Fake real projective spaces

This page has not been refereed. The information given here might be incomplete or provisional.

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$${{Stub}} == Introduction == <wikitex>; A '''fake real projective space''' is a manifold homotopy equivalent to real projective space. Equivalently, it is the orbit space of a free involution on a (homotopy) sphere. </wikitex> == Construction and examples == <wikitex>; Besides ordinary real projective spaces, the construction of tame free circle actions on odd-dimensional spheres (fake complex projective spaces) gives free involutions on odd-dimensional spheres. Free involutions on Brieskorn spheres. {{cite|Cappell&Shaneson1976a}} give examples of exotic smooth $\Rr P^4$ and {{cite|Fintushel&Stern1983}} for fake smooth $\Rr P^{4k}$. Suspension: In the topological case the join of a free involution on $S^n$ with the free involution on $S^0$ is a free involution on $S^{n+1}$. </wikitex> == Invariants == <wikitex>; Splitting invariants. Browder-Livesay invariant. Rho/Eta invariant. </wikitex> == Classification == ==== Homotopy classification ==== <wikitex>; The orbit space of a free involution on $S^n$ is homotopy equivalent to $\Rr P^n$. </wikitex> ==== Homeomorphism classification of topological actions ==== <wikitex>; Thus for a homeomorphism classification for $n\ge 5$ we compute the structure set of $\Rr P^n$ using the surgery exact sequence. {{beginthm|Theorem}} For $n\ge 5$: $$ 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. $$ {{endthm}} </wikitex> ==== Proof ==== <wikitex>; {{beginthm|Proposition}} $ [ \Rr P^n, G/Top] \cong \Zz_2^k$ where $k=\lfloor \frac{n}{2} \rfloor $. {{endthm}} This follows from Sullivan's result about the $-local structure of $G/Top$ (see {{cite|Madsen&Milgram1979|Remark 4.36}}), using the Puppe sequence for $\Rr P^{n-1}\subseteq \Rr P^n$ and induction. {{beginthm|Proposition}} The normal invariant of a homotopy projective space is the restriction of the normal invariant of its suspension. {{endthm}} {{beginthm|Proposition}} The $\rho$-invariant is injective on the fibers of $S(\Rr P^{4k+3})\to [\Rr P^{4k+3},G/Top]$. {{endthm}} {{beginthm|Proposition|{{cite|Wall1999|Theorem 13.A.1}}}} The $L$-groups of $\Zz_2$ are: $$\begin{array}{c|cccc} n\text{ mod }4 & 0 & 1 & 2 & 3 \ \hline L_n(\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} $$ {{endthm}} {{beginthm|Proposition}} The surgery obstruction of $\Rr P^n\to G/Top$ equals the obstruction for its restriction $\Rr P^{n-1}\to G/Top$ for $n$ congruent to 0 and -1 modulo 4. {{endthm}} Recall that the action of the $L$-groups on the structure set uses the plumbing construction, in particular the action of the image of $L_n(1)$ is by connected sum with a homotopy sphere, which is trivial in the topological case. One obtains the diagram (taken from {{cite|Lopez de Medrano1971}}) $$ \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 } $$ in which the preceding propositions determine all maps. </wikitex> == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}} [[Category:Manifolds]]\Rr P^4$ and [Fintushel&Stern1983] for fake smooth $\Rr P^{4k}$$\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$:

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

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:

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

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.

5 Further discussion

...

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 $4k$$4k$-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