Fake complex projective spaces
Contents |
1 Introduction
A fake complex projective space is a topological manifold which is homotopy equivalent to a complex projective space for some .
2 Construction and examples
Given a free tame action of the circle on a -sphere, the orbit space is a fake . 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 of is in bijection to the set of free tame circle actions on modulo -equivariant homeomorphism.
Given a homotopy equivalence , we can suspend to obtain a fake as follows: Denote by the disk bundle of the canonical complex line bundle over . 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
Obviously the homology and homotopy groups of a fake complex projective space are isomorphic to the ones of the . Different fake complex projective spaces may be distinguished using the so-called splitting invariants. More precisely, for any , there is a function
from the surgery structure set of to 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.
The -invariant of a free tame circle action on may be explicitly expressed in terms of the splitting invariants of the corresponding homotopy equivalence :
Theorem 3.3. For , we have
where , and .
4 Classification/Characterization
The surgery structure set of may be completely described using the splitting invariants:
Theorem 4.1. The cartesian product
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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