# Smooth Structures on $M\times\mathbb{S}^{k}$

Samik Basu  Stat Math Unit, Indian Statistical Institute, Kolkata-700108,
India<EMAIL_ADDRESS>,  Ramesh Kasilingam  Department of Mathematics,
Indian Institute of Technology Madras, Chennai-600036, India
<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Ankur Sarkar  The
Institute of Mathematical Sciences, A CI of Homi Bhabha National Institute,
Chennai-600113, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

This paper explores various differentiable structures on the product manifold
$M\times\mathbb{S}^{k}$, where $M$ is either a 4-dimensional closed oriented
manifold or a simply connected 5-dimensional closed manifold. We identify the
possible stable homotopy types of $M$ and use it to calculate the concordance
inertia group and the concordance structure set of $M\times\mathbb{S}^{k}$ for
$1\leq k\leq 10.$ These calculations enable us to further classify all
manifolds that are homeomorphic to $\mathbb{C}P^{2}\times\mathbb{S}^{k}$, up
to diffeomorphism, for each $4\leq k\leq 6$.

###### Key words and phrases:

Concordance inertia group, Concordance smooth structure set, Normal invariant

###### 2020 Mathematics Subject Classification:

Primary : 57R55, 55P42 ; Secondary : 57R65, 55Q45

##  1\. Introduction

We work in the smooth category of closed, connected, oriented smooth manifolds
and orientation preserving maps. One of the main interests in the smooth
category is the classification of all smooth manifolds up to diffeomorphism.
The discovery of exotic spheres by Milnor in 1956 [19] has sparked global
interest and intensive research into the existence and non-existence of exotic
smooth structures. In this paper, we turn our attention to the investigation
of exotic smooth structures on the product manifold $M\times\mathbb{S}^{k},$
where $M$ is either a $4$-dimensional manifold or a simply connected
$5$-dimensional manifold with $1\leq k\leq 10.$ Focusing on product manifolds,
extensive studies have been conducted to classify smooth structures on
$\mathbb{S}^{p}\times\mathbb{S}^{q}$ [17, 18, 35, 10, 43],
$\mathbb{C}P^{2}\times\mathbb{S}^{7}$ [4], and
$\mathbb{C}P^{2}\times\mathbb{S}^{3}$ [44]. The inertia groups of these
manifolds have been identified as key elements in these studies.

Recall that the inertia group $I(M)$ of a closed smooth oriented $n$-manifold
$M$ is the subgroup of the Kervaire-Milnor group $\Theta_{n}$ [19] consisting
of homotopy $n$-spheres $\Sigma$ such that $M\\#\Sigma$ is diffeomorphic to
$M$ via an orientation-preserving diffeomorphism. Generally, there are no
systematic methods available for determining the inertia group of a manifold.
Therefore, one might consider studying a specific subgroup of $I(M)$, referred
to as the concordance inertia group of $M$ which is defined as follows

######  Definition 1.1.

Let $M$ be a smooth $n$-manifold. Consider the pair $(N,f),$ where $N$ is a
smooth manifold and $f:N\to M$ is a homeomorphism. We say that two pairs
$(N_{1},f_{1})$ and $(N_{2},f_{2})$ are concordant if there exist a
diffeomorphism $\Phi:N_{1}\to N_{2}$ and a homeomorphism
$H:N_{1}\times[0,1]\to M\times[0,1]$ such that $H|_{N_{1}\times\\{0\\}}=f_{1}$
and $H|_{N_{1}\times\\{1\\}}=f_{2}\circ\Phi.$ We denote the set of all such
concordance classes by $\mathcal{C}(M)$ and represent the concordance class of
$(N,f)$ in $\mathcal{C}(M)$ as $[(N,f)].$ The base point of $\mathcal{C}(M)$
is the concordance class $[(M,Id_{M})]$ of the identity map $Id_{M}:M\to M$.

The concordance inertia group $I_{c}(M)$ of $M$ is the subgroup of $I(M)$
consisting of homotopy $n$-spheres $\Sigma$ such that $(M,Id_{M})$ and
$(M\\#\Sigma,h_{\Sigma})$ are concordant, where $h_{\Sigma}:M\\#\Sigma\to M$
is the standard homeomorphism.

Following Kirby and Siebenmann [20, Page 194], we have that there exists a
connected $H$-space $Top/O$ such that there is a bijection between
$\mathcal{C}(N)$ and $[N,Top/O]$ for any closed smooth $n$-manifold $N$ with
$n\geq 5$. Furthermore, the concordance class of $(N,Id_{N})$ corresponds to
the homotopy class of the constant map under this bijection. Additionally,
since $Top/O$ is an infinite loop space [5], it enables the computation of
$I_{c}(N)$ and $\mathcal{C}(N)$ by establishing the stable homotopy type of
$N$. In particular, if $N=M\times\mathbb{S}^{k},$ where $M$ is either a
$4$-dimensional manifold or a simply connected $5$-dimensional manifold, the
concordance inertia group $I_{c}(N)$ is explicitly computed using the stable
decomposition of the manifold $M$. This decomposition is achieved through
certain cohomology operations (see Section 3.1 and 4.1). To be more precise,
we prove the following :

######  Theorem A.

Let $M$ be a closed, smooth, oriented $n$-manifold. The following statements
hold.

  1. (1)

Suppose $M$ is a non-spin $4$-manifold. Then

    1. (a)

For $3\leq k\leq 16,$ we have

$I_{c}(M\times\mathbb{S}^{k})=\begin{cases}\mathbb{Z}/2,&\mbox{if
$k=5,6,7,11,$ or $13$}.\\\ \mathbb{Z}/2\oplus\mathbb{Z}/2,&\mbox{if $k=14$}\\\
0&\mbox{otherwise}.\end{cases}$

    2. (b)

For $k=17,18,8n-2~{}(n\geq 3)$, $\mathbb{Z}/2\subseteq
I_{c}(M\times\mathbb{S}^{k})$.

  2. (2)

If $M$ is a simply connected non-spin $5$-manifold, then

    1. (a)

$I_{c}(M\times\mathbb{S}^{k})=0$ for all $2\leq k\leq 10,$ excluding
$k=4,5,6,$ and $10.$

    2. (b)

For $k=4,5,6,$ and $10,$

      1. (i)

If a least positive integer $r$ exists such that $\mathbb{Z}/{2^{r}}\subseteq
H_{2}(M;\mathbb{Z}),$ then:

        1. (A)

$I_{c}(M\times\mathbb{S}^{4})=\begin{cases}\mathbb{Z}/2\oplus\mathbb{Z}/2,&\mbox{if
$r=1$ or $2$}\\\ \mathbb{Z}/2,&\mbox{if $r\geq 3$}\end{cases}$

        2. (B)

$I_{c}(M\times\mathbb{S}^{k})=\mathbb{Z}/2$ for $k=5,10.$

        3. (C)

$I_{c}(M\times\mathbb{S}^{6})=\begin{cases}\mathbb{Z}/4,&\mbox{if $r=1$ }\\\
\mathbb{Z}/2,&\mbox{if $r\geq 2$}.\end{cases}$

      2. (ii)

If $H_{2}(M;\mathbb{Z})$ does not contain $2$-torsion, then
$I_{c}(M\times\mathbb{S}^{k})=\mathbb{Z}/2.$

  3. (3)

If $M$ is either a spin $4$-manifold or a simply connected spin $5$ manifold,
then $I_{c}(M\times\mathbb{S}^{k})=0$ for all $k\geq 1.$

(Theorem A is a synthesis of Theorem 3.5 and Theorem 4.3).

The techniques involved in the proof of Theorem A are explicit calculations
that make use of the non-triviality of certain elements in the stable homotopy
groups. Following these calculations, we also determine the concordance smooth
structure set of $M\times\mathbb{S}^{k}$ based on the stable homotopy type of
aforementioned manifolds $M$ (refer to Propositions 3.9 and 4.4). In
particular, we apply these computations to the manifolds
$\mathbb{C}P^{2}\times\mathbb{S}^{k},$ where $4\leq k\leq 6$, and then
identify the orbits of elements of
$\mathcal{C}(\mathbb{C}P^{2}\times\mathbb{S}^{k})$ under the action of the
group of self-homotopy equivalences. As a result of these computations, we are
able to establish the following classification result.

######  Theorem B.

Suppose $N$ is a closed smooth oriented manifold that is homeomorphic to
$\mathbb{C}P^{2}\times\mathbb{S}^{k}.$ Then

  * (a)

For $k=4$ and $5,$ $N$ is oriented diffeomorphic to
$\mathbb{C}P^{2}\times\mathbb{S}^{k}.$

  * (b)

When $k=6,$ the manifold $N$ is oriented diffeomorphic to exactly one of the
manifolds
$\mathbb{C}P^{2}\times\mathbb{S}^{6},(\mathbb{C}P^{2}\times\mathbb{S}^{6})\\#\Sigma_{\alpha_{1}},$
or $(\mathbb{C}P^{2}\times\mathbb{S}^{6})\\#(\Sigma_{\alpha_{1}})^{-1},$ where
$\Sigma_{\alpha_{1}}\in\Theta_{10}$ is the exotic sphere of order $3$.

The following corollary is an immediate consequence of the theorem above.

######  Corollary.

For $k=4$ and $5,$ the inertia group of $\mathbb{C}P^{2}\times\mathbb{S}^{k}$
equals $\Theta_{4+k}$. On the other hand, for $k=6$, the inertia group of
$\mathbb{C}P^{2}\times\mathbb{S}^{k}$ equals $\mathbb{Z}/2.$

###  1.1. Notation

  * •

Let $O_{n}$ be the orthogonal group, $PL_{n}\subset O_{n}$ is the group of
piece-wise linear homeomorphisms, $Top_{n}\subset O_{n}$ is the group of self
homeomorphisms of $\mathbb{R}^{n}$ preserving origin, and $G_{n}$ denote the
topological monoid of self homotopy equivalences
$\mathbb{S}^{n-1}\to\mathbb{S}^{n-1}.$ Denote by $O=\varinjlim~{}O_{n}$,
$PL=\varinjlim PL_{n}$, $Top=\varinjlim Top_{n},$ and $G=\varinjlim~{}G_{n}$
[21, 22, 24].

  * •

Let $G/O$ be the homotopy fiber of the canonical map $BO\to BG$ between the
classifying spaces for stable vector bundles and stable spherical fibrations
[24, §2 and §3], and $G/PL$ be the homotopy fiber of the canonical map $BPL\to
BG$ between the classifying spaces for $PL$ $\mathbb{\mathbb{R}}^{n}$-bundles
and stable spherical fibrations [34]. Again $Top/O$ is the homotopy fibre of
the map $BO\to BTop$ between the classifying spaces for stable vector bundles
and stable topological $\mathbb{R}^{n}$-bundles [20, Theorem 10.1 Essay IV].

  * •

There are standard fibre sequences $\cdots\to\Omega
G/Top\xrightarrow{w}Top/O\xrightarrow{\psi}G/O\xrightarrow{\phi}G/Top$ [20]
and $O\xrightarrow{\Omega
J}G\xrightarrow{j}G/O\xrightarrow{i}BO\xrightarrow{J}BG$ [7] where
$J:\pi_{n}(SO)\to\pi_{n}^{s}$ is the classical $J$-homomorphism.

  * •

We write $X_{(p)}$ for localization of the space $X$ at prime $p.$ We use the
notation $f_{(p)}$ when a map $f$ is localized at $p$ [40].

  * •

$\Sigma^{n}M(A)$ denotes the Moore spectrum.

  * •

$f_{M}:M\to\mathbb{S}^{n}$ denotes the degree one map for an oriented
$n$-manifold $M.$

###  1.2. Organization

In section §2, we discuss the homotopy decomposition of
$\mathcal{C}(M\times\mathbb{S}^{k})$ and present some results related to
$I_{c}(M\times\mathbb{S}^{k})$. In section §3, we provide a stable
decomposition of a closed, oriented $4$-manifold $M$, based on whether it
admits a spin or non-spin structure. Using these decompositions, we compute
both the concordance inertia group and the concordance smooth structure set of
$M\times\mathbb{S}^{k}$ for $1\leq k\leq 10$. In section §4, we determine the
stable homotopy type of a simply connected, closed, oriented $5$-manifold $M$
and extend our calculations to the product manifold $M\times\mathbb{S}^{k}$.
Finally, in section §5, we apply our calculations to classify all manifolds
that are homeomorphic to $\mathbb{C}P^{2}\times\mathbb{S}^{k},$ where $4\leq
k\leq 6$, up to diffeomorphism.

###  1.3. Acknowledgements

The third author expresses gratitude to Douglas C. Ravenel and Diarmuid
Crowley for their valuable assistance in clarifying doubts through email.

##  2\.  Concordance inertia groups of the product manifolds
$M\times\mathbb{S}^{k}$

We start by recalling some facts about the Kervaire-Milnor group $\Theta_{m}$.

######  Definition 2.1.

[19]

  * (a)

A homotopy $m$-sphere $\Sigma^{m}$ is an oriented smooth closed manifold
homeomorphic to the standard unit sphere $\mathbb{S}^{m}$ in
$\mathbb{R}^{m+1}.$

  * (b)

A homotopy $m$-sphere $\Sigma^{m}$ is said to be exotic if it is not
diffeomorphic to $\mathbb{S}^{m}$.

  * (c)

Two homotopy $m$-spheres $\Sigma_{1}^{m}$ and $\Sigma_{2}^{m}$ are said to be
equivalent if there exists an orientation preserving diffeomorphism
$f:\Sigma_{1}^{m}\to\Sigma_{2}^{m}$.

The set of equivalence classes of homotopy $m$-spheres is denoted by
$\Theta_{m}$. Each $\Theta_{m}$, $(m\neq 4)$ is a finite abelian group under
the connected sum operation.

Assume $m\geq 5$ and consider the obvious forgetting information map
$\mathcal{C}(\mathbb{S}^{m})\to\Theta_{m}$, which is bijective. Observe that
the abelian group structure on $\Theta_{m}$ given by the connected sum agrees
with the one given by Kirby-Siebenmann [20] via the identification
$\mathcal{C}(\mathbb{S}^{m})=\pi_{m}(Top/O)$. Let
$f_{N}:N^{m}\to\mathbb{S}^{m}$ be a degree one map and note that $f_{N}$ is
well-defined up to homotopy. Composition with $f_{N}$ defines a homomorphism
$f_{N}^{*}:\pi_{m}(Top/O)\to[N^{m},Top/O].$ In terms of the Kirby-Siebenmann
identifications $\Theta_{m}=\pi_{m}(Top/O)$ and
$\mathcal{C}(N^{m})=[N^{m},Top/O]$, $f_{N}^{*}$ becomes
$[\Sigma^{m}]\to[(N\\#\Sigma^{m},h_{\Sigma^{m}})]$, where
$h_{\Sigma^{m}}:N^{m}\\#\Sigma^{m}\to N^{m}$ is the canonical homeomorphism.
Therefore, the concordance inertia group $I_{c}(N^{m})$ can be identified with
the kernel
$\mathit{ker}\left(\Theta_{m}\xrightarrow{(f_{N})^{*}}\mathcal{C}(N^{m})\right).$
Specifically, for $N=M\times\mathbb{S}^{k},$ we have the following :

######  Lemma 2.2.

Let $M$ be a closed oriented smooth $n$-manifold and
$\Sigma^{k}f_{M}:\Sigma^{k}M\to\mathbb{S}^{n+k}$ be the $k$-fold suspension of
the degree one map $f_{M}:M\to\mathbb{S}^{n}.$ Then the concordance inertia
group $I_{c}\left(M\times\mathbb{S}^{k}\right)$ is equal to
$\mathit{ker}\left(\Theta_{n+k}\xrightarrow{(\Sigma^{k}f_{M})^{*}}[\Sigma^{k}M,Top/O]\right)$
for $n+k\geq 5.$

###### Proof.

We observe that there is a split short exact sequence along an $H$-group $Y$

$0\to[\Sigma^{k}M,Y]\xrightarrow{p^{*}}[M\times\mathbb{S}^{k},Y]\xrightarrow{i^{*}}[M\vee\mathbb{S}^{k},Y]\to
0$ (1)

which is induced from the cofiber sequence

$M\vee\mathbb{S}^{k}\xhookrightarrow{i}M\times\mathbb{S}^{k}\xrightarrow{p}M\times\mathbb{S}^{k}/M\vee\mathbb{S}^{k}\simeq\Sigma^{k}M.$

In particular, by using the decomposition (1) for $Y=Top/O$, we have the
following commutative diagram

${{[\Sigma^{k}\mathbb{S}^{n},Top/O]}}$${{\Theta_{n+k}}}$${{[M\times\mathbb{S}^{k},Top/O]}}$${{[\Sigma^{k}\mathbb{S}^{n},Top/O]}}$${{[\mathbb{S}^{n}\times\mathbb{S}^{k},Top/O]}}$$\scriptstyle{\cong}$$\scriptstyle{(f_{M\times\mathbb{S}^{k}})^{*}}$$\scriptstyle{(f_{\mathbb{S}^{n}\times\mathbb{S}^{k}})^{*}}$$\scriptstyle{f_{M}^{*}\oplus
Id_{\mathbb{S}^{k}}^{*}\oplus(\Sigma^{k}f_{M})^{*}}$

where the map
$(f_{\mathbb{S}^{n}\times\mathbb{S}^{k}})^{*}:\Theta_{n+k}\to[\mathbb{S}^{n}\times\mathbb{S}^{k},Top/O]$
is an isomorphism onto the summand $[\Sigma^{k}\mathbb{S}^{n},Top/O]$ [35].
From the right triangle, we deduce that the kernel of the map
$(f_{M\times\mathbb{S}^{k}})^{*}$ is isomorphic to the kernel of the component
$(\Sigma^{k}f_{M})^{*}$. This concludes the proof of the lemma.

