# 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.

Suspension: In the topological case the join of a free involution on $S^n$$ {{Stub}} == 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. == 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. 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}. == Invariants == ; Splitting invariants. Browder-Livesay invariant. == Classification/Characterization == ==== Homotopy classification ==== ; The orbit space of a free involution on S^n is homotopy equivalent to \Rr P^n. <\wikitex> ==== 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. {{beginthm|Theorem|{{cite|Lopez de Medrano1971}}}} For n\ge 5: 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. {{endthm}} ==== Proof ==== ; {{beginthm|Proposition}} [ \Rr P^n, G/Top] \cong \Zz_2^k where k=\lfloor \frac{n}{2} \rfloor . {{endthm}} This follows from Sullivan's result about the -local structure of G/Top (see {{cite|Madsen&Milgram1979|Remark 4.36}}), using the Puppe sequence for \Rr P^{n-1}\subseteq \Rr P^n and induction. {{beginthm|Proposition}} The normal invariant of a homotopy projective space is the restriction of the normal invariant of its suspension. {{endthm}} {{beginthm|Proposition}} The \rho-invariant is injective on the fibers of S(\Rr P^{4k+3})\to [\Rr P^{4k+3},G/Top]. {{endthm}} {{beginthm|Proposition|{{cite|Wall1999|Theorem 13.A.1}}}} The L-groups of \Zz_2 are: \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} {{endthm}} {{beginthm|Proposition}} 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. {{endthm}} 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 {{cite|Lopez de Medrano1971}}) \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. == Further discussion == ; ... == References == {{#RefList:}} [[Category:Manifolds]]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.

## 4 Classification/Characterization

#### 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$. <\wikitex>

#### 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 [Lopez de Medrano1971]. For $n\ge 5$$n\ge 5$:

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

#### 4.2 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:

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

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