Fake complex projective spaces

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[edit] 1 Introduction

A fake complex projective space is a topological manifold which is homotopy equivalent to a complex projective space \Cc P^n for some n. The classification of these spaces was one of the early milestones in surgery theory.

This class of manifolds is of interest for many reasons. On the one hand, they are related to certain (topological) circle actions on spheres. Moreover, they lead to easy-to-handle examples of non-tangential homotopy equivalences. In addition certain fake complex projective spaces are part of a generating set of the topological oriented cobordism groups.

[edit] 2 Construction and examples

Recall that a free circle action on a manifold is called tame if the orbit space is a manifold again. 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:

Proposition 2.1 [Wall1999, chapter 14C]. The surgery structure set of \Cc P^n is in bijection to the set of free tame circle actions on S^{2n+1} modulo equivariant homeomorphism.

Edmonds [Edmonds1977] has shown that the latter set agrees with the set of all free circle actions on S^{2n+1} modulo a weaker relation called concordance, provided n\geq 3.

[edit] 2.1 The suspension map

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 [Sullivan1996]

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

[edit] 2.2 The Madsen-Milgram construction

The previous construction has been generalized as follows [Madsen&Milgram1979]: Suppose that the map f we started with is just a degree one normal map, without assuming that it is a homotopy equivalence. We can still pull back the disk bundle p over \Cc P^n along f to obtain a disk bundle E' over M. The induced map E'\to E may now fail to be a homotopy equivalence, but it is a degree one normal map which restricts to a degree one normal map \gamma\colon \partial E'\to \partial E\cong S^{2n+1} on the boundary. As \partial E' is null-bordant in E, the surgery obstruction of \gamma is zero. An additional argument shows that \gamma is bordant to a homotopy equivalence \Sigma^{2n+1}\to S^{2n+1} via a normal cobordism W such that

\displaystyle  E'\cup_{\partial E'} W \to E\cup_{S^{2n+1}} S^{2n+1}\times I

is a homotopy equivalence. Then, coning off \Sigma^{2n+1} and S^{2n+1} produces a homotopy equivalence g\colon N\to \Cc P^{n+1} from some closed (2n+2)-manifold N to \Cc P^{n+1}. The map g has the pleasant feature that it is transverse to \Cc P^n\subset \Cc P^{n+1} and the restriction of g to a degree one normal map g^{-1}(\Cc P^n)\to \Cc P^n is the map f we started with.

Hence, the suspension map \Sigma extends to a map

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

which is split injective. In fact, the following holds:

Theorem 2.2. For n\geq 2, the map \Sigma' is a bijection.

This follows from the classification \mathcal{N}(\Cc P^n) as described below: Both the domain and the target of \Sigma' are completely described by the first n splitting invariants, and they remain unchanged under the Madsen-Milgram construction.

[edit] 3 Invariants

[edit] 3.1 Splitting invariants

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

\displaystyle  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).

Proposition 3.1. 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\}.

This is immediate from the construction of the suspension map. Using the classification of fake complex projective spaces described in the next section, it follows:

Corollary 3.2. Let n\geq 3. Then the suspension map is injective, and its image is given by the homotopy equivalences f\colon M\to \Cc P^{n+1} whose highest splitting invariant is zero.

[edit] 3.2 Rational Pontryagin classes

An interesting feature of fake complex projective spaces is that their stable tangent bundle may differ from the one of the standard \Cc P^n. Given a homotopy equivalence f\colon M\to \Cc P^n, in theory the total Hirzebruch \mathcal{L}-class \mathcal{L}(M)\in H^{4*}(M;\Qq) may be computed inductively from the splitting invariants using the formula [Madsen&Milgram1979, Theorem 4.9]

\displaystyle s_{4i}(f) = \langle\mathcal{L}(\Cc P^{2i}) \cdot \bigl(\sum_{k\geq 1} K_{4k}(f)\vert_{\Cc P^{2i}}\bigr), [\Cc P^{2i}]\rangle,

where, by [Madsen&Milgram1979, Corollary 4.22], we have

\displaystyle K_{4k}(f) = \frac18 \mathcal{L}_{4k}((f^{-1})^* TM - T\Cc P^n)\in H^{4k}(\Cc P^n;\Qq)\cong \Qq\quad (k\leq n/2).

[edit] 3.3 The rho-invariant

The rho-invariant of a free tame circle action on S^{2n-1} may be explicitly expressed in terms of the splitting invariants of the corresponding homotopy equivalence f\colon M\to \Cc P^{n-1}:

Theorem 3.3 [Wall1999, Theorem 14C.4]. For t\in S^1, we have

\displaystyle \rho(t) = q(t)^n + \sum_{r=1}^{[n/2]-1} 8 \cdot s_{4r}(f) \cdot \bigl(q(t)^{n-2r} - q(t)^{n-2r-2}\bigr)\in \Cc,

where q(t)= (1+t)/(1-t), and s_{4r}(f)\in L_{4r}(\Zz)\cong \Zz.

[edit] 4 Classification/Characterization

The surgery classification of fake complex projective spaces was initiated by Brumfiel [Brumfiel1969a].

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

Theorem 4.1 [Wall1999, Theorem 14C.2]. Let n\geq 3. 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)

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.

The proof of Theorem 4.1 is surgery-theoretic. In fact the splitting invariants are defined more generally on the set of normal invariants \mathcal{N}(\Cc P^n), where the surgery obstruction \sigma=:s_{2n} may be non-zero. Theorem 4.1 therefore follows immediately by applying the surgery exact sequence to the following homotopy-theoretic computation, which goes back to Sullivan [Sullivan1996].

Theorem 4.2 [Wall1999, Lemma 14C.1]. The cartesian product

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

is a bijection.

[edit] 5 Further discussion

Fake complex projective spaces are interesting for the study of the topological oriented cobordism ring. In fact, we have

Theorem 5.1 [Madsen&Milgram1979, chapter 8]. A set of generators for the topological oriented cobordism ring modulo torsion is contained in the set consisting of the index 8 Milnor manifolds, the differentiable generators and the exotic complex projective spaces.

An interesting question is which of the fake complex projective spaces are smoothable. Sullivan [Sullivan1996] gave some examples of both smoothable and non-smoothable fake \Cc P^n's. Weinberger [Weinberger1990] proved that for any smooth manifold M, the image of the forgetful map \mathcal{S}^{\Diff}(M) \to \mathcal{S}^{\Top}(M) contains a subgroup of finite index; in particular there are infinitely many smoothable fake complex projective spaces.

[edit] 6 References

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