∎

By combining Kirby and Siebenmann result and the split short exact sequence
(1), we obtain the following corollary.

######  Corollary 2.3.

For any closed oriented smooth manifold $M,$

$\mathcal{C}(M\times\mathbb{S}^{k})\cong[M,Top/O]\oplus\pi_{k}(Top/O)\oplus[\Sigma^{k}M,Top/O],$

where $\mathit{dim}(M)+k\geq 5.$

Now the next lemma provides a pathway to interpret the concordance inertia
group of $M\times\mathbb{S}^{k},$ if the stable homotopy type of $M$ is known.

######  Lemma 2.4.

Let $M$ be an $n$-dimensional, closed, oriented, and smooth manifold. Suppose
$M$ is stably homotopy equivalent to $X\vee Z$, where $X$ is the cofibre of a
certain map $f:\mathbb{S}^{n-1}\to\bigvee\limits_{i=1}^{w}A_{i}$ with $w\geq
2$, and the dimensions of both $\bigvee\limits_{i=1}^{w}A_{i}$ and $Z$ are
less than $n$. Then, for any integer $k\geq 1$ such that $n+k\geq 5,$ the
concordance inertia group $I_{c}(M\times\mathbb{S}^{k})$ is isomorphic to

$\mathit{Im}\left(\bigoplus\limits_{i=1}^{w}[\Sigma^{k+1}A_{i},Top/O]\xrightarrow{\bigoplus\limits_{i=1}^{w}(\Sigma^{k+1}f_{i})^{*}}\pi_{n+k}(Top/O)\right),$

where $f_{i}=Pr_{i}\circ f,1\leq i\leq w$ are the compositions of $f$ with the
projection maps $Pr_{i}$ from $\bigvee\limits_{i=1}^{w}A_{i}$ onto its
$i^{th}$ factor.

###### Proof.

By assumption, there exists a positive integer $j$ such that for any $l\geq
j,$ we have a homotopy equivalence
$\phi_{l}:\Sigma^{l}M\to\Sigma^{l}X\vee\Sigma^{l}Z.$ Since $Top/O$ is an
infinite loop space, there exists a topological space $Y$ such that
$\Omega^{j}Y=Top/O$. Now by Lemma 2.2, we have that

$I_{c}(M\times\mathbb{S}^{k})\cong\mathit{Ker}\left(\pi_{n+k+j}(Y)\xrightarrow{(\Sigma^{k+j}f_{M})^{*}}[\Sigma^{k+j}M,Y]\right).$

Observe that the $(k+j)$-fold suspension of the degree one map
$\Sigma^{k+j}f_{M}:\Sigma^{k+j}M\to\mathbb{S}^{n+k+j}$ is homotopic to the
composition $\lambda\circ\Sigma^{k+j}f_{X}\circ Pr\circ\phi_{k+j}$, where
$Pr:\Sigma^{k+j}X\vee\Sigma^{k+j}Z\to\Sigma^{k+j}X$ is the the projection map,
$\Sigma^{k+j}f_{X}:\Sigma^{k+j}X\to\Sigma^{k+j}\mathbb{S}^{n}$ is the
$(k+j)$-fold suspension of the pinch map onto the top cell
$f_{X}:X\to\mathbb{S}^{n}$, and
$\lambda:\mathbb{S}^{n+k+j}\to\mathbb{S}^{n+k+j}$ is either the identity or
the reflection map. This implies that

$I_{c}(M\times\mathbb{S}^{k})\cong\mathit{Ker}\left(\pi_{n+k+j}(Y)\xrightarrow{(\Sigma^{k+j}f_{X})^{*}}[\Sigma^{k+j}X,Y]\right).$

This can be equivalently expressed as the image of the homomorphism
$\bigoplus\limits_{i=1}^{w}(\Sigma^{k+j+1}f_{i})^{*}:\bigoplus\limits_{i=1}^{w}[\Sigma^{k+j+1}A_{i},Y]\to\pi_{n+k+j}(Y),$
by using the long exact sequence induced from the cofibre sequence
$\Sigma^{k+j}\mathbb{S}^{n-1}\xrightarrow{\Sigma^{k+j}f}\Sigma^{k+j}(\bigvee\limits_{i=1}^{w}A_{i})\hookrightarrow\Sigma^{k+j}X$
along $Y$. Consequently, the latter group is isomorphic to the image of the
map
$\bigoplus\limits_{i=1}^{w}(\Sigma^{k+1}f_{i})^{*}:\bigoplus\limits_{i=1}^{w}[\Sigma^{k+1}A_{i},Top/O]\to\Theta_{n+k}.$
This completes the proof. ∎

We notice from the above lemma that the calculation of the concordance inertia
group of $M\times\mathbb{S}^{k}$ depends on the stable homotopy type of $M.$
Further, if $M$ has a stable homotopy decomposition $M\simeq X\vee Y$ as
described in Lemma 2.4, then it suffices to determine the image of each
component in the suspension of the top cell attaching map of $X$ at the
$Top/O$ level. In order to make explicit computations, we require the
knowledge of a few stable homotopy groups and their mapping cones.

$\pi_{0}^{s}=\mathbb{Z},~{}\pi_{1}^{s}=\mathbb{Z}/2\\{\eta\\},~{}\pi_{2}^{s}=\mathbb{Z}/2\\{\eta^{2}\\},~{}\pi_{3}^{s}=\mathbb{Z}/{24}\\{\tau\\},~{}\pi_{4}^{s}=0,~{}\pi_{5}^{s}=0,$

where $(\tau)_{(2)}=\nu,$ and $(\tau)_{(3)}=\alpha_{1}.$ Let $b\in\mathbb{Z}$
be a non-zero integer. The mapping cone of $b:\mathbb{S}^{0}\to\mathbb{S}^{0}$
in spectra is denoted by the Moore spectrum $M(\mathbb{Z}/b)$. Its homotopy
groups in the above degrees are described as

$\pi_{0}(M(\mathbb{Z}/b))=\mathbb{Z}/b,~{}\pi_{1}(M(\mathbb{Z}/b))=\begin{cases}\mathbb{Z}/2\\{i\circ\eta\\}&\mbox{if
$b$ is even}\\\ 0&\mbox{ otherwise,}\end{cases}$
$\pi_{2}(M(\mathbb{Z}/b))=\begin{cases}\mathbb{Z}/4\\{\tilde{\eta}_{b}\\}&\mbox{if
$2\mid b$ but $4\nmid b$}\\\
\mathbb{Z}/2\\{\tilde{\eta}_{b}\\}\oplus\mathbb{Z}/2\\{i\circ\eta^{2}\\}&\mbox{if
}4\mid b\\\ 0&\mbox{otherwise,}\end{cases}$
$\pi_{3}(M(\mathbb{Z}/b))=\begin{cases}\mathbb{Z}/\gcd(24,b)\\{i\circ\tau\\}\oplus\mathbb{Z}/2\\{\tilde{\eta_{b}^{2}}\\}&\mbox{if
}2\mid b\\\ \mathbb{Z}/\gcd(24,b)\\{i\circ\tau\\}&\mbox{if }2\nmid
b,\end{cases}$
$~{}\pi_{4}(M(\mathbb{Z}/b))=\mathbb{Z}/\gcd(24,b)\\{\tilde{\tau}\\},~{}\pi_{5}(M(\mathbb{Z}/b))=0.$

In the above $\tilde{\eta}_{b}$ (respectively $\tilde{\tau}$) refers to a
homotopy class which gives $\eta$ (respectively $\tau$) after composition onto
$\mathbb{S}^{1}$. We recall that the Hopf map $\eta$ is detected by $Sq^{2}$,
and that the mapping cone $Cone(\eta)\simeq\Sigma^{-2}\mathbb{C}P^{2}$. The
square of $\eta$, $\eta^{2}$ is detected by the secondary cohomology operation
[13, 26]

$\Theta:\mbox{Ker}(Sq^{2})\cap\mbox{Ker}(Sq^{3})\Big{(}\subset
H^{i}(X;\mathbb{Z}/2)\Big{)}\to H^{i+3}(X;\mathbb{Z}/2),$

associated to the Adem relation $Sq^{2}Sq^{2}=Sq^{3}Sq^{1}$ in the Steenrod
algebra.

Additionally, we need to understand the homotopy class of the map between
$\Sigma^{n}M(\mathbb{Z}/{p^{r}})$ and $\Sigma^{n}M(\mathbb{Z}/{p^{t}}),$ for
some $r,t\in\mathbb{N}.$

Let’s consider $\chi^{r}_{t}:\mathbb{Z}/{p^{r}}\to\mathbb{Z}/{p^{t}}$ as the
generator of $\mathit{Hom}(\mathbb{Z}/{p^{r}},\mathbb{Z}/{p^{t}})$. Here,
$\chi^{r}_{t}(1)=1,$ if $r\geq t$ and $\chi^{r}_{t}(1)=p^{t-r},$ if $t\geq r.$
In this case, there exists a unique element

$B(\chi^{r}_{t}):M(\mathbb{Z}/{p^{r}})\to M(\mathbb{Z}/{p^{t}})$ (2)

such that the map induced by $\Sigma^{n}B(\chi^{r}_{t})$ at the $n$-th
homology level is $\chi^{r}_{t}$ for each $n\geq 3$ [8]. Moreover,
$(\Sigma^{n}B(\chi^{r}_{t}))_{*}:\pi_{n+2}(\Sigma^{n}M(\mathbb{Z}/{p^{r}}))\to\pi_{n+2}(\Sigma^{n}M(\mathbb{Z}/{p^{t}}))$
is given by

$(\Sigma^{n}B(\chi^{r}_{t}))_{*}(\tilde{\eta}_{2^{r}})=\tilde{\eta}_{2^{t}}~{}\text{and}~{}(\Sigma^{n}B(\chi^{r}_{r}))_{*}(\tilde{\eta}_{2^{r}})=\tilde{\eta}_{2^{r}}$
(3)

Also, $B(\chi^{r}_{t})$ satisfies the following relations (cf. [8]):

$\Sigma^{n}B(\chi^{r}_{t})\circ i_{n}=\left\\{\begin{array}[]{ll}i_{n},&r\geq
t;\\\ 2^{t-r}i_{n},&r\leq
t.\end{array}\right.~{}~{}q_{n+1}\circ\Sigma^{n}B(\chi^{r}_{t})=\left\\{\begin{array}[]{ll}2^{r-t}q_{n+1},&r\geq
t;\\\ q_{n+1},&r\leq t.\end{array}\right.$ (4)

where $i_{n}:\mathbb{S}^{n}\hookrightarrow\Sigma^{n}M(\mathbb{Z}/{p^{s}})$ and
$q_{n+1}:\Sigma^{n}M(\mathbb{Z}/{p^{s}})\to\mathbb{S}^{n+1}$ are the inclusion
and pinching map for $\Sigma^{n}M(\mathbb{Z}/{p^{s}})$ and any integer $s\geq
1$ respectively.

##  3\. Smooth structures on $M^{4}\times\mathbb{S}^{k}$

Let $M$ be a closed, smooth, and oriented 4-manifold. In this section, we
first determine the stable homotopy type of $M$. Following this, we compute
$I_{c}(M\times\mathbb{S}^{k})$ and $\mathcal{C}(M\times\mathbb{S}^{k})$. By
applying Poincaré duality and the Universal Coefficient Theorem, we obtain

$H_{l}(M;\mathbb{Z})=\begin{cases}\oplus\mathbb{Z}^{m}\oplus\bigoplus\limits_{j=1}^{n}\mathbb{Z}/{b_{j}}&$l=1$\\\
\oplus\mathbb{Z}^{d}\oplus\bigoplus\limits_{j=1}^{n}\mathbb{Z}/{b_{j}}&$l=2$\\\
\oplus\mathbb{Z}^{m}&$l=3$\\\ \mathbb{Z}&$l=0,4$\\\ 0&\text{otherwise}\\\
\end{cases}$ (5)

where $m,d\geq 0$ are any integers and each $b_{j}$ is a prime power. We make
certain choices in the above decomposition. Suppose that $u_{1},\cdots,u_{m}$
be a basis of the free part of $H_{1}(M;\mathbb{Z})$, and $v_{1},\cdots,v_{d}$
be a basis of the free part of $H_{2}(M;\mathbb{Z})$. Choose a basis
$\mu_{1},\cdots,\mu_{m}$ of $H^{1}(M;\mathbb{Z})\cong\mathbb{Z}^{m}$ such that
$\langle\mu_{i},u_{j}\rangle=\delta_{ij}$. The basis of $H_{3}(M;\mathbb{Z})$
is then fixed as $[M]\cap\mu_{1},\cdots,[M]\cap\mu_{m}$. Let $a_{i}$ denote a
generator of the $\mathbb{Z}/b_{i}$ factor of $H_{1}(M;\mathbb{Z})$. These
come equipped with maps $\hat{q_{i}}:M\to K(\mathbb{Z}/b_{i},1)$ such that
$a_{i}$ maps to $1$ in $H_{1}(K(\mathbb{Z}/b_{i},1);\mathbb{Z})$, and the
$u_{j}$ map to $0$. Define $\alpha_{i}\in H^{2}(M)$ as the image

$1\in\mathbb{Z}/b_{i}\cong
H^{2}(K(\mathbb{Z}/b_{i},1);\mathbb{Z})\stackrel{{\scriptstyle\hat{q_{i}}^{\ast}}}{{\to}}H^{2}(M;\mathbb{Z}).$

Define $k_{i}=[M]\cap\alpha_{i}$. With $\mathbb{Z}/2$-coefficients, define
$\nu_{i}$ to be a dual basis of $v_{i}$ which evaluates to $0$ on the torsion
classes. If $2\mid b_{i}$, let $c_{i}\in H_{2}(M;\mathbb{Z}/2)$ denote a
choice of a class such that $\hat{q_{i}}_{\ast}(c_{i})$ is a generator of
$H_{2}(K(\mathbb{Z}/b_{i},1);\mathbb{Z}/2)$. In this case, also define
$\chi_{i}\in H^{2}(M;\mathbb{Z}/2)$ by the equation

$\langle\chi_{i},k_{j}\rangle=\delta_{ij},~{}\langle\chi_{i},v_{j}\rangle=0,~{}\langle\chi_{i},c_{j}\rangle=0.$

We thus have for some $r\leq n$,

$H^{2}(M;\mathbb{Z}/2)\cong\mathbb{Z}/2\\{\nu_{1},\cdots,\nu_{d},\alpha_{1},\cdots,\alpha_{r},\chi_{1},\cdots,\chi_{r}\\}.$

In terms of the cup product $H^{2}(M;\mathbb{Z}/2)\otimes
H^{2}(M;\mathbb{Z}/2)\to H^{4}(M;\mathbb{Z}/2)\cong\mathbb{Z}/2$, we have

$\alpha_{i}\nu_{j}=0,~{}\chi_{i}\alpha_{j}=\delta_{ij}.$

###  3.1. The stable homotopy type of an orientable $4$-manifold

We use a minimal cell structure on the stable homotopy type of $M$ [14, §4.C].
The $2$-skeleton is then clearly homotopy equivalent to

$M^{(2)}\simeq\bigvee_{i=1}^{m}\mathbb{S}^{1}\vee\bigvee_{j=1}^{n}\left(\Sigma
M(\mathbb{Z}/b_{j})\vee\mathbb{S}^{2}\right)\vee\bigvee_{l=1}^{d}\mathbb{S}^{2}.$

We now consider the attachment of the $3$-cells which are stably a sum of maps
onto the wedge summands. We further note that $Sq^{2}:H^{1}(M;\mathbb{Z}/2)\to
H^{3}(M;\mathbb{Z}/2)$ equals to zero. This means that the attaching map does
not involve any factor of $\eta$. It follows that the $3$-skeleton has the
following stable homotopy type.

