that fix the base point $x_{0}$. Sending a diffeomorphism $h\in Diff(N,x_{0})$
to the tangent map of $h$ at $x_{0}$ yields a representation
$e_{N}:Diff(N,x_{0})\rightarrow GL(n)$, where $GL(n)$ is the general linear
group of order $n$. The manifold $N$ is called $\Sigma$-stable if the induced
action on the fundamental groups

>
> $e_{N\ast}:\pi_{1}(Diff(N,x_{0}),id)\rightarrow\pi_{1}(GL(n),I_{n})=\mathbb{Z}_{2}$,
> $n\geq 3$,

surjects. The difference between the two manifolds $\Sigma_{0}N$ and
$\Sigma_{1}N$ is quite subtle, as indicated by our first result.

Theorem A. If a manifold $N\in\mathcal{T}_{n}$ is $\Sigma$-stable, then
$\Sigma_{0}N$ $=\Sigma_{1}N$.

As supplements to Theorem A we shall show in Section 3 that manifolds of the
form $S^{k}\times M$ with $k\geq 3$ are all $\Sigma$-stable, while
$\Sigma_{1}\mathbb{H}P^{2}$ $\neq\Sigma_{0}\mathbb{H}P^{2}$ for the
quaternionic projective plane $\mathbb{H}P^{2}$.

For a regular circle action on a manifold $E\in\mathcal{T}_{n}$ regard the
quotient map $p:E\rightarrow B:=E/S^{1}$ as an oriented circle bundle over
$B$. For a manifold $N\in\mathcal{T}_{n-1}$ let $B\\#N$ be the connected sum,
$f_{N}:B\\#N\rightarrow B$ be the canonical projection, and denote the induced
bundle $f_{N}^{\ast}E$ by

  1. (1.3)

$p_{N}:E_{N}\rightarrow$ $B\\#N$.

One main result expresses the total space $E_{N}$ in term of $E$ and $N$.

Theorem B. Let $S^{1}\times E\rightarrow E$ be a regular circle action on an
$1$-connected manifold $E\in T_{n}$ with quotient $B$, where $n\geq 5$. Then
for any $N\in\mathcal{T}_{n-1}$

  1. (1.4)

$E_{N}=\left\\{\begin{tabular}[]{l}$E\\#\Sigma_{0}N$ {if} $w_{2}(B)\neq
0${;}\\\ $E\\#\Sigma_{1}N$ {if} $w_{2}(B)=0${.}\end{tabular}\right.$

In addition, if $p^{\ast}(w_{2}(B))=w_{2}(E)\neq 0$, then

  1. (1.5)

$E_{N}=E\\#\Sigma_{0}N=E\\#\Sigma_{1}N$,

where $w_{2}(M)$ denotes the second Stiefel–Whitney class of $M$.

In Theorem B the assumption of the $1$-connectness of $E$ is essential. For an
integer $q\geq 2$ let $\pi:E\rightarrow\mathbb{C}P^{2}$ be the oriented circle
bundle over the complex projective plane with Euler class $q\cdot x$, where
$x\in H^{2}(\mathbb{C}P^{2})$ is a generator. Then $E$ is the $5$-dimensional
lens space $L^{5}(q)$ whose fundamental group is cyclic of order $q$. It can
be shown that $E_{N}$ is homeomorphic to neither $E\\#\Sigma_{0}N$ nor
$E\\#\Sigma_{1}N$, unless $N$ is homeomorphic to $S^{4}$.

Theorem B provides a method to construct new manifolds, which can be furnished
with the regular circle actions inherited from the known ones.

Corollary 1.3. Let $E\in\mathcal{T}_{n}$ be an $1$-connected manifold which
has a regular circle action with quotient $B$, where $n\geq 5$.

Then, for any $N\in\mathcal{T}_{n-1}$ one of the two connected sums
$E\\#\Sigma_{i}N$ admits a regular circle action with quotient
$B\\#N$.$\square$

Example 1.4. Let $p:S^{2n+1}\rightarrow\mathbb{C}P^{n}$ be the Hopf fibration
over the complex projective $n$-space $\mathbb{C}P^{n}$, $n\geq 2$. By
Corollary 1.3 for any $N\in\mathcal{T}_{2n}$ either $\Sigma_{0}N$ or
$\Sigma_{1}N$ admits a regular circle action with quotient
$\mathbb{C}P^{n}\\#N$. In particular, the homotopy groups of the suspensions
$\Sigma_{i}N$ are

  1. (1.6)

$\pi_{r}(\Sigma_{i}N)=\left\\{\begin{array}[]{c}\pi_{r}(N)\text{ }{if}\text{
}r=1\text{ }{or}\text{ }2{,}\\\ \pi_{r}(\mathbb{C}P^{n}\\#N)\text{ }{if}\text{
}r\geq 3{.}\end{array}\right.$

In general, by a fake complex projective $n$-space we mean a smooth manifold
$F_{\mathbb{C}P^{n}}$ homotopy equivalent to the complex projective space
$\mathbb{C}P^{n}$, $n\geq 2$. For the classifications of such manifolds, see
[21, Theorem 4.9] and [26, Chapter 14C]. In particular, if we let
$p:E\rightarrow F_{\mathbb{C}P^{n}}$ be the oriented circle bundle on
$F_{\mathbb{C}P^{n}}$ whose Euler class generates the group
$H^{2}(F_{\mathbb{C}P^{n}})=\mathbb{Z}$, then $E$ is a homotopy
$(2n+1)$-spheres. In addtion, since the second Stiefel-Whitney class is a
homotopy invariant of smooth manifolds, we have
$w_{2}(F_{\mathbb{C}P^{n}})\neq 0$ if and only if $n$ even. Thus, Theorem B
implies that, for any fake complex projective space $F_{\mathbb{C}P^{n}}$ and
any $N\in\mathcal{T}_{2n}$, the (topological) manifold $\Sigma_{i}N$ admits a
smooth structure and a regular circle action, whose quotient is diffeomorphic
to $F_{\mathbb{C}P^{n}}\\#N$, where $n=i\mathop{\mathrm{m}od}2$.$\square$

Example 1.5. For an integer $n\geq 2$ let $S^{2}\widetilde{\times}S^{n}$ be
the only nontrivial smooth $S^{n}$-bundle over $S^{2}$. Take $B:=S^{2}\times
S^{n-2}$ or $S^{2}\widetilde{\times}S^{n-2}$ with $n\geq 4$, and let
$p:E\rightarrow B$ be the circle bundle over $B$ whose Euler class is the
generator of $H^{2}(B)$ corresponding to the orientation class of the factor
$S^{2}\subset B$. Since the total space $E=S^{3}\times S^{n-2}$ is
$1$-connected, $w_{2}(E)=0$, and since $w_{2}(B)\neq 0$ if and only if
$B=S^{2}\widetilde{\times}S^{n-2}$, we conclude by Corollary 1.3 that for any
$N\in\mathcal{T}_{n-1}$ with $n\geq 5$, both manifolds $S^{3}\times
S^{n-2}\\#\Sigma_{i}N$ admit regular circle actions with quotients
$S^{2}\widetilde{\times}S^{n-2}\\#N$ and $S^{2}\times S^{n-2}\\#N$,
respectively.$\square$

Combined with general properties of the operations $\Sigma_{i}$ developed in
Section 3, Theorem B is applicable to classify those manifolds that admit
regular circle actions. In the 1960’s Smale and Barden [2, 24] classified all
the 1-connected 5-manifolds in term of a collection of five basic ones
$\\{S^{2}\times S^{3},S^{2}\widetilde{\times}S^{3},W,M_{k},X_{2^{i}}\\}$, see
Section 4 for the constructions of these manifolds. On the other hand, in [14,
16, 17] R. Goldstein and L. Lininger studied the problem of classifying the
1-connected 6-manifolds that admit regular circle actions. Our next result
completes the subject (compare with [16, Theorem 4, Theorem 6]).

Theorem C. If $M$ is an $1$-connected $6$-manifold that admits regular circle
actions, then

  1. (1.7)

$M=\left\\{\begin{tabular}[]{l}$S^{3}\times S^{3}\\#_{r}\Sigma_{0}(S^{2}\times
S^{3})\\#_{1\leq j\leq t}\Sigma_{1}M_{k_{j}}\\#\Sigma_{1}H$ {if} $w_{2}(M)\neq
0${,}\\\ $S^{3}\times S^{3}\\#_{r}\Sigma_{0}(S^{2}\times S^{3})\\#_{1\leq
j\leq t}\Sigma_{i}M_{k_{j}}$ {if} $w_{2}(M)=0,$\end{tabular}\right.$

where $H\in\\{S^{2}\widetilde{\times}S^{3},W,X_{k}\\}$,
$\Sigma_{0}(S^{2}\times S^{3})=S^{3}\times S^{3}\\#S^{2}\times S^{4}$, and
where $\\#_{0}(\cdots)$ means $S^{6}$.

