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 for some . 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 -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.
Edmonds [Edmonds1977] has shown that the latter set agrees with the set of all free circle actions on modulo a weaker relation called concordance, provided .
[edit] 2.1 The suspension 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 PoincarĂ© 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 [Sullivan1996]
[edit] 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. For , 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.
[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 . 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. Let . Then the suspension map is injective, and its image is given by the homotopy equivalences 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 . 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
[edit] 3.3 The rho-invariant
The rho-invariant of a free tame circle action on may be explicitly expressed in terms of the splitting invariants of the corresponding homotopy equivalence :
[edit] 4 Classification/Characterization
The surgery classification of fake complex projective spaces was initiated by Brumfiel [Brumfiel1969a].
The surgery structure set of may be completely described using the splitting invariants:
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, which goes back to Sullivan [Sullivan1996].
[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 's. Weinberger [Weinberger1990] proved that for any smooth manifold , the image of the forgetful map contains a subgroup of finite index; in particular there are infinitely many smoothable fake complex projective spaces.
[edit] 6 References
- [Brumfiel1969a] G. Brumfiel, Differentiable -actions on homotopy spheres, unpublished Princeton notes (1969).
- [Edmonds1977] A. L. Edmonds, Taming free circle actions, Proc. Amer. Math. Soc. 62 (1977), no.2, 337–343. MR0431230 (55 #4231) Zbl 0362.57017
- [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
- [Sullivan1996] D. P. Sullivan, Triangulating and smoothing homotopy equivalences and homeomorphisms. Geometric Topology Seminar Notes, 1 (1996), 69–103. MR1434103 (98c:57027) Zbl 0871.57021
- [Wall1999] C. T. C. Wall, Surgery on compact manifolds, American Mathematical Society, Providence, RI, 1999. MR1687388 (2000a:57089) Zbl 0935.57003
- [Weinberger1990] S. Weinberger, On smooth surgery, Comm. Pure Appl. Math. 43 (1990), no.5, 695–696. MR1057237 (91c:57037) Zbl 0715.57011