Smooth structures on $\mathbb{C}P^{m}$ for $5\leq m\leq 8$

TL;DR

This paper classifies smooth manifolds homeomorphic to complex projective spaces for dimensions 5 through 8, analyzing their diffeomorphism types and tangential structures.

Contribution

It provides a classification of smooth structures on $ ext{CP}^m$ for $m=5,6,7,8$, including computations of tangential structure sets and existence of exotic manifolds.

Findings

Classified all smooth manifolds homeomorphic to $ ext{CP}^m$ for $m=5,6,7,8$.

Computed the smooth tangential structure set for $ ext{CP}^7$ and $ ext{CP}^8$.

Demonstrated existence of a tangentially homotopy equivalent but non-homeomorphic manifold to $ ext{CP}^8$.

Abstract

We classify up to diffeomorphism all smooth manifolds homeomorphic to the complex projective m-space $\mathbb{C}P^{m}$ for $m = 5, 6, 7$ and $8$. As an application, for $m = 7$ and $8$, we compute the smooth tangential structure set of $\mathbb{C}P^{m}$ and obtain a bound on the number of smooth homotopy complex projective m-spaces with given Pontryagin classes up to orientation-preserving diffeomorphism. We also show that there exists a smooth manifold which is tangentially homotopy equivalent but not homeomorphic to $\mathbb{C}P^{8}$.

Equations

$$\mathcal{C}(M)\cong [M,Top/O]$$

$$f_M^{\ast }:[S^m,Top/O]\to [M^m,Top/O],$$

$$\left([{\mathbb{S}}^m,Top/O]\stackrel{f_M^{\ast }}{⟶}[M,Top/O]\right)$$

$${\varphi }_{\ast }:[\mathbb{C}{P}^m,SF]\to [\mathbb{C}{P}^m,F/O]$$

$${\psi }_{\ast }:[\mathbb{C}{P}^m,Top/O]\to [\mathbb{C}{P}^m,F/O]$$

$$[\mathbb{C}{P}^{m-1},F/O]\cong {\mathbb{Z}}^{[\frac{m-1}{2}]}\oplus [\mathbb{C}{P}^{m-1},SF],$$

$${\mathbb{Z}}^{[\frac{m-1}{2}]}\subset \mathrm{Image}\left([\mathbb{C}{P}^m,F/O]\stackrel{i^{\ast }}{⟶}[\mathbb{C}{P}^{m-1},F/O]\right)$$

$$[\mathbb{C}{P}^5,SF]\cong {\mathbb{Z}}_2\oplus {\mathbb{Z}}_2\oplus {\mathbb{Z}}_3\mathrm{and}[\mathbb{C}{P}^6,SF]\cong {\mathbb{Z}}_2\oplus {\mathbb{Z}}_3.$$

$$\sigma :[\mathbb{C}{P}^m,F/O]\to {\Theta }_{2m+1}$$

$$[\mathbb{C}{P}^m,F/O]\stackrel{\sigma }{⟶}{\Theta }_{2m+1}\stackrel{{\psi }_{\ast }}{⟶}{\pi }_{2m+1}(F/O)\cong {\pi }_{2m+1}^s/\mathrm{Im}\mathrm{J}=\mathrm{Coker}(J_{2m+1})$$

$$[\mathbb{C}{P}^m,F/O]\stackrel{p^{\ast }}{⟶}[{\mathbb{S}}^{2m+1},F/O]$$

$$\sigma :[\mathbb{C}{P}^m,Top/O]\to {\Theta }_{2m+1}$$

$$[\mathbb{C}{P}^m,Top/O]\stackrel{p^{\ast }}{⟶}[{\mathbb{S}}^{2m+1},Top/O]$$

$$\sigma ([\mathbb{C}{P}^m,SF])\subseteq \mathrm{Coker}(J_{2m+1})\subset {\Theta }_{2m+1}=b{P}_{2m+2}\oplus \mathrm{Coker}(J_{2m+1}).$$

$${\mathbb{S}}^{2m-1}\stackrel{p}{⟶}\mathbb{C}{P}^{m-1}\stackrel{i}{⟶}\mathbb{C}{P}^m\stackrel{f_{\mathbb{C}{P}^m}}{⟶}{\mathbb{S}}^{2m}:$$

$$\cdots ⟶[{\mathbb{S}}^{2m},Top/O]\stackrel{f_{\mathbb{C}{P}^m}^{\ast }}{⟶}[\mathbb{C}{P}^m,Top/O]\stackrel{i^{\ast }}{⟶}[\mathbb{C}{P}^{m-1},Top/O]\stackrel{p^{\ast }}{⟶}[{\mathbb{S}}^{2m-1},Top/O],$$

$$[{\Sigma }^{2m}]\mapsto [\mathbb{C}{P}^m\#{\Sigma }^{2m}].$$

$$\left([{\mathbb{S}}^{2m},Top/O]\stackrel{f_{\mathbb{C}{P}^m}^{\ast }}{⟶}[\mathbb{C}{P}^m,Top/O]\right)$$

$$0\to {\Theta }_{10}\to \mathcal{C}(\mathbb{C}{P}^5)\to \mathcal{C}(\mathbb{C}{P}^4)\to 0,$$

$$0\to \mathcal{C}(\mathbb{C}{P}^6)\to \mathcal{C}(\mathbb{C}{P}^5)\to \mathrm{Im}({\mathrm{p}}^{\ast })\to 0,$$

$$0\to {\Theta }_{14}\to \mathcal{C}(\mathbb{C}{P}^7)\to \mathrm{Ker}({\mathrm{p}}^{\ast })\to 0,$$

$$g:\mathbb{C}{P}^4\stackrel{f_{\mathbb{C}{P}^4}}{⟶}{\mathbb{S}}^8\stackrel{\Sigma }{⟶}Top/O,$$

$${\mathbb{S}}^9\stackrel{p}{⟶}\mathbb{C}{P}^4\stackrel{f_{\mathbb{C}{P}^4}}{⟶}{\mathbb{S}}^8\stackrel{\Sigma }{⟶}Top/O.$$

$$0\to {\Theta }_{10}\to [\mathbb{C}{P}^5,Top/O]\to [\mathbb{C}{P}^4,Top/O]\to 0,$$

$$i^{\ast }:[\mathbb{C}{P}^6,Top/O]⟶[\mathbb{C}{P}^5,Top/O]$$

$${\psi }_{\ast }:[\mathbb{C}{P}^6,Top/O]⟶[\mathbb{C}{P}^6,F/O]\cong {\mathbb{Z}}^3\oplus {\mathbb{Z}}_2\oplus {\mathbb{Z}}_3$$

$$i^{\ast }:[\mathbb{C}{P}^6,Top/O]⟶[\mathbb{C}{P}^5,Top/O]$$

$$0\to [\mathbb{C}{P}^6,Top/O]\to [\mathbb{C}{P}^5,Top/O]\to \mathrm{Im}({\mathrm{p}}^{\ast })={\mathbb{Z}}_2\to 0,$$

$$\begin{array}{ccccccc}0000& & & & & & \\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ {\mathbb{Z}}_2\cong [{\mathbb{S}}^{14},Top/O]& \stackrel{f_{\mathbb{C}{P}^7}}{\to }& [\mathbb{C}{P}^7,Top/O]& \stackrel{i^{\ast }}{\to }& [\mathbb{C}{P}^6,Top/O]& \stackrel{p^{\ast }}{\to }& [{\mathbb{S}}^{13},Top/O]\cong {\mathbb{Z}}_3\\ \downarrow {\psi }_{\ast }& & \downarrow {\psi }_{\ast }& & \downarrow {\psi }_{\ast }& & \cong \downarrow {\psi }_{\ast }\\ {\mathbb{Z}}_2\oplus {\mathbb{Z}}_2\cong [{\mathbb{S}}^{14},F/O]& \stackrel{f_{\mathbb{C}{P}^7}}{\to }& [\mathbb{C}{P}^7,F/O]& \stackrel{i^{\ast }}{\to }& [\mathbb{C}{P}^6,F/O]& \stackrel{p^{\ast }}{\to }& [{\mathbb{S}}^{13},F/O]\\ \cong \uparrow {\varphi }_{\ast }& & \uparrow {\varphi }_{\ast }& & \uparrow {\varphi }_{\ast }& & \cong \uparrow {\varphi }_{\ast }\\ {\mathbb{Z}}_2\oplus {\mathbb{Z}}_2\cong [{\mathbb{S}}^{14},SF]& \stackrel{f_{\mathbb{C}{P}^7}}{\to }& [\mathbb{C}{P}^7,SF]& \stackrel{i^{\ast }}{\to }& [\mathbb{C}{P}^6,SF]& \stackrel{p^{\ast }}{\to }& [{\mathbb{S}}^{13},SF]\\ \uparrow & & \uparrow & & \uparrow & & \uparrow \\ 0000& & & & & & \end{array}$$

