Fake real projective spaces

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{{Stub}}
{{Stub}}
== Introduction ==
== Introduction ==
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Besides ordinary real projective spaces, the construction of tame free circle actions on
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.
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 $S^{n+1}$.
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 $S^{n+1}$.
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== Invariants ==
== Invariants ==
<wikitex>;
<wikitex>;
Splitting invariants. Browder-Livesay invariant.
+
Splitting invariants. Browder-Livesay invariant. Rho/Eta invariant.
</wikitex>
</wikitex>
== Classification/Characterization ==
+
== Classification ==
==== Homeomorphism classification of topological actions ====
+
==== Homotopy classification ====
<wikitex>;
<wikitex>;
The orbit space of a free involution on $S^n$ is homotopy equivalent to $\Rr P^n$.
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.
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}}}}
+
{{beginthm|Theorem}}
For $n\ge 5$:
For $n\ge 5$:
$$ S(\Rr P^n)=\left\{ \begin{array}{cl} \Zz_2^{2k-1} & \text{ for } n=4k \\
$$ S(\Rr P^n)=\left\{ \begin{array}{cl} \Zz_2^{2k-1} & \text{ for } n=4k \\
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This follows from Sullivan's result about the $2$-local structure of $G/Top$ (see {{cite|Madsen&Milgram1979|Remark 4.36}}),
This follows from Sullivan's result about the $2$-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.
using the Puppe sequence for $\Rr P^{n-1}\subseteq \Rr P^n$ and induction.
{{beginthm|Proposition}}
{{beginthm|Proposition}}
The normal invariant of a homotopy projective space is the restriction of the normal
The normal invariant of a homotopy projective space is the restriction of the normal
invariant of its suspension.
invariant of its suspension.
{{endthm}}
{{endthm}}
{{beginthm|Proposition}}
{{beginthm|Proposition}}
The $\rho$-invariant is injective on the fibers of $S(\Rr P^{4k+3})\to [\Rr P^{4k+3},G/Top]$.
The $\rho$-invariant is injective on the fibers of $S(\Rr P^{4k+3})\to [\Rr P^{4k+3},G/Top]$.
{{endthm}}
{{endthm}}
{{beginthm|Proposition|{{cite|Wall1999|Theorem 13.A.1}}}}
{{beginthm|Proposition|{{cite|Wall1999|Theorem 13.A.1}}}}
The $L$-groups of $\Zz_2$ are:
The $L$-groups of $\Zz_2$ are:
$$\begin{array}{c|cccc} n\text{ mod }4 & 0 & 1 & 2 & 3 \\
$$\begin{array}{c|cccc} n\text{ mod }4 & 0 & 1 & 2 & 3 \\
\hline
\hline
L_n(\Z_2,w=1) & \Zz\oplus \Zz & 0 & \Zz_2 & \Zz_2 \\
+
L_n(\Zz_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} $$
+
L_n(\Zz_2,w=-1) & \Zz_2 & 0 & \Zz_2 & 0 \end{array} $$
{{endthm}}
{{endthm}}
{{beginthm|Proposition}}
{{beginthm|Proposition}}
The surgery obstruction of $\Rr P^n\to G/Top$ equals the obstruction for its restriction
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.
$\Rr P^{n-1}\to G/Top$ for $n$ congruent to 0 and -1 modulo 4.
{{endthm}}
{{endthm}}
Recall that the action of the $L$-groups on the structure set uses the plumbing construction,
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,
in particular the action of the image of $L_n(1)$ is by connected sum with a homotopy sphere,
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One obtains the diagram (taken from {{cite|Lopez de Medrano1971}})
One obtains the diagram (taken from {{cite|Lopez de Medrano1971}})
$$
$$
\xymatrix{
\xymatrix{
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$$
$$
in which the preceding propositions determine all maps.
in which the preceding propositions determine all maps.
</wikitex>
</wikitex>
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== References ==
== References ==
{{#RefList:}}
{{#RefList:}}
<!-- Please modify these headings or choose other headings according to your needs. -->
[[Category:Manifolds]]
[[Category:Manifolds]]

Latest revision as of 13:12, 12 December 2010

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

Contents

[edit] 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.

[edit] 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}.

[edit] 3 Invariants

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

[edit] 4 Classification

[edit] 4.1 Homotopy classification

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

[edit] 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.

[edit] 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(\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 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.

[edit] 5 Further discussion

...

[edit] 6 References

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