Fake complex projective spaces

1 Introduction

A fake complex projective space is a topological manifold which is homotopy equivalent to a complex projective space $\Cc P^n$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == Introduction == <wikitex>; A fake complex projective space is a topological manifold which is homotopy equivalent to a complex projective space $\Cc P^n$ for some $n$. </wikitex> == Construction and examples == <wikitex>; Given a free tame action of the circle on a $(2n-1)$-sphere, the orbit space is a fake $\Cc P^n$. On the other hand, if $M$ is a closed manifold, any homotopy equivalence $M\to \Cc P^n$ induces a principal $S^1$-bundle over $M$ whose total space is homeomorphic to $S^{2n-1}$. We obtain the following result: {{beginthm|Proposition}} The surgery structure set of $\Cc P^n$ is in bijection to the set of free tame circle actions on $S^{2n-1}$ modulo $S^1$-equivariant homeomorphism. {{endthm}} Given a homotopy equivalence $f\colon M\to \Cc P^n$, we can ''suspend'' $f$ to obtain a fake $\Cc P^{n+1}$ as follows: Denote by $p\colon E\to \Cc P^n$ the disk bundle of the canonical complex line bundle over $\Cc P^n$. Notice that $\partial E\cong S^{2n+1}$, and we obtain $\Cc P^{n+1}$ by glueing a $(2n+2)$-disk to $E$ along the boundary. Let $E':= f^* E$ be the total space of the disk bundle pulled back bundle from $p$ using $f$. The homotopy equivalence $f$ induces a homotopy equivalence $\partial E'\to \partial E\cong S^{2n+1}$. By the Poincar&eacute; conjecture, $\partial E'$ is therefore homeomorphic to $S^{2n+1}$; hence glueing the cone of $\partial E'$ onto $E'$ produces a $(2n+1)$-manifold $N$ equipped with a homotopy equivalence to $\Cc P^{n+1}$. In fact, this construction defines a suspension map $$\Sigma\colon \mathcal{S}(\Cc P^n)\to \mathcal{S}(\Cc P^{n+1}).$$ </wikitex> == Invariants == <wikitex>; Obviously the homology and homotopy groups of a fake complex projective space are isomorphic to the ones of the $\Cc P^n$. Different fake complex projective spaces may be distinguished using the so-called splitting invariants. More precisely, for any $i\in\{1,\dots, n-1\}$, there is a function $$ s_{2i}\colon \mathcal{S}(\Cc P^n)\to L_{2i}(\Zz) $$ from the surgery structure set of $\Cc P^n$ to the $L$-groups of the integers, where $L_{2i}(Z)\cong Z$ if $i$ is even, and $L_{2i}(Z)\cong \Zz/2$ if $i$ is odd. The splitting invariant $s_{2i}$ is defined as follows: Given an element $x\in\mathcal{S}(\Cc P^n)$, represent it by homotopy equivalence $f\colon M\to \Cc P^n$ which is transverse to $\Cc P^i\subset \Cc P^n$. The restriction of $f$ to a map $g\colon f^{-1}(\Cc P^i)\to \Cc P^i$ may fail to be a homotopy equivalence, but it is still a degree one normal map. Hence the surgery obstruction $\sigma(g)\in L_{2i}(\Zz)$ of $g$ is defined. Let $s_{2i}(x):=\sigma(g)$. {{beginthm|Proposition}} Under the suspension map $\Sigma\colon \mathcal{S}(\Cc P^n)\to\mathcal{S}(\Cc P^{n+1})$, the splitting invariants $s_{2i}$ remain unchanged for $i\in\{1,\dots n-1\}$. {{endthm}} Using the classification of fake complex projective spaces described in the next section, it follows: {{beginthm|Corollary}} The suspension map is injective. Its image is given by the homotopy equivalences $f\colon M\to \Cc P^{n+1}$ whose highest splitting invariant is zero. {{endthm}} </wikitex> == Classification/Characterization == <wikitex>; The surgery structure set of $\Cc P^n$ may be completely described using the splitting invariants: {{beginthm|Theorem}} The cartesian product $$\prod_{i=1}^{n-1} s_{2i}\colon \mathcal{S}(\Cc P^n) \to \prod_{i=1}^{n-1} L_{2i}(\Zz)$$ is a bijection. {{endthm}} Thus, all possible combination of splitting invariants are realized by elements in the structure set, and two elements of the structure set agree if and only if all the splitting invariants agree. </wikitex> == Further discussion == <wikitex>; ... </wikitex>\Cc P^n$ for some $n$$n$.

2 Construction and examples

Given a free tame action of the circle on a $(2n-1)$$(2n-1)$-sphere, the orbit space is a fake $\Cc P^n$$\Cc P^n$. On the other hand, if $M$$M$ is a closed manifold, any homotopy equivalence $M\to \Cc P^n$$M\to \Cc P^n$ induces a principal $S^1$$S^1$-bundle over $M$$M$ whose total space is homeomorphic to $S^{2n-1}$$S^{2n-1}$. We obtain the following result:

Proposition 2.1. The surgery structure set of $\Cc P^n$$\Cc P^n$ is in bijection to the set of free tame circle actions on $S^{2n-1}$$S^{2n-1}$ modulo $S^1$$S^1$-equivariant homeomorphism.