Let $T^{k}=S^{1}\times\cdots\times S^{1}$ ($k$-copies) be the $k$-dimensional
torus group with classifying space $BT^{k}$. For a (topological) manifold $M$,
and a set $\left\\{\alpha_{1},\cdots,\alpha_{k}\right\\}\subset H^{2}(M)$ of
$k$ cohomology classes of degree $2$, let
$f_{i}:M\rightarrow\mathbb{C}P^{\infty}$ be the classifying map of
$\alpha_{i}$, and let

  1. (1.8)

$\pi:M(\alpha_{1},\cdots,\alpha_{k})\rightarrow M$

be the principal $T^{k}$-bundle on $M$ whose classifying map is
$f=\left(f_{1},\cdots,f_{k}\right):M\rightarrow
BT^{k}=\mathbb{C}P^{\infty}\times\cdots\times\mathbb{C}P^{\infty}$
($k$-copies). Theorem B may be iterated to calculate the homeomorphism type of
the total space $M(\alpha_{1},\cdots,\alpha_{k})$ in term of $M$. We
illustrate the calculation for the cases where $M$ is an $1$-connected
$4$-manifold (not necessarily smoothable).

For an integer $k\geq 1$ define the sequence
$(b_{1},\cdots,b_{\left[\frac{k}{2}\right]})$ of integers by

>
> $b_{i}:=(k-1)\binom{k-1}{i}-\binom{k-1}{i+1}+(k-1)\binom{k-1}{i-1}-\binom{k-1}{i-2})$,
> $1\leq i\leq\left[\frac{k}{2}\right]$.

Consequently, let $Q_{k}$ be the connected sum of products of spheres

  1. (1.9)

$Q_{k}:=\\#_{c_{1}}(S^{{\small 3}}\times S^{{\small
k+1}})\\#_{c_{2}}(S^{{\small 4}}\times S^{{\small
k}})\\#_{c_{3}}\cdots\\#_{c_{r}}(S^{r{\small+2}}\times S^{{\small
k-}r{\small+2}})$, $r=\left[\frac{k}{2}\right]$,

where $c_{i}=b_{i\text{ }}$with the only exception that
$c_{r}=\frac{1}{2}b_{r}$ if $k\equiv 0\mathop{\mathrm{m}od}2$.

Orlik and Raymond [25, p.553] (Davis and Januszkiewicz [7, Example 1.20])
proved that any toric $4$-manifold is (non-equivariantly) homeomorphic to a
connected sum of copies of $\mathbb{C}P^{2},\overline{\mathbb{C}P}^{2}$ and
$S^{2}\times S^{2}$. According to Buchstaber and Panov [4, Theorem 4.6.12]
with $n=2$ (e.g. [3, Theorem 6.3], [18]), if $M$ is a toric $4$-manifold and
if $\left\\{\alpha_{1},\cdots,\alpha_{k}\right\\}$ is the canonical basis of
$H^{2}(M)$, then the manifold $M(\alpha_{{\small 1}},\cdots,\alpha_{{\small
k}})$ is diffeomorphic to $Q_{k}$. We extend this phenomina to all
$1$-connected topological $4$-manifolds.

Theorem D. If $M$ is an $1$-connected $4$-manifold and if
$\left\\{\alpha_{1},\cdots,\alpha_{k}\right\\}$ is a basis of $H^{2}(M)$, then
$M(\alpha_{{\small 1}},\cdots,\alpha_{{\small k}})$ is diffeomorphic to
$Q_{k}$.

In particular, any $1$-connected $4$-manifold is the quotient of a
(topological) principal $T^{k}$-action on the single manifold $Q_{k}$, where
$k=\beta_{2}(M)$.$\square$

By the homotopy sequence of the bundle $\pi$ one has

> $\pi_{r}(M(\alpha_{{\small 1}},\cdots,\alpha_{{\small
> k}}))=\left\\{\begin{tabular}[]{l}$0$ for $r=0,1,2$;\\\ $\pi_{r}(M)$ for
> $r\geq 3$ (induced by $\pi_{\ast}$).\end{tabular}\right.$

With $M(\alpha_{{\small 1}},\cdots,\alpha_{{\small k}})\cong Q_{k}$ by Theorem
C we obtain that

Corollary 1.6. The $2$–connected cover of any $1$-connected $4$-manifold $M$
is homeomorphic to $Q_{k}$, where $k=\beta_{2}(M)$.

In particular, if $3\leq i\leq k+3$ the homotopy group $\pi_{i}(M)$ can be
expressed by that of spheres by the formula

  1. (1.10)

$\pi_{i}(M)=\pi_{i}(\underset{{\small c}_{{\small
1}}}{{\small\vee}}(S^{{\small 3}}\vee S^{{\small k+1}})\underset{{\small
c}_{2}}{{\small\vee}}(S^{{\small 4}}\vee S^{{\small k}})\underset{{\small
c}_{{\small 3}}}{{\small\vee}}\cdots\underset{{\small
c}_{r}}{{\small\vee}}(S^{r{\small+2}}\vee S^{{\small k-}r{\small+2}}))$,

as well as the Hilton-Milnor’s formula [30, p.533].$\square$

The map $\pi$ induces the fibration $\Omega\pi:\Omega M(\alpha_{{\small
1}},\cdots,\alpha_{{\small k}})\rightarrow\Omega M$ in the loop spaces, where
$\Omega M(\alpha_{{\small 1}},\cdots,\alpha_{{\small k}})\cong\Omega Q_{k}$ by
Theorem C. Since $\Omega Q_{k}$ is $1$-connected, while the fiber $\Omega
T^{k}$ is homotopy equivalent to the Eilenberg-McLane space
$K(\oplus_{k}\mathbb{Z},0)$, we get

Corollary 1.7. For any $1$-connected $4$-manifold $M$ the universal cover of
the loop space $\Omega M$ is $\Omega Q_{k}$, where $k=\beta_{2}(M)$.$\square$

The remaining sections of the paper are organized as follows. Section 2
introduces isotopy invariants for the framed circles on a manifold, and
establishes Theorem B. Section 3 develops preliminary properties of the
suspension operators $\Sigma_{i}$, and shows Theorem A. Sections 4 and 5 are
devoted to the proofs of Theorem C and D, respectively.

##  2 Invariants of framed circles on a manifold.

For an orientation preserving smooth embedding $F:S^{1}\times
D^{n-1}\rightarrow M$ into a manifold $M\in\mathcal{T}_{n}$ we set
$S^{1}=F(S^{1}\times\\{0\\})\subset M$, and call the pair $(S^{1},F)$ a framed
circle on $M$. Two framed circles $(S^{1},F)$ and $(S^{1},F^{\prime})$ are
called isotopic if there is a diffeomorphism $H$ of $M\times I$ such that

> $H(x,0)=(F(x),0)$ and $H(x,1)=(F^{\prime}(x),1)$, $x\in S^{1}\times
> D^{n-1}$.

Let $[S^{1},M]^{fr}$ be the set of isotopy classes $[S^{1},F]$ of framed
circles on $M$, and write

> $\gamma:$ $[S^{1},M]^{fr}\rightarrow\pi_{1}(M)$,
> $\gamma([S^{1},F])=[S^{1}]$,

for the projection of isotopy classes to homotopy classes. Observe that if
$\widetilde{\alpha}$ is the diffeomorphism of $S^{1}\times D^{n-1}$ in (1.1),
then a framed circle $(S^{1},F)$ on $M$ determines a new one
$(S^{1},F\widetilde{\alpha})$.

Lemma 2.1. If $n\geq 5$ the map $\gamma$ surjects. Moreover, for any framed
circle $(S^{1},F)$ on $M$ we have
$\gamma^{-1}[S^{1}]=\\{[S^{1},F],[S^{1},F\widetilde{\alpha}]\\}$.

Proof. With $n\geq 5$ any map $S^{1}\rightarrow M$ is homotopic to a smooth
embedding, and any two homotopic embeddings $S^{1}\subset M$ are isotopic (by
a result of Whitney). Furthermore, with both $S^{1}$ and $M$ oriented, the
normal disc bundle $N(S^{1})$ of an embedding $S^{1}\subset M$ has the induced
orientation, hence has a trivialization $N(S^{1})\cong S^{1}\times D^{n-1}$,
showing $\gamma$ surjects.

As the group of trivializations of $N(S^{1})$ is
$\pi_{1}(SO(n-1))=\mathbb{Z}_{2}$, which is generated by $\alpha$, the pre-
image $\gamma^{-1}[S^{1}]$ of any $S^{1}\subset M$ contains at most two
elements: if one of them is $[S^{1},F]$, the other one must be
$[S^{1},F\widetilde{\alpha}]$.$\square$

By the proof of Lemma 2.1 we may consider a framed circle $(S^{1},F)$ as a
tubular neighborhood of the embedding $S^{1}\subset M$; project on the factor
$D^{n-1}$, and shrink the boundary to a point $\infty$, giving a sphere
$S^{n-1}$, and finally extend the map of the tubular neighborhood to $M$ by
mapping the complement to $\infty$. We have defined a map $M\rightarrow
S^{n-1}$ whose homotopy class is denoted by
$\gamma_{1}(S^{1},F)\in[M,S^{n-1}]$. Combining Lemma 2.1 with the Thom-
Pontryagin theorem [19, §7] one sees that the map

> $(\gamma,\gamma_{1}):[S^{1},M]^{fr}\rightarrow\pi_{1}(M)\times[M,S^{n-1}]$

injects. We write $\varepsilon(S^{1},F)=0$ or $1$ to denote whether
$\gamma_{1}(S^{1},F)=0$ or not.