$M^{(3)}\simeq\bigvee_{i=1}^{m}\left(\mathbb{S}^{1}\vee\mathbb{S}^{3}\right)\vee\bigvee_{j=1}^{n}(\Sigma
M(\mathbb{Z}/{b_{j}})\vee\Sigma^{2}M(\mathbb{Z}/{b_{j}}))\vee\bigvee_{l=1}^{d}\mathbb{S}^{2}.$
(6)

Finally, the stable homotopy type of $M$ is determined by the attaching map
$\phi_{3}:\mathbb{S}^{3}\to M^{(3)}$ for the $4$-cell of $M$. The formula for
the homology implies that the attaching map becomes trivial after quotienting
out the $2$-skeleton. Thus, we consider the factorization of $\phi_{3}$ as

$\mathbb{S}^{3}\stackrel{{\scriptstyle\phi}}{{\to}}M^{(2)}\to M^{(3)}.$

The possibilities for the map $\phi$ are computed via two observations. As
both $Sq^{2}$ and $Sq^{3}$ are identically the $0$-operation on a
$1$-dimensional class, we have $\Theta:H^{1}(M;\mathbb{Z}/2)\to
H^{4}(M;\mathbb{Z}/2)$ is $0$. This implies that $\phi$ maps trivially onto
the factors given by $\mathbb{S}^{1}$ and the $0$-cell of $\Sigma
M(\mathbb{Z}/b_{j})$. Therefore, the non-trivial factors of the map $\phi$ may
only be
1) As a non-zero multiple of $\eta$ onto some of the $d$-copies of
$\mathbb{S}^{2}$.
2) As a non-zero multiple of $i\circ\eta$ onto one of the copies of
$\Sigma^{2}M(\mathbb{Z}/b_{j})$.
3) As a non-zero multiple of $\tilde{\eta}_{b_{j}}$ onto one of the copies of
$\Sigma M(\mathbb{Z}/b_{j})$.
If any of the cases above occur, the operation
$Sq^{2}:H^{2}(M;\mathbb{Z}/2)\to H^{4}(M;\mathbb{Z}/2)$ is non-trivial, which
means by Wu’s formula that the manifold is not spin. We also observe that if
$2\mid b_{j}$ but $4\nmid b_{j}$, the last case cannot arise for a
$4$-manifold, as then the composition

$H^{1}(M;\mathbb{Z}/2)\stackrel{{\scriptstyle
Sq^{1}}}{{\to}}H^{2}(M;\mathbb{Z}/2)\stackrel{{\scriptstyle
Sq^{2}}}{{\to}}H^{4}(M;\mathbb{Z}/2)$

would be non-trivial. However, this composition maps a class
$\alpha\mapsto\alpha^{4}$, and we then have $Sq^{1}(\alpha^{3})=\alpha^{4}$,
which contradicts $w_{1}(M)=0$ by Wu’s formula. In fact, we will now observe
that 3) cannot occur at all and if 2) occurs, 1) will also occur. The latter
case is clear as the cohomology classes $\chi_{j}$ which evaluate non-
trivially on the factors corresponding to the $\Sigma^{2}M(\mathbb{Z}/b_{j})$
factors for $2\mid b_{j}$ are not integral, and the fact that $w_{2}(M)$ is
the $\pmod{2}$ reduction of an integral class [38, Page 170] and [41].

Suppose that the composite

$\mathbb{S}^{3}\to M^{(3)}\stackrel{{\scriptstyle q_{j}}}{{\to}}\Sigma
M(\mathbb{Z}/b_{j})$

equals $\tilde{\eta}$ for some $j$ such that $2\mid b_{j}$. Viewing unstably,
the map $q_{j}$ yields $\hat{q_{j}}:M^{(3)}\to K(\mathbb{Z}/b_{j},1)$ which
classifies
$1\in\mbox{Hom}(\mathbb{Z}/b_{j},\mathbb{Z}/b_{j})\subset\mbox{Hom}(H_{1}(M),\mathbb{Z}/b_{j})$.
As $Sq^{2}(\alpha)=\alpha^{2}$ for a degree $2$ class $\alpha$, the assumption
implies that the class $a_{j}\in H^{2}(M;\mathbb{Z}/2)$ corresponding to the
generator of
$\mbox{Ext}(\mathbb{Z}/b_{j},\mathbb{Z}/2)\Big{(}\subset\mbox{Ext}(H_{1}(M),\mathbb{Z}/2)\subset
H^{2}(M;\mathbb{Z}/2)\Big{)}$ satisfies $a_{j}^{2}$ is the generator of
$H^{4}(M;\mathbb{Z}/2)\cong\mathbb{Z}/2$. The entire information fits into the
following diagram.

$\textstyle{H^{2}(K(\mathbb{Z}/b_{j},1);\mathbb{Z}/2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Sq^{2}}$$\scriptstyle{\hat{q}^{\ast}}$$\textstyle{H^{2}(M;\mathbb{Z}/2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Sq^{2}}$$\textstyle{H^{2}(M;\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(~{})^{2}}$$\textstyle{H^{4}(K(\mathbb{Z}/b_{j},1);\mathbb{Z}/2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{q}^{\ast}}$$\textstyle{H^{4}(M;\mathbb{Z}/2)}$$\textstyle{H^{4}(M;\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

As
$\mbox{Ext}(\mathbb{Z}/b_{j},\mathbb{Z})\to\mbox{Ext}(\mathbb{Z}/b_{j},\mathbb{Z}/2)$
is surjective, the class $a_{j}$ lifts to a $b_{j}$-torsion class in
$H^{2}(M;\mathbb{Z})$. Therefore, it’s square is $0$, and thus
$Sq^{2}(a_{j})=0$. This rules out the option 3) for an orientable non-spin
manifold $M^{4}$.

We conclude that for a non-spin $4$-manifold $M^{4}$, $\phi$ may map by $\eta$
to more than one copy of $\mathbb{S}^{2}$, as well as the maps of the type
(2). However, we may change the wedge sum decomposition of $M^{(3)}$, so that
$\phi$ maps to exactly one copy of $\mathbb{S}^{2}.$ This is done by
considering the homotopy class $\mathbb{S}^{2}\to M^{(3)}$ as the sum of the
non-trivial $\mathbb{S}^{2}$-factors and the inclusions coming from the
$M(\mathbb{Z}/b_{j})$-factors of type (2). This means that after such a choice
$\phi$ maps by $\eta$ to exactly one copy of $\mathbb{S}^{2}$.

Therefore, we obtain exactly two cases (noting that
$Cone(\eta:\mathbb{S}^{3}\to\mathbb{S}^{2})\simeq\mathbb{C}P^{2}$).

  1. (a)

If the manifold $M$ is spin, then the stable homotopy type of $M$ is

$M\simeq\mathbb{S}^{4}\vee\bigvee_{i=1}^{m}\left(\mathbb{S}^{1}\vee\mathbb{S}^{3}\right)\vee\bigvee_{j=1}^{n}(\Sigma
M(\mathbb{Z}/{b_{j}})\vee\Sigma^{2}M(\mathbb{Z}/{b_{j}}))\vee\bigvee_{l=1}^{d}\mathbb{S}^{2}.$
(7)

  2. (b)

If the manifold $M$ is non-spin, then its stable homotopy type is

$M\simeq\mathbb{C}P^{2}\vee\bigvee_{i=1}^{m}\left(\mathbb{S}^{1}\vee\mathbb{S}^{3}\right)\vee\bigvee_{j=1}^{n}(\Sigma
M(\mathbb{Z}/{b_{j}})\vee\Sigma^{2}M(\mathbb{Z}/{b_{j}}))\vee\bigvee_{l=1}^{d-1}\mathbb{S}^{2}.$
(8)

###  3.2. Concordance classes for $M^{4}\times\mathbb{S}^{k}$

From the above section, we have $M\simeq M^{(3)}\cup_{\phi_{3}}\mathbb{D}^{4}$
as an object in the stable homotopy category, where $\phi_{3}$ is null
homotopic if $M$ is a spin manifold and $\phi_{3}=i_{d}\circ\eta$ if $M$ is a
non-spin manifold. Here $i_{d}$ refers to the map which is the inclusion of
the $d^{th}$ $\mathbb{S}^{2}$ factor. This, together with Lemma 2.4, allows us
to compute $I_{c}(M\times\mathbb{S}^{k})$ by analyzing only the image of
$\eta^{*}:\pi_{k}(Top/O)\to\pi_{k+1}(Top/O).$ In particular, we prove the
following lemma.

######  Lemma 3.1.

The image of $\eta^{*}:\pi_{k}(Top/O)\to\pi_{k+1}(Top/O)$ is given in the
following table:

$k=$ | 1 to 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$Im(\eta^{*})$ | 0 | 0 | $\mathbb{Z}/2$ | $\mathbb{Z}/2$ | $\mathbb{Z}/2$ | $0$ | $0$ | $0$ | $\mathbb{Z}/2$ | $0$ | $\mathbb{Z}/2$ | $\mathbb{Z}/2\oplus\mathbb{Z}/2$ | $0$ | $0$

###### Proof.

The proof of the lemma follows from the calculations of stable homotopy groups
given in [31, 42]. ∎

From the proof of the above Lemma, we observe the following:

######  Remark 3.2.

The image of $\eta^{*}:\pi_{k}(Top/O)\to\pi_{k+1}(Top/O)$ is generated by the
elements $\eta\epsilon,\eta\mu,\eta\kappa,\eta\eta^{*},4\nu^{*},$ and
$\eta\bar{\mu}$ of $\pi_{k+1}^{s}$ for $k=8,9,14,16,$ and $17$ respectively.

######  Corollary 3.3.

  * (i)

The map $(\eta^{2})^{*}:\pi_{k}(Top/O)\to\pi_{k+2}(Top/O)$ is zero for $7\leq
k\leq 19$, except for $k=9$ and $16.$

  * (ii)

The image of the map $(\eta^{2})^{*}:\pi_{k}(Top/O)\to\pi_{k+2}(Top/O)$ is
$\mathbb{Z}/2$ for $k=9$ and $16.$

Using the long exact sequence induced from the cofibre sequence
$\mathbb{S}^{0}\xrightarrow{\times
2^{r}}\mathbb{S}^{0}\xhookrightarrow{i_{0}}M(\mathbb{Z}/{2^{r}})$ along
$Top/O$ and Lemma 3.1, we have the following result.

######  Corollary 3.4.

The image of
$(i_{3+k}\circ\eta)^{*}:[\Sigma^{3+k}M(\mathbb{Z}/2^{r}),Top/O]\to\pi_{4+k}(Top/O)$
is

  * (i)

$0$ for $k=1,2,3,4,8,9,10,12,15,16.$

  * (ii)

$\mathbb{Z}/2$ for $k=5,6,7,11,13.$

  * (iii)

$\mathbb{Z}/2\oplus\mathbb{Z}/2$ for $k=14.$

where
$i_{3+k}:\mathbb{S}^{3+k}\hookrightarrow\Sigma^{3+k}M(\mathbb{Z}/{2^{r}})$ is
the inclusion map.

We are now ready to establish the computations of
$I_{c}(M\times\mathbb{S}^{k})$ for $3\leq k\leq 16.$

######  Theorem 3.5.

Let $M$ be a closed, oriented, smooth $4$-manifold.

  1. (i)

If $M$ is a spin manifold, then $I_{c}(M\times\mathbb{S}^{k})=0$ for $k\geq
3.$

  2. (ii)

If $M$ is a non-spin manifold, then

     * (a)

$I_{c}(M\times\mathbb{S}^{k})=0$ for $k=3,4,8,9,10,12,15,16.$

     * (b)

$I_{c}(M\times\mathbb{S}^{k})=\mathbb{Z}/2$ for $k=5,6,7,11,13.$

     * (c)

$I_{c}(M\times\mathbb{S}^{k})=\mathbb{Z}/2\oplus\mathbb{Z}/2$ for $k=14$.

###### Proof.

The proof is obtained by applying the stable splittings (7), and (8) for $M,$
in combination with Lemma 2.4, and Lemma 3.1. ∎

######  Proposition 3.6.

Let $M$ be a non-spin, closed, oriented, smooth $4$ dimensional manifold. Then
$\mathbb{Z}/2\subseteq I_{c}(M\times\mathbb{S}^{k})$ for $k=17,18,8n-2(n\geq
3).$

###### Proof.

From [31, Table A3.3] and [1, Theorem 1.3, and 1.4], it can be obtained that
for $k=17,18$, or $8n-2(n\geq 3)$, there exists an element of order 2, denoted
as $x_{3+k}$, in $\pi_{3+k}^{s}/{\text{Im}(J_{3+k})}$ such that $\eta\circ
x_{3+k}$ is non-zero in $\pi_{4+k}^{s}/{\text{Im}(J_{4+k})}$. The existence of
such elements, along with the surjectivity of the induced map
$\psi^{*}:\Theta_{3+k}\to\pi_{3+k}^{s}/{\text{Im}(J_{3+k})}$, implies that the
image of $\eta^{*}:\pi_{3+k}(Top/O)\to\pi_{4+k}(Top/O)$ includes an exotic
$(4+k)$-sphere corresponding to the element $\eta\circ x_{3+k}$. The
proposition then follows from the stable splitting (8) and Lemma 2.4. ∎

We now proceed to calculate $\mathcal{C}(M^{4}\times\mathbb{S}^{k})$. It
follows from Corollary 2.3 that it is sufficient to compute the summand
$[\Sigma^{k}M,Top/O]$, which can be obtained from either the stable splitting
(7) or (8) by examining the same computations for each component. Therefore,
we first compute the component $[\Sigma^{k}\mathbb{C}P^{2},Top/O]$. Consider
the cofiber sequence for the map

$\mathbb{S}^{3}\xrightarrow{\eta_{2}}\mathbb{S}^{2}\xhookrightarrow{i}\mathbb{C}P^{2}\xrightarrow{f_{\mathbb{C}P^{2}}}\Sigma\mathbb{S}^{3}\xrightarrow{\Sigma\eta_{2}}\Sigma\mathbb{S}^{2}\hookrightarrow\cdots$
(9)

This sequence induces a long exact sequence

$\cdots\to\pi_{3+k}(X)\xrightarrow{\eta^{*}}\pi_{4+k}(X)\xrightarrow{(\Sigma^{k}f_{\mathbb{C}P^{2}})^{*}}[\Sigma^{k}\mathbb{C}P^{2},X]\xrightarrow{(\Sigma^{k}i)^{*}}\pi_{2+k}(X)\xrightarrow{\eta^{*}}\pi_{3+k}(X)\to\cdots$
(10)

where $\Sigma^{k}\eta_{2}=\eta$ for $k\geq 1$ [42] and $X$ is any infinite
loop space.