Full text

1. Introduction

A smooth homotopy complex projective $m$-space is a closed smooth manifold homotopy equivalent to the complex projective $m$-space $\mathbb{C}P^{m}$ . The study of smooth homotopy complex projective spaces has been done by means of classifying differentiable free $\mathbb{S}^{1}$-actions on homotopy spheres [10, 11, 19, 20]. D. Sullivan [31] later classified PL homotopy complex projective spaces as an application of his characteristic variety theorem. The surgery theoretic diffeomorphism classification of smooth homotopy complex projective $m$-spaces for $3\leq m\leq 6$ was given by Brumfiel [5, 7]. For $m=3$ and $4$, the diffeomorphism classification of smooth manifolds homeomorphic to $\mathbb{C}P^{m}$ was obtained in [24] (see [19, 10] for the case $m=3$).

In this paper we classify, up to (unoriented) diffeomorphism and orientation-preserving diffeomorphism, all smooth manifolds homeomorphic to $\mathbb{C}P^{m}$ for $m=5,6,7$ and $8$ (see Theorems 3.3 and 3.5) by computing the group $\mathcal{C}(\mathbb{C}P^{m})$ of concordance classes of smooth structures on $\mathbb{C}P^{m}$ (see Definition 2.3 and Theorem 2.7). For $m=7$ and $8$, we compute the smooth tangential structure set of $\mathbb{C}P^{m}$ by using the tangential surgery exact sequence (see Theorem 4.2) and show that the number of smooth homotopy complex projective $m$-spaces with given Pontryagin classes up to orientation-preserving diffeomorphism does not exceed eight (see Theorem 4.4). We also show that, up to orientation-preserving diffeomorphism, there are at most four smooth manifolds which are tangentially homotopy equivalent to $\mathbb{C}P^{8}$ but not homeomorphic to $\mathbb{C}P^{8}$ (see Corollary 4.7(ii)).

Organisation of the paper: In Section 2, we introduce some preliminaries from [5, 7] and compute the group $\mathcal{C}(\mathbb{C}P^{m})$ using the computation of the stable homotopy groups of spheres ([32], [27]). Classification results are discussed in Section 3. The classifications are achieved by computing the action of self-homeomorphisms on $\mathcal{C}(\mathbb{C}P^{m})$. Finally, in Section 4, we recall the tangential surgery exact sequence ([18], [2, §8]) and study some results regarding the classification of smooth homotopy complex projective $m$-spaces with the same Pontryagin classes as $\mathbb{C}\mathbb{P}^{m}$ up to orientation-preserving diffeomorphism.

Notation: Denote by $O=\underset{n\to\infty}{\mathit{colim}}~{}O_{n}$, $Top=\underset{n\to\infty}{\mathit{colim}}~{}Top_{n}$, $F=\underset{n\to\infty}{\mathit{colim}}~{}F_{n}$ and $SF=\underset{n\to\infty}{\mathit{colim}}~{}SF_{n}$ ([13, 16]), the direct limit of the groups of orthogonal transformations, self-homeomorphisms of $\mathbb{R}^{n}$ preserving the origin, base point preserving self-homotopy equivalences of $\mathbb{S}^{n}$, and base point preserving self-homotopy equivalences of degree one of $\mathbb{S}^{n}$ respectively. Let $F/O$ be the homotopy fibre of the canonical map $BO\to BF$ between the classifying spaces for stable vector bundles and stable spherical fibrations (see [33, Lemma 10.6], [17, §2, §3]) and $Top/O$ be the homotopy fibre of the canonical map $BO\to BTop$ between the classifying spaces for stable vector bundles and stable topological $\mathbb{R}^{n}$-bundles (see [15, Theorem 10.1 Essay IV]). If $A$ is an abelian group, then $A_{(2)}$ denotes its localization at the prime $2$; that is, $A_{(2)}=A\otimes\mathbb{Z}[\frac{1}{3},\frac{1}{5},\cdots]$. We will use ${\rm Im(f)}$ and ${\rm Ker(f)}$ to denote the image and kernel of the group homomorphism $f$, and let $p_{i}(\xi)$ denote the $i^{th}$ (integral) Pontryagin class of the vector bundle $\xi$ over a smooth manifold. From now on all manifolds will be oriented and connected, and all homeomorphisms and diffeomorphisms are assumed to preserve orientation, unless otherwise stated.

2. Group of Concordance Classes of Smooth Structures

We recall some terminology from [14]:

Definition 2.1**.**

• (a)

A homotopy $m$-sphere $\Sigma^{m}$ is a closed smooth manifold homotopy equivalent 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 $\mathbb{S}^{m}$.

Definition 2.2**.**

Define the $m$-th group of smooth homotopy spheres $\Theta_{m}$ as follows.

Elements are oriented $h$-cobordism classes $[\Sigma]$ of homotopy $m$-spheres $\Sigma$, where $\Sigma$ and $\Sigma^{\prime}$ are called oriented $h$-cobordant if there is an oriented $h$-cobordism $(W,\partial_{0}W,\partial_{1}W)$ together with orientation-preserving diffeomorphisms $\Sigma\to\partial_{0}W$ and $(-\Sigma^{\prime})\to\partial_{1}W$. (Here $(-\Sigma^{\prime})$ is obtained from $\Sigma^{\prime}$ by reversing the orientation.) The addition is given by the connected sum. The zero element is represented by $[\mathbb{S}^{m}]$ and the inverse of $[\Sigma]$ is given by $[-\Sigma]$. M. Kervaire and J. Milnor [14] showed that each $\Theta_{m}$ is a finite abelian group $(m\geq 1)$; in particular, $\Theta_{m}\cong\mathbb{Z}_{2}$, where $m=8,14,16$, and $\Theta_{10}\cong\mathbb{Z}_{6}$. For $m\geq 5$, the $h$-cobordism theorem ([30]) implies that the group $\Theta_{m}$ is the same as the set of all oriented diffeomorphism classes of smooth structures on $\mathbb{S}^{m}$.

Definition 2.3**.**

Let $M$ be a smooth manifold. Let $(N,f)$ be a pair consisting of a smooth manifold $N$ together with a homeomorphism $f:N\to M$. Two such pairs $(N_{1},f_{1})$ and $(N_{2},f_{2})$ are concordant provided there exists a diffeomorphism $g:N_{1}\to N_{2}$ such that the composition $f_{2}\circ g$ is topologically concordant to $f_{1}$, i.e., there exists a homeomorphism $F:N_{1}\times[0,1]\to M\times[0,1]$ such that $F_{|N_{1}\times 0}=f_{1}$ and $F_{|N_{1}\times 1}=f_{2}\circ g$. The set of all such concordance classes is denoted by $\mathcal{C}(M)$. The concordance class of $(N,f)$ is denoted by $[N,f]$ and the class $[M,Id]$ of the identity $Id:M\to M$ can be considered as the base point of $\mathcal{C}(M)$.

Observe that there is a homeomorphism $h:M^{m}\#\Sigma^{m}\to M^{m}$ $(m\geq 5)$ which is the inclusion map outside of homotopy sphere $\Sigma^{m}$ and well defined up to topological concordance. We denote the class in $\mathcal{C}(M)$ of $(M^{m}\#\Sigma^{m},h)$ by $[M^{m}\#\Sigma^{m}]$. (Note that $[M^{m}\#\mathbb{S}^{m}]$ is the class of $(M^{m},Id)$.)

The key to analyzing $\mathcal{C}(M)$ is the following result.

Theorem 2.4**.**

(Kirby and Siebenmann, [15, p.194])* Let $M$ be a smooth manifold of dimension at least 5, then there is a bijection*

[TABLE]

which takes the base point $[M,Id]$ to the homotopy class of the constant map.