Lemma 2.2. If $M\in\mathcal{T}_{n}$ is $1$-connected with $n\geq 5$, the map
$\varepsilon:[S^{1},M]^{fr}\rightarrow\mathbb{Z}_{2}$ injects, and satisfies
the following relations:

i) If $\varepsilon(S^{1},F)=1$ then
$\varepsilon(S^{1},F\widetilde{\alpha})=0$;

ii) If $w_{2}(M)=0$ and $\varepsilon(S^{1},F)=0$ then
$\varepsilon(S^{1},F\widetilde{\alpha})=1$;

iii) If $w_{2}(M)\neq 0$ and $\varepsilon(S^{1},F)=0$ then
$\varepsilon(S^{1},F\widetilde{\alpha})=0$.

In addition, for any $[S^{1},F]\in[S^{1},M]^{fr}$ we can write $M=M\\#S^{n}$
in which the sphere $S^{n}$ has the decomposition

  1. (2.1)

$S^{n}=\left\\{\begin{tabular}[]{l}$D^{2}\times S^{n-2}\cup_{id}(S^{1}\times
D^{n-1})$ {if} $\varepsilon(S^{1},F)=0${;}\\\ $D^{2}\times
S^{n-2}\cup_{\tau}(S^{1}\times D^{n-1})$ {if}
$\varepsilon(S^{1},F)=1${,}\end{tabular}\right.$

such that the component $S^{1}\times D^{n-1}$ is identical to $F$.

Proof. Let $S^{1}\subset$ $M$ be an oriented circle on $M$. Since $M$ is
$1$-connected with dimension $n\geq 5$ there is an embedded disk $D^{2}\subset
M$ with $\partial D^{2}=S^{1}\subset M$. Taking a tubular neighborhood $D^{n}$
of the disc one finds a frame $F$ of the circle $S^{1}$ so that
$\mathop{\mathrm{I}m}F\subset\overset{\circ}{D^{n}}$. That is, the manifold
$M$ has a decomposition $M=M\\#S^{n}$ such that $(S^{1},F)$ is a framed circle
on the sphere $S^{n}$.

For $M=S^{n}$ we have $[M,S^{n-1}]=\mathbb{Z}_{2}$, while the map
$\varepsilon$ is one to one, implying that $S^{n}$ has the two decompositions
given in (2.1). Finally, the relations i)-iii) have been shown by Goldstein
and Linger [17, Lemma 1].$\square$

Corollary 2.3. If $M\in\mathcal{T}_{n}$ is $1$-connected with $n\geq 5$, then
$[S^{1},M]^{fr}=\mathbb{Z}_{2}$ or $\\{0\\}$, where
$[S^{1},M]^{fr}=\mathbb{Z}_{2}$ happens if and only if $w_{2}(M)=0$.$\square$

For a $N\in\mathcal{T}_{n-1}$ we furnish the product $S^{1}\times N$ with the
induced orientation, and let $(D^{n-1},0)\subset(N,x_{0})$ be an orientation
preserving embedded disc centered at $x_{0}$. Then the natural inclusion
$F_{0}:S^{1}\times D^{n-1}\subset S^{1}\times N$ gives rise to the canonical
frame of the oriented circle $S^{1}\times\\{x_{0}\\}$ on $S^{1}\times N$, for
which $\gamma(S^{1}\times\\{x_{0}\\},F_{0})$ is the generator of the first
summand of $\pi_{1}(S^{1}\times N)=\mathbb{Z}\oplus\pi_{1}(N)$ that
corresponds to the orientation on $S^{1}\times\\{x_{0}\\}$, while
$\gamma_{1}(S^{1}\times\\{x_{0}\\},F_{0})$ is the homotopy class of the
composition

> $c:S^{1}\times N\rightarrow N\rightarrow
> N/(N-\overset{\circ}{D_{n}})=S^{n-1}$.

For any frame $F$ of the circle $S^{1}\times\\{x_{0}\\}$ on $S^{1}\times N$ we
write $\delta(S^{1}\times\\{x_{0}\\},F)=0$ or $1$ to denote
$\gamma_{1}(S^{1}\times\\{x_{0}\\},F)=[c]$ or not. From Lemma 2.1 we get

Lemma 2.4. If $n\geq 4$ and if $F$ is a frame of the oriented circle
$S^{1}\times\\{x_{0}\\}$ on $S^{1}\times N$, then
$\delta(S^{1}\times\\{x_{0}\\},F)=1$ implies that
$\delta(S^{1}\times\\{x_{0}\\},F\widetilde{\alpha})=0.\square$

Let $(S^{1},F)$ be a framed circle on a $1$-connected manifold
$M\in\mathcal{T}_{n}$ with $n\geq 5$, and let $F^{\prime}$ be one of the two
possible frames of the oriented circle $S^{1}\times\\{x_{0}\\}$ on
$S^{1}\times N$. The tunnel sum $\circ$ of $M$ and $S^{1}\times N$ along the
framed circles $(S^{1},F)$ and $(S^{1}\times\\{x_{0}\\},F^{\prime})$ is the
adjoint manifold

  1. (2.2)

$(M,F)\circ(S^{1}\times
N,F^{\prime}):=(M-\overset{\circ}{\mathop{\mathrm{I}m}F})\cup_{f\circ
f^{{}^{\prime}-1}}(S^{1}\times
N-\overset{\circ}{\mathop{\mathrm{I}m}F^{\prime}})$.

where $f:=F\mid S^{1}\times S^{n-2}$, $f^{\prime}:=F^{\prime}\mid S^{1}\times
S^{n-2}$. Useful properties of the operation $\circ$ are collected in the
following lemma.

Lemma 2.5. The tunnel sum operation $\circ$ satisfies the following relations

  1. (2.3)

$(M,F)\circ(S^{1}\times N,F^{\prime})=(M,F\widetilde{\alpha})\circ(S^{1}\times
N,F^{\prime}\widetilde{\alpha})$;

  2. (2.4)

$(M,F)\circ(S^{1}\times
N,F^{\prime})=\left\\{\begin{tabular}[]{l}$M\\#\Sigma_{0}N$ {if}
$\varepsilon(S^{1},F)=\delta(S^{1},F^{\prime})$;\\\ $M\\#\Sigma_{1}N$ {if}
$\varepsilon(S^{1},F)\neq\delta(S^{1},F^{\prime})$.\end{tabular}\right.$

In addition,

  1. (2.5)

$M\\#\Sigma_{0}N=M\\#\Sigma_{1}N$ if $w_{2}(M)\neq 0$.

Proof. Since the generator $\alpha\in\pi_{1}(SO(n-1))$ is of order $2$ the
relation (2.3) follows from the definition of $\circ$ in (2.2).

To show (2.4) we can assume by Lemma 2.2 that $M=S^{n}$, and take the
decomposition of $S^{n}$ relative to the index $\varepsilon(S^{1},F)$ as in
(2.1). If

$\varepsilon(S^{1},F)=\delta(S^{1},F^{\prime})=0$ (resp. if
$\varepsilon(S^{1},F)=1$ but $\delta(S^{1},F^{\prime})=0$)

one gets by (1.2) the desired relation

> $(S^{n},F)\circ(S^{1}\times N,F^{\prime})=D^{{\small 2}}\times S^{{\small
> n-1}}\cup_{id}(S^{{\small
> 1}}\times(N-\overset{\circ}{D^{n-1}}))=\Sigma_{0}N$
>
> (resp. $=D^{{\small 2}}\times S^{{\small n-1}}\cup_{\tau}(S^{{\small
> 1}}\times(N-\overset{\circ}{D^{n-1}})=\Sigma_{1}N$).

Granted with (2.3) the remaining case of (2.4)

> $\varepsilon(S^{1},F)=\varepsilon(S^{1},F^{\prime})=1$ (resp.
> $\varepsilon(S^{1},F)=0$ but $\varepsilon(S^{1},F^{\prime})=1$)

can be reduced to the case calculated above, by composing both $F$ and
$F^{\prime}$ with the diffeomorphism $\widetilde{\alpha}$ of $S^{{\small
1}}\times D^{n-1}$, then applying the relations i) of Lemma 2.2 to obtain the
result.

Finally, the relations in (2.5) is shown by iii) of Lemma 2.2.$\square$

We conclude this section by showing Theorem B.

Proof of Theorem B. Let $p:E\rightarrow B$ an oriented circle bundle over a
manifold $B\in\mathcal{T}_{n-1}$, in which the total space $E$ is
$1$-connected and $\dim E=n\geq 5$. Take an orientation preserving embedding
$i:D^{n-1}\subset B$ and set $i(0)=b\in B$. Then the restricted bundle $E\mid
D^{n-1}$ is a frame $(p^{-1}(b),F_{b})$ of the oriented circle
$p^{-1}(b)\subset E$, to be called the canonical frame of $p^{-1}(b)$. Recall
by [17, Theorem 8] that the map $\varepsilon:B\rightarrow\\{0,1\\}$ by
$\varepsilon(b):=\varepsilon(p^{-1}(b),F_{b})$ is constant, and satisfies that

  1. (2.6)

$\varepsilon(p^{-1}(b),F_{b})=0$ or $1$ if $w_{2}(B)\neq 0$ or $w_{2}(B)=0$.

On the other hand, let $p^{\prime}:S^{1}\times N\rightarrow N$ be the trivial
bundle on an $N\in\mathcal{T}_{n-1}$. Take an orientation preserving embedding
$i:D^{n-1}\subset N$ and set $x_{0}:=i(0)\in N$. Since the restriction
$p^{\prime}\mid D^{n-1}$ is precisely the canonical frame
$(p^{\prime-1}(x_{0}),F_{0})$ of the oriented circle
$p^{\prime-1}(x_{0})=S^{1}\times x_{0}$ we have

  1. (2.7)

$\delta(p^{\prime-1}(x_{0}),F_{0})=0$ (by the definition).