######  Proposition 3.7.

  1. (1)

$[\Sigma^{2}\mathbb{C}P^{2},Top/O]\cong 0.$

  2. (2)

$[\Sigma^{3}\mathbb{C}P^{2},Top/O]\cong\Theta_{7}.$

  3. (3)

$[\Sigma^{4}\mathbb{C}P^{2},Top/O]\cong\Theta_{8}.$

  4. (4)

There is a non-split short exact sequence

$0\to\mathbb{Z}/2\oplus\mathbb{Z}/2\to[\Sigma^{5}\mathbb{C}P^{2},Top/O]\to\Theta_{7}\to
0.$

  5. (5)

$[\Sigma^{6}\mathbb{C}P^{2},Top/O]\cong\mathbb{Z}/3\subset\Theta_{10}.$

  6. (6)

There is a short exact sequence

$0\to\mathbb{Z}/{496}\to[\Sigma^{7}\mathbb{C}P^{2},Top/O]\to\mathbb{Z}/2\oplus\mathbb{Z}/2\to
0.$

  7. (7)

$[\Sigma^{8}\mathbb{C}P^{2},Top/O]\cong\mathbb{Z}/3\subset\Theta_{10}.$

  8. (8)

$[\Sigma^{9}\mathbb{C}P^{2},Top/O]\cong\Theta_{13}\oplus\Theta_{11}.$

  9. (9)

$[\Sigma^{10}\mathbb{C}P^{2},Top/O]\cong\Theta_{14}.$

###### Proof.

In this proof, we make use of the long exact sequence (10) for $X$ being
either $Top/O$, $G/O$, or $G/Top$. We also use the long exact sequence derived
from the fibre sequence
$\Omega(G/Top)\xrightarrow{\omega}Top/O\xrightarrow{\psi}G/O\xrightarrow{\phi}G/Top.$

Statements (1) to (9), excluding (4) and (6), are direct consequences of the
long exact sequence (10) and Lemma 3.1. For the statements (4) and (6), we
derive the following short exact sequences :

$0\to\mathbb{Z}/2\oplus\mathbb{Z}/2\to[\Sigma^{5}\mathbb{C}P^{2},Top/O]\to\Theta_{7}\to
0$ (11)

and

$0\to\mathbb{Z}/{496}\to[\Sigma^{7}\mathbb{C}P^{2},Top/O]\to\mathbb{Z}/2\oplus\mathbb{Z}/2\to
0,$ (12)

where $\mathbb{Z}/2\oplus\mathbb{Z}/2\subset\Theta_{9}$ and
$\mathbb{Z}/{496}\subset\Theta_{11}.$ We first show that the short exact
sequence (11) does not split. Consider the following commutative diagram

${0}$${0}$${{\mathbb{Z}/2}}$${{[\Sigma^{6}\mathbb{C}P^{2},G/Top]}}$${{\mathbb{Z}}}$${0}$${{\mathbb{Z}/2}}$${{\bigoplus\limits_{i=1}^{3}\mathbb{Z}/2}}$${{[\Sigma^{5}\mathbb{C}P^{2},Top/O]}}$${{\mathbb{Z}/28}}$${{0}}$${{\mathbb{Z}\oplus\mathbb{Z}/2}}$${{\mathbb{Z}/2\oplus\mathbb{Z}/2}}$${{[\Sigma^{5}\mathbb{C}P^{2},G/O]}}$${0}$${0}$${0}$$\scriptstyle{(\Sigma^{6}f_{\mathbb{C}P^{2}})^{*}}$$\scriptstyle{\omega_{*}}$$\scriptstyle{(\Sigma^{6}i)^{*}}$$\scriptstyle{\omega_{*}}$$\scriptstyle{\omega_{*}}$$\scriptstyle{\eta^{*}}$$\scriptstyle{(\Sigma^{5}f_{\mathbb{C}P^{2}})^{*}}$$\scriptstyle{\psi_{*}}$$\scriptstyle{(\Sigma^{5}i)^{*}}$$\scriptstyle{\psi_{*}}$$\scriptstyle{\eta^{*}}$$\scriptstyle{(\Sigma^{5}f_{\mathbb{C}P^{2}})^{*}}$

where the top row splits at $[\Sigma^{6}\mathbb{C}P^{2},G/Top]$ and
$\psi_{*}:[\Sigma^{5}\mathbb{C}P^{2},Top/O]\to[\Sigma^{5}\mathbb{C}P^{2},G/O]$
is surjective as $[\Sigma^{5}\mathbb{C}P^{2},G/Top]\cong 0.$ Since the map
$\eta^{\ast}:\pi_{8}(G/O)\to\pi_{9}(G/O)$ is surjective and $\pi_{7}(G/O)=0$,
we get from the bottom row of the diagram that
$[\Sigma^{5}\mathbb{C}P^{2},G/O]$ is zero. Consequently, we deduce that the
map
$\omega_{\ast}:[\Sigma^{6}\mathbb{C}P^{2},G/Top]\to[\Sigma^{5}\mathbb{C}P^{2},Top/O]$
is also surjective. By combining this with the fact
$[\Sigma^{6}\mathbb{C}P^{2},G/Top]=\mathbb{Z}\oplus\mathbb{Z}_{2}$ and the
commutativity of the top rectangle, we conclude that
$[\Sigma^{5}\mathbb{C}P^{2},Top/O]\cong\mathbb{Z}/2\oplus\mathbb{Z}/56$. This
confirms that the short exact sequence (11) does not split, thereby completing
the proof of Statement (4). It should be noted that Statement (6) directly
follows from the sequence (12). This concludes the proof of the proposition. ∎

By using the group structure of $\Theta_{m}$ $(1\leq m\leq 18)$ localized at
the odd primes and the long exact sequence induced from the cofibre sequence
$\mathbb{S}^{0}\xrightarrow{p^{r}}\mathbb{S}^{0}\xhookrightarrow{i_{0}}M(\mathbb{Z}/{p^{r}})$
along $Top/O,$ we get the following lemma :

######  Lemma 3.8.

Given any positive integer $r,$ the following holds:

  * (a)

$[\Sigma^{k}M(\mathbb{Z}/{3^{r}}),Top/O]=0,$ for $1\leq k\leq 17$ with $k\neq
9,10,12,$ and $13.$

  * (b)

$[\Sigma^{k}M(\mathbb{Z}/3^{r}),Top/O]$ equals $\mathbb{Z}/3,$ for
$k=9,10,12,$ and $13.$

  * (c)

$[\Sigma^{k}M(\mathbb{Z}/{7^{r}}),Top/O]=0$ for $1\leq k\leq 17$ with $k\neq
6,7.$

  * (d)

$[\Sigma^{k}M(\mathbb{Z}/{7^{r}}),Top/O]$ equals $\mathbb{Z}/7,$ for $k=6$ and
$7.$

Let $p$ be $2$, $3$, or $7$ and let $l_{p}$ represent the number of
$p$-torsion summands of $H_{2}(M;\mathbb{Z})$ in (5).

######  Proposition 3.9.

Let $M$ be a $4$-dimensional closed, oriented manifold with homology as in
(5). Then

  * (i)

$[\Sigma
M,Top/O]\cong\bigoplus\limits_{j=1}^{2l_{2}}\pi_{3}(Top/O)\oplus\bigoplus\limits_{l=1}^{d}\pi_{3}(Top/O).$

  * (ii)

$[\Sigma^{2}M,Top/O]\cong\bigoplus\limits_{i=1}^{m}\pi_{3}(Top/O)\oplus\bigoplus\limits_{j=1}^{l_{2}}\pi_{3}(Top/O).$

  * (iii)

$[\Sigma^{3}M,Top/O]\cong\Theta_{7}.$

  * (iv)

$[\Sigma^{4}M,Top/O]\cong\Theta_{8}\oplus\bigoplus\limits_{i=1}^{m}\Theta_{7}\oplus\bigoplus\limits_{j=1}^{l_{2}}[\Sigma^{6}M(\mathbb{Z}/{2^{r_{j}}}),Top/O]\oplus\bigoplus\limits_{j=1}^{l_{7}}\mathbb{Z}/7.$

  * (v)

$[\Sigma^{9}M,Top/O]\cong\Theta_{13}\oplus\bigoplus\limits_{i=1}^{m}\Theta_{10}\oplus\bigoplus\limits_{j=1}^{n}\left([\Sigma^{10}M(\mathbb{Z}/{b_{j}}),Top/O]\oplus[\Sigma^{11}M(\mathbb{Z}/{b_{j}}),Top/O]\right)\oplus\bigoplus\limits_{l=1}^{d}\Theta_{11}.$

  * (vi)

$[\Sigma^{10}M,Top/O]\cong\Theta_{14}\oplus\bigoplus\limits_{i=1}^{m}\left(\Theta_{11}\oplus\Theta_{13}\right)\oplus\bigoplus\limits_{j=1}^{n}[\Sigma^{11}M(\mathbb{Z}/{b_{j}}),Top/O]\oplus\bigoplus\limits_{j=1}^{l_{3}}\Theta_{13}.$

  * (vii)

    * (a)

$[\Sigma^{5}M,Top/O]\cong\Theta_{9}\oplus\bigoplus\limits_{i=1}^{m}\Theta_{8}\oplus\bigoplus\limits_{j=1}^{l_{2}}[\Sigma^{6}M(\mathbb{Z}/{2^{r_{j}}}),Top/O]\oplus\bigoplus\limits_{j=1}^{2l_{7}}\mathbb{Z}/7\\\
\oplus\bigoplus\limits_{j=1}^{l_{2}}[\Sigma^{7}M(\mathbb{Z}/{2^{r_{j}}}),Top/O]\oplus\bigoplus\limits_{l=1}^{d}\Theta_{7},$
if $M$ is a spin manifold.

    * (a)

$[\Sigma^{5}M,Top/O]\cong\mathbb{Z}/{56}\oplus\mathbb{Z}/2\oplus\bigoplus\limits_{i=1}^{m}\Theta_{8}\oplus\bigoplus\limits_{j=1}^{l_{2}}[\Sigma^{6}M(\mathbb{Z}/{2^{r_{j}}}),Top/O]\oplus\bigoplus\limits_{j=1}^{2l_{7}}\mathbb{Z}/7\\\
\oplus\bigoplus\limits_{j=1}^{l_{2}}[\Sigma^{7}M(\mathbb{Z}/{b_{j}}),Top/O]\oplus\bigoplus\limits_{l=1}^{d-1}\Theta_{7},$
if $M$ is a non-spin manifold.

  * (viii)

    * (a)

if $M$ is a spin manifold, then
$[\Sigma^{6}M,Top/O]\cong\Theta_{10}\oplus\bigoplus\limits_{i=1}^{m}\left(\Theta_{7}\oplus\Theta_{9}\right)\oplus\\\
\bigoplus\limits_{j=1}^{l_{2}}[\Sigma^{7}M(\mathbb{Z}/{2^{r_{j}}}),Top/O]\oplus\bigoplus\limits_{j=1}^{l_{7}}\mathbb{Z}/7\oplus\bigoplus\limits_{j=1}^{l_{2}}[\Sigma^{8}M(\mathbb{Z}/{2^{r_{j}}}),Top/O]\oplus\bigoplus\limits_{l=1}^{d}\Theta_{8}.$

    * (b)

if $M$ is a non-spin manifold, then
$[\Sigma^{6}M,Top/O]\cong\mathbb{Z}/3\oplus\bigoplus\limits_{i=1}^{m}\left(\Theta_{7}\oplus\Theta_{9}\right)\oplus\\\
\bigoplus\limits_{j=1}^{l_{2}}[\Sigma^{7}M(\mathbb{Z}/{2^{r_{j}}}),Top/O]\oplus\bigoplus\limits_{j=1}^{l_{7}}\mathbb{Z}/7\oplus\bigoplus\limits_{j=1}^{l_{2}}[\Sigma^{8}M(\mathbb{Z}/{2^{r_{j}}}),Top/O]\oplus\bigoplus\limits_{l=1}^{d-1}\Theta_{8}.$

  * (ix)

    * (a)

$[\Sigma^{7}M,Top/O]\cong\Theta_{11}\oplus\bigoplus\limits_{i=1}^{m}\left(\Theta_{8}\oplus\Theta_{10}\right)\oplus\bigoplus\limits_{j=1}^{l_{2}}[\Sigma^{8}M(\mathbb{Z}/{2^{r_{j}}}),Top/O]\oplus\\\
\bigoplus\limits_{j=1}^{l_{2}}[\Sigma^{9}M(\mathbb{Z}/{2^{r_{j}}}),Top/O]\oplus\bigoplus\limits_{j=1}^{l_{3}}\mathbb{Z}/3\oplus\bigoplus\limits_{l=1}^{d}\Theta_{9}$
for a spin manifold $M.$

    * (b)

$[\Sigma^{7}M,Top/O]\cong[\Sigma^{7}\mathbb{C}P^{2},Top/O]\oplus\bigoplus\limits_{i=1}^{m}\left(\Theta_{8}\oplus\Theta_{10}\right)\oplus\bigoplus\limits_{j=1}^{l_{2}}[\Sigma^{8}M(\mathbb{Z}/{2^{r_{j}}}),Top/O]\\\
\oplus\bigoplus\limits_{j=1}^{l_{2}}[\Sigma^{9}M(\mathbb{Z}/{2^{r_{j}}}),Top/O]\oplus\bigoplus\limits_{j=1}^{l_{3}}\mathbb{Z}/3\oplus\bigoplus\limits_{l=1}^{d-1}\Theta_{9}$
for a non-spin manifold $M.$

  * (x)

    * (a)

if $M$ is a spin manifold, then
$[\Sigma^{8}M,Top/O]\cong\bigoplus\limits_{i=1}^{m}\left(\Theta_{9}\oplus\Theta_{11}\right)\oplus\\\
\bigoplus\limits_{j=1}^{l_{2}}[\Sigma^{9}M(\mathbb{Z}/{2^{r_{j}}}),Top/O]\oplus\bigoplus\limits_{j=1}^{l_{3}}\mathbb{Z}/3\oplus\bigoplus\limits_{j=1}^{n}[\Sigma^{10}M(\mathbb{Z}/{b_{j}}),Top/O]\oplus\bigoplus\limits_{l=1}^{d}\Theta_{10}.$

    * (b)

if $M$ is a non-spin manifold, then
$[\Sigma^{8}M,Top/O]\cong\mathbb{Z}/3\oplus\bigoplus\limits_{i=1}^{m}\left(\Theta_{9}\oplus\Theta_{11}\right)\oplus\\\
\bigoplus\limits_{j=1}^{l_{2}}[\Sigma^{9}M(\mathbb{Z}/{2^{r_{j}}}),Top/O]\oplus\bigoplus\limits_{j=1}^{l_{3}}\mathbb{Z}/3\oplus\bigoplus\limits_{j=1}^{n}[\Sigma^{10}M(\mathbb{Z}/{b_{j}}),Top/O]\oplus\bigoplus\limits_{l=1}^{d-1}\Theta_{10}.$

###### Proof.

The computations of $[\Sigma^{k}M,Top/O]$ for $1\leq k\leq 10$ can be
immediately obtained from the stable splittings (7) and (8) by applying
Proposition 3.7 and Lemma 3.8. ∎

##  4\. Smooth structures on $M^{5}\times\mathbb{S}^{k}$

Let $M$ be a simply connected, closed, oriented $5$ dimensional manifold. Note
that by using Poincaré duality and the Universal coefficient theorem, $M$ has
the following homology group

$H_{l}(M;\mathbb{Z})=\begin{cases}\mathbb{Z}&\text{l=0,5}\\\
\mathbb{Z}^{d}\oplus\bigoplus\limits_{i=1}^{t}\mathbb{Z}/{p_{i}^{r_{i}}}&\text{l=2}\\\
\mathbb{Z}^{d}&\text{l=3}\\\ 0&\text{otherwise}\\\ \end{cases}$ (13)

where $d,t,r_{j}(1\leq j\leq t)$ are positive integers and $p_{i}$ are prime
numbers.