Let $f_{M}:M^{m}\to S^{m}$ be a degree one map. Note that $f_{M}$ is well-defined up to homotopy. Composition with $f_{M}$ defines a homomorphism

[TABLE]

and in terms of the identifications

$\Theta_{m}=[S^{m},Top/O]$ and $\mathcal{C}(M^{m})=[M^{m},Top/O]$

given by Theorem 2.4, $f_{M}^{*}$ becomes $[\Sigma^{m}]\mapsto[M^{m}\#\Sigma^{m}]$. Note that the kernel

[TABLE]

can be identified with a subgroup of $\Theta_{m}$, called the concordance inertia group of $M$, consisting of those $\Sigma\in\Theta_{m}$ such that $(M,Id)$ and $(M\#\Sigma,Id)$ are concordant.

We recall some facts about complex projective spaces which will be used later. There are $H$-spaces $SF$, $F/O$ and $H$-space maps $\phi:SF\to F/O$, $\psi:Top/O\to F/O$ ([1]) such that

[TABLE]

is a monomorphism for all $m\geq 1$ ([5, Corollary 2.1 ] or [28, Lemma 2.7]) and

[TABLE]

is also a monomorphism for all $m\geq 1$ ([28, Lemma 2.6]). Since Brumfiel ([5, p.400], [7, p.12, Corollary 2.3]) has shown that there is a splitting

[TABLE]

where

[TABLE]

and the torsion subgroup of $[\mathbb{C}P^{m},F/O]$ is identified with $[\mathbb{C}P^{m},SF]$. Computations of the groups $[\mathbb{C}P^{m},SF]$ have been made by Brumfiel [5, Table 8.1, p.56] for $m\leq 6$. In particular, for $m=5$ and $6$, we have

[TABLE]

Note from (2.2) that $[\mathbb{C}P^{m},Top/O]$ can be considered as a subgroup of $[\mathbb{C}P^{m},SF]$. In [5, $\S$ 3], Brumfiel defined an invariant

[TABLE]

such that the composition

[TABLE]

coincides with the map

[TABLE]

induced by the hopf fibration $p:\mathbb{S}^{2m+1}\to\mathbb{C}P^{m}$. In particular, the map

[TABLE]

coincides with the map

[TABLE]

induced by the hopf fibration $p:\mathbb{S}^{2m+1}\to\mathbb{C}P^{m}$. It was furthermore proved that, for $m=4k-1$ $(k\geq 1)$,

[TABLE]

In this section, we compute the group $\mathcal{C}(\mathbb{C}P^{m})\cong[\mathbb{C}P^{m},Top/O]$ by considering the following long exact sequence associated to the cofiber sequence

[TABLE]

[TABLE]

where the homomorphism $f^{*}_{\mathbb{C}P^{m}}:[\mathbb{S}^{2m},Top/O]\to[\mathbb{C}P^{m},Top/O]$ is given by

[TABLE]

It follows from [23, Theorem 4.2] that the kernel

[TABLE]

equals the inertia group of $\mathbb{C}P^{m}$, where the inertia group of $\mathbb{C}P^{m}$ is defined to be the subgroup of $\Theta_{2m}$ cosisting of all homotopy $2m$-spheres $\Sigma^{2m}$ such that $\mathbb{C}P^{m}\#\Sigma^{2m}$ is orientation-preserving diffeomorphic to $\mathbb{C}P^{m}$. Using this fact together with [12, Theorem 1] , we have the following result.

Theorem 2.5**.**

If $m\leq 8$, the map $f^{*}_{\mathbb{C}P^{m}}:[\mathbb{S}^{2m},Top/O]\to[\mathbb{C}P^{m},Top/O]$ is a monomorphism .

To compute the group $\mathcal{C}(\mathbb{C}P^{m})$ we will mostly rely on the following Proposition and Theorem 2.5.

Proposition 2.6**.**

[21, p.190, Proof of (5.1)]** Let $p:\mathbb{S}^{2m+1}\to\mathbb{C}P^{m}$ be the Hopf fibration and $f_{\mathbb{C}P^{m}}:\mathbb{C}P^{m}\to\mathbb{S}^{2m}$ be a degree one map. Then, the map $f_{\mathbb{C}P^{m}}\circ p:\mathbb{S}^{2m+1}\to\mathbb{S}^{2m}$ is null-homotopic if and only if $m$ is even.

Theorem 2.7**.**

• (i)

There is a split short exact sequence

[TABLE]

where $\Theta_{10}\cong\mathbb{Z}_{6}$ and the natural map $f^{*}_{\mathbb{C}P^{4}}:\Theta_{8}\cong\mathbb{Z}_{2}\to\mathcal{C}(\mathbb{C}P^{4})$ is bijective.

• (ii)

There is a split short exact sequence

[TABLE]

where the map $p^{*}:\mathcal{C}(\mathbb{C}P^{5})\to\mathbb{Z}_{2}\subset\mathcal{C}(\mathbb{S}^{11})$ is induced by the Hopf fibration $p:\mathbb{S}^{11}\to\mathbb{C}P^{5}$.

• (iii)

There is a split short exact sequence

[TABLE]

where $\Theta_{14}\cong\mathbb{Z}_{2}$ and the map $p^{*}:\mathcal{C}(\mathbb{C}P^{6})\to\mathcal{C}(\mathbb{S}^{13})\cong\mathbb{Z}_{3}$ induced by the Hopf fibration $p:\mathbb{S}^{13}\to\mathbb{C}P^{6}$ is surjective.

Proof.

(i). Since $f^{*}_{\mathbb{C}P^{4}}:[\mathbb{S}^{8},Top/O]\to[\mathbb{C}P^{4},Top/O]$ is an isomorphism by [24, Theorem 2.3] and $[\mathbb{S}^{8},Top/O]\cong\Theta_{8}\cong\mathbb{Z}_{2}$, the non-trivial element in $[\mathbb{C}P^{4},Top/O]$ is represented by a map

[TABLE]

where $\Sigma:\mathbb{S}^{8}\to Top/O$ represents the exotic $8$-sphere in $\Theta_{8}$. Therefore, the effect of $p^{*}:[\mathbb{C}P^{4},Top/O]{\longrightarrow}[\mathbb{S}^{9},Top/O]$ induced by the Hopf fibration $p:\mathbb{S}^{9}\to\mathbb{C}P^{4}$ on the homotopy class $[g]$ is represented by the map

[TABLE]

Since $f_{\mathbb{C}P^{4}}\circ p:\mathbb{S}^{9}\to\mathbb{S}^{8}$ is null-homotopic by Proposition 2.6, it follows that $p^{*}\left([g]\right)=0$ and hence $p^{*}:[\mathbb{C}P^{4},Top/O]{\longrightarrow}[\mathbb{S}^{9},Top/O]$ is the zero homomorphism. Therefore, from the exact sequence (2.7) and Theorem 2.5, it follows that there is an exact sequence

[TABLE]

which splits by the fact (2.4). This proves (i).

(ii). In the exact sequence (2.7), where $m=6$, we use the fact $[\mathbb{S}^{12},Top/O]\cong\Theta_{12}=0$, to deduce that

[TABLE]

is injective. Observe that $[\mathbb{C}P^{5},Top/O]\cong\mathbb{Z}_{6}\oplus\mathbb{Z}_{2}$ by (i). Now if

[TABLE]

is an isomorphism, it follows from (2.4) and (2.3) that the map

[TABLE]

induced by the natural map $\psi:Top/O\to F/O$ cannot be injective, which contradicts the fact (2.2). Hence the map

[TABLE]

cannot be surjective. Therefore, from the long exact sequence (2.7), it follows that there is a short exact sequence

[TABLE]

where the map $p^{*}:[\mathbb{C}P^{5},Top/O]\to[\mathbb{S}^{11},Top/O]\cong\Theta_{11}=\mathbb{Z}_{992}$ is induced by the Hopf fibration $p:\mathbb{S}^{11}\to\mathbb{C}P^{5}$. This, together with the fact $[\mathbb{C}P^{5},Top/O]\cong\mathbb{Z}_{2}\oplus\mathbb{Z}_{3}\oplus\mathbb{Z}_{2}$, implies that the exact sequence is split. This proves (ii).