Consider now the canonical projection $f_{N}:B\\#N\rightarrow B$, and let
$p_{N}:E_{N}\rightarrow$ $B\\#N$ be the induced bundle $f_{N}^{\ast}E$. Then
the total space $E_{N}$ admits the decomposition into tunnel sum

> $E_{N}=(E,F_{b})\circ(S^{1}\times N,F_{x_{0}})$,

where $b\in B$, $x_{0}\in N$, and where both $(p^{-1}(b),F_{b})$ and
$(S^{1}\times\\{x_{0}\\},F_{x_{0}})$ are canonical. Granted with (2.6) and
(2.7), the formulae (1.4) and (1.5) of Theorem B have been shown by (2.4) and
(2.5), respectively.$\square$

##  3 Properties of the suspension operations

This section develops general properties of the suspension operations
$\Sigma_{i}$.

3.1. Relationship between $\Sigma_{1}N$ and $\Sigma_{0}N$. We begin with a
proof of Theorem A.

Proof of Theorem A. Let $N\in\mathcal{T}_{n}$ be a manifold with base point
$x_{0}\in N$, and let $F_{0}$ be the canonical frame of the oriented circle
$S^{1}\times\\{x_{0}\\}$ on $S^{1}\times N$. A map $\beta:S^{1}\rightarrow
Diff(N,x_{0})$ gives rise to a diffeomorphism of $S^{1}\times N$

> $\widetilde{\beta}:S^{1}\times N\rightarrow S^{1}\times N$ by
> $\widetilde{\beta}(z,x)=(z,\beta(z)\cdot x)$

that fixes $S^{1}\times x_{0}$ pointwisely. In particular, in addition to
$F_{0}$ the composition $\widetilde{\beta}\circ F_{0}$ is a second frame of
the circle $S^{1}\times\\{x_{0}\\}$. Assume that the manifold $N$ is
$\Sigma$-stable, and that $e_{N}\circ\beta=\alpha$. Then
$\widetilde{\beta}\circ F_{0}=F_{0}\widetilde{\alpha}$. In particular,
surgering along $S^{1}\times x_{0}$ with respect to the two frames $F_{0}$ and
$\widetilde{\beta}\circ F_{0}$, we obtain from $\widetilde{\beta}$ a desired
diffeomorphism $\Sigma_{0}N\rightarrow\Sigma_{1}N$.$\square$

For a second manifold $M\in\mathcal{T}_{m}$ with a base point $y_{0}\in M$ the
representation $e_{N\times M}$ satisfies the commutative diagram

>
> $\begin{array}[]{ccc}Diff(N,\\{x_{0}\\})&\overset{e_{N}}{\rightarrow}&GL(n)\\\
> i\downarrow&&j\downarrow\\\ Diff(N\times
> M,\\{x_{0},y_{0}\\})&\overset{e_{N\times
> M}}{\rightarrow}&GL(n+m)\end{array}$,

where $i(h)=h\times id_{M}$ and $j(g)=g\oplus I_{m}$. As the inclusion $j$ is
$(n-1)$-homotopy equivalent we get

Corollary 3.1. If $N\in\mathcal{T}_{n}$ with $n\geq 3$ is $\Sigma$-stable,
then the product $N\times M$ is also $\Sigma$-stable. In particular,
$\Sigma_{0}(N\times M)=\Sigma_{1}(N\times M)$ for any
$M\in\mathcal{T}_{m}$.$\square$

3.2. The diffeomorphism types of $\Sigma_{i}N$. For a manifold $N\in T_{n}$
with an orientation preserving embedding $(D^{n},0)\subset(N,x_{0})$ set
$W_{N}:=D^{2}\times(N-\overset{\circ}{D^{n}})$. In addition to (1.2) we have

  1. (3.1)

$\Sigma_{0}N=\partial W_{N}$.

This relation is useful to calculate the diffeomorphism types of
$\Sigma_{i}N$. We present such examples.

Proposition 3.2. The sphere $S^{n}$ is $\Sigma$-stable with
$\Sigma_{0}S^{n}=\Sigma_{1}S^{n}=S^{n+1}$.

In particular, for any $p\leq q$ with $q\geq 3$ we have

  1. (3.2)

$\Sigma_{0}(S^{p}\times S^{q})=\Sigma_{1}(S^{p}\times S^{q})=S^{p}\times
S^{q+1}\\#S^{p+1}\times S^{q}$.

Proof. Let $\\{e_{1},\cdots,e_{n+1}\\}$ be the canonical orthonormal basis of
$\mathbb{R}^{n+1}$ and take $e_{n+1}$ as the base point $x_{0}$ of $S^{n}$.
The generator $\alpha\in\pi_{1}(SO(n))$ gives rise to the loop on
$Diff(S^{n},x_{0})$

> $\beta:S^{1}\rightarrow Diff(S^{n},x_{0})$, $\beta(z)(x)=\alpha(z)x$, $z\in
> S^{1}$, $x\in S^{n}$,

that satisfies $e_{S^{n}}(\beta)=\alpha$, implying that $S^{n}$ is
$\Sigma$-stable. Moreover, with $W_{S^{n}}=D^{n+2}$ we have
$\Sigma_{0}S^{n}=\partial W_{S^{n}}=S^{n+1}$ by (3.1), showing the first part
of the proposition.

By (1.2) the manifold $\Sigma_{0}(S^{p}\times S^{q})$ is obtained by
performing surgery along the oriented circle $S^{1}\times x_{0}\subset$
$S^{1}\times(S^{p}\times S^{q})$ ($x_{0}\in S^{p}\times S^{q}$) with trivial
framing, implying

> $\Sigma_{0}(S^{p}\times S^{q})=S^{p}\times S^{q+1}\\#S^{p+1}\times S^{q}$.

With $S^{p}\times S^{q}$ being $\Sigma$-stable by Corollary 3.1, we get the
equalities in (3.2) by Theorem A.$\square$

Let $\mathbb{F}P^{n}$ be the $n$-dimensional projective space of the
$1$-dimensional $\mathbb{F}$-subspaces of the vector space $\mathbb{F}^{n+1}$,
where $\mathbb{F}$ is either the field $\mathbb{R}$ of reals, the field
$\mathbb{C}$ of complexes, or the algebra $\mathbb{H}$ of quaternions. The
canonical Hopf line bundle over $\mathbb{F}P^{n}$ is
$\lambda_{\mathbb{F}}^{n}:=\\{(x,v)\in\mathbb{F}P^{n}\times\mathbb{F}^{n+1}\mid
v\in x\\}$. In accordance to $\mathbb{F}=\mathbb{R},\mathbb{C}$ or
$\mathbb{H}$ the real reduction of $\lambda_{\mathbb{F}}^{n}$ is an Euclidean
vector bundle over $\mathbb{F}P^{n}$ of real dimension $1$, $2$ or $4$,
respectively. For an Euclidean vector bundle $\xi$ over a CW-complex $B$ write
$D(\xi)$ and $S(\xi)$ to denote the total spaces of the associated disc-bundle
and sphere-bundle, respectively.

Proposition 3.3.
$\Sigma_{0}\mathbb{F}P^{n}=S(\lambda_{\mathbb{F}}^{n-1}\oplus\varepsilon^{2})$,
where $\varepsilon^{2}$ denotes the 2-dimensional trivial bundle over
$\mathbb{F}P^{n-1}$.

In particular, the homotopy groups of $\Sigma_{0}\mathbb{F}P^{n}$ are

  1. (3.3)

$\pi_{r}(\Sigma_{0}\mathbb{F}P^{n})=$
$\pi_{r}(\mathbb{F}P^{n-1})\oplus\pi_{r}(S^{d})$,

where $d=2$, $3$ or $5$ in accordance to $\mathbb{F}=\mathbb{R},\mathbb{C}$ or
$\mathbb{H}$.

Proof. Set $r:=\dim_{\mathbb{R}}\mathbb{F}$. In view of $\partial
D(\lambda_{\mathbb{F}}^{n-1})=S^{rn-1}$ the manifold $\mathbb{F}P^{n}$ has the
decomposition $\mathbb{F}P^{n}=D(\lambda_{\mathbb{F}}^{n-1})\cup_{id}D^{rn}$,
which implies that

>
> $W_{\mathbb{F}P^{n}}=D^{2}\times(\mathbb{F}P^{n}-\overset{\circ}{D^{rn}})=D^{2}\times
> D(\lambda_{\mathbb{F}}^{n-1})=D(\lambda_{\mathbb{F}}^{n-1}\oplus\varepsilon^{2})$.

One obtains $\Sigma_{0}\mathbb{F}P^{n}=\partial
W_{\mathbb{F}P^{n}}=S(\lambda_{\mathbb{F}}^{n-1}\oplus\varepsilon^{2})$ by
(3.1). Since $\Sigma_{0}\mathbb{F}P^{n}$ is the total space of a $d$-sphere
bundle over $\mathbb{F}P^{n-1}$ which has a section, we obtain (3.3) by the
homotopy sequence of fiber bundles.$\square$

Example 3.4. Let $\Omega_{n}^{Spin}$ be the $n$-dimensional Spin bordism
group. Then $\Sigma_{0}\mathbb{H}P^{2}=0\in\Omega_{9}^{Spin}$ by
$\Sigma_{0}\mathbb{H}P^{2}=\partial W_{\mathbb{H}P^{2}}$. On the other hand,
there exists a surgery along the circle $S^{1}\times\\{x_{0}\\}\subset
S^{1}\times\mathbb{H}P^{2}$ such that the resulting manifold $M$ satisfies
$M\neq 0\in\Omega_{9}^{Spin}$ (e.g. [1, p.258], [6]). That is

> $M\neq\Sigma_{0}\mathbb{H}P^{2}$ and $M=\Sigma_{1}\mathbb{H}P^{2}$.

