# Fake real projective spaces

## 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$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\Rr P^4$ and [Fintushel&Stern1983] for fake smooth $\Rr P^{4k}$$\Rr P^{4k}$.

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

## 3 Invariants

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

## 4 Classification

#### 4.1 Homotopy classification

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

#### 4.2 Homeomorphism classification of topological actions

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

Theorem 4.1. For $n\ge 5$$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$$[ \Rr P^n, G/Top] \cong \Zz_2^k$ where $k=\lfloor \frac{n}{2} \rfloor$$k=\lfloor \frac{n}{2} \rfloor$.

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

Proposition 4.5 [Wall1999, Theorem 13.A.1]. The $L$$L$-groups of $\Zz_2$$\Zz_2$ are: $\displaystyle \begin{array}{c|cccc} n\text{ mod }4 & 0 & 1 & 2 & 3 \\ \hline L_n(\Zz_2,w=1) & \Zz\oplus \Zz & 0 & \Zz_2 & \Zz_2 \\ L_n(\Zz_2,w=-1) & \Zz_2 & 0 & \Zz_2 & 0 \end{array}$

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

Recall that the action of the $L$$L$-groups on the structure set uses the plumbing construction, in particular the action of the image of $L_n(1)$$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.

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