(iii). Finally we consider the case $m=7$. We prove that the map $p^{*}:[\mathbb{C}P^{6},Top/O]\to[\mathbb{S}^{13},Top/O]$ induced by the Hopf map $p:\mathbb{S}^{13}\to\mathbb{C}P^{6}$ is surjective. We use the following commutative diagram

[TABLE]

where the rows are part of the long exact sequences obtained from the cofiber sequence

[TABLE]

and the vertical maps $\psi_{*}:[\mathbb{S}^{13},Top/O]\to[\mathbb{S}^{13},F/O]$ and $\phi_{*}:[\mathbb{S}^{n},SF]\to[\mathbb{S}^{n},F/O]$ $(n=13,14)$ are isomorphism by the results of Kervaire and Milnor ([14]). Since

[TABLE]

by (ii) and the fact (2.4), it follows that the image of the vertical map

[TABLE]

is $\phi_{*}([\mathbb{C}P^{6},SF])$. Therefore, a simple diagram chase in the right rectangle of the diagram shows that, in order to prove the map $p^{*}:[\mathbb{C}P^{6},Top/O]\to[\mathbb{S}^{13},Top/O]$ is surjective, it suffices to prove the surjectivity of the map

[TABLE]

which follows from [7, Lemma I.9(iv)]. Therefore it follows from the long exact sequence (2.7) and Theorem 2.5 that there is an exact sequence

[TABLE]

Since $[\mathbb{C}P^{7},Top/O]\to[\mathbb{C}P^{7},F/O]$ is injective by the fact (2.2) and

[TABLE]

([5, pp. 55, Table 7.5]), the exact sequence splits. This proves (iii).

∎

We now compute the group $[\mathbb{C}P^{m},SF]$ for $m=7$ and $8$, which will be used later.

Proposition 2.8**.**

There is a split short exact sequence

[TABLE]

where $[\mathbb{S}^{14},SF]\cong\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ and the map

[TABLE]

induced by the Hopf fibration $p:\mathbb{S}^{13}\to\mathbb{C}P^{6}$ is surjective.

Proof.

From the surgery exact sequences of $\mathbb{C}P^{7}$ and $\mathbb{S}^{14}$, we get the following commutative diagram ([8, Lemma 3.4]):

[TABLE]

where $\mathcal{S}_{\rm Diff}(\mathbb{C}P^{m})$ is the smooth structure set of $\mathbb{C}P^{m}$ (see e.g. [4, 26, 22, 33]) and the map $f_{\mathbb{C}P^{7}}^{*}:\Theta_{14}\to\mathcal{S}_{\rm Diff}(\mathbb{C}P^{7})$ given by $[\Sigma]\mapsto[\mathbb{C}P^{7}\#\Sigma]$ is injective by Theorem 2.5. A diagram chase in (2.9) now shows that the map $f_{\mathbb{C}P^{7}}:[\mathbb{S}^{14},F/O]\to[\mathbb{C}P^{7},F/O]$ is injective. By using this fact and the commutative diagram (2.8), it follows that the map $f_{\mathbb{C}P^{7}}:[\mathbb{S}^{14},SF]\to[\mathbb{C}P^{7},SF]$ is also injective. We have observed in the proof of Theorem 2.7(iii) that the map

[TABLE]

is surjective. Therefore, from the long exact sequence appearing in the last row of the diagram (2.8) and the fact $[\mathbb{C}P^{6},SF]\cong\mathbb{Z}_{2}\oplus\mathbb{Z}_{3}$, it follows that there is a short exact sequence

[TABLE]

Note from the proof of Theorem 2.7(iii) that

[TABLE]

and hence the above short exact sequence is split. This completes the proof. ∎

Theorem 2.9**.**

• (i)

There is a split short exact sequence

[TABLE]

where $\Theta_{16}\cong\mathbb{Z}_{2}$ and the map $p^{*}:\mathcal{C}(\mathbb{C}P^{7})\to\mathbb{Z}_{2}\subset\mathcal{C}(\mathbb{S}^{15})$ is induced by the Hopf fibration $p:\mathbb{S}^{15}\to\mathbb{C}P^{7}$.

• (ii)

There is a split short exact sequence

[TABLE]

where the map $f_{\mathbb{C}P^{8}}^{*}:[\mathbb{S}^{16},SF]\cong\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\to\mathbb{Z}_{2}\subset[\mathbb{C}P^{8},SF]$ is induced by a degree one map $f_{\mathbb{C}P^{8}}:\mathbb{C}P^{8}\to\mathbb{S}^{16}$ and the map $p^{*}:[\mathbb{C}P^{7},SF]\to\mathbb{Z}_{2}\subset[\mathbb{S}^{15},SF]$ is induced by the Hopf fibration $p:\mathbb{S}^{15}\to\mathbb{C}P^{7}$.

Proof.

(i): We proceed to show that the image of the map $p^{*}:[\mathbb{C}P^{7},Top/O]\to[\mathbb{S}^{15},Top/O]$ induced by the Hopf fibration $p:\mathbb{S}^{15}\to\mathbb{C}P^{7}$ is $\mathbb{Z}_{2}$ by considering the following diagram as in (2.8) for the inclusion $i:\mathbb{C}P^{7}\hookrightarrow\mathbb{C}P^{8}$:

[TABLE]

where the map $\psi_{*}:bP_{16}\oplus{\rm Coker}(J_{15})\cong[\mathbb{S}^{15},Top/O]\to{\rm Coker}(J_{15})$ is the Kervaire-Milnor map, which can be identified with the map $[\mathbb{S}^{15},Top/O]\to[\mathbb{S}^{15},F/O]$ induced by the map $Top/O\to F/O$ ([6]) and $[\mathbb{C}P^{7},Top/O]\cong\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ by Theorem 2.7(iii). Note from the fact (2.6) that the image

[TABLE]

Since the non-trivial element $\Sigma^{14}\in\Theta_{14}$ corresponds to the element $\kappa\in\pi_{14}^{s}$ (see [27]), it follows from a simple diagram chase in the left rectangle of the diagram (2.8) that the image of $\Sigma^{14}$ and $\kappa$ in $[\mathbb{C}P^{7},F/O]$ are equal, that is,

[TABLE]

where $f^{*}_{\mathbb{C}P^{7}}\left(\kappa\right)\in[\mathbb{C}P^{7},SF]$ and $f^{*}_{\mathbb{C}P^{7}}\left(\Sigma^{14}\right)\in[\mathbb{C}P^{7},Top/O].$ This implies that $\phi_{*}\left(f^{*}_{\mathbb{C}P^{7}}\left(\kappa\right)\right)$ lies in the image $\left([\mathbb{C}P^{7},Top/O]\stackrel{{\scriptstyle\psi_{*}}}{{\longrightarrow}}[\mathbb{C}P^{7},F/O]\right).$ Therefore it follows from a diagram chase in the right rectangle of the diagram (2.10) that, in order to show the homomorphism $p^{*}:[\mathbb{C}P^{7},Top/O]\to{\rm Coker}(J_{15})$ sends the element $f^{*}_{\mathbb{C}P^{7}}(\Sigma^{14})$ to $\eta\circ\kappa\in{\rm Coker}(J_{15})$, it suffices to show that the composition

[TABLE]

sends the element $f^{*}_{\mathbb{C}P^{7}}(\kappa)$ to $\eta\circ\kappa\in{\rm Coker}(J_{15}).$ Consider the map

[TABLE]

This composite is $\eta$ by Proposition 2.6. Thus the induced map

[TABLE]

is multiplication by $\eta$, where

[TABLE]

and

[TABLE]

([32, p. 189]). Using this, we see that the image of $(f_{\mathbb{C}P^{7}}\circ p)^{*}:\pi_{14}^{s}\to\pi_{15}^{s}$ is generated by the following two elements :

[TABLE]

Since $\sigma^{2}\circ\eta=0$ by [32, Theorem 14.1 (i), p.190] and $\kappa\circ\eta=\eta\circ\kappa\neq 0\in{\rm Coker}(J_{15})$ ([27, Theorem 1.1.14, p.5]). Therefore, the image of $(p^{*}\circ f^{*}_{\mathbb{C}P^{7}}):\pi_{14}^{s}\to\pi_{15}^{s}$ is generated by $\eta\circ\kappa$, where $p^{*}\left(f^{*}_{\mathbb{C}P^{7}}(\kappa)\right)=\eta\circ\kappa$. This implies that the composition

[TABLE]

sends the element $f^{*}_{\mathbb{C}P^{7}}(\kappa)$ to $\eta\circ\kappa\in{\rm Coker}(J_{15})$. Hence, the image of the map $p^{*}:[\mathbb{C}P^{7},Top/O]\to[\mathbb{S}^{15},Top/O]$ is $\mathbb{Z}_{2}\{\eta\circ\kappa\}.$ Now, from the exact sequence (2.7) and Theorem 2.5, where $m=8$, it follows that there is a short exact sequence

[TABLE]

Again by Table 7.5 in [5, pp. 55], $[\mathbb{C}P^{8},F/O]_{(2)}\cong\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ and hence the short exact sequence splits. This proves (i).