In particular, the manifold $\mathbb{H}P^{2}$ is not $\Sigma$-stable by
Theorem A.$\square$

Example 3.5. Formula (1.10) indicates that Theorem B may be useful to deduce
the presentation of homotopy groups of certain manifolds. Combining (1.6) and
(3.3) we obtain further evidences:

>
> $\pi_{r}(\mathbb{C}P^{2n}\\#\mathbb{R}P^{4n})=\left\\{\begin{array}[]{c}\mathbb{Z}_{2}\text{
> if }r=1\text{,}\\\ \pi_{r}(S^{4n-1})\oplus\pi_{r}(S^{2})\text{ if }r\neq
> 1\end{array}\right.$;
>
>
> $\pi_{r}(\mathbb{C}P^{2n}\\#\mathbb{C}P^{2n})=\left\\{\begin{array}[]{c}\mathbb{Z}\oplus\mathbb{Z}\text{
> if }r=2\\\ \pi_{r}(S^{4n-1})\oplus\pi_{r}(S^{3})\text{ if }r\neq
> 2\end{array}\right.$;
>
>
> $\pi_{r}(\mathbb{C}P^{2n}\\#\mathbb{H}P^{n})=\left\\{\begin{array}[]{c}\mathbb{Z}\text{
> if }r=2,4\\\ \pi_{r}(S^{4n-1})\oplus\pi_{r}(S^{5})\text{ if }r\neq
> 2,4\end{array}\right.$.$\square$

Example 3.6. It is clear by (1.2) that the operations $\Sigma_{i}$ applies to
the orientable surfaces to yield the correspondences
$\Sigma_{i}:\mathcal{T}_{2}\rightarrow\mathcal{T}_{3}$.

Let $M_{g}\in\mathcal{T}_{2}$ be the orientable surface with genus $g\geq 1$,
and let $D^{2}\subset M_{g}$ be an embedded disc. Then the fundamental group
$\pi_{1}(M_{g}-\overset{\circ}{D^{2}})$ is the free group generated by the
standard elements

> $a_{1},b_{1},\cdots a_{g},b_{g}\in\pi_{1}(M_{g}-\overset{\circ}{D^{2}})$.

Applying the Van Kampen Theorem to (1.2) one gets the presentations of the
fundamental groups $\pi_{1}(\Sigma_{i}M_{g})$ in term of generators and
relations

> $\pi_{1}(\Sigma_{0}M_{g})=\left\langle a_{1},b_{1},\cdots
> a_{g},b_{g}\right\rangle$;
>
> $\pi_{1}(\Sigma_{1}M_{g})=\left\langle a_{1},b_{1},\cdots a_{g},b_{g},z\mid
> a_{i}z=za_{i},b_{i}z=zb_{i},\right.$
>
> $\qquad\qquad\qquad\left.z^{-1}=a_{1}b_{1}a_{1}^{-1}b_{1}^{-1}\cdots
> a_{n}b_{n}a_{n}^{-1}b_{n}^{-1}\right\rangle$.

Geometrically, $\Sigma_{0}M_{g}=S^{1}\times S^{2}\\#\cdots\\#S^{1}\times
S^{2}$ ($2g$-copies), while $\Sigma_{1}M_{g}$ is the total space of the
oriented circle bundle over $M_{g}$ whose Euler class is the orientation class
on $M_{g}$ (e.g. Freedman [10]).$\square$

3.3. General properties of the operation $\Sigma_{i}$. Theorem B is applicable
to deduce general properties of the operations $\Sigma_{i}$. We present two
such examples.

Corollary 3.7. For any sequence $N_{1},\cdots,N_{k}\in T_{n-1}$ of
$1$-connected manifolds with ${n\geq 5}$, we have

  1. (3.4)

$\Sigma_{i}(N_{1}\\#\cdots\\#N_{k})=\Sigma_{i}N_{1}\\#\cdots\\#\Sigma_{i}N_{k}$,
$i=0,1$.

Proof. As example we show (3.4) for the case $i=1$. By the definition of the
space $E_{N}$ in (1.3) we get $E_{N\\#M}=(E_{N})_{M}$. In addition, the Van-
Kampen Theorem [11, p.43] implies that $\pi_{1}(\Sigma_{i}N)=\pi_{1}(N)$ for
any $N\in T_{n}$. In particular, with $N_{1},\cdots,N_{k}\in T_{n-1}$ all
1-connected Theorem B can be applied repeatedly to show that

> $E_{N_{1}\\#\cdots\\#N_{k}}=E\\#\Sigma_{1}N_{1}\\#\cdots\\#\Sigma_{1}N_{k}$
> if $w_{2}(B)=0$.

Taking $p$ to be the circle bundle $E=S^{3}\times S^{n-2}\rightarrow
S^{2}\times S^{n-2}$ mentioned in Example 1.5 we get by Theorem B that

> $S^{3}\times S^{n-2}\\#\Sigma_{1}(N_{1}\\#\cdots\\#N_{k})=S^{3}\times
> S^{n-2}\\#\Sigma_{1}N_{1}\\#\cdots\\#\Sigma_{1}N_{k}$.

Surgeries on both sides to kill the homotopy class represented by the
embedding $S^{3}\subset$ $S^{3}\times S^{n-2}$ shows (3.4).$\square$

A manifold $N$ is called a homology $n$-sphere if its integral cohomology is
isomorphic to that of $S^{n}$. The operators $\Sigma_{i}$ enables one to
produce higher dimensional homology spheres from the lower dimensional ones.
For a homology $n$-sphere $N$ consider the circle bundle in Example 1.5:

> $p_{N}:S^{3}\times S^{n-2}\\#\Sigma_{i}N\rightarrow
> S^{2}\widetilde{\times}S^{n-2}\\#N$ (resp. $S^{2}\times S^{n-2}\\#N$).

By the Gysin sequence of $p_{N}$ [20, p.143] with integer coefficients we get
$H^{\ast}(S^{3}\times S^{n-2}\\#\Sigma_{i}N)=H^{\ast}(S^{3}\times S^{n-2})$.
This shows that

Corollary 3.8. If $N$ is a homology $n$-sphere with $n\geq 4$, then both
$\Sigma_{i}N$, $i=0,1$, are homology $(n+1)$-sphere.$\square$

3.4. Invariants of $\Sigma_{i}M$. We conclude this section by expressing the
topological invariants (i.e. homology, cohomology, characteristic classes) of
the suspensions $\Sigma_{i}N$ in term of that of $N$. Recall that the reduced
suspension $SX$ of a CW-complex $X$ is the quotient space $S^{1}\times
X/(S^{1}\vee X)$ [11, p.223]. Let $\widetilde{H_{\ast}}$ (resp.
$\widetilde{H^{\ast}}$) denotes the reduced homology (resp. cohomology) of
topological spaces.

Proposition 3.9. The homology and cohomology of the suspensions $\Sigma_{i}N$
have the following presentations

  1. (3.5)

$H_{\ast}(\Sigma_{i}N)=\widetilde{H_{\ast}}(SN)\oplus
H_{\ast}(N-\left\\{x_{0}\right\\})$;

  2. (3.6)

$H^{\ast}(\Sigma_{i}N)=\widetilde{H^{\ast}}(SN)\oplus
H^{\ast}(N-\left\\{x_{0}\right\\})$.

In addition, corresponding to the splitting (3.6), the Stiefel-Whitney classes
$w_{r}$ and Pontryagin classes $p_{r}$ of $\Sigma_{i}N$ are given by

  1. (3.7)

$w_{r}(\Sigma_{i}N)=0\oplus w_{r}(N-\left\\{x_{0}\right\\})$;
$p_{r}(\Sigma_{i}N)=0\oplus p_{r}(N-\left\\{x_{0}\right\\})$.

Proof. It suffices to show (3.6) and (3.7). In term of (1.2) we set

> $A:=S^{1}\times(N-\overset{\circ}{D^{n}})$, $B:=D^{2}\times S^{n-1}$

to get $\Sigma_{i}N=A\cup B$, $A\cap B=S^{1}\times S^{n-1}$, and the Mayer-
Vietoris sequence

$\cdots\rightarrow H^{r-1}(A\cap
B)\overset{\delta}{\rightarrow}H^{r}(\Sigma_{i}N)\overset{i_{1}^{\ast}\oplus
i_{2}^{\ast}}{\rightarrow}H^{r}(A)\oplus
H^{r}(B)\overset{j_{1}^{\ast}-j_{2}^{\ast}}{\rightarrow}H^{r}(A\cap
B)\rightarrow\cdots$.

