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
Tex syntax erroris a closed manifold, any homotopy equivalence induces a principal -bundle over
Tex syntax errorwhose 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.
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.
Theorem 3.1. 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.
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 .
4 Classification/Characterization
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5 Further discussion
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