(ii): Note from the proof of (i) and the fact (2.6) that the image

[TABLE]

Therefore, from Proposition 2.8, it follows that the kernel

[TABLE]

Since the $J$-homomorphism $J_{16}:[\mathbb{S}^{16},SO]=\mathbb{Z}_{2}\to[\mathbb{S}^{16},SF]$ is a monomorphism ([27, Theorem 1.1.13, p.4]), it follows from a diagram chase on the left two columns of the digram (2.10) using Theorem 2.5 that the image

[TABLE]

Now, the conclusion of (ii) follows from the long exact sequence appearing in the second row from the bottom of the diagram (2.10) and the fact

[TABLE]

This completes the proof. ∎

Remark 2.10**.**

We express the group $\mathcal{C}(\mathbb{C}P^{m})$ for $5\leq m\leq 8$ using the notation in [5, §8], where if $\alpha\in\pi_{2k}(Top/O)$, then $\alpha_{m}\in[\mathbb{C}P^{m},Top/O]$, $m\geq k$, denotes an extension of the composition $\mathbb{C}P^{k}\stackrel{{\scriptstyle f^{*}_{\mathbb{C}P^{k}}}}{{\longrightarrow}}\mathbb{S}^{2k}\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}Top/O$ to $\mathbb{C}P^{m}\to Top/O$.

• (1)

Since the natural map $f^{*}_{\mathbb{C}P^{4}}:\Theta_{8}\to\mathcal{C}(\mathbb{C}P^{4})$ is an isomorphism, $\Theta_{8}\cong\mathbb{Z}_{2}\{\varepsilon\}$ and $\Theta_{10}\cong\mathbb{Z}_{2}\{\mu_{10}\}\oplus\mathbb{Z}_{3}\{\beta_{1}\}$, it follows from Theorem 2.7(i) that

[TABLE]

• (2)

It follows from Theorem 2.7(ii) that the image of $\mathcal{C}(\mathbb{C}P^{6})$ in $\mathcal{C}(\mathbb{C}P^{5})$ is the kernel

[TABLE]

where $\Sigma^{11}$ is the Milnor generator of $bP_{12}=[\mathbb{S}^{11},Top/O]$. By [5, Proposition 8.12], the fact (2.5) and (1), we have that the kernel

[TABLE]

Hence, $\mathcal{C}(\mathbb{C}P^{6})=\mathbb{Z}_{3}\{(\beta_{1})_{6}\}\oplus\mathbb{Z}_{2}\{\varepsilon_{6}\}$.

• (3)

Since $\Theta_{14}\cong\mathbb{Z}_{2}\{\kappa\}$, it follows from Theorem 2.7(iii) and (2) that $\mathcal{C}(\mathbb{C}P^{7})=\mathbb{Z}_{2}\{\kappa_{7}\}\oplus\mathbb{Z}_{2}\{\varepsilon_{7}\}$.

• (4)

Note from the proof of Theorem 2.9(i) that

[TABLE]

sends the element $f^{*}_{\mathbb{C}P^{7}}(\Sigma^{14})$ to $\eta\circ\kappa\in{\rm Coker}(J_{15}),$ where $\Sigma^{14}$ is the non-trivial element of $\Theta_{14}$, which corresponds to $\kappa\in\pi_{14}^{s}$. By (3), the kernel of

[TABLE]

is $\mathbb{Z}_{2}\{\varepsilon_{7}\}$. Therefore, it follows from Theorem 2.9(i) using the fact $\Theta_{16}\cong\mathbb{Z}_{2}\{\eta^{*}\}$ that

[TABLE]

3. Classification up to diffeomorphism

In this section, we classify, up to (unoriented) diffeomorphism and orientation-preserving diffeomorphism, all closed smooth manifolds homeomorphic to $\mathbb{C}P^{m}$ for $5\leq m\leq 8$ by using the computations in the previous section. For a smooth manifold $M^{m}$ $(m\geq 5)$, the group of concordance classes of self homeomorphisms (resp. orientation-preserving homeomorphisms) of $M^{m}$ is denoted by $MCG^{\pm}(M)$ (resp. $MCG^{+}(M)$). Denote by $\mathcal{M}^{+}_{\rm Diff}(M)$ the set of oriented diffeomorphism classes of smooth manifolds homeomorphic to $M$. There is an obvious forgetful map

[TABLE]

which descends to define a bijection

[TABLE]

where the group $MCG^{+}(M)$ acts on $\mathcal{C}(M)$ via post-composition :

[TABLE]

Next we recall a reformulation of the action (3.2) via homotopy theory. Suppose $\alpha_{0}$ is the basic smoothing of $M$; i.e., under the isomorphism between the set of concordance classes of smoothings of $M$ and the group $[M,Top/O]$, $\alpha_{0}$ corresponds to the identity element. lf $(N,f)$ is a smoothing of $M$ representing the concordance class $\alpha\in[M,Top/O]$ and $[h]\in MCG^{+}(M)$, then the concordance class of $(N,h\circ f)$ is, denoted by $h\cdot\alpha\in[M,Top/O]$, given by the formula

[TABLE]

where $h^{*}:[M,Top/O]\to[M,Top/O]$ is induced by the self homeomorphism $h:M\to M$ such that $h^{*}(\alpha)=\alpha\circ h$ and the operation “+” refers to the abelian group structure in $[M,Top/O]$ induced by the Whitney sum in $Top/O$ (see [29, p.143]) or [8, Proposition 3.10].

We now compute the action (3.3) for $M=\mathbb{C}P^{m}$. By [31, Theorem 8], it follows that for $m>2$,

[TABLE]

and

[TABLE]

where $[Id]$ and $[C]$ are respectively concordance classes of the identity map $Id:\mathbb{C}P^{m}\to\mathbb{C}P^{m}$ and the conjugation map $C:\mathbb{C}P^{m}\to\mathbb{C}P^{m},~{}~{}[z_{0},z_{1},\cdots,z_{m}]\to[\bar{z}_{0},\bar{z}_{1},\cdots,\bar{z}_{m}]$. Since the conjugation map $C:\mathbb{C}P^{m}\to\mathbb{C}P^{m}$ is a diffeomorphism, then the smooth structures $(\mathbb{C}P^{m},C)$ and $(\mathbb{C}P^{m},Id)$ are concordant. Therefore, from the formula (3.3), the action of $[C]\in MCG^{+}(\mathbb{C}P^{m})$ on an arbitrary smoothing $\alpha$ is given by

[TABLE]

Thus, to compute $\mathcal{M}^{+}_{\rm Diff}(\mathbb{C}P^{m})$, it suffices to determine the map

[TABLE]

Remark 3.1**.**

Recall that there is a natural map $f^{*}_{\mathbb{C}P^{m}}:[\mathbb{S}^{2m},Top/O]\to[\mathbb{C}P^{m},Top/O]$, where $f_{\mathbb{C}P^{m}}:\mathbb{C}P^{m}\to\mathbb{S}^{2m}$ is a degree one map, and the map $p^{*}:[\mathbb{C}P^{m},Top/O]\to[\mathbb{S}^{2m+1},Top/O]$ is induced by the Hopf fibration $p:\mathbb{S}^{2m+1}\to\mathbb{C}P^{m}$. We now have the following observations.

• (1.)

