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 \Rr P^4 and [Fintushel&Stern1983] for fake smooth \Rr P^{4k}.

Suspension: In the topological case the join of a free involution on S^n with the free involution on S^0 is a free involution on S^{n+1}.

3 Invariants

Splitting invariants. Browder-Livesay invariant. Rho/Eta invariant.

4 Classification/Characterization

4.1 Homotopy classification

The orbit space of a free involution on S^n is homotopy equivalent to \Rr P^n.

4.2 Homeomorphism classification of topological actions

Thus for a homeomorphism classification for n\ge 5 we compute the structure set of \Rr P^n using the surgery exact sequence.

Theorem 4.1. For n\ge 5:

\displaystyle  S(\Rr P^n)=\left\{ \begin{array}{cl} \Zz_2^{2k-1}         & \text{ for } n=4k \\                                         \Zz_2^{2k}           & \text{ for } n=4k+1, 4k+2 \\                                          \Zz_2^{2k}\oplus \Zz & \text{ for } n=4k+3  \end{array}\right.

4.3 Proof

Proposition 4.2. [ \Rr P^n, G/Top] \cong \Zz_2^k where k=\lfloor \frac{n}{2} \rfloor.

This follows from Sullivan's result about the 2-local structure of G/Top (see [Madsen&Milgram1979, Remark 4.36]), using the Puppe sequence for \Rr P^{n-1}\subseteq \Rr P^n 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 \rho-invariant is injective on the fibers of S(\Rr P^{4k+3})\to [\Rr P^{4k+3},G/Top].

Proposition 4.5 [Wall1999, Theorem 13.A.1]. The L-groups of \Zz_2 are:

\displaystyle \begin{array}{c|cccc} n\text{ mod }4 & 0 & 1 & 2 & 3 \\                         \hline                         L_n(\Z_2,w=1)  & \Zz\oplus \Zz & 0 & \Zz_2 & \Zz_2 \\                         L_n(\Z_2,w=-1) & \Zz_2 & 0 & \Zz_2 & 0 \end{array}

Proposition 4.6. The surgery obstruction of \Rr P^n\to G/Top equals the obstruction for its restriction \Rr P^{n-1}\to G/Top for n congruent to 0 and -1 modulo 4.

Recall that the action of the L-groups on the structure set uses the plumbing construction, in particular the action of the image of L_n(1) is by connected sum with a homotopy sphere, which is trivial in the topological case.

One obtains the diagram (taken from [Lopez de Medrano1971])

\displaystyle  \xymatrix{ \Zz_2           \ar[r] & S(\Rr P^{4k+1}) \ar[d]  \ar[r]  &  [\Rr P^{4k+1},G/Top]          \ar[r]  & 0     \\ 0               \ar[r] & S(\Rr P^{4k+2}) \ar[d]  \ar[r]  &  [\Rr P^{4k+2},G/Top]   \ar[u] \ar[r]  & \Zz_2 \\ \Zz\oplus \Zz   \ar[r] & S(\Rr P^{4k+3}) \ar[d]  \ar[r]  &  [\Rr P^{4k+3},G/Top]   \ar[u] \ar[r]  & \Zz_2 \\ 0               \ar[r] & S(\Rr P^{4k+4}) \ar[d]  \ar[r]  &  [\Rr P^{4k+4},G/Top]   \ar[u] \ar[r]  & \Zz_2 \\ \Zz_2           \ar[r] & S(\Rr P^{4k+5})         \ar[r]  &  [\Rr P^{4k+5},G/Top]   \ar[u] \ar[r]  & 0 }

in which the preceding propositions determine all maps.

5 Further discussion


6 References

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