###  4.1. The stable homotopy type of a simply connected $5$-manifold

We may now consider the minimal cell structure on $M$ [14, Propositon 4C.1],
for which the stable homotopy type of the $3$-skeleton $M^{(3)}$ of $M$ is
given by

$M^{(3)}\simeq\bigvee_{i=1}^{d}\left(\mathbb{S}^{2}\vee\mathbb{S}^{3}\right)\vee\bigvee_{j=1}^{t}\Sigma^{2}M(\mathbb{Z}/p_{j}^{r_{j}}).$
(14)

Therefore, $M\simeq M^{(3)}\cup_{g}\mathbb{D}^{5}$ where $g:\mathbb{S}^{4}\to
M_{(3)}$ is the attaching map of the top cell. Hence we have a cofibre
sequence

$\mathbb{S}^{4}\xrightarrow{g}M^{(3)}\xhookrightarrow{i}M\xrightarrow{f_{M}}\mathbb{S}^{5}\to\cdots$
(15)

of spectra. We divide the possibilities into two cases according to whether
$M$ is spin or not. As in the $4$-dimensional case, we observe that the stable
attaching map is $0$ if $M$ is spin.

######  Proposition 4.1.

Let $M$ be a simply connected spin $5$-manifold with homology as in (13),
then, we have a stable equivalence

$M\simeq\mathbb{S}^{5}\vee\bigvee_{i=1}^{d}\left(\mathbb{S}^{2}\vee\mathbb{S}^{3}\right)\vee\bigvee_{j=1}^{t}\Sigma^{2}M(\mathbb{Z}/p_{j}^{r_{j}})$

###### Proof.

If $M$ is a spin manifold, then from the results of Smale and Barden we see
that $M$ is completely clasified as a connected sum of some atomic pieces.
These atomic pieces are $\mathbb{S}^{2}\times\mathbb{S}^{3},\mathbb{S}^{5}$
and $M_{k}$ with $H_{2}(M_{k})=\mathbb{Z}/k\oplus\mathbb{Z}/k,k\neq 1$ and
$\infty$ [2, Page-373]. We have to show that $\eta^{2}$ does not appear in the
attaching map in each case, and this will imply the proposition. This is clear
for $\mathbb{S}^{2}\times\mathbb{S}^{3}$ and $\mathbb{S}^{5}$. We now prove
this for $M_{k}$ from an explicit cell complex construction of it following
[2, Page-373].

Let $N$ be the manifold
$(\mathbb{S}^{2}\times\mathbb{S}^{2})\\#(\mathbb{S}^{2}\times\mathbb{S}^{2})$.
Write $N=\partial D_{i}$ for $i=1,2$ with each

$D_{i}\cong\mathbb{D}^{3}\times\mathbb{S}^{2}\setminus(\mathbb{D}^{2}\times\mathbb{D}^{2})\cup_{\partial\mathbb{D}^{4}\times\mathbb{D}^{1}}\mathbb{D}^{3}\times\mathbb{S}^{2}\setminus(\mathbb{D}^{2}\times\mathbb{D}^{2}).$

Now consider $\phi:N\to N$ which is described by the matrix $A_{k}$ on the
$2$-skeleton.

$A_{k}=\begin{bmatrix}1&0&0&0\\\ 0&1&k&0\\\ 0&0&1&0\\\ -k&0&0&1\end{bmatrix}.$

The manifold $M_{k}=D_{1}\cup_{\phi:N\to N}D_{2}$. Note that
$\mathbb{D}^{3}\times\mathbb{S}^{2}$ is obtained out of
$\mathbb{S}^{2}\times\mathbb{S}^{2}$ by attaching a $3$-cell via the inclusion
of the first factor (so that the space following the attachment is
$\mathbb{S}^{2}\times\mathbb{S}^{2}/(\mathbb{S}^{2}\times\ast)\simeq\mathbb{S}^{2}\vee\mathbb{S}^{4}$,
followed by a $5$-cell along the copy of $\mathbb{S}^{4}$. Let $u_{i}$, $1\leq
i\leq 4$, be the $4$ $2$-spheres in $N$. Proceeding along the same idea,
$D_{i}$ may be built from $N$ by attaching $3$-cells along $u_{1}$ and $u_{3}$
(so that $Cone(\mathbb{S}^{2}\vee\mathbb{S}^{2}\stackrel{{\scriptstyle
u_{1}\vee
u_{3}}}{{\to}}N)\simeq\mathbb{S}^{2}\vee\mathbb{S}^{2}\vee\mathbb{S}^{4}$),
followed by a $5$-cell attachment along the copy of $\mathbb{S}^{4}$. Finally
putting all together, we have that $M_{k}$ is constructed out of $N$ by
attaching $3$-cells along $u_{1}$, $u_{3}$, $u_{1}-ku_{4}$, $ku_{2}+u_{3}$,
followed by $2$ $5$-cells along the copy of $\mathbb{S}^{4}$. One clearly
obtains that the stable homotopy type of $M_{k}$ is

$\Sigma^{2}M(\mathbb{Z}/k)\vee\Sigma^{2}M(\mathbb{Z}/k)\vee\mathbb{S}^{5}.$

This completes the proof.

∎

It now remains to consider the case where $M$ is not spin. This means that we
have a non-trivial second Stiefel Whitney class, which by Wu’s formula implies
that $\eta$ appears in the attaching map of the top cell. If the map $\eta$
hits the $\mathbb{S}^{3}$-factor as in the $4$-dimensional case, it cones off
to give a copy of the suspension of $\mathbb{C}P^{2}$. However $\eta$ may also
map via $\tilde{\eta}_{2^{s}}$ for some $s\in\mathbb{N}$ onto the Moore
spectrum part. If in addition, in the expression of the attaching map

$\mathbb{S}^{4}\to\bigvee_{i=1}^{d}\left(\mathbb{S}^{2}\vee\mathbb{S}^{3}\right)\vee\bigvee_{j=1}^{t}\Sigma^{2}M(\mathbb{Z}/p_{j}^{r_{j}})$

there is a factor of $\mathbb{S}^{2}$ which receives a map $\eta^{2}$, we may
change the homotopy class of $\mathbb{S}^{3}\to M^{(4)}$ or
$\Sigma^{2}M(\mathbb{Z}/2^{s})\to M^{(4)}$ to make that factor $0$. More
explicitly, we add composites

$\mathbb{S}^{3}\stackrel{{\scriptstyle\eta}}{{\to}}\mathbb{S}^{2},~{}~{}~{}\Sigma^{2}M(\mathbb{Z}/2^{s})\stackrel{{\scriptstyle\hat{\eta}}}{{\to}}\mathbb{S}^{2},$

where the map $\hat{\eta}$ is given as
$\Sigma^{2}M(\mathbb{Z}/2^{s})\to\mathbb{S}^{3}\stackrel{{\scriptstyle\eta}}{{\to}}\mathbb{S}^{2}$.
On composing with $\eta:\mathbb{S}^{4}\to\mathbb{S}^{3}$ (and respectively,
$\tilde{\eta}_{2^{s}}:\mathbb{S}^{4}\to\Sigma^{2}M(\mathbb{Z}/2^{s})$), we may
adjust the $\eta^{2}$ in the expression to $0$. Thus, we may have two cases
both of which may occur as demonstrated in the examples of [2, Page-373].
Thus, if $M$ is not spin, the stable homotopy type is either

$M\simeq\Sigma\mathbb{C}P^{2}\vee\bigvee_{i=1}^{d-1}\left(\mathbb{S}^{2}\vee\mathbb{S}^{3}\right)\vee\mathbb{S}^{2}\vee\bigvee_{j=1}^{t}\Sigma^{2}M(\mathbb{Z}/p_{j}^{r_{j}})$
(16)

or

$M\simeq
Cone(\tilde{\eta}_{2^{r}})\vee\bigvee_{i=1}^{d}\left(\mathbb{S}^{2}\vee\mathbb{S}^{3}\right)\vee\bigvee_{j=2}^{t}\Sigma^{2}M(\mathbb{Z}/p_{j}^{r_{j}})$
(17)

where $p_{1}^{r}=2^{r}$ is the factor which supported the
$\tilde{\eta}_{2^{r}}$ map. The value of $r$ denotes the smallest power of $2$
present in the Moore spectrum segment.

###  4.2. Concordance classes of $M^{5}\times\mathbb{S}^{k}$

It follows from Lemma 2.4 that the concordance inertia group
$I_{c}(M^{5}\times\mathbb{S}^{k})$ can be identified with the trivial group,
or
$Im\left(\pi_{4+k}(Top/O)\xrightarrow{\eta^{*}}\pi_{5+k}(Top/O)\right),$ or
$Im\left([\Sigma^{3+k}M(\mathbb{Z}/{2^{r}}),Top/O]\xrightarrow{(\tilde{\eta}_{2^{r}})^{*}}\pi_{5+k}(Top/O)\right),$
based on the stable decomposition given in Proposition 4.1 and (16), (17).
Note that the image of $\eta^{*}$ is given in Lemma 3.1. So we compute the
image of
$(\tilde{\eta}_{2^{r}})^{*}:[\Sigma^{3+k}M(\mathbb{Z}/{2^{r}}),Top/O]\to\pi_{5+k}(Top/O).$
We frequently use the long exact sequence

$\cdots\xrightarrow{\times
2^{r}}\pi_{4+k}(Top/O)\xrightarrow{(q_{4+k})^{*}}[\Sigma^{3+k}M(\mathbb{Z}/{2^{r}}),Top/O]\xrightarrow{(i_{3+k})^{*}}\pi_{3+k}(Top/O)\xrightarrow{\times
2^{r}}\cdots$ (18)

induced from the cofibre sequence $\mathbb{S}^{3+k}\xrightarrow{\times
2^{r}}\mathbb{S}^{3+k}\xhookrightarrow{i_{3+k}}\Sigma^{3+k}M(\mathbb{Z}/{2^{r}})\xrightarrow{q_{4+k}}\mathbb{S}^{4+k}\cdots$
throughout the proof of the following lemma.

######  Lemma 4.2.

The following statements hold.

  * (a)

For each $r\geq 1$, the image of the map
$(\tilde{\eta}_{2^{r}})^{*}:[\Sigma^{3+k}M(\mathbb{Z}/{2^{r}}),Top/O]\to\pi_{5+k}(Top/O)$
is

    * (i)

trivial for $k=1,2,3,7,8,9.$

    * (ii)

$\mathbb{Z}/2$ for $k=5,10.$

  * (b)

The image of
$(\tilde{\eta}_{2^{r}})^{*}:[\Sigma^{7}M(\mathbb{Z}/{2^{r}}),Top/O]\to\pi_{9}(Top/O)$
equals

    * (i)

$\mathbb{Z}/2\oplus\mathbb{Z}/2$ if $r=1$ and $2.$

    * (ii)

$\mathbb{Z}/2$ if $r\geq 3.$

  * (c)

The image of
$(\tilde{\eta}_{2^{r}})^{*}:[\Sigma^{9}M(\mathbb{Z}/{2^{r}}),Top/O]\to\pi_{11}(Top/O)$
equals

    * (i)

$\mathbb{Z}/4$ if $r=1.$

    * (ii)

$\mathbb{Z}/2$ if $r\geq 2.$

###### Proof.

We examine each value of $k$ one by one to prove the result. By the definition
of $\tilde{\eta}_{2^{r}},$ we have the following commutative diagram

${{\pi_{4+k}(Top/O)}}$${{[\Sigma^{3+k}M(\mathbb{Z}/{2^{r}}),Top/O]}}$${{\pi_{5+k}(Top/O)}}$$\scriptstyle{(q_{4+k})^{*}}$$\scriptstyle{\eta^{*}}$$\scriptstyle{(\tilde{\eta}_{2^{r}})^{*}}$
(19)

For $k=1,$ and $7$, the result is immediate due to the triviality of
$\pi_{5+k}(Top/O).$ Additionally, from the exact sequence (18), it can be
observed that the groups $[\Sigma^{5}M(\mathbb{Z}/{2^{r}}),Top/O]$ and
$[\Sigma^{11}M(\mathbb{Z}/{2^{r}}),Top/O]_{(3)}$ are both trivial.
Consequently, the map
$(\tilde{\eta}_{2^{r}})^{*}:[\Sigma^{3+k}M(\mathbb{Z}/{2^{r}}),Top/O]\to\pi_{5+k}(Top/O)$
is trivial for $k=2$ and $8.$ This concludes the cases $k=2$ and $8.$

For both cases $k=3$ and $9$, the map
$(\eta)^{*}:\pi_{4+k}(Top/O)\to\pi_{5+k}(Top/O)$ is trivial as per Lemma 3.1.
Moreover, the map
$(q_{4+k})^{*}:\pi_{4+k}(Top/O)\to[\Sigma^{3+k}M(\mathbb{Z}/2^{r}),Top/O]$ is
surjective, as seen from the exact sequence (18). The combination of these
observations with the commutativity of the diagram (19) implies that the
result holds for these cases.

Consider the cases $k=5$ and $10$ : In these particular cases, Lemma 3.1
implies that the image of the map
$\eta^{*}:\pi_{4+k}(Top/O)\to\pi_{5+k}(Top/O)$ is $\mathbb{Z}/2$. From the
sequence (18), we can deduce the following :

  * •

The map $q_{9}^{*}:\pi_{9}(Top/O)\to[\Sigma^{8}M(\mathbb{Z}/{2^{r}}),Top/O]$
is injective, and $[\Sigma^{8}M(\mathbb{Z}/{2^{r}}),Top/O]_{(3)}$ is trivial.

  * •

The map
$q_{14}^{*}:\pi_{14}(Top/O)\to[\Sigma^{13}M(\mathbb{Z}/{2^{r}}),Top/O]$ is an
isomorphism.

By combining these observations in the commutative diagram (19), we can obtain
the desired result.