If the map $C:\mathbb{C}P^{m}\to\mathbb{C}P^{m}$ has degree one, then $f_{\mathbb{C}P^{m}}\circ C:\mathbb{C}P^{m}\to\mathbb{S}^{2m}$ and $f_{\mathbb{C}P^{m}}:\mathbb{C}P^{m}\to\mathbb{S}^{2m}$ are homotopic and hence they induce the same map from $[\mathbb{S}^{2m},Top/O]$ to $[\mathbb{C}P^{m},Top/O]$. Therefore, we have that

[TABLE]

This implies that all elements in the image $\left([\mathbb{S}^{2m},Top/O]\stackrel{{\scriptstyle f^{*}_{\mathbb{C}P^{m}}}}{{\longrightarrow}}[\mathbb{C}P^{m},Top/O]\right)$ are fixed by the map $C^{*}:[\mathbb{C}P^{m},Top/O]\to[\mathbb{C}P^{m},Top/O]$.

• (2.)

If the map $C:\mathbb{C}P^{m}\to\mathbb{C}P^{m}$ has degree $-1$, then $f_{\mathbb{C}P^{m}}\circ C:\mathbb{C}P^{m}\to\mathbb{S}^{2m}$ and $\rho\circ f_{\mathbb{C}P^{m}}:\mathbb{C}P^{m}\to\mathbb{S}^{2m}$, where $\rho:\mathbb{S}^{2m}\to\mathbb{S}^{2m}$ is a standard reflection, are homotopic and hence the map $C^{*}:[\mathbb{C}P^{m},Top/O]\to[\mathbb{C}P^{m},Top/O]$ sends $f^{*}_{\mathbb{C}P^{m}}\left(\Sigma\right)$ to $-\left(f^{*}_{\mathbb{C}P^{m}}\left(\Sigma\right)\right)$.

• (3.)

If $m$ is odd, then $C^{*}:[\mathbb{C}P^{m},Top/O]\to[\mathbb{C}P^{m},Top/O]$ maps the kernel

[TABLE]

onto itself. This is immediate from the following commutative diagram :

[TABLE]

where the commutativity of the diagram follows from [5, Proposition 1.1 and §4].

Since $[\mathbb{C}P^{3},Top/O]=\{[\mathbb{C}P^{3}]\}$ and $[\mathbb{C}P^{4},Top/O]=\{[\mathbb{C}P^{4}\#\Sigma]~{}~{}|~{}~{}\Sigma\in\Theta_{8}\}\cong\mathbb{Z}_{2}$ (see [24, Theorem 2.3]), it follows that $C^{*}:[\mathbb{C}P^{m},Top/O]\to[\mathbb{C}P^{m},Top/O]$ is the identity for $m=3$ and $4$. Now we prove the following

Theorem 3.2**.**

• (i)

The map

[TABLE]

where $[\mathbb{C}P^{5},Top/O]=\mathbb{Z}_{2}\{(\mu_{10})_{5}\}\oplus\mathbb{Z}_{3}\{(\beta_{1})_{5}\}\oplus\mathbb{Z}_{2}\{\varepsilon_{5}\}$, is given by

[TABLE]

• (ii)

The map

[TABLE]

where $[\mathbb{C}P^{6},Top/O]=\mathbb{Z}_{3}\{(\beta_{1})_{6}\}\oplus\mathbb{Z}_{2}\{\varepsilon_{6}\}$, is given by

[TABLE]

• (iii)

The map

[TABLE]

where $[\mathbb{C}P^{7},Top/O]=\mathbb{Z}_{2}\{\kappa_{7}\}\oplus\mathbb{Z}_{2}\{\varepsilon_{7}\}$, is given by

[TABLE]

• (iv)

The map

[TABLE]

where $[\mathbb{C}P^{8},Top/O]=\mathbb{Z}_{2}\{\eta_{8}^{*}\}\oplus\mathbb{Z}_{2}\{\varepsilon_{8}\}$, is given either by

[TABLE]

Proof.

(i): By Remark 2.10(1) and (2),

[TABLE]

where $(\mu_{10})_{5}$ and $(\beta_{1})_{5}$ belong to the image

[TABLE]

and $\varepsilon_{5}$ lies in the kernel of $p^{*}:[\mathbb{C}P^{5},Top/O]\to[\mathbb{S}^{11},Top/O]$. It follows immediately from Remark 3.1(2)and (3) that $C^{*}:[\mathbb{C}P^{5},Top/O]\to[\mathbb{C}P^{5},Top/O]$ is given by

[TABLE]

(ii): The conclusion follows from the following commutative diagram and (i)

[TABLE]

where the rows exact (see the proof of Theorem 2.7(ii)) and

[TABLE]

by Remark 2.10(2).

(iii): By Remark 2.10(3) and (4), $\mathcal{C}(\mathbb{C}P^{7})=\mathbb{Z}_{2}\{\kappa_{7}\}\oplus\mathbb{Z}_{2}\{\varepsilon_{7}\}$, where $\kappa_{7}$ belongs to the image of $f^{*}_{\mathbb{C}P^{7}}:[\mathbb{S}^{14},Top/O]\to[\mathbb{C}P^{7},Top/O]$ and $\varepsilon_{7}$ belongs to the kernel of

[TABLE]

Therefore the conclusion follows from Remark 3.1(2) and (3).

(iv): Since $[\mathbb{C}P^{8},Top/O]=\mathbb{Z}_{2}\{\eta_{8}^{*}\}\oplus\mathbb{Z}_{2}\{\varepsilon_{8}\}$, where $\eta_{8}^{*}$ belongs to the image of $f^{*}_{\mathbb{C}P^{8}}:[\mathbb{S}^{16},Top/O]\to[\mathbb{C}P^{8},Top/O]$ (see Remark 2.10(4)). Therefore, from Remark 3.1(1), the conclusion follows. This completes the proof. ∎

Applying Theorem 3.2 and the fact (3.5) to the action (3.6), we obtain

Theorem 3.3**.**

• (i)

$\mathcal{M}^{+}_{\rm Diff}(\mathbb{C}P^{5})=\mathbb{Z}_{2}\{(\mu_{10})_{5}\}\oplus\mathbb{Z}_{3}\{(\beta_{1})_{5}\}\oplus\mathbb{Z}_{2}\{\varepsilon_{5}\},$* where*

[TABLE]

• (ii)

$\mathcal{M}^{+}_{\rm Diff}(\mathbb{C}P^{6})=\{[\mathbb{C}P^{6}],[(\beta_{1})_{6}],[\varepsilon_{6}],[(\beta_{1})_{6}+\varepsilon_{6}]\},$* where the orientation-preserving diffeomorphism classes*

[TABLE]

and $[(\beta_{1})_{6}+\varepsilon_{6}]=\{(\beta_{1})_{6}+\varepsilon_{6},-\left((\beta_{1})_{6}+\varepsilon_{6}\right)\}$.

• (iii)

$\mathcal{M}^{+}_{\rm Diff}(\mathbb{C}P^{7})=\mathbb{Z}_{2}\{\kappa_{7}\}\oplus\mathbb{Z}_{2}\{\varepsilon_{7}\},$* where $\mathbb{Z}_{2}\{\kappa_{7}\}=\{[\mathbb{C}P^{7}\#\Sigma^{14}]~{}~{}|~{}~{}\Sigma^{14}\in\Theta_{14}\}.$*

• (iv)

$\mathcal{M}^{+}_{\rm Diff}(\mathbb{C}P^{8})=\mathbb{Z}_{2}\{\eta_{8}^{*}\}\oplus\mathbb{Z}_{2}\{\varepsilon_{8}\}$* or $\{[\mathbb{C}P^{8}],[\eta_{8}^{*}],[\varepsilon_{8}]\},$ where the orientation-preserving diffeomorphism classes $[\eta_{8}^{*}]=\{\eta_{8}^{*}\}$ and $[\varepsilon_{8}]=\{\varepsilon_{8},\eta_{8}^{*}+\varepsilon_{8}\}$.*

Remark 3.4**.**

Let $\mathcal{M}^{\pm}_{\rm Diff}(\mathbb{C}P^{m})$ denote the set of (unoriented) diffeomorphism classes of smooth manifolds homeomorphic to $\mathbb{C}P^{m}$. Then $\mathcal{M}^{\pm}_{\rm Diff}(\mathbb{C}P^{m})=\mathcal{M}^{+}_{\rm Diff}(\mathbb{C}P^{m})/\{\pm 1\},$ where the group $\mathbb{Z}_{2}=\{\pm 1\}$ acts on $\mathcal{M}^{+}_{\rm Diff}(\mathbb{C}P^{m})$ by reversing orientation.