Given a homotopy equivalence $f\colon M\to \Cc P^n$$f\colon M\to \Cc P^n$, we can suspend $f$$f$ to obtain a fake $\Cc P^{n+1}$$\Cc P^{n+1}$ as follows: Denote by $p\colon E\to \Cc P^n$$p\colon E\to \Cc P^n$ the disk bundle of the canonical complex line bundle over $\Cc P^n$$\Cc P^n$. Notice that $\partial E\cong S^{2n+1}$$\partial E\cong S^{2n+1}$, and we obtain $\Cc P^{n+1}$$\Cc P^{n+1}$ by glueing a $(2n+2)$$(2n+2)$-disk to $E$$E$ along the boundary. Let $E':= f^* E$$E':= f^* E$ be the total space of the disk bundle pulled back bundle from $p$$p$ using $f$$f$. The homotopy equivalence $f$$f$ induces a homotopy equivalence $\partial E'\to \partial E\cong S^{2n+1}$$\partial E'\to \partial E\cong S^{2n+1}$. By the Poincaré conjecture, $\partial E'$$\partial E'$ is therefore homeomorphic to $S^{2n+1}$$S^{2n+1}$; hence glueing the cone of $\partial E'$$\partial E'$ onto $E'$$E'$ produces a $(2n+1)$$(2n+1)$-manifold $N$$N$ equipped with a homotopy equivalence to $\Cc P^{n+1}$$\Cc P^{n+1}$.

In fact, this construction defines a suspension map

$\displaystyle \Sigma\colon \mathcal{S}(\Cc P^n)\to \mathcal{S}(\Cc P^{n+1}).$ $$\displaystyle \Sigma\colon \mathcal{S}(\Cc P^n)\to \mathcal{S}(\Cc P^{n+1}).$$

3 Invariants

Obviously the homology and homotopy groups of a fake complex projective space are isomorphic to the ones of the $\Cc P^n$$\Cc P^n$. Different fake complex projective spaces may be distinguished using the so-called splitting invariants. More precisely, for any $i\in\{1,\dots, n-1\}$$i\in\{1,\dots, n-1\}$, there is a function

$\displaystyle s_{2i}\colon \mathcal{S}(\Cc P^n)\to L_{2i}(\Zz)$ $$\displaystyle s_{2i}\colon \mathcal{S}(\Cc P^n)\to L_{2i}(\Zz)$$

from the surgery structure set of $\Cc P^n$$\Cc P^n$ to the $L$$L$-groups of the integers, where $L_{2i}(Z)\cong Z$$L_{2i}(Z)\cong Z$ if $i$$i$ is even, and $L_{2i}(Z)\cong \Zz/2$$L_{2i}(Z)\cong \Zz/2$ if $i$$i$ is odd.

The splitting invariant $s_{2i}$$s_{2i}$ is defined as follows: Given an element $x\in\mathcal{S}(\Cc P^n)$$x\in\mathcal{S}(\Cc P^n)$, represent it by homotopy equivalence $f\colon M\to \Cc P^n$$f\colon M\to \Cc P^n$ which is transverse to $\Cc P^i\subset \Cc P^n$$\Cc P^i\subset \Cc P^n$. The restriction of $f$$f$ to a map $g\colon f^{-1}(\Cc P^i)\to \Cc P^i$$g\colon f^{-1}(\Cc P^i)\to \Cc P^i$ may fail to be a homotopy equivalence, but it is still a degree one normal map. Hence the surgery obstruction $\sigma(g)\in L_{2i}(\Zz)$$\sigma(g)\in L_{2i}(\Zz)$ of $g$$g$ is defined. Let $s_{2i}(x):=\sigma(g)$$s_{2i}(x):=\sigma(g)$.

Proposition 3.1. Under the suspension map $\Sigma\colon \mathcal{S}(\Cc P^n)\to\mathcal{S}(\Cc P^{n+1})$$\Sigma\colon \mathcal{S}(\Cc P^n)\to\mathcal{S}(\Cc P^{n+1})$, the splitting invariants $s_{2i}$$s_{2i}$ remain unchanged for $i\in\{1,\dots n-1\}$$i\in\{1,\dots n-1\}$.

Using the classification of fake complex projective spaces described in the next section, it follows:

Corollary 3.2. The suspension map is injective. Its image is given by the homotopy equivalences $f\colon M\to \Cc P^{n+1}$$f\colon M\to \Cc P^{n+1}$ whose highest splitting invariant is zero.

4 Classification/Characterization

The surgery structure set of $\Cc P^n$$\Cc P^n$ may be completely described using the splitting invariants:

Theorem 4.1. The cartesian product

$\displaystyle \prod_{i=1}^{n-1} s_{2i}\colon \mathcal{S}(\Cc P^n) \to \prod_{i=1}^{n-1} L_{2i}(\Zz)$ $$\displaystyle \prod_{i=1}^{n-1} s_{2i}\colon \mathcal{S}(\Cc P^n) \to \prod_{i=1}^{n-1} L_{2i}(\Zz)$$

is a bijection.

Thus, all possible combination of splitting invariants are realized by elements in the structure set, and two elements of the structure set agree if and only if all the splitting invariants agree.

5 Further discussion

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