Fake real projective spaces
(Created page with "<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - Fo...") |
m (→Proof) |
||
(4 intermediate revisions by 2 users not shown) | |||
Line 1: | Line 1: | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
{{Stub}} | {{Stub}} | ||
== Introduction == | == Introduction == | ||
Line 22: | Line 10: | ||
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 $S^{n+1}$. | ||
Line 28: | Line 19: | ||
== Invariants == | == Invariants == | ||
<wikitex>; | <wikitex>; | ||
− | Splitting invariants. Browder-Livesay invariant. | + | Splitting invariants. Browder-Livesay invariant. Rho/Eta invariant. |
</wikitex> | </wikitex> | ||
− | == Classification | + | == Classification == |
− | ==== | + | ==== 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 | + | {{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 \\ | ||
Line 53: | Line 46: | ||
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(\ | + | L_n(\Zz_2,w=1) & \Zz\oplus \Zz & 0 & \Zz_2 & \Zz_2 \\ |
− | L_n(\ | + | L_n(\Zz_2,w=-1) & \Zz_2 & 0 & \Zz_2 & 0 \end{array} $$ |
{{endthm}} | {{endthm}} | ||
− | |||
{{beginthm|Proposition}} | {{beginthm|Proposition}} | ||
The surgery obstruction of $\Rr P^n\to G/Top$ equals the obstruction for its restriction | The surgery obstruction of $\Rr P^n\to G/Top$ equals the obstruction for its restriction | ||
$\Rr P^{n-1}\to G/Top$ for $n$ congruent to 0 and -1 modulo 4. | $\Rr P^{n-1}\to G/Top$ for $n$ congruent to 0 and -1 modulo 4. | ||
{{endthm}} | {{endthm}} | ||
− | |||
Recall that the action of the $L$-groups on the structure set uses the plumbing construction, | Recall that the action of the $L$-groups on the structure set uses the plumbing construction, | ||
in particular the action of the image of $L_n(1)$ is by connected sum with a homotopy sphere, | in particular the action of the image of $L_n(1)$ is by connected sum with a homotopy sphere, | ||
Line 84: | Line 69: | ||
One obtains the diagram (taken from {{cite|Lopez de Medrano1971}}) | One obtains the diagram (taken from {{cite|Lopez de Medrano1971}}) | ||
− | |||
$$ | $$ | ||
\xymatrix{ | \xymatrix{ | ||
Line 95: | Line 79: | ||
$$ | $$ | ||
in which the preceding propositions determine all maps. | in which the preceding propositions determine all maps. | ||
− | |||
</wikitex> | </wikitex> | ||
Line 105: | Line 88: | ||
== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
− | |||
− | |||
[[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 and [Fintushel&Stern1983] for fake smooth .
Suspension: In the topological case the join of a free involution on with the free involution on is a free involution on .
[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 is homotopy equivalent to .
[edit] 4.2 Homeomorphism classification of topological actions
Thus for a homeomorphism classification for we compute the structure set of using the surgery exact sequence.
Theorem 4.1. For :
[edit] 4.3 Proof
Proposition 4.2. where .
This follows from Sullivan's result about the -local structure of (see [Madsen&Milgram1979, Remark 4.36]), using the Puppe sequence for 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 -invariant is injective on the fibers of .
Proposition 4.5 [Wall1999, Theorem 13.A.1]. The -groups of are:
Proposition 4.6. The surgery obstruction of equals the obstruction for its restriction for congruent to 0 and -1 modulo 4.
Recall that the action of the -groups on the structure set uses the plumbing construction, in particular the action of the image of is by connected sum with a homotopy sphere, which is trivial in the topological case.
One obtains the diagram (taken from [Lopez de Medrano1971])
in which the preceding propositions determine all maps.
[edit] 5 Further discussion
...
[edit] 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 -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