• (1).

Since $\mathbb{C}P^{2n}$ does not admit a self-homeomorphism reversing orientation, it follows that the group $\mathbb{Z}_{2}=\{\pm 1\}$ acts trivially on $\mathcal{M}^{+}_{\rm Diff}(\mathbb{C}P^{m})$ and hence

[TABLE]

• (2).

Recall that there is a natural action of $MCG^{\pm}(\mathbb{C}P^{2n+1})=\{[Id],[C]\}$ on $\mathcal{C}(\mathbb{C}P^{2n+1})$ given by post-composition, where the action of $[C]\in MCG^{\pm}(\mathbb{C}P^{m})$ on an arbitrary smoothing $[(N,f)]$ is given by $[C]\cdot[N,f]=[-N,C\circ f]$. Here $-N$ is the manifold $N$ with reversed orientaion. Observe from (3.5) and (3.1) that the forgetful map

[TABLE]

is bijective. With this identification, the action of $MCG^{\pm}(\mathbb{C}P^{2n+1})=\{[Id],[C]\}$ on $\mathcal{C}(\mathbb{C}P^{2n+1})$ factors over the action of the group $\mathbb{Z}_{2}=\{\pm 1\}$ on the set $\mathcal{M}^{+}_{\rm Diff}(\mathbb{C}P^{2n+1})$. Indeed, there is a bijection

[TABLE]

Note that for any $\alpha\in\mathcal{C}(\mathbb{C}P^{2n+1})$, the action of $[C]\in MCG^{\pm}(\mathbb{C}P^{m})$ on $\alpha$ is given by the formula (3.6).

Putting together Remark 3.4, Theorems 3.3 and 3.2, we get :

Theorem 3.5**.**

• (i)

$\mathcal{M}^{\pm}_{\rm Diff}(\mathbb{C}P^{5})=\{[\mathbb{C}P^{5}],[(\mu_{10})_{5}],[(\beta_{1})_{5}],[\varepsilon_{5}],[(\mu_{10})_{5}+(\beta_{1})_{5}],[(\beta_{1})_{5}+\varepsilon_{5}],[(\mu_{10})_{5}+\varepsilon_{5}],[(\mu_{10})_{5}+(\beta_{1})_{5}+\varepsilon_{5}]\},$* where the (unoriented) diffeomorphism classes*

[TABLE]

[TABLE]

[TABLE]

and $[(\mu_{10})_{5}+(\beta_{1})_{5}+\varepsilon_{5}]=\{(\mu_{10})_{5}+(\beta_{1})_{5}+\varepsilon_{5},(\mu_{10})_{5}-(\beta_{1})_{5}+\varepsilon_{5}\}$.

• (ii)

$\mathcal{M}^{\pm}_{\rm Diff}(\mathbb{C}P^{6})=\{[\mathbb{C}P^{6}],[(\beta_{1})_{6}],[\varepsilon_{6}],[(\beta_{1})_{6}+\varepsilon_{6}]\},$* where the (unoriented) diffeomorphism classes*

[TABLE]

and $[(\beta_{1})_{6}+\varepsilon_{6}]=\{(\beta_{1})_{6}+\varepsilon_{6},-\left((\beta_{1})_{6}+\varepsilon_{6}\right)\}$.

• (iii)

$\mathcal{M}^{\pm}_{\rm Diff}(\mathbb{C}P^{7})=\mathbb{Z}_{2}\{\kappa_{7}\}\oplus\mathbb{Z}_{2}\{\varepsilon_{7}\},$* where $\mathbb{Z}_{2}\{\kappa_{7}\}=\{[\mathbb{C}P^{7}\#\Sigma^{14}]~{}~{}|~{}~{}\Sigma^{14}\in\Theta_{14}\}.$*

• (iv)

$\mathcal{M}^{\pm}_{\rm Diff}(\mathbb{C}P^{8})=\mathbb{Z}_{2}\{\eta_{8}^{*}\}\oplus\mathbb{Z}_{2}\{\varepsilon_{8}\}$* or $\{[\mathbb{C}P^{8}],[\eta_{8}^{*}],[\varepsilon_{8}]\},$ where the (unoriented) diffeomorphism classes $[\eta_{8}^{*}]=\{\eta_{8}^{*}\}$ and $[\varepsilon_{8}]=\{\varepsilon_{8},\eta_{8}^{*}+\varepsilon_{8}\}$.*

Remark 3.6**.**

It follows from Theorems 3.3(i) and 3.5(i) that $\mathbb{C}P^{5}\#\Sigma^{10}$ and $\mathbb{C}P^{5}\#(-\Sigma^{10})$ are diffeomorphic but not orientation-preserving diffeomorphic, where $\Sigma^{10}$ corresponds to the element $\beta_{1}\in\Theta_{10}\cong\mathbb{Z}_{2}\{\mu_{10}\}\oplus\mathbb{Z}_{3}\{\beta_{1}\}$.

4. The smooth tangential structure set $\mathcal{S}^{t}_{\rm Diff}(\mathbb{C}P^{m})$

In this section, for $m=7$ and $8$, we compute the smooth tangential structure set of $\mathbb{C}P^{m}$ and obtain a bound on the number of smooth homotopy complex projective $m$-spaces with given Pontryagin classes up to orientation-preserving diffeomorphism. Recall from [2, §8] that a tangential homotopy structure on an $n$-manifold $M$, with or without boundary, is a triple $(N,f,\hat{f})$ such that $f:N\to M$ is a homotopy equivalence and $\hat{f}:\tau_{N}\to\tau_{M}$ is an isomorphism of stable tangent bundles that covers $f$. Two such structures $(N_{1},f_{1},\hat{f}_{1})$, $(N_{2},f_{2},\hat{f}_{2})$ are said to be equivalent if $f_{1}$ and $f_{2}\circ h$ are homotopic through maps of pairs $(N_{1},\partial N_{1})\to(M,\partial M)$ for some diffeomorphism $h$. We denote the set of equivalence classes of tangential homotopy structures on $M$ by $\mathcal{S}_{\rm Diff}^{t}(M)$. Recall that a homotopy equivalence $f:N\to M$ is called tangential if $f^{*}(\tau_{M})$ is stably equivalent to $\tau_{N}$. Therefore, every tangential homotopy structure on $M$ can be regarded as a pair $(N,f)$, where $f:N\to M$ is a tangential homotopy equivalence.

The main tool for calculating the smooth tangential structure set $\mathcal{S}_{\rm Diff}^{t}(M)$ of an $n$-dimensional simply connected closed smooth manifold $M$ where $n\geq 5$ is the surgery exact sequence

[TABLE]

(see also [9, §6]). Since $L_{{\rm odd}}(e)=0$ ([33]), the tangential surgery exact sequence (4.1) for $M=\mathbb{C}P^{m}$ runs as follows:

[TABLE]

where $\sigma_{1}$ is a homomorphism, but $\sigma_{2}$ is not [31]. Computations of the groups $[\mathbb{C}P^{m},SF]$ and of $\sigma:[\mathbb{C}P^{m},SF]{\longrightarrow}L_{2m}(e)$ have been made by Brumfiel [5, 7] for $m\leq 6$. For $m=3$, $\sigma$ is an isomorphism. For $m=4,5,6$, $\sigma=0$. We now have the following

Proposition 4.1**.**

The surgery obstruction $\sigma_{1}:[\mathbb{C}P^{7},SF]\to\mathbb{Z}_{2}$ is a non zero homomorphism.

Proof.

This follows immediately from the commutative diagram with exact rows (2.9) using the fact that $[\mathbb{C}P^{m},SF]$ is the torsion subgroup of $[\mathbb{C}P^{m},F/O]$. ∎

Theorem 4.2**.**

• (i)

$\mathcal{S}_{\rm Diff}^{t}(\mathbb{C}P^{7})\cong\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$.

• (ii)

$\mathcal{S}_{\rm Diff}^{t}(\mathbb{C}P^{8})\cong\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$.