The case $k=4$: Since
$[\Sigma^{7}M(\mathbb{Z}/{2^{r}}),Top/O]\cong[\Sigma^{7}M(\mathbb{Z}/{2^{r}}),PL/O]$
and $\pi_{9}(Top/O)\cong\pi_{9}(PL/O)$, we can identify the image of
$(\tilde{\eta}_{2^{r}})^{*}:[\Sigma^{7}M(\mathbb{Z}/{2^{r}}),Top/O]\to\pi_{9}(Top/O)$
with the image of
$(\tilde{\eta}_{2^{r}})^{*}:[\Sigma^{7}M(\mathbb{Z}/{2^{r}}),PL/O]\to\pi_{9}(PL/O)$.
Therefore, we proceed to compute the image of the map
$(\tilde{\eta}_{2^{r}})^{*}:[\Sigma^{7}M(\mathbb{Z}/{2^{r}}),PL/O]\to\pi_{9}(PL/O)$.
Consider first $r=1$ and then from [9, Page-29, Corollary 1.6.12], we have the
following split short exact sequence

$0\to Ext(\mathbb{Z}/2,\pi_{8}(PL/O))\to[\Sigma^{7}M(\mathbb{Z}/{2}),PL/O]\to
Hom(\mathbb{Z}/2,\pi_{7}(PL/O))\to 0.$ (20)

Furthermore, it follows that the map
$(\tilde{\eta}_{2})^{*}:[\Sigma^{7}M(\mathbb{Z}/{2}),PL/O]\to\pi_{9}(PL/O)$
fits into the following diagram

${{[\Sigma^{7}M(\mathbb{Z}/2),PL/O]}}$${{\pi_{9}(PL/O)}}$${{\Gamma_{9}(PL/O)}}$$\scriptstyle{(\tilde{\eta}_{2})^{*}}$$\scriptstyle{\cong}$$\scriptstyle{(\xi)^{*}}$

where $\Gamma_{9}(PL/O)$ is the image of the map
$\pi_{9}(PL/O^{(8)})\to\pi_{9}(PL/O^{(9)})$ induced by the inclusion
$PL/O^{(8)}\hookrightarrow PL/O^{(9)}$ and the map
$(\xi)^{*}:[\Sigma^{7}M(\mathbb{Z}/2),PL/O]\to\Gamma_{9}(PL/O)$ is an
isomorphism given by [9, Page-264, Theorem 8.3.7]. This, together with the
fact
$[\Sigma^{7}M(\mathbb{Z}/{2^{r}}),PL/O]\cong\mathbb{Z}/2\oplus\mathbb{Z}/2$
provided by (20) implies that the image of the map
$(\tilde{\eta}_{2})^{*}:[\Sigma^{7}M(\mathbb{Z}/2),PL/O]\to\pi_{9}(PL/O)$ is
$\mathbb{Z}/2\oplus\mathbb{Z}/2.$ This concludes the statement (b)(i).
Let’s now assume that $r\geq 2$ and consider the following diagram of homotopy
cofibrations:

${{\mathbb{S}^{7}}}$${{\mathbb{S}^{7}}}$${{\Sigma^{7}M(\mathbb{Z}/{2^{r}})}}$${{\mathbb{S}^{8}}}$${{\mathbb{S}^{8}}}$${{\mathbb{S}^{7}}}$${{\mathbb{S}^{7}}}$${{\Sigma^{7}M(\mathbb{Z}/2)}}$${{\mathbb{S}^{8}}}$${{\mathbb{S}^{8}}}$$\scriptstyle{\times
2^{r}}$$\scriptstyle{i_{7}}$$\scriptstyle{q_{8}}$$\scriptstyle{\times
2^{r}}$$\scriptstyle{\times 2}$$\scriptstyle{i_{7}}$$\scriptstyle{\times
2^{r-1}}$$\scriptstyle{q_{8}}$$\scriptstyle{\Sigma^{7}B(\chi^{1}_{r})}$$\scriptstyle{\times
2}$$\scriptstyle{\times 2^{r-1}}$ (21)

where
$\Sigma^{7}B(\chi^{1}_{r}):\Sigma^{7}M(\mathbb{Z}/{2})\to\Sigma^{7}M(\mathbb{Z}/{2^{r}})$
is the map given in (2) and it satisfies the relations (4). This induces a
diagram of long exact sequences at the $PL/O.$

${{\pi_{8}(PL/O)}}$${{[\Sigma^{7}M(\mathbb{Z}/{2^{r}}),PL/O]}}$${{\pi_{7}(PL/O)}}$${{\pi_{7}(PL/O)}}$${{\pi_{8}(PL/O)}}$${{[\Sigma^{7}M(\mathbb{Z}/2),PL/O]\cong\mathbb{Z}/2\oplus\mathbb{Z}/2}}$${{\pi_{7}(PL/O)}}$${{\pi_{7}(PL/O)}}$$\scriptstyle{0}$$\scriptstyle{(q_{8})^{*}}$$\scriptstyle{(\Sigma^{7}B(\chi^{1}_{r}))^{*}}$$\scriptstyle{(i_{7})^{*}}$$\scriptstyle{\times
2^{r-1}}$$\scriptstyle{\times
2^{r}}$$\scriptstyle{0}$$\scriptstyle{(q_{8})^{*}}$$\scriptstyle{(i_{7})^{*}}$$\scriptstyle{\times
2}$

From this, we can easily deduce the following:

  * •

If $r\geq 3,$ then the image of
$(\Sigma^{7}B(\chi^{1}_{r}))^{*}:[\Sigma^{7}M(\mathbb{Z}/{2^{r}}),PL/O]\to[\Sigma^{7}M(\mathbb{Z}/2),PL/O]$
is $\mathbb{Z}/2.$

  * •

If $r=2,$ then the map
$(\Sigma^{7}B(\chi^{1}_{r}))^{*}:[\Sigma^{7}M(\mathbb{Z}/{4}),PL/O]\to[\Sigma^{7}M(\mathbb{Z}/2),PL/O]$
has image $\mathbb{Z}/2\oplus\mathbb{Z}/2.$

Combining these observations in the following commutative diagram given by the
relation (3)

${{[\Sigma^{7}M(\mathbb{Z}/{2^{r}}),PL/O]}}$${{[\Sigma^{7}M(\mathbb{Z}/2),PL/O]\cong\mathbb{Z}/2\oplus\mathbb{Z}/2}}$${{\pi_{9}(PL/O)}}$$\scriptstyle{(\Sigma^{7}B(\chi^{1}_{r}))^{*}}$$\scriptstyle{(\tilde{\eta}_{2^{r}})^{*}}$$\scriptstyle{(\tilde{\eta}_{2})^{*}}$

establishes the statement (b)(ii).

The case $k=6$: In this case, it follows from the long exact sequence (18) for
$r=1$ and [12, Lemma 2.3] that there is a non-split sequence

$0\to\mathbb{Z}_{2}\subset\pi_{10}(Top/O)\to[\Sigma^{9}M(\mathbb{Z}/{2}),Top/O]=\mathbb{Z}/4\oplus\mathbb{Z}/2\oplus\mathbb{Z}/2\to\pi_{9}(Top/O)\to
0,$ (22)

where the map
$(q_{10})^{*}:\pi_{10}(Top/O)\to[\Sigma^{9}M(\mathbb{Z}/2),Top/O]$ sends the
generator $\mu\circ\eta$ to the element of order $2$ in the summand
$\mathbb{Z}/4$. Further, from Lemma 3.1, the image of
$\eta^{*}:\pi_{10}(Top/O)\to\pi_{11}(Top/O)$ is $\mathbb{Z}/2$. Combining
these facts in the diagram (19) implies that the image of
$(\tilde{\eta}_{2})^{*}:[\Sigma^{9}M(\mathbb{Z}/2),Top/O]\to\pi_{11}(Top/O)$
is $\mathbb{Z}/4.$ This completes the proof of the statement (c)(i). Assume
$r\geq 2$ and consider the following diagram

${{0}}$${{\mathbb{Z}/2}}$${{[\Sigma^{9}M(\mathbb{Z}/{2^{r}}),Top/O]}}$${{\pi_{9}(Top/O)}}$${{0}}$${{0}}$${{\mathbb{Z}/2}}$${{[\Sigma^{9}M(\mathbb{Z}/2),Top/O]\cong\mathbb{Z}/4\oplus\mathbb{Z}/2\oplus\mathbb{Z}/2}}$${{\pi_{9}(Top/O)}}$${{0}}$$\scriptstyle{(\Sigma^{9}B({}^{1}\chi_{r}))^{*}}$$\scriptstyle{(i_{9})^{*}}$$\scriptstyle{0}$$\scriptstyle{0}$$\scriptstyle{(i_{9})^{*}}$$\scriptstyle{0}$

where the first and second rows are induced by the cofibre sequences of the
mapping cones $\Sigma^{9}M(\mathbb{Z}/2^{r})$ and $\Sigma^{9}M(\mathbb{Z}/2)$
respectively as in (21). From this diagram, we can deduce that
$[\Sigma^{9}M(\mathbb{Z}/{2^{r}}),Top/O]\cong\bigoplus\limits_{i=1}^{4}\mathbb{Z}/2$.
Now apply the same argument as in the case $r=1$ using the diagram (19), we
get that the image of
$(\tilde{\eta}_{2^{r}})^{*}:[\Sigma^{9}M(\mathbb{Z}/2^{r}),Top/O]\to\pi_{11}(Top/O)$
is $\mathbb{Z}/2.$ This proves the statement (c)(ii).

This finishes the proof of the lemma. ∎

The following theorem determines the concordance inertia group of
$M^{5}\times\mathbb{S}^{k}.$

######  Theorem 4.3.

Let $M$ be a simply connected, closed, oriented $5$-manifold with homology of
the form (13). The following statements hold.

  1. (1)

If $M$ is a spin manifold, then $I_{c}(M\times\mathbb{S}^{k})=0$ for $k\geq
1.$

  2. (2)

Suppose $M$ is a non-spin manifold.

    1. (a)

$I_{c}(M\times\mathbb{S}^{k})=0$ for $k=2,3,7,8,$ and $9.$

    2. (b)

For $k=4,5,6,$ and $10,$

      1. (i)

if $M$ satisfies (16), then $I_{c}(M\times\mathbb{S}^{k})=\mathbb{Z}/2.$

      2. (ii)

if $M$ satisfies (17), then

        1. (A)

$I_{c}(M\times\mathbb{S}^{4})=\begin{cases}\mathbb{Z}/2\oplus\mathbb{Z}/2,&\mbox{
if $r=1$ or $2.$}\\\ \mathbb{Z}/2,&\mbox{if $r\geq 3.$}\end{cases}$

        2. (B)

$I_{c}(M\times\mathbb{S}^{5})=\mathbb{Z}/2.$

        3. (C)

$I_{c}(M\times\mathbb{S}^{6})=\begin{cases}\mathbb{Z}/4,&\mbox{if $r=1.$}\\\
\mathbb{Z}/2,&\mbox{if $r\geq 2.$}\end{cases}$

        4. (D)

$I_{c}(M\times\mathbb{S}^{10})=\mathbb{Z}/2.$

where $r$ is the least positive integer such that $\mathbb{Z}/{2^{r}}$ is in
the torsion part of $H_{2}(M;\mathbb{Z}).$

###### Proof.

The results (1) and (2) are derived from Corollary 2.4 by using Proposition
4.1 and the stable splitting (16), (17), along with the calculations from
Lemmas 3.1 and 4.2. ∎

As a direct consequence of the stable homotopy type of $M$ described in
Proposition 4.1 and (16), (17), we have the following:

######  Proposition 4.4.

Let $M$ be a closed, oriented, simply-connected $5$-dimensional manifold with
homology of the form (13). The following holds:

  * (a)

If $M$ is a spin manifold, then $[\Sigma^{k}M,Top/O]$ is isomorphic to

$\pi_{5+k}(Top/O)\oplus\\\
\bigoplus\limits_{i=1}^{d}\left(\pi_{2+k}(Top/O)\oplus\pi_{4+k}(Top/O)\right)\oplus\bigoplus\limits_{j=1}^{t}[\Sigma^{2+k}M(\mathbb{Z}/p_{j}^{r_{j}}),Top/O].$

  * (b)

If $M$ is a non-spin manifold, then $[\Sigma^{k}M,Top/O]$ is isomorphic to
either

$[\Sigma^{k+1}\mathbb{C}P^{2},Top/O]\oplus\bigoplus\limits_{i=1}^{d-1}\left(\pi_{2+k}(Top/O)\oplus\pi_{3+k}(Top/O)\right)\oplus\bigoplus\limits_{j=1}^{t}[\Sigma^{2+k}M(\mathbb{Z}/{p_{j}}^{r_{j}}),Top/O]$

or

$[\Sigma^{k}Cone(\tilde{\eta}_{2^{r}}),Top/O]\oplus\bigoplus\limits_{i=1}^{d}\left(\pi_{2+k}(Top/O)\oplus\pi_{3+k}(Top/O)\right)\oplus\bigoplus\limits_{j=2}^{t}[\Sigma^{2+k}M(\mathbb{Z}/{p_{j}}^{r_{j}}),Top/O]$

In conclusion, according to Corollary 2.3, if $M$ is either a 4-dimensional
manifold or a simply connected 5-dimensional manifold, then the smooth
concordance structure set $\mathcal{C}(M\times\mathbb{S}^{k})$ can be
determined using Proposition 3.9 and Proposition 4.4.

##  5\. Smooth structures on $\mathbb{C}P^{2}\times\mathbb{S}^{k}$

In this section, we use the computations from the previous sections to provide
a diffeomorphism classification of smooth structures on
$\mathbb{C}P^{2}\times\mathbb{S}^{k},$ where $4\leq k\leq 6.$ To do this, we
recall the surgery exact sequence and some related concepts from surgery
theory [25, 30, 10, 11].

######  Definition 5.1 (Relative Smooth Structure Set).

Consider a compact smooth manifold $M$ of dimension $n$ with possibly empty
boundary $\partial M.$ The relative smooth structure set of $M,$

$\mathcal{S}^{Diff}(M~{}\operatorname{rel}~{}\partial
M):=\\{(N,f):f:(N,\partial N)\to(M,\partial M)\\}/{\simeq}$

consists of the equivalence classes of pairs $(N,f),$ where $N$ is a compact
smooth manifold with boundary $\partial N$ and $f:(N,\partial N)\to(M,\partial
M)$ is a homotopy equivalence such that the restircted map $f|_{\partial
N}:\partial N\to\partial M$ is a diffeomorphism. Two pairs $(N_{1},f_{1})$ and
$(N_{2},f_{2})$ are equivalent if there exists a diffeomorphism
$H:(N_{1},\partial N_{1})\to(N_{2},\partial N_{2})$ such that $f_{2}\circ H$
is homotopic to $f_{1}$ rel $\partial N_{1}.$ The equivalence class of $(N,f)$
is denoted by $[(N,f)].$ The base point of
$\mathcal{S}^{Diff}(M\operatorname{rel}\partial M)$ is the equivalence class
of $(M,Id).$

If $\partial M$ is empty, we abbreviate
$\mathcal{S}^{Diff}(M):=\mathcal{S}^{Diff}(M~{}\operatorname{rel}~{}\varnothing).$

The main tool for calculating the smooth structure set
$\mathcal{S}^{Diff}(M~{}\operatorname{rel}~{}\partial M)$ of an
$n$-dimensional manifold $M$ with $n\geq 5$ is the surgery exact sequence

$\cdots\to
L_{n+1}(\pi_{1}(M))\xrightarrow{\omega_{\operatorname{rel}}^{Diff}}\mathcal{S}^{Diff}(M~{}\operatorname{rel}~{}\partial
M)\xrightarrow{\eta_{\operatorname{rel}}^{Diff}}[M/{\partial
M},G/O]\xrightarrow{\Theta_{\operatorname{rel}}^{Diff}}L_{n}(\pi_{1}(M)),$
(23)

Observe that there is a forgetful map
$\mathcal{F}:\mathcal{C}(M)\to\mathcal{S}^{Diff}(M)$ such that the following
diagram commutes.

${{\mathcal{S}^{Diff}(M)}}$${{[M,G/O]}}$${{\mathcal{C}(M)}}$$\scriptstyle{\eta^{Diff}}$$\scriptstyle{\mathcal{F}}$$\scriptstyle{\psi_{*}}$

Recall some facts from [15, 3, 37, 4] related to smooth and tangential
structure sets, which will be used later in this section.

Consider $E_{1}(\mathbb{C}P^{q})$ to be the path-component of the space of all
self maps of $\mathbb{C}P^{q}$ that include $Id_{\mathbb{C}P^{q}},$ and let
$F_{\mathbb{S}^{1}}(\mathbb{C}^{q+1})$ represent the space of all
$\mathbb{S}^{1}$-equivariant self maps of $\mathbb{S}^{2q+1}.$ Additionally,
there is a stabilization map $s_{q+1}:F_{\mathbb{S}^{1}}(\mathbb{C}^{q+1})\to
F_{\mathbb{S}^{1}}(\mathbb{C}^{q+2}),$ and we define $F_{\mathbb{S}^{1}}$ as
the limit of these stabilizations.