For an $M\in\mathcal{T}_{m}$ let $\omega_{M}\in H^{m}(M)$ be the orientation
class. Then the group $H^{\ast}(A\cap B)$ is torsion free with basis
$\\{1,\omega_{S^{1}},\omega_{S^{n-1}},\omega_{S^{1}\times S^{n-1}}\\}$ in
which $\omega_{S^{1}}\in\mathop{\mathrm{I}m}j_{1}^{\ast}$,
$\omega_{S^{n-1}}\in\mathop{\mathrm{I}m}j_{2}^{\ast}$. In particular, the
sequence can be simplified as the one with only three terms:

  1. (3.8)

$0\rightarrow\mathbb{Z}\\{\omega_{S^{1}\times
S^{n-1}}\\}\overset{\delta}{\rightarrow}H^{\ast}(\Sigma_{i}N)\overset{i_{1}^{\ast}}{\rightarrow}\frac{H^{\ast}(S^{1}\times(N-\left\\{x_{0}\right\\}))}{\mathbb{Z}\\{\omega_{S^{1}}\cdot
1\\}}\rightarrow 0$,

where $\mathbb{Z}\\{\omega\\}$ denotes the cyclic group with generator
$\omega$, $N-\overset{\circ}{D^{n}}$ is replaced by $N-\left\\{x_{0}\right\\}$
by the homotopy invariance of cohomology, and with
$H^{n+1}(S^{1}\times(N-\left\\{x_{0}\right\\}))=0$, the map $\delta$ is an
isomorphism onto $H^{n+1}(\Sigma_{i}N)=\mathbb{Z}$. One obtains (3.6) from
(3.8), under the convention that the orientation class
$\omega_{\Sigma_{i}N}=\delta(\omega_{S^{1}\times S^{n-1}})$ corresponds to the
top degree generator $s\wedge\omega_{N}$ of
$\widetilde{H^{n+1}}(SN)=\mathbb{Z}$.

Let $TM$ be the tangent bundle of a smooth manifold $M$. Then the inclusion
$i_{1}$ satisfies that $i_{1}^{\ast}T(\Sigma_{i}N)=\varepsilon^{1}\oplus
T(N-\left\\{x_{0}\right\\})$. Formula (3.7) follows from the naturality of the
characteristic classes, together with the exactness of (3.8).$\square$

##  4 Circle actions on 1-connected 6-manifolds

In [14, 16, 17] R. Goldstein and L. Lininger studied the problem of
classifying those 1-connected 6-manifolds that admit regular circle actions.
In this section we simplify their approach, and make the main results of [14,
17] precise (see Theorem C).

Let $A$ be an abelian group. An element $\omega\in A$ is called primitive if
the cyclic subgroup generated by $\omega$ is a direct summand of the free part
$A/TorA$ of $A$. By the homotopy sequence of fiber bundles one gets (e.g. [8])

Lemma 4.1. Let $M$ be an $1$-connected $6$-manifolds that has a regular circle
action with quotient map $\pi:$ $M\rightarrow M/S^{1}$. Then

i) $M/S^{1}$ is an $1$-connected $5$-manifolds;

ii) The Euler class $\omega\in H^{2}(M/S^{1})$ of $\pi$ is primitive.$\square$

Let $\gamma$ be the Hopf complex line bundle over $S^{2}$, and consider the
handlebodies $A:=S^{2}\times D^{3}$, $B:=D(\gamma\oplus\mathbb{R})$ of
dimension $5$. For a connected manifold $V$ with non-empty boundary $\partial
V$ write $V\cdot V$ to denote the boundary connected sum of two copies of $V$.
As examples, it follows from $\partial A=S^{2}\times S^{2}$ and $\partial
B=\mathbb{C}P^{2}\\#\overline{\mathbb{C}P}^{2}$ that

> $\partial(A\cdot A)=S^{2}\times S^{2}\\#S^{2}\times S^{2}$, $\partial(B\cdot
> B)=\mathbb{C}P^{2}\\#\overline{\mathbb{C}P}^{2}\\#\mathbb{C}P^{2}\\#\overline{\mathbb{C}P}^{2}$.

Denote by $\\{a,b\\}$, $\\{a_{1},b_{1},a_{2},b_{2}\\}$ and
$\\{x_{1},y_{1},x_{2},y_{2}\\}$ the canonical bases of the second cohomologies
of the boundaries $\partial B$, $\partial(A\cdot A)$ and $\partial(B\cdot B)$,
respectively. With these convention we introduced in the table below a
collection of five $1$-connected $5$-manifolds

  1. (4.1)

$S^{2}\times S^{3}$ | $S^{2}\widetilde{\times}S^{3}$ | $W$ | $M_{k}$ | $X_{2^{i}}$
---|---|---|---|---
$A\cup_{id}A$ | $B\cup_{id}B$ | $B\cup_{\tau}B$ | $A\cdot A\cup_{f_{k}}A\cdot A$ | $B\cdot B\cup_{g_{k}}B\cdot B$

where $\tau\in Diff(\partial B)$, $f_{k}\in Diff(\partial(A\cdot A))$, and
$g_{k}\in Diff(\partial(B\cdot B))$ are the diffeomorphisms whose actions on
the second homologies are (e.g. [29])

> i) $(\tau_{\ast}(a),\tau_{\ast}(b))=(a,-b)$;
>
> ii)
> $(f_{k\ast}(a_{1}),f_{k\ast}(b_{1}),f_{k\ast}(a_{2}),f_{k\ast}(b_{2}))=(a_{1},b_{1},a_{2},b_{2})\cdot
> C_{k}$,
>
> iii)
> $(g_{k\ast}(x_{1}),g_{k\ast}(y_{1}),g_{k\ast}(x_{2}),g_{k\ast}(y_{2}))=(x_{1},y_{1},x_{2},y_{2})\cdot
> L_{k}$,

and where $C_{k}$ and $L_{k}$ denote the $4\times 4$ matrices

> $C_{k}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&k&0\\\ 0&0&1&0\\\
> -k&0&0&1\end{array}\right)$ and
> $L_{k}=\left(\begin{array}[]{cccc}1&2^{k}&2^{k}&0\\\ 2^{k}&1&0&-2^{k}\\\
> -2^{k}&0&1&2^{k}\\\ 0&2^{k}&2^{k}&1\end{array}\right)$,

respectively. By the Mayer-Vietoris sequence one gets (e.g. [5])

Lemma 4.2. Let $M$ be one of the five manifolds in (4.1).

i) The second cohomology $H_{2}(M)$ is

> $\qquad H_{2}(S^{2}\times
> S^{3})=H_{2}(S^{2}\widetilde{\times}S^{3})=\mathbb{Z}$;
> $H_{2}(W)=\mathbb{Z}_{2}$;
>
> $\qquad H_{2}(M_{k})=\mathbb{Z}_{k}\oplus\mathbb{Z}_{k}$;
> $H_{2}(X_{k})=\mathbb{Z}_{2^{k}}\oplus\mathbb{Z}_{2^{k}}$;

ii) $w_{2}(M)\neq 0$ implies that
$M\in\\{S^{2}\widetilde{\times}S^{3},W,X_{k}\\}.$

In term of the five manifolds constructed in (4.1) the classification of Smale
and Barden on the $1$-connected $5$-manifolds [2, 24] may be rephrased as
follows.

Lemma 4.3. For any 1-connected 5-manifold $N$ one has

  1. (4.2)

$N=\left\\{\begin{tabular}[]{l}$\\#_{r}S^{2}\times S^{3}\\#_{1\leq i\leq
t}M_{k_{i}}$ if $w_{2}(N)=0${;}\\\ $\\#_{r}S^{2}\times S^{3}\\#_{1\leq j\leq
t}M_{k_{j}}\\#H$ if $w_{2}(N)\neq 0${,}\end{tabular}\right.$

where $H\in\\{S^{2}\widetilde{\times}S^{3},W,X_{k}\\}$, $\\#_{0}(\cdots)$
implies $S^{5}$.

In addition, for any primitive element $\omega\in H^{2}(N)$ one can modify the
decomposition (4.2) as

  1. (4.3)

$N=\left\\{\begin{tabular}[]{l}$S^{2}\times S^{3}\\#N^{\prime}${,} {if}
$\omega\neq w_{2}(N)\mathop{\mathrm{m}od}2${;}\\\
$S^{2}\widetilde{\times}S^{3}\\#N^{\prime}$ {with} $w_{2}(N^{\prime})=0${,}
{if} $\omega\equiv w_{2}(N)\mathop{\mathrm{m}od}2${,}\end{tabular}\right.$

such that $\omega$ is a generator of the first summand in
$H_{2}(N)=H_{2}(S^{2})\oplus H_{2}(N^{\prime})$.$\square$

Proof of Theorem C. For an $1$-connected $6$-manifold $M$ with a regular
circle action let $\pi:$ $M\rightarrow N:=M/S^{1}$ be the quotient map. Since
the Euler class $\omega\in H^{2}(N)$ of $\pi$ is primitive, we can take a
decomposition of $N$ relative to $\omega$ as in (4.3), to get by (1.4) that
$M=S^{3}\times S^{3}\\#\Sigma_{i}N^{\prime}$, where $i=0$ or $1$ in accordance
to $\omega\equiv w_{2}(N)\mathop{\mathrm{m}od}2$, or $\omega\neq
w_{2}(N)\mathop{\mathrm{m}od}2$. Decomposing $N^{\prime}$ in the form of
(4.2), and using (3.4) to expand $\Sigma_{i}N^{\prime}$, we obtain formula
(1.7), together with $\Sigma_{0}(S^{2}\times S^{3})=\Sigma_{1}(S^{2}\times
S^{3})=S^{3}\times S^{3}\\#S^{2}\times S^{4}$ by (3.2).$\square$

