Fake real projective spaces
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== Classification/Characterization == | == Classification/Characterization == | ||
+ | ==== Homotopy classification ==== | ||
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+ | The orbit space of a free involution on $S^n$ is homotopy equivalent to $\Rr P^n$. | ||
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==== Homeomorphism classification of topological actions ==== | ==== Homeomorphism classification of topological actions ==== | ||
<wikitex>; | <wikitex>; | ||
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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. | ||
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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. | ||
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{{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}} | ||
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{{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}} | ||
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{{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: | ||
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L_n(\Z_2,w=-1) & \Zz_2 & 0 & \Zz_2 & 0 \end{array} $$ | L_n(\Z_2,w=-1) & \Zz_2 & 0 & \Zz_2 & 0 \end{array} $$ | ||
{{endthm}} | {{endthm}} | ||
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{{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}} | ||
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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}}) | ||
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$$ | $$ | ||
\xymatrix{ | \xymatrix{ | ||
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$$ | $$ | ||
in which the preceding propositions determine all maps. | in which the preceding propositions determine all maps. | ||
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</wikitex> | </wikitex> | ||
Revision as of 14:07, 2 December 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
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 with the free involution on is a free involution on .
3 Invariants
Splitting invariants. Browder-Livesay invariant.
4 Classification/Characterization
4.1 Homotopy classification
The orbit space of a free involution on is homotopy equivalent to . <\wikitex>
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 [Lopez de Medrano1971]. For :
4.2 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.
5 Further discussion
...
6 References
- [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