######  Fact 5.2.

  * (a)

There is a group homomorphism
$\alpha_{\mathbb{S}^{k}}:\pi_{k}(E_{1}(\mathbb{C}P^{q}))\to\mathcal{E}(\mathbb{C}P^{q}\times\mathbb{S}^{k})$
for all $k,q\geq 1,$ where $\mathcal{E}(X)$ denotes the group of all homotopy
classes of based self homotopy equivalences of $X.$

  * (b)

For $q\geq 1$ and $k\geq 2,$ there is a group homomorphism
$\Psi:\pi_{k}(E_{1}(\mathbb{C}P^{q}))\to\mathcal{S}^{Diff}(\mathbb{C}P^{q}\times\mathbb{D}^{k}~{}\operatorname{rel}~{}\mathbb{C}P^{q}\times\mathbb{S}^{k-1}),$
and a canonical base-point preserving map
$\Gamma:\mathcal{S}^{Diff}(\mathbb{C}P^{q}\times\mathbb{D}^{k}~{}\operatorname{rel}~{}\mathbb{C}P^{q}\times\mathbb{S}^{k-1})\to\mathcal{S}^{Diff}(\mathbb{C}P^{q}\times\mathbb{S}^{k}).$
Moreover, the composition $\Gamma\circ\Psi$ sends the element
$\alpha\in\pi_{k}(E_{1}(\mathbb{C}P^{q}))$ to the element
$[(\mathbb{C}P^{q}\times\mathbb{S}^{k},f)]$ in
$\mathcal{S}^{Diff}(\mathbb{C}P^{q}\times\mathbb{S}^{k}),$ where $f$
represents a tangential self homotopy equivalence of
$\mathbb{C}P^{q}\times\mathbb{S}^{k}$ that comes from $\alpha.$

  * (c)

The homotopy groups $\pi_{k}(F_{\mathbb{S}^{1}}(\mathbb{C}^{q+1}))$ and
$\pi_{k}(E_{1}(\mathbb{C}P^{q}))$ are isomorphic for all $q\geq 1$ and $k\geq
2.$

  * (d)

For all $q\geq 1,$ $\pi_{k}(F_{\mathbb{S}^{1}}(\mathbb{C}^{q+1}))$ is
isomorphic to $\pi_{k}(F_{\mathbb{S}^{1}})$ if $1\leq k\leq 2q$ [4,
Proposition 6.1].

  * (e)

$\pi_{k}(F_{\mathbb{S}^{1}})\cong\pi_{k}^{s}(\Sigma\mathbb{C}P^{\infty})\oplus\pi_{k-1}^{s}$
for $k\geq 1.$

  * (f)

Let $\alpha\in\pi_{k-1}^{s}$ correspond to the element
$f\in\pi_{k}(F_{\mathbb{S}^{1}}).$ Then the normal invariant of $f$ is the
image of the composition
$\mathbb{S}^{k}\wedge\mathbb{C}P_{+}^{\infty}\xrightarrow{\alpha\wedge
Id_{\mathbb{C}P_{+}^{\infty}}}\mathbb{S}^{1}\wedge\mathbb{C}P_{+}^{\infty}\xrightarrow{t}\mathbb{S}^{0}$
under the map
$j_{*}:[\Sigma^{k}\mathbb{C}P_{+}^{\infty},G]\to[\Sigma^{k}\mathbb{C}P_{+}^{\infty},G/O],$
where $t:\mathbb{S}^{1}\wedge\mathbb{C}P_{+}^{\infty}\to\mathbb{S}^{0}$ is the
Umkehr map [3]. Further, we note that $t\circ(\alpha\wedge
Id_{\mathbb{C}P_{+}^{\infty}})=\alpha\circ\Sigma^{k-1}t.$

Let’s turn our attention to the product manifold
$\mathbb{C}P^{2}\times\mathbb{S}^{4}$. By applying the previously mentioned
facts in conjunction with surgery theory, we prove the following.

######  Lemma 5.3.

There exists a tangential homotopy equivalence
$f_{2\nu}:\mathbb{C}P^{2}\times\mathbb{S}^{4}\to\mathbb{C}P^{2}\times\mathbb{S}^{4},$
for which the normal invariant is non-zero. In particular, the normal
invariant of $f_{2\nu}$ equals
$f_{\mathbb{C}P^{2}\times\mathbb{S}^{4}}^{*}([\epsilon]),$ where $[\epsilon]$
represents the generator of $2$-torsion of
$\pi_{8}(G/O)\subseteq[\mathbb{C}P^{2}\times\mathbb{S}^{4},G/O].$

###### Proof.

By Fact 5.2(d) and (e), we get
$\pi_{4}(F_{\mathbb{S}^{1}}(\mathbb{C}^{3}))\cong\pi_{4}(F_{\mathbb{S}^{1}})\cong\mathbb{Z}/{8}\\{\nu\\}\oplus\mathbb{Z}/3\\{\alpha_{1}\\}.$
Let
$f_{2\nu}:\mathbb{C}P^{2}\times\mathbb{S}^{4}\to\mathbb{C}P^{2}\times\mathbb{S}^{4}$
be a self homotopy equivalence that is induced by the element
$2\nu\in\pi_{4}(F_{\mathbb{S}^{1}}(\mathbb{C}^{3})).$ Now it follows from Fact
5.2(f) together with $\pi_{4}^{s}=0$ that the normal invariant
$\eta^{Diff}(f_{2\nu})$ is the image of the composition map
$(2\nu)\circ\Sigma^{3}t:\Sigma^{4}\mathbb{C}P^{2}\to\mathbb{S}^{0}$ in
$[\Sigma^{4}\mathbb{C}P^{2},G/O]\subset[\mathbb{C}P^{2}\times\mathbb{S}^{4},G/O].$
Since
$((2\nu)\circ\Sigma^{3}t)|_{\Sigma^{4}\mathbb{C}P^{1}}:\Sigma^{4}\mathbb{C}P^{1}\to\mathbb{S}^{0}$
is trivial, it suffices to compute the Toda bracket $<\eta,\nu,2\nu>$ to
determine whether the composition
$(2\nu)\circ\Sigma^{3}t:\Sigma^{4}\mathbb{C}P^{2}\to\mathbb{S}^{0}$ is trivial
or not. By [42, Page-189], $<\eta,\nu,2\nu>=\\{\bar{\nu},\epsilon\\}$ which
projects non-trivially to the class $[\epsilon]$ in $\pi_{8}(G/O).$ This
implies that $\eta^{Diff}(f_{2\nu})$ is the image of the class $[\epsilon]$
under the map
$(\Sigma^{4}f_{\mathbb{C}P^{2}})^{*}:\pi_{8}(G/O)\to[\Sigma^{4}\mathbb{C}P^{2},G/O].$
Now the result follows from the following commutative diagram

${{[\Sigma^{4}\mathbb{C}P^{2},G/O]}}$${{\pi_{8}(G/O)}}$${{[\mathbb{C}P^{2}\times\mathbb{S}^{4},G/O]}}$$\scriptstyle{p^{*}}$$\scriptstyle{(\Sigma^{4}f_{\mathbb{C}P^{2}})^{*}}$$\scriptstyle{(f_{\mathbb{C}P^{2}\times\mathbb{S}^{4}})^{*}}$

where the map
$p^{*}:[\Sigma^{4}\mathbb{C}P^{2},G/O]\to[\mathbb{C}P^{2}\times\mathbb{S}^{4},G/O]$
is injective. ∎

######  Theorem 5.4.

Let $N$ be a closed oriented smooth manifold homeomorphic to
$\mathbb{C}P^{2}\times\mathbb{S}^{4}$. Then $N$ is oriented diffeomorphic to
$\mathbb{C}P^{2}\times\mathbb{S}^{4}.$

###### Proof.

Suppose we have a homeomorphism $g:N\to\mathbb{C}P^{2}\times\mathbb{S}^{4}$.
This yields an element
$[(N,g)]\in\mathcal{C}(\mathbb{C}P^{2}\times\mathbb{S}^{4}).$ Consider the
following commutative diagram

${{0}}$${{0}}$${{\Theta_{8}}}$${{\mathcal{C}(\mathbb{C}P^{2}\times\mathbb{S}^{4})\cong[\Sigma^{4}\mathbb{C}P^{2},Top/O]}}$${{\pi_{8}(G/O)}}$${{[\mathbb{C}P^{2}\times\mathbb{S}^{4},G/O]}}$${{[\Sigma^{4}\mathbb{C}P^{2},G/O]}}$$\scriptstyle{(f_{\mathbb{C}P^{2}\times\mathbb{S}^{4}})^{*}}$$\scriptstyle{\cong}$$\scriptstyle{\psi_{*}}$$\scriptstyle{\psi_{*}}$$\scriptstyle{(\Sigma^{4}f_{\mathbb{C}P^{2}})^{*}}$$\scriptstyle{(f_{\mathbb{C}P^{2}\times\mathbb{S}^{4}})^{*}}$$\scriptstyle{p^{*}}$

where the induced degree one map
$f^{*}_{\mathbb{C}P^{2}\times\mathbb{S}^{4}}:\Theta_{8}\to\mathcal{C}(\mathbb{C}P^{2}\times\mathbb{S}^{4})$
is an isomorphism by Corollary 2.3 and Proposition 3.7(3), and the map
$(\Sigma^{4}f_{\mathbb{C}P^{2}})^{*}:\pi_{8}(G/O)\to[\Sigma^{4}\mathbb{C}P^{2},G/O]$
is injective as $\pi_{7}(G/O)=0$. From this diagram, it’s clear that the
normal invariant $\eta^{Diff}(g)$ is either trivial or equal to
$(f_{\mathbb{C}P^{2}\times\mathbb{S}^{4}})^{*}([\epsilon]).$ If
$\eta^{Diff}(g)$ equals
$(f_{\mathbb{C}P^{2}\times\mathbb{S}^{4}})^{*}([\epsilon])$ then by applying
Lemma 5.3 and the composition formula for normal invariants, we find that

$\displaystyle\eta^{Diff}(f_{2\nu}\circ g)=$
$\displaystyle\eta^{Diff}(f_{2\nu})+(f_{2\nu}^{-1})^{*}\eta^{Diff}(g)$
$\displaystyle=$
$\displaystyle(f_{\mathbb{C}P^{2}\times\mathbb{S}^{4}})^{*}([\epsilon])\pm(f_{\mathbb{C}P^{2}\times\mathbb{S}^{4}})^{*}([\epsilon])=0.$

This implies that $(N,f_{2\nu}\circ g)$ and
$(\mathbb{C}P^{2}\times\mathbb{S}^{4},Id)$ represent the same element in
$\mathcal{S}^{Diff}(\mathbb{C}P^{2}\times\mathbb{S}^{4})$ from the surgery
exact sequence (23). This concludes the proof of the theorem. ∎

Next, let’s focus on the normal invariants associated with the smooth
structures of $\mathbb{C}P^{2}\times\mathbb{S}^{5}$. We initiate our
discussion by proving the following:

######  Lemma 5.5.

The map
$\psi_{*}:[\mathbb{C}P^{2}\times\mathbb{S}^{5},Top/O]\to[\mathbb{C}P^{2}\times\mathbb{S}^{5},G/O]$
is trivial.

###### Proof.

Since the short exact sequence (1) splits along both $Top/O$ and $G/O,$ there
is a commutative diagram where each of the three components of
$[\mathbb{C}P^{2}\times\mathbb{S}^{5},Top/O]$ is mapped into the corresponding
component of $[\mathbb{C}P^{2}\times\mathbb{S}^{5},G/O]$ under the induced
homomorphism by the canonical map $\psi:Top/O\to G/O.$ From this diagram, by
using the facts that both $[\Sigma^{5}\mathbb{C}P^{2},G/O]$ and
$[\mathbb{C}P^{2},Top/O]$ are zero, we find that the map
$\psi_{*}:[\mathbb{C}P^{2}\times\mathbb{S}^{5},Top/O]\to[\mathbb{C}P^{2}\times\mathbb{S}^{5},G/O]$
is trivial, as desired. ∎

######  Theorem 5.6.

If a closed smooth oriented manifold $N$ is homeomorphic to
$\mathbb{C}P^{2}\times\mathbb{S}^{5},$ then it is oriented diffeomorphic to
$\mathbb{C}P^{2}\times\mathbb{S}^{5}.$

###### Proof.

Let $g:N\to\mathbb{C}P^{2}\times\mathbb{S}^{5}$ be a homeomorphism. We show
that the pairs $(N,g)$ and $(\mathbb{C}P^{2}\times\mathbb{S}^{5},Id)$
represent the same element in
$\mathcal{S}^{Diff}(\mathbb{C}P^{2}\times\mathbb{S}^{5}).$ By applying Lemma
5.5, we find that the normal invariant $\eta^{Diff}(g)=0.$ Now it follows from
the smooth surgery exact sequence (23) for
$\mathbb{C}P^{2}\times\mathbb{S}^{5}$ that there exists $\Sigma\in
bP_{10}=\mathbb{Z}/2$ such that $(N,g)$ and
$(\mathbb{C}P^{2}\times\mathbb{S}^{5}\\#\Sigma,h)$ are equivalent in
$\mathcal{S}^{Diff}(\mathbb{C}P^{2}\times\mathbb{S}^{5}),$ where
$h:(\mathbb{C}P^{2}\times\mathbb{S}^{5})\\#\Sigma\to\mathbb{C}P^{2}\times\mathbb{S}^{5}$
is the canonical homeomorphism. We claim that $\Sigma$ is (oriented)
diffeomorphic to the standard $9$-sphere, which is equivalent to prove that
$\Theta^{Diff}:[\Sigma(\mathbb{C}P^{2}\times\mathbb{S}^{5}),G/O]\to L_{10}(e)$
is non-zero.