##  5 Torus bundles over 4-manifolds

The connected sums furnishes the set $\mathcal{T}_{n}$ with the structure of a
semi-abelian group, in which the sphere $S^{n}$ represents the null element.
Let $\mathcal{S}_{n}\subset\mathcal{T}_{n}$ be the semi-subgroup generated by
the product $S^{r}\times S^{n-r}$ of spheres. Since every $M\in$
$\mathcal{S}_{n}$ has the unique expression

> $M=\\#_{a_{1}}S^{1}\times
> S^{n-1}\\#_{a_{2}}{\small\cdots}\\#_{a_{[\frac{n}{2}]}}S^{[\frac{n}{2}]}\times
> S^{n-[\frac{n}{2}]}$,

in which the sequence of integers $\\{a_{1},\cdots,a_{[\frac{n}{2}]}\\}$ is
determined by the Poincarè polynomial

>
> $P_{t}(M)=1+a_{1}(t^{1}+t^{n-1})+\cdots+a_{\left[\frac{n}{2}\right]}(t^{\left[\frac{n}{2}\right]}+t^{n-\left[\frac{n}{2}\right]})+t^{n}$,

of $M$, we have

Lemma 5.1. The correspondence $P_{t}:\mathcal{S}_{n}\rightarrow\mathbb{Z}[t]$
injects.$\square$

We state two additional lemmas required by showing Theorem C.

Lemma 5.2. Let $M$ be a topological manifold whose second cohomology
$H^{2}(M)$ is torsion free with basis
$\left\\{\alpha_{1},\cdots,\alpha_{k}\right\\}$. Then, in the notion of (1,8),

i) $M(\alpha_{1},\cdots,\alpha_{k})\cong
M(\alpha_{1}^{\prime},\cdots,\alpha_{k}^{\prime})$ for any basis
$\left\\{\alpha_{1}^{\prime},\cdots,\alpha_{k}^{\prime}\right\\}$ of
$H^{2}(M)$;

ii) $M\in\mathcal{S}_{n}$ implies that
$M(\alpha_{1},\cdots,\alpha_{k})\in\mathcal{S}_{n+k}$, where $n\geq 5$.

Lemma 5.3. If $N:=\\#_{1\leq i\leq k-1}(S^{2}\times S^{3})$ and if
$\left\\{\omega_{1},\cdots,\omega_{k-1}\right\\}$ is the basis of $H^{2}(N)$
Kronecker dual to the canonical embeddings $S^{2}\subset N$, then
$N(\omega_{1},\cdots,\omega_{k-1})=Q_{k}$, where $Q_{k}$ is the manifold
defined by (1.9).

We postpone the proofs of Lemmas 5.2 and 5.3 for the moment, and proceed to
show how them lead to Theorem D.

Proof of Theorem D. Let $M$ be an $1$-connected $4$-manifold with
$\beta_{2}(M)=k$. According to [8, Corollary 3] there is a primitive element
$e\in H^{2}(M)$ so that the total space of the circle bundle
$\pi:M(e)\rightarrow M$ with Euler class $e$ is

> $M(e)=\\#_{1\leq i\leq k-1}(S^{2}\times S^{3})$ ($=N$ in the notation of
> Lemma 5.3).

In term of the Gysin sequence of $\pi$ one finds a basis
$\left\\{\alpha_{1},\cdots,\alpha_{k}\right\\}$ of $H^{2}(M)$ such that
$\alpha_{k}=e$, $\pi^{\ast}(\alpha_{i})=\omega_{i}$, $1\leq i\leq k-1$, where
$\\{\omega_{1},\cdots,\omega_{k-1}\\}$ is the basis of $H^{2}(M(e))$ stated in
Lemma 5.3. Then $M(\alpha_{1},\cdots,\alpha_{k})=$
$N(\omega_{1},\cdots,\omega_{k-1})$ in the notion of (1.8). We get
$M(\alpha_{1},\cdots,\alpha_{k})=Q_{k}$ by Lemma 5.3.

Finally, suppose that
$\left\\{\alpha_{1}^{\prime},\cdots,\alpha_{k}^{\prime}\right\\}$ is an
arbitrary basis of $H^{2}(M)$. By i) of Lemma 5.2 we have
$M(\alpha_{1}^{\prime},\cdots,\alpha_{k}^{\prime})=M(\alpha_{1},\cdots,\alpha_{k})$.
This completes the proof of Theorem D.$\square$

It remains to show Lemmas 5.2 and 5.3.

Proof of Lemma 5.2. Let $M$ be a topological manifold whose second cohomology
$H^{2}(M)$ is torsion free, and let
$\left\\{\alpha_{1},\cdots,\alpha_{k}\right\\}$ and
$\left\\{\alpha_{1}^{\prime},\cdots,\alpha_{k}^{\prime}\right\\}$ be two bases
of $H^{2}(M)$. Expressing the basis elements $\alpha_{i}$’s in the
$\alpha_{i}^{\prime}$’s yields a $k\times k$ integral matrix
$C=(c_{ij})_{k\times k}$ such that

  1. (5.1)

$\left\\{\alpha_{1},\cdots,\alpha_{k}\right\\}=\left\\{\alpha_{1}^{\prime},\cdots,\alpha_{k}^{\prime}\right\\}\cdot
C,$ $\det(C)=\pm 1$.

Let $f$ and $f^{\prime}:M\rightarrow BT^{k}$ be respectively the classifying
maps of the two bases.

Regard $S^{1}$ as the group of unit complexes $z$, and let
$B_{g}:BT^{k}\rightarrow BT^{k}$ be induced by the group isomorphism

> $g:T^{k}\rightarrow T^{k}$, $g(z_{1},\cdots,z_{k})=(z_{1}^{c_{11}}\cdots
> z_{k}^{c_{1k}},\cdots,z_{1}^{c_{k1}}\cdots z_{k}^{c_{kk}})$.

With $f^{\ast}=f^{\prime\ast}B_{g}^{\ast}$ by (5.1) we can assume, by the
homotopy invariance of the induced bundles, that $f=B_{g}\circ f^{\prime}$.
Since $B_{g}$ admits a homotopy inverse we get a desired homeomorphism
$M(\alpha_{1},\cdots,\alpha_{k})\cong
M(\alpha_{1}^{\prime},\cdots,\alpha_{k}^{\prime})$, showing the first part of
Lemma 5.2.

For the second part of the lemma we can assume by $M\in\mathcal{S}_{n}$ and
$n\geq 5$ that

  1. (5.2)

$M=\\#_{k}(S_{i}^{2}\times S_{i}^{n-2})\\#_{1\leq j\leq r}(S_{j}^{k_{j}}\times
S_{j}^{n-k_{j}})$, $k_{j}\leq n-k_{j}$, $k_{j}\neq 2$.

where $S_{t}^{p}\times S_{t}^{n-p}=S^{p}\times S^{n-p}$. Set for each $1\leq
i\leq k$ that

> $g_{i}:M\overset{q_{i}}{\rightarrow}S_{i}^{2}\times
> S_{i}^{n-2}\overset{p_{1}}{\rightarrow}S^{2}$,
> $\omega_{i}:=g_{i}^{\ast}(\omega)\in H^{2}(M)$,

where $q_{i}$ is the quotient of $M$ by the summands other than the component
$S_{i}^{2}\times S_{i}^{n-2}$, $p_{1}$ is the projection onto the first
factor, and where $\omega\in H^{2}(S^{2})=\mathbb{Z}$ is the orientation
class. According to assertion i) it suffices to show
$M(\omega_{1},\cdots,\omega_{k})\in\mathcal{S}_{n+k}$. This can be done by
induction on $k$.

Let $\pi_{1}$ be the circle bundle $M(\omega_{1})\rightarrow M$ over $M$ with
Euler class $\omega_{1}$. Then, in the notation of (1.8),

>
> $M(\omega_{1},\cdots,\omega_{k})=M(\omega_{1})(\pi_{1}^{\ast}(\omega_{2}),\cdots,\pi_{1}^{\ast}(\omega_{k-1}))$,

where
$\left\\{\pi_{1}^{\ast}(\omega_{2}),\cdots,\pi_{1}^{\ast}(\omega_{k-1})\right\\}$
is a basis of $H^{2}(M(\omega_{1}))$. It follows from Example 1.5 and (5.2)
that

> $M(\omega_{1})=(S^{3}\times S^{n-2})\\#_{k-1}\Sigma_{1}(S_{i}^{2}\times
> S_{i}^{n-2})\\#_{r}\Sigma_{1}(S_{j}^{k_{j}}\times S_{j}^{n-k_{j}})$,

showing $M(\omega_{1})\in\mathcal{S}_{n+1}$ by (3.2). This completes the
inductive procedure of showing part ii).$\square$

Proof of Lemma 5.3. Suppose that $N=\\#_{1\leq i\leq k-1}(S_{i}^{2}\times
S_{i}^{3})$, and that $\\{\omega_{1},\cdots,\omega_{k-1}\\}$ is the basis of
$H^{2}(N)$ stated in Lemma 5.3. Then the cohomology $H^{\ast}(N)$ has the
basis

  1. (5.3)

$\left\\{1,\omega_{1},\cdots,\omega_{k-1},y_{1},\cdots,y_{k-1},z\right\\}$,
$\deg\omega_{i}=2$, $\deg y_{i}=3$, $\deg z=5$,

that is subject to the multiplicative relations

  1. (5.4)

$\omega_{i}\cdot\omega_{j}=0;$ $y_{i}\cdot y_{j}=0;\quad\omega_{i}\cdot
y_{j}=\delta_{ij}\cdot z$, $1\leq i,j\leq k-1$,

where $\delta_{ij}$ denotes the Kronecker symbol.

