Fake complex projective spaces
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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. | 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}} | {{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 Poincaré 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 | ||
+ | $$\Sigma\colon \mathcal{S}(\Cc P^n)\to \mathcal{S}(\Cc P^{n+1}).$$ | ||
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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. | 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 $s_{2i}(x):=\sigma(g)$. |
− | The | + | |
− | $$\ | + | {{beginthm|Proposition}} |
− | + | Under the suspension map $\Sigma\colon \mathcal{S}(\Cc P^n)\to\mathcal{S}(\Cc P^{n+1}$, the splitting invariants $s_i$ remain unchanged for $i\in\{1,\dots n-1\}$. | |
{{endthm}} | {{endthm}} | ||
− | + | Using the classification of fake complex projective spaces described in the next section, it follows: | |
− | + | {{beginthm|Corollary}} | |
+ | The suspension map is injective. Its image is given by the homotopy equivalences $f\colon M\to \Cc P^{n+1}$ whose highest splitting invariant is zero. | ||
+ | {{endthm}} | ||
</wikitex> | </wikitex> | ||
== Classification/Characterization == | == Classification/Characterization == | ||
<wikitex>; | <wikitex>; | ||
− | . | + | The surgery structure set of $\Cc P^n$ may be completely described using the splitting invariants: |
+ | |||
+ | {{beginthm|Theorem}} | ||
+ | The cartesian product | ||
+ | $$\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> | </wikitex> | ||
Revision as of 14:33, 7 June 2010
Contents |
1 Introduction
Tex syntax errorfor some .
2 Construction and examples
Tex syntax error. On the other hand, if is a closed manifold, any homotopy equivalence induces a principal -bundle over whose total space is homeomorphic to . We obtain the following result:
Proposition 2.1.
The surgery structure set ofTex syntax erroris in bijection to the set of free tame circle actions on modulo -equivariant homeomorphism.
Tex syntax error. Notice that , and we obtain by glueing a -disk to along the boundary. Let be the total space of the disk bundle pulled back bundle from using . The homotopy equivalence induces a homotopy equivalence . By the Poincaré conjecture, is therefore homeomorphic to ; hence glueing the cone of onto produces a -manifold equipped with a homotopy equivalence to .
In fact, this construction defines a suspension map
3 Invariants
Tex syntax error. Different fake complex projective spaces may be distinguished using the so-called splitting invariants. More precisely, for any , there is a function
Tex syntax errorto the -groups of the integers, where if is even, and if is odd.
The splitting invariant is defined as follows: Given an element , represent it by homotopy equivalence which is transverse to . The restriction of to a map may fail to be a homotopy equivalence, but it is still a degree one normal map. Hence the surgery obstruction of is defined. Let .
Proposition 3.1. Under the suspension map , the splitting invariants remain unchanged for .
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 whose highest splitting invariant is zero.
4 Classification/Characterization
Tex syntax errormay be completely described using the splitting invariants:
Theorem 4.1. The cartesian product
Tex syntax error
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
...