Consider the following commutative diagram

${{\mathbb{Z}/2\\{\nu^{2}\\}\cong\pi_{6}(G/O)}}$${{[\Sigma(\mathbb{C}P^{2}\times\mathbb{S}^{5}),G/O]}}$${{[\mathbb{C}P^{2}\times\mathbb{S}^{5}\times\mathbb{S}^{1},G/O]}}$${{L_{10}(e)}}$${{L_{10}(\mathbb{Z})}}$${{\mathcal{S}^{Diff}(\mathbb{C}P^{2}\times\mathbb{S}^{5})}}$$\scriptstyle{(Pr)^{*}}$$\scriptstyle{p^{*}}$$\scriptstyle{\Theta^{Diff}}$$\scriptstyle{\Theta^{Diff}}$$\scriptstyle{\omega^{diff}}$$\scriptstyle{\cong}$

where the map
$p^{*}:[\Sigma(\mathbb{C}P^{2}\times\mathbb{S}^{5}),G/O]\to[\mathbb{C}P^{2}\times\mathbb{S}^{5}\times\mathbb{S}^{1},G/O]$
is injective by (1), the induced map
$(Pr)^{*}:\pi_{6}(G/O)\to[\Sigma(\mathbb{C}P^{2}\times\mathbb{S}^{5}),G/O]$
from the projection map
$Pr:\Sigma(\mathbb{C}P^{2}\times\mathbb{S}^{5})\simeq\Sigma\mathbb{C}P^{2}\vee\Sigma\mathbb{S}^{5}\vee\Sigma^{6}\mathbb{C}P^{2}\to\Sigma\mathbb{S}^{5}$
is injective, and $L_{10}(e)\to L_{10}(\mathbb{Z})$ is the isomorphism given
in [39]. Using the above commutative diagram, to show the non-triviality of
$\Theta^{Diff}:[\Sigma(\mathbb{C}P^{2}\times\mathbb{S}^{5}),G/O]\to
L_{10}(e),$ it is enough to prove that the composition $\Theta^{Diff}\circ
p^{*}\circ(Pr)^{*}:\pi_{6}(G/O)\to L_{10}(\mathbb{Z})$ is non-trivial. Let
$f:\mathbb{C}P^{2}\times\mathbb{S}^{5}\times\mathbb{S}^{1}\to G/O$ be the
image of $\nu^{2}\in\pi_{6}(G/O)$ under the induced map $p^{*}\circ
Pr^{*}:\pi_{6}(G/O)\to[\mathbb{C}P^{2}\times\mathbb{S}^{5}\times\mathbb{S}^{1},G/O].$

Then, the surgery obstruction of $f$ is given by [33]

$\displaystyle\eta^{Diff}(f_{2\nu}\circ g)=$ $\displaystyle\langle
V(\mathbb{C}P^{2}\times\mathbb{S}^{5}\times\mathbb{S}^{1})^{2}f^{*}(\mathcal{K}_{6}),\;[\mathbb{C}P^{2}\times\mathbb{S}^{5}\times\mathbb{S}^{1}]\rangle$
$\displaystyle=$
$\displaystyle\langle(1+x^{2})f^{*}(\mathcal{K}_{6}),\;[\mathbb{C}P^{2}\times\mathbb{S}^{5}\times\mathbb{S}^{1}]\rangle$
$\displaystyle=$ $\displaystyle\langle
x^{2}f^{*}(\mathcal{K}_{6}),\;[\mathbb{C}P^{2}\times\mathbb{S}^{5}\times\mathbb{S}^{1}]\rangle,$
(24)

where
$(V(\mathbb{C}P^{2}\times\mathbb{S}^{5}\times\mathbb{S}^{1}))^{2}=1+x^{2}$ is
the square of total Wu class of the manifold
$\mathbb{C}P^{2}\times\mathbb{S}^{5}\times\mathbb{S}^{1},x$ is the generator
of $H^{2}(\mathbb{C}P^{2};\mathbb{Z}/2)$ and $\mathcal{K}_{6}\in
H^{6}(G/O;\mathbb{Z}/2)$ is the smooth Kervaire class. Observe that
$f^{*}(\mathcal{K}_{6})=p^{*}\circ(Pr)^{*}\circ(\nu^{2})^{*}(\mathcal{K}_{6})\neq
0,$ as $(\nu^{2})^{*}(\mathcal{K}_{6})\neq 0.$ Now it follows from (5) that
$\Theta^{Diff}(f)\neq 0$ in $L_{10}(e).$ This completes the proof. ∎

We now proceed to give the classification of smooth structures of
$\mathbb{C}P^{2}\times\mathbb{S}^{6}.$ Recall that if $M$ is a closed smooth
$n$-manifold and $k\geq 1,$ it is well known that the relative tangential
structure set

$\mathcal{S}_{k}^{Diff}(M):=\mathcal{S}^{Diff}(M\times\mathbb{D}^{k}~{}\text{rel}~{}M\times\mathbb{S}^{k-1})$

has a group structure [29, 32, 6, 4]. Moreover, when $n+k\geq 5$, a relative
surgery exact sequence of groups exists [37, Page-195], which can be expressed
as

$\cdots\to
L_{n+k+1}(\pi_{1}(M))\xrightarrow{\omega_{\operatorname{rel}}}\mathcal{S}_{k}^{Diff}(M)\xrightarrow{\eta_{\operatorname{rel}}}[\Sigma^{k}M_{+},G/O]\xrightarrow{\Theta_{\operatorname{rel}}}L_{n+k}(\pi_{1}(M)),$
(25)

where
$\eta_{\operatorname{rel}}:\mathcal{S}_{k}^{Diff}(M)\to[\Sigma^{k}M_{+},G/O]$
and $\Theta_{\operatorname{rel}}:[\Sigma^{k}M_{+},G/O]\to L_{n+k}(\pi_{1}(M))$
are group homomorphisms.
Let’s also revisit some notations. The group of homotopy classes of based
self-homotopy equivalences is denoted by $\mathcal{E}(M\times N)$. Now,
consider the subgroup $\mathcal{E}_{M}(M\times N)$, whose elements are
represented by maps $f:X\times Y\to X\times Y$ that fix $X$. In other words,
these maps take the form $f=(p_{M},p_{N}\circ f),$ where $p_{M}$ and $p_{N}$
are the projection maps from $M\times N$ to $M$ and $N$ respectively.
Similarly, we can define the subgroup $\mathcal{E}_{N}(M\times N).$ We are now
going to examine the product manifold $\mathbb{C}P^{2}\times\mathbb{S}^{6}$
and establish the following.

######  Theorem 5.7.

Let $f\in\mathcal{E}(\mathbb{C}P^{2}\times\mathbb{S}^{6})$. Then $f$ is
homotopic to a diffeomorphism.

To prove the Theorem 5.7, we require the following lemma.

######  Lemma 5.8.

Every self homotopy equivalence of $\mathbb{C}P^{2}\times\mathbb{S}^{6}$
induced by an element of $\pi_{6}(F_{\mathbb{S}^{1}}(\mathbb{C}^{3}))$ is
homotopic to a diffeomorphism.

###### Proof.

By Fact 5.2 (b) and (c), it suffices to show that the homomorphism
$\Psi:\pi_{6}(F_{\mathbb{S}^{1}}(\mathbb{C}^{3}))\to\mathcal{S}_{6}^{Diff}(\mathbb{C}P^{2})$
is trivial. Consider the following commutative diagram [37, Theorem 3.9]

${{0}}$${{[\Sigma^{6}\mathbb{C}P_{+}^{2},PL]}}$${{[\Sigma^{6}\mathbb{C}P_{+}^{2},G]}}$${{[\Sigma^{6}\mathbb{C}P_{+}^{2},G/PL]}}$${{0}}$${{\mathcal{S}_{6}^{Diff}(\mathbb{C}P^{2})}}$${{[\Sigma^{6}\mathbb{C}P_{+}^{2},G/O]}}$${{L_{10}(e)}}$$\scriptstyle{H}$$\scriptstyle{\beta}$$\scriptstyle{j_{*}}$$\scriptstyle{\sigma}$$\scriptstyle{\eta_{\operatorname{rel}}}$$\scriptstyle{\Theta_{\operatorname{rel}}}$

where the first row is the exact sequence induced from the fibre sequence
$\cdots\to\Omega(G/PL)\to PL\to G\to G/PL$ and the second row corresponds to
the relative version of surgery exact sequence (25). Observe that the
restricted maps $\sigma|_{\pi_{6}(G/PL)}:\pi_{6}(G/PL)\to L_{10}(e)$,
$j_{*}|_{\pi_{6}^{s}}:\pi_{6}^{s}\to\pi_{6}(G/O)$ and
$\beta|_{\pi_{6}^{s}}:\pi_{6}^{s}\to[\mathbb{S}^{6},G/PL]\subset[\Sigma^{6}\mathbb{C}P_{+}^{2},G/PL]$
are all isomorphisms. Therefore, from the above diagram, the surgery
obstruction map $\Theta_{\operatorname{rel}}|_{\pi_{6}(G/O)}:\pi_{6}(G/O)\to
L_{10}(e)$ is non-zero, and the image of the map
$\eta_{\operatorname{rel}}:\mathcal{S}_{6}^{Diff}(\mathbb{C}P^{2})\to[\Sigma^{6}\mathbb{C}P_{+}^{2},G/O]$
is contained in the component $[\Sigma^{6}\mathbb{C}P^{2},G/O].$ Hence, by
using the fact that $\pi_{6}(F_{\mathbb{S}^{1}}(\mathbb{C}^{3}))$ is finite
group [4, Page-29], we deduce that the image of the composition
$\eta_{\operatorname{rel}}\circ\Psi:\pi_{6}(F_{\mathbb{S}^{1}}(\mathbb{C}^{3}))\to[\Sigma^{6}\mathbb{C}P^{2},G/O]$
is contained in the torsion subgroup of $[\Sigma^{6}\mathbb{C}P^{2},G/O]$,
which is equal to $\mathbb{Z}_{3}$. However, it follows from [36] that the
composition
$\eta_{\operatorname{rel}}\circ\Psi:\pi_{6}(F_{\mathbb{S}^{1}}(\mathbb{C}^{3}))\to[\Sigma^{6}\mathbb{C}P^{2},G/O]$
fits into the following commutative diagram

${{\pi_{6}(F_{\mathbb{S}^{1}}(\mathbb{C}^{3}))}}$${{[\Sigma^{6}\mathbb{C}P_{+}^{2},G/O]}}$${{\pi_{6}(F_{\mathbb{S}^{1}}(\mathbb{C}^{4}))\cong\pi_{6}(F_{\mathbb{S}^{1}})}}$${{[\Sigma^{6}\mathbb{C}P_{+}^{3},G/O]}}$$\scriptstyle{\eta_{\operatorname{rel}}\circ\Psi}$$\scriptstyle{s_{3}}$$\scriptstyle{\eta_{\operatorname{rel}}\circ\Psi}$$\scriptstyle{(\Sigma^{6}i\vee
id)^{*}}$

where
$s_{3}:\pi_{6}(F_{\mathbb{S}^{1}}(\mathbb{C}^{3}))\to\pi_{6}(F_{\mathbb{S}^{1}})$
is the stabilization map and the right vertical arrow is induced from the
inculsion
$\Sigma^{6}i:\Sigma^{6}\mathbb{C}P^{2}\hookrightarrow\Sigma^{6}\mathbb{C}P^{3}$
along $G/O.$ Since $\pi_{6}(F_{\mathbb{S}^{1}})\cong\mathbb{Z}/2$ [27], the
diagram implies that the image of
$\eta_{\operatorname{rel}}\circ\Psi:\pi_{6}(F_{\mathbb{S}^{1}}(\mathbb{C}^{3}))\to[\Sigma^{6}\mathbb{C}P_{+}^{2},G/O]$
is trivial. Now, by using the injectivity of the map
$\mathcal{S}_{6}^{Diff}(\mathbb{C}P^{2})\to[\Sigma^{6}\mathbb{C}P_{+}^{2},G/O]$,
we conclude that the homomorphism
$\Psi:\pi_{6}(F_{\mathbb{S}^{1}}(\mathbb{C}^{3}))\to\mathcal{S}_{6}^{Diff}(\mathbb{C}P^{2})$
is trivial. This completes the proof. ∎

###### Proof. of Theorem 5.7.

We begin by observing that for any self homotopy equivalence
$g:\mathbb{C}P^{2}\times\mathbb{S}^{6}\to\mathbb{C}P^{2}\times\mathbb{S}^{6},$
the composition map $p_{X}\circ g\circ i_{X}:X\to X$ is a homotopy equivalence
for $X\in\\{\mathbb{C}P^{2},\mathbb{S}^{6}\\}$, where
$i_{X}:X\hookrightarrow\mathbb{C}P^{2}\times\mathbb{S}^{6}$ is the inclusion.
Now applying [28, Theorem 2.5], there is a decomposition

$\mathcal{E}(\mathbb{C}P^{2}\times\mathbb{S}^{6})=\mathcal{E}_{\mathbb{C}P^{2}}(\mathbb{C}P^{2}\times\mathbb{S}^{6})\cdot\mathcal{E}_{\mathbb{S}^{6}}(\mathbb{C}P^{2}\times\mathbb{S}^{6}).$
(26)

Further, from [28, Proposition 2.3]
$\mathcal{E}_{\mathbb{C}P^{2}}(\mathbb{C}P^{2}\times\mathbb{S}^{6})$ and
$\mathcal{E}_{\mathbb{S}^{6}}(\mathbb{C}P^{2}\times\mathbb{S}^{6})$ fit into
the following two split short exact sequences

$0\to[\mathbb{C}P^{2},E_{1}(\mathbb{S}^{6})]\to\mathcal{E}_{\mathbb{C}P^{2}}(\mathbb{C}P^{2}\times\mathbb{S}^{6})\to\mathcal{E}(\mathbb{S}^{6})\to
0$

and

$0\to\pi_{6}(E_{1}(\mathbb{C}P^{2}))\to\mathcal{E}_{\mathbb{S}^{6}}(\mathbb{C}P^{2}\times\mathbb{S}^{6})\to\mathcal{E}(\mathcal{C}P^{2})\to
0,$

where
$[\mathbb{C}P^{2},E_{1}(\mathbb{S}^{6})]\cong[\mathbb{C}P^{2},SG_{7}]\cong 0$.
We also observe that any self homotopy equivalence of
$\mathbb{C}P^{2}\times\mathbb{S}^{6}$ induced from the factor
$\mathcal{E}(\mathbb{S}^{6})$ or $\mathcal{E}(\mathbb{C}P^{2})$ is represented
by a diffeomorphism. Furthermore, Lemma 5.8 assures the same for the self
homotopy equivalences corresponding to elements of
$\pi_{6}(E_{1}(\mathbb{C}P^{2}))\cong\pi_{6}(F_{\mathbb{S}^{1}}(\mathbb{C}^{3})).$
By combining these in the decomposition (26) and using the composition formula
for normal invariants, we deduce that the normal invariant of any self
homotopy equivalence of $\mathbb{C}P^{2}\times\mathbb{S}^{6}$ is zero. This
concludes the proof of Theorem 5.7. ∎

Recall that the homotopy inertia group $I_{h}(M)$ of a manifold $M,$ is
defined as the collection of all $\Sigma\in I(M)$ for which a diffeomorphism
$M\\#\Sigma\to M$ exists that is homotopic to the standard homeomorphism
$h_{\Sigma}:M\\#\Sigma\to M.$ Clearly, $I_{c}(M)\subseteq I_{h}(M).$ The next
lemma confirms that they are indeed equal for the product manifold
$\mathbb{C}P^{q}\times\mathbb{S}^{2k}.$

######  Lemma 5.9.

For every $q,k\geq 1$, we have
$I_{h}(\mathbb{C}P^{q}\times\mathbb{S}^{2k})=I_{c}(\mathbb{C}P^{q}\times\mathbb{S}^{2k})$.

###### Proof.

This follows quickly from [16, Corollary 3.2]. ∎

######  Theorem 5.10.

The inertia group of $\mathbb{C}P^{2}\times\mathbb{S}^{6}$ is $\mathbb{Z}/2.$

###### Proof.

It follows from Theorem 3.5 and Lemma 5.9 that
$I_{h}(\mathbb{C}P^{2}\times\mathbb{S}^{6})=\mathbb{Z}/2$. Hence the
conclusion immediately obtained from Theorem 5.7. ∎

######  Theorem 5.11.

If $N$ if a closed oriented smooth manifold that is homeomorphic to
$\mathbb{C}P^{2}\times\mathbb{S}^{6},$ then $N$ is oriented diffeomorphic to
exactly one of the manifolds
$\mathbb{C}P^{2}\times\mathbb{S}^{6},(\mathbb{C}P^{2}\times\mathbb{S}^{6})\\#\Sigma_{\alpha_{1}}$,
or $(\mathbb{C}P^{2}\times\mathbb{S}^{6})\\#(\Sigma_{\alpha_{1}})^{-1},$ where
$\Sigma_{\alpha_{1}}\in\Theta_{10}$ is the exotic sphere of order $3.$

###### Proof.

It follows from the decomposition (1) and the proof of Proposition 3.7(5) that

$[\mathbb{C}P^{2}\times\mathbb{S}^{6},Top/O]\cong[\Sigma^{6}\mathbb{C}P^{2},Top/O]\cong\mathbb{Z}/3$

and the induced degree one map
$f^{*}_{\mathbb{C}P^{2}\times\mathbb{S}^{6}}:\Theta_{10}\to[\mathbb{C}P^{2}\times\mathbb{S}^{6},Top/O]$
is onto. This implies that the concordance structure set
$\mathcal{C}(\mathbb{C}P^{2}\times\mathbb{S}^{6})$ is equal to

$\\{\mathbb{C}P^{2}\times\mathbb{S}^{6}\\#\Sigma~{}|~{}\Sigma\in\mathbb{Z}/3\subset\Theta_{10}\\}.$

The theorem now follows from Theorem 5.10. ∎

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