Fake real projective spaces
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1 Introduction
A fake real projective space is a manifold homotopy equivalent to real projective space. Equivalently, it is the orbit space of a free involution on a (homotopy) sphere.
2 Construction and examples
Besides ordinary real projective spaces, the construction of tame free circle actions on odd-dimensional spheres (fake complex projective spaces) gives free involutions on odd-dimensional spheres. Free involutions on Brieskorn spheres. [Cappell&Shaneson1976a] give examples of exotic smooth and [Fintushel&Stern1983] for fake smooth .
Suspension: In the topological case the join of a free involution on with the free involution on is a free involution on .
3 Invariants
Splitting invariants. Browder-Livesay invariant. Rho/Eta invariant.
4 Classification
4.1 Homotopy classification
The orbit space of a free involution on is homotopy equivalent to .
4.2 Homeomorphism classification of topological actions
Thus for a homeomorphism classification for we compute the structure set of using the surgery exact sequence.
Theorem 4.1. For :
4.3 Proof
Proposition 4.2. where .
This follows from Sullivan's result about the -local structure of (see [Madsen&Milgram1979, Remark 4.36]), using the Puppe sequence for and induction.
Proposition 4.3. The normal invariant of a homotopy projective space is the restriction of the normal invariant of its suspension.
Proposition 4.4. The -invariant is injective on the fibers of .
Proposition 4.5 [Wall1999, Theorem 13.A.1]. The -groups of are:
Proposition 4.6. The surgery obstruction of equals the obstruction for its restriction for congruent to 0 and -1 modulo 4.
Recall that the action of the -groups on the structure set uses the plumbing construction, in particular the action of the image of is by connected sum with a homotopy sphere, which is trivial in the topological case.
One obtains the diagram (taken from [Lopez de Medrano1971])
in which the preceding propositions determine all maps.
5 Further discussion
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6 References
- [Cappell&Shaneson1976a] S. E. Cappell and J. L. Shaneson, Some new four-manifolds, Ann. of Math. (2) 104 (1976), no.1, 61–72. MR0418125 (54 #6167) Zbl 0345.57003
- [Fintushel&Stern1983] R. Fintushel and R. J. Stern, Smooth free involutions on homotopy -spheres, Michigan Math. J. 30 (1983), no.1, 37–51. MR694927 (84f:57025) Zbl 0543.57023
- [Lopez de Medrano1971] S. López de Medrano, Involutions on manifolds, Springer-Verlag, 1971. MR0298698 (45 #7747) Zbl 0214.22501
- [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
- [Wall1999] C. T. C. Wall, Surgery on compact manifolds, American Mathematical Society, Providence, RI, 1999. MR1687388 (2000a:57089) Zbl 0935.57003