Proof.

Statements (i) and (ii) follow from the exact sequence (4.2), Proposition 2.8, Theorem 2.9(ii), Proposition 4.1 and [7, Lemma I.5]. ∎

For a smooth manifold $M$, we let $\mathcal{M}^{+}_{\rm hDiff}(M)$ denote the set of (oriented) diffeomorphism classes of smooth manifolds in the homotopy type of $M$ and is identified as the quotient $\mathcal{S}_{\rm Diff}(M)/{\rm haut^{+}(M)},$ where ${\rm haut^{+}(M)}$, the group of homotopy classes of orientation-preserving self homotopy equivalences of $M$ acts on $\mathcal{S}_{\rm Diff}(M)$ by post-composition. Recall that the surgery exact sequence appearing in the bottom row of the diagram (2.9) for $\mathbb{C}P^{m}$ gives an inclusion of $\mathcal{S}_{\rm Diff}(\mathbb{C}P^{m})$ in $[\mathbb{C}P^{m},F/O]$ and $\mathcal{S}_{\rm Diff}(\mathbb{C}P^{m})$ is identified with the kernel

[TABLE]

There is also an exact sequence

[TABLE]

induced by the fibrations $SF\to F/O\to BSO\to BSF,$ where $\widetilde{KO}^{0}(\mathbb{C}P^{m})$ is reduced real $K$-theory, and if $f:M\to\mathbb{C}P^{m}$ represents an element of $\mathcal{S}_{\rm Diff}(\mathbb{C}P^{m})$, then its image under the map $i_{*}:[\mathbb{C}P^{m},F/O]\to\widetilde{KO}^{0}(\mathbb{C}P^{m})$ is given by

[TABLE]

(see [5, §2]). Thus, given $f:M\to\mathbb{C}P^{m}$ in $\mathcal{S}_{\rm Diff}(\mathbb{C}P^{m})$, we have from the exact sequence (4.3) that $f^{*}(\tau_{\mathbb{C}P^{m}})=\tau_{M}$ if and only if $(f^{-1})^{*}\tau_{M}-\tau_{\mathbb{C}P^{m}}=0\in\widetilde{KO}^{0}(\mathbb{C}P^{m})$ or, equivalently, if and only if the map $f:M\to\mathbb{C}P^{m}$ represents an element of $[\mathbb{C}P^{m},SF]$.

Theorem 4.3**.**

The number of elements in $\mathcal{M}^{+}_{\rm hDiff}(\mathbb{C}P^{m})$ $(m\geq 3)$ with a given set of Pontryagin classes is bounded by the order of the group $[\mathbb{C}P^{m},SF]$.

Proof.

Let $[f:M\to\mathbb{C}P^{m}]$, $[g:N\to\mathbb{C}P^{m}]\in$ $\mathcal{M}^{+}_{\rm hDiff}(\mathbb{C}P^{m})$ such that $M$ and $N$ have the same Pontryagin classes; that is, there exists a homotopy equivalence $h:M\to N$ such that $h^{*}(p_{i}(\tau_{N}))=p_{i}(\tau_{M})$ $(\forall i)$. Therefore, we have

[TABLE]

By [31, Theorem 8(i)], $h\circ f^{-1}$ is homotopic either to $g^{-1}$ or to $g^{-1}\circ C$, where $C:\mathbb{C}P^{m}\to\mathbb{C}P^{m}$ is the conjugation map. Since

[TABLE]

is the identity (see [5, p.35]), we have

[TABLE]

Hence,

[TABLE]

Now by [3, Lemma 2.25], $(f^{-1})^{*}(\tau_{M})-(g^{-1})^{*}(\tau_{N})$ lies in the torsion subgroup of $\widetilde{KO}^{0}(\mathbb{C}P^{m})$. Since the image

[TABLE]

is a free subgroup of $\widetilde{KO}^{0}(\mathbb{C}P^{m})$ (see [5, Corollary 2.1]), it follows that

[TABLE]

in $\widetilde{KO}^{0}(\mathbb{C}P^{m})$. Therefore, the elements $[f:M\to\mathbb{C}P^{m}]$ and $[g:N\to\mathbb{C}P^{m}]$ have the same image in $\widetilde{KO}^{0}(\mathbb{C}P^{m})$. Thus, the set of elements in $\mathcal{M}^{+}_{\rm hDiff}(\mathbb{C}P^{m})$ with the same Pontryagin classes have the same image in $\widetilde{KO}^{0}(\mathbb{C}P^{m})$. Now it follows from the exact sequence (4.3) that the number of such elements in $[\mathbb{C}P^{m},F/O]$ is bounded by the order of the group $[\mathbb{C}P^{m},SF]$. This completes the proof. ∎

An immediate consequence of Proposition 2.8, Theorems 2.9(ii) and 4.3 is the following

Theorem 4.4**.**

For $m=7$ and $8$, the number of elements in $\mathcal{M}^{+}_{\rm hDiff}(\mathbb{C}P^{m})$ with given Pontryagin classes does not exceed eight.

Note that there is a natural map $F_{t}:\mathcal{S}^{t}(\mathbb{C}P^{m})\to\mathcal{S}_{\rm Diff}(\mathbb{C}P^{m})$ which is injective. The next result describes that the image of $F_{t}$ may be identified with the set of elements in $\mathcal{S}_{\rm Diff}(\mathbb{C}P^{m})$ with the same Pontryagin classes as $\mathbb{C}\mathbb{P}^{m}$.

Theorem 4.5**.**

Let $M$ be a smooth manifold and $f:M\to\mathbb{C}P^{m}$ be a homotopy equivalence that preserves Pontryagin classes. Then the map $f:M\to\mathbb{C}P^{m}$ is a tangential homotopy equivalence.

Proof.

Since the map $f:M\to\mathbb{C}P^{m}$ preserves Pontryagin classes, it follows from the proof of Theorem 4.3 that $(f^{-1})^{*}(\tau_{M})-\tau_{\mathbb{C}P^{m}}=0\in\widetilde{KO}^{0}(\mathbb{C}P^{m})$ and hence the map $f:M\to\mathbb{C}P^{m}$ is a tangential homotopy equivalence. ∎

Recall that the canonical map $\psi_{*}:[\mathbb{C}P^{m},Top/O]\to[\mathbb{C}P^{m},F/O]$ factors as

[TABLE]

where the forgetful map $F_{C}:[\mathbb{C}P^{m},Top/O]\to\mathcal{S}_{\rm Diff}(\mathbb{C}P^{m})$ is injective by [25, Theorem 3.1], and the image $\left([\mathbb{C}P^{m},Top/O]\stackrel{{\scriptstyle\psi_{*}}}{{\longrightarrow}}[\mathbb{C}P^{m},F/O]\right)\subseteq[\mathbb{C}P^{m},SF]$ (see 2.2 and 2.3). Therefore, the image

[TABLE]

Now, from the exact sequences (4.2) there is an inclusion

[TABLE]

This, together with Theorems 2.7(iii), 2.9(i) and 4.2, implies the following.

Theorem 4.6**.**

• (i)

The map $\mathcal{C}(\mathbb{C}P^{7})\to\mathcal{S}^{t}(\mathbb{C}P^{7})$ is bijective.

• (ii)

The map $\mathcal{C}(\mathbb{C}P^{8})\to\mathcal{S}^{t}(\mathbb{C}P^{8})$ is injective but not surjective. In particular, there are exactly four elements in $\mathcal{S}^{t}(\mathbb{C}P^{8})$ which do not belong to the image of the map $\mathcal{C}(\mathbb{C}P^{8})\to\mathcal{S}^{t}(\mathbb{C}P^{8})$.

As a consequence of Theorem 4.6, we have

Corollary 4.7**.**

• (i)

Let $M$ be a smooth manifold homotopy equivalent to $\mathbb{C}P^{7}$. Then $M$ is homeomorphic to $\mathbb{C}P^{7}$ if and only if $M$ is tangentially homotopy equivalent to $\mathbb{C}P^{7}$.

• (ii)

Up to orientation-preserving diffeomorphism, there are at most four smooth manifolds which are tangentially homotopy equivalent to $\mathbb{C}P^{8}$ but not homeomorphic to $\mathbb{C}P^{8}$.