Let $\pi:N(\omega_{1},\cdots,\omega_{k-1})\rightarrow N$ be the principal
$T^{k-1}$-bundle over $N$. Then $N(\omega_{1},\cdots,\omega_{k-1})\in S_{4+k}$
by ii) of Lemma 5.2. According to Lemma 5.1 it suffices to show

  1. (5.5)

$P_{t}(N(\omega_{1},\cdots,\omega_{k-1}))=P_{t}(Q_{k})$,

where the polynomial $P_{t}(Q_{k})$ is given by (1.9). To this end we make use
of the Leray-Serre spectral sequence $\\{E_{r}^{\ast,\ast},d_{r}\\}$ of $\pi$
[30, p.645].

Since both cohomologies $H^{\ast}(N)$ and $H^{\ast}(T^{k-1})$ are torsion free
while $N$ is $1$-connected, the Leray-Serre theorem tells that
$E_{2}^{\ast,\ast}=H^{\ast}(N)\otimes H^{\ast}(T^{k-1})$. Moreover, there are
unique elements $t_{1},\cdots,t_{k-1}\in H^{1}(T^{k-1})$ such that
$H^{\ast}(T^{k-1})=\Lambda(t_{1},\cdots,t_{k-1})$ and that $d_{2}(1\otimes
t_{i})=\omega_{i}\otimes 1$, implying

  1. (5.6)

$d_{2}(x\otimes t_{i})=\omega_{i}\cdot x\otimes 1$, $d_{2}(x\otimes 1)=0$,
$x\in H^{\ast}(N)$.

For a multi-index $I\subseteq\\{1,\cdots,k-1\\}$ we set $t_{I}:=\Pi_{i\in
I}t_{i}$, and let $B_{i}\subset E_{2}^{\ast,\ast}$, $1\leq i\leq 4$, be the
subgroup with the basis

> $B_{1}:\\{1\otimes t_{I}\mid I\subseteq\\{1,\cdots,k-1\\}\\}$;
>
> $B_{2}:\\{1,\omega_{i}\otimes 1,\omega_{i}\otimes t_{I}\mid
> i\in\\{1,\cdots,k-1\\},I\subseteq\\{1,\cdots,k-1\\}\\}$;
>
> $B_{3}:\\{y_{i}\otimes 1,y_{i}\otimes t_{I}\mid
> i\in\\{1,\cdots,k-1\\},I\subseteq\\{1,\cdots,k-1\\}\\}$;
>
> $B_{4}:\\{z\otimes 1,z\otimes t_{I}\mid I\subseteq\\{1,\cdots,k-1\\}\\}$.

Then $E_{2}^{\ast,\ast}=B_{1}\oplus B_{2}\oplus B_{3}\oplus B_{4}$ by (5.3).
Moreover, the actions of $d_{2}$ on the basis elements $x\in B_{i}$ are
determined by (5.4) and (5.6) as

$d_{2}(x)=\left\\{\begin{tabular}[]{l}$0$ if $x=\omega_{i}\otimes
1,y_{i}\otimes 1,z\otimes t_{I},\omega_{i}\otimes t_{I},$ $y_{j}\otimes t_{J}$
with $j\notin J$;\\\ $\underset{1\leq s\leq
t}{\Sigma}(-1)^{s-1}\omega_{i_{s}}\otimes t_{I_{s}}$ if $x=1\otimes t_{I}$
with $I=\\{i_{1},\cdots,i_{t}\\}$;\\\ $(-1)^{s-1}z\otimes t_{I_{s}}$ if
$x=y_{i}\otimes t_{I}$ with $i=i_{s}\in
I=\\{i_{1},\cdots,i_{t}\\}$,\end{tabular}\right.$

where $I_{s}$ is the multi-index obtained from $I$ by deleting the $s$-th
coordinate $i_{s}\in I$. In particular

> i) $d_{2}(B_{2})=0$ while $d_{2}$ maps $B_{1}$ monomorphically into $B_{2}$;
>
> ii) $d_{2}(B_{4})=0$ and $d_{2}(B_{3})\subset B_{4}$ with
> $B_{4}/d_{2}(B_{3})=\\{z\otimes t_{\\{1,\cdots,k-1\\}}\\}$.

Consequently,

  1. (5.7)

$E_{3}^{\ast,\ast}=B_{2}/d_{2}(B_{1})\oplus
B_{4}/d_{2}(B_{3})\oplus\overline{B}_{3}$ with
$\overline{B}_{3}:=\ker[d_{2}:B_{3}\rightarrow B_{4}]$;

  2. (5.8)

$E_{3}^{\ast,\ast}=E_{\infty}^{\ast,\ast}=H^{\ast}(N(\omega_{1},\cdots,\omega_{k-1}))$.

For a finitely generated graded abelian group $A=A^{0}\oplus A^{1}\oplus
A^{2}\oplus\cdots$ define the Poincarè polynomial $P_{t}(A)$ by

> $P_{t}(A):=a_{0}+a_{1}t+a_{2}t^{2}+\cdots$, $a_{i}=\dim
> A^{i}\otimes\mathbb{R}$.

As examples

> $P_{t}(B_{1})=\underset{1\leq i\leq k-1}{\Sigma}\binom{k-1}{i}t^{i}$;
> $P_{t}(B_{2})=1+(k-1)\underset{0\leq i\leq
> k-1}{\Sigma}\binom{k-1}{i}t^{i+2}$;
>
> $P_{t}(B_{3})=(k-1)\underset{0\leq i\leq k-1}{\Sigma}\binom{k-1}{i}t^{i+3}$;
> $P_{t}(B_{4})=\underset{0\leq i\leq k-1}{\Sigma}\binom{k-1}{i}t^{i+5}$.

These imply by (5.7) that $P_{t}(E_{3}^{\ast,\ast})=P_{t}(Q_{k})$. We get
(5.5) from (5.8).$\square$

##  6 Concluding Remarks

The definitions, results and calculations of the present paper are likely to
be extended in certain cases. We discuss such examples.

6.1. According to Duan-Liang [8, Corollary 2] and Theorem C, the
classification of the $1$-connected $5$\- and $6$-dimensional manifolds that
admit regular circle actions has been completed. In addition, Montgemory-Yang
[22] showed that among all the $28$ homotopy $7$-spheres there are precisely
$10$ admit regular circle actions, which has been extended by Jiang [12] to
the classification of regular circle actions on the $2$-connected
$7$-manifolds, both in the topological and smooth categories. Hopefully, the
suspension operations, combined with Theorem B, could provide a general tool
to deal with the classification problems in higher dimensional situations.

For instance, let $\mathcal{V}_{2n}\subset\mathcal{T}_{2n}$ (resp.
$\mathcal{V}_{2n+1}\subset\mathcal{T}_{2n+1}$) be the subset of
$(n-1)$-connected $2n$-manifolds (resp. the subset of $(n-1)$-connected
$(2n+1)$-manifolds). The operators $\Sigma_{i}$ clearly satisfy the relation
$\Sigma_{i}(\mathcal{V}_{2n})\subset\mathcal{V}_{2n+1}$. On the other hand in
[27, 28] Wall classified the manifolds of $\mathcal{V}_{m}$ ($m\geq 8$) in
term of a set of invariants. It is hopeful to describe the map
$\Sigma_{i}:\mathcal{V}_{2n}\rightarrow\mathcal{V}_{2n+1}$ in term of Wall’s
invariants, so that a classification for those manifolds of
$\mathcal{V}_{2n+1}$ that admit regular circle actions is attainable.

6.2. The operations $\Sigma_{i}:\mathcal{T}_{n}\rightarrow\mathcal{T}_{n+1}$
admit a natural generalization. Namely, for each pair
$(\alpha,M)\in\pi_{k}(SO(n))\times\mathcal{T}_{n}$ take an orientation
persevering embedding $(D^{n},0)\subset(M,x_{0})$ centered at a base point
$x_{0}$, surgery along the oriented sphere $S^{k}\times\\{x_{0}\\}\subset
S^{k}\times M$ corresponding to the framing $\alpha\in\pi_{k}(SO(n))$, and
denote the resulting manifold by $\Sigma_{\alpha}M$ to get the pairing

> $\Sigma:\pi_{k}(SO(n))\times\mathcal{T}_{n}\rightarrow\mathcal{T}_{n+k}$,
> $(\alpha,M)\rightarrow\Sigma_{\alpha}M$.

This operation can be use to construct $S^{k}$-bundles over the connected sum
$B\\#N$ from the known ones $E\rightarrow B$, where $\dim B=\dim N$. In
particular, when $k=3$ it can be applied to construct or classify
$S^{3}$-actions on smooth manifolds.

Acknowledements. This paper is based on a seminar talk given at Institute of
Mathematics, Chinese Academy of Sciences in 2016. The author would like to
thank Drs. Yang Su, Feifei Fan, Xueqi Wang for useful communication.

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Haibao Duan

<EMAIL_ADDRESS>
Academy of Mathematics and Systems Sciences, Beijing 100190;

School of Mathematical Sciences, UCAS, Beijing 100049.