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 [Wall1999, chapter 14C]. The surgery structure set of is in bijection to the set of free tame circle actions on modulo equivariant homeomorphism.
2.1 The supension map
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 [Sullivan1967]
2.2 The Madsen-Milgram construction
The previous construction has been generalized as follows [Madsen&Milgram1979]: Suppose that the map 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 over along to obtain a disk bundle over . The induced map may now fail to be a homotopy equivalence, but it is a degree one normal map which restricts to a degree one normal map on the boundary. As is null-bordant in , the surgery obstruction of is zero. An additional argument shows that is bordant to a homotopy equivalence via a normal cobordism such that
is a homotopy equivalence. Then, coning off and produces a homotopy equivalence from some closed -manifold to . The map has the pleasant feature that it is transverse to and the restriction of to a degree one normal map is the map we started with.
Hence, the suspension map extends to a map
which is split injective. In fact, the following holds:
Theorem 2.2. The map is a bijection.
This follows from the classification as described below: Both the domain and the target of are completely described by the first splitting invariants, and they remain unchanged under the Madsen-Milgram construction.
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 .
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. The suspension map is injective. Its image is given by the homotopy equivalences whose highest splitting invariant is zero.
An interesting feature of fake complex projective spaces is that their stable tangent bundle may differ from the one of the standard . Given a homotopy equivalence , in theory the total Hirzebruch -class may be computed inductively from the splitting invariants using the formula [Madsen&Milgram1979, Theorem 4.9]
where, by [Madsen&Milgram1979, Corollary 4.22], we have
The -invariant of a free tame circle action on may be explicitly expressed in terms of the splitting invariants of the corresponding homotopy equivalence :
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.
The proof of Theorem 4.1 is surgery-theoretic. In fact the splitting invariants are defined more generally on the set of normal invariants , where the surgery obstruction may be non-zero. Theorem 4.1 therefore follows immediately by applying the surgery exact sequence to the following homotopy-theoretic computation:
Theorem 4.2. The cartesian product
is a bijection.
5 Further discussion
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6 Refereces
- [Madsen&Milgram1979] I. Madsen and R. J. Milgram, The classifying spaces for surgery and cobordism of manifolds, Princeton University Press, Princeton, N.J., 1979. MR548575 (81b:57014) Zbl 0446.57002
- [Sullivan1967] D. Sullivan, On the Hauptvermutung for manifolds, Bull. Amer. Math. Soc. 73 (1967), 598–600. MR0212811 (35 #3676) Zbl 0153.54002
- [Wall1999] C. T. C. Wall, Surgery on compact manifolds, American Mathematical Society, Providence, RI, 1999. MR1687388 (2000a:57089) Zbl 0935.57003