Petrie conjecture

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{{Stub}}== Introduction ==
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{{Stub}}
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== Problem ==
<wikitex>;
<wikitex>;
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The Petrie conjecture was formulated in the following context: suppose that a Lie group $G$ acts smoothly on a closed smooth manifold $M$, what constraints does this place on the topology of $M$ in general and on the Pontrjagin classes of $M$ in particular.
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If a compact Lie group $G$ acts smoothly and non-trivially on a closed smooth manifold $M$, what constraints does this place on the topology of $M$ in general and on the Pontrjagin classes of $M$ in particular?
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In the case where $M$ is [[Fake complex projective spaces|homotopy equivalent to $\CP^n$]], $M \simeq \CP^n$, Petrie {{cite|Petrie1972}} restricted his attention to actions of the Lie group
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$S^1 = \{z \in \mathbb{C} \ \colon \ |z| = 1\}$ and proved that if the fixed point set of the action consists only of isolated fixed points, then the Pontrjagin classes of $M$ are determined by the representations of $S^1$ at the fixed points. Motivated by this result, Petrie {{cite|Petrie1972}} posed the following conjecture.
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Petrie restricted his attention to smooth actions of the Lie group $S^1$, the circle {{cite|Petrie1972}} (or more generally, the torus $T^k$ for $k \geq 1$ {{cite|Petrie1973}})
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{{beginthm|Conjecture|(Petrie conjecture)}}
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on closed smooth manifolds $M$ which are [[Fake complex projective spaces|homotopy equivalent to $\CP^n$]].
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Suppose that $S^1$ acts smoothly and non-trivially on a closed smooth $2n$-manifold $M \simeq \CP^n$. Then the total Pontrjagin class $p(M)$ of $M$ agrees with that of $\CP^n$, i.e.,
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$p(M) = (1+x^2)^{n}$ for a generator $x$ of $H^2(M; \mathbb{Z})$.
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{{endthm}}
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</wikitex>
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{{beginthm|Conjecture|{{cite|Petrie1972}}}}
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== Progress to date ==
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Suppose that $M$ is a closed smooth $2n$-manifold homotopy equivalent to $\CP^n$ and that $S^1$ acts smoothly and non-trivially on $M$. Then the total Pontrjagin class $p(M)$ of $M$ agrees with that of $\CP^n$, i.e., for a generator $x \in H^2(M; \mathbb{Z})$, $$p(M) = (1+x^2)^{n+1}.$$
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<wikitex>;
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As of December 21, 2010, the Petrie conjecture has not been confirmed in general. However, it has been proven in the following special cases.
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* Petrie {{cite|Petrie1973}} has verified his conjecture under the assumption that the manifold $M \simeq \CP^n$ admits a smooth action of the torus $T^n$.
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* By the work of {{cite|Dejter1976}}, the Petrie conjecture is true if $\dim M = 6$, i.e., $M \simeq \CP^3$ and hence, if $\dim M \leq 6$.
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* Related results go back to {{cite|Musin1978}} and {{cite|Musin1980}}, in particular, the latter work shows that the Petrie conjecture holds if $\dim M = 8$, i.e., $M \simeq \CP^4$.
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* According to {{cite|Hattori1978}}, the Petrie conjecture holds if $M$ admits an invariant almost complex structure with the first Chern class $c_1(M) = (n+1)x$.
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* Other special cases where the Petrie conjecture holds are described by {{cite|Wang1975}}, {{cite|Yoshida1975/76}}, {{cite|Iberkleid1978}}.
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* By {{cite|Tsukada&Washiyama1979}} and {{cite|Masuda1981}}, the Petrie conjecture is true if the fixed point set consists of three or four connected components.
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* Masuda {{cite|Masuda1983}} proved the Petrie conjecture in the case where $M$ admits a specific smooth action of $T^k$ for $k \geq 2$.
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* The work of {{cite|James1985}} confirms the result of {{cite|Musin1980}} that the Petrie conjecture is true if $\dim M = 8$, i.e., $M \simeq \CP^4$.
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* According to {{cite|Dessai2002}}, the Petrie conjecture holds if $M$ admits an appropriate smooth action of $Pin(2)$ and $\dim M \leq 22$.
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* It follows from {{cite|Dessai&Wilking2004}} that the Petrie conjecture holds if $M$ admits a smooth action of $T^k$ and $\dim M \leq 8k-4$.
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</wikitex>
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== Further discussion ==
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<wikitex>;
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Masuda and Suh {{cite|Masuda&Suh2008}} posed the following question about the invariance of Pontrjagin classes for toric $2n$-manifolds.
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{{beginthm|Question}}
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For two toric $2n$-manifolds with isomorphic cohomology rings, is it true that any isomorphism between the cohomology rings preserves the Pontrjagin classes of the two manifolds?
{{endthm}}
{{endthm}}
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Petrie {{cite|Petrie1973}} has shown that the total Pontrjagin class of $M$ agrees with that of $\CP^n$ if the action of $S^1$ on $M$ extends to a smooth action of $T^n$.
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A symplectic version of the Petrie conjecture is discussed by Tolman {{cite|Tolman2010}}. In particular, the following question has been posed.
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The Petrie conjecture has not been confirmed in general, but if $\dim M \leq 8$, the statement is true by the work of {{cite|Dejter1976}} and {{cite|James1985}}.
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{{beginthm|Question}}
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According to {{cite|Hattori1978}}, the conjecture is also true if $M$ admits an invarint almost complex structure with appropriate first Chern class. Another special cases
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If the circle $S^1$ acts in a Hamiltonian way on a compact symplectic manifold $M$ with $H^{2i}(M;\Rr) \cong H^{2i}(\CP^n; \Rr)$ for all $i \geq 0$,
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where the Petrie conjecture holds are described by {{cite|Wang1975}}, {{cite|Yoshida1976}}, {{cite|Iberkleid1978}}, {{cite|Muslin1978}}, {{cite|Tsukada&Washiyama1979}}, and
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is it true that $H^{j}(M;\Zz) \cong H^{j}(\CP^n; \Zz)$ for all $j \geq 0$? Is the total Chern class of $M$ determined by the cohomology ring $H^*(M;\Zz)$?
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{{cite|Masuda1981}}. Moreover, by the work of {{cite|Dessai2002}}, the Petrie conjecture holds if $\dim M \leq 22$ and the action of $S^1$ on $M$ extends to an appropriate action of $Pin(2)$.
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{{endthm}}
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Similarly, the work of {{cite|Dessai&Wilking2004}} confirms the Petrie conjecture under the assumption that $\dim M \leq 8k-4$ and the action of $S^1$ on $M$ extends to an effective smooth action of $T^k$
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</wikitex>
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for $k \geq 1$.
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== References ==
== References ==
{{#RefList:}}
{{#RefList:}}
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[[Category:Problems]]
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[[Category:Group actions on manifolds]]
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</wikitex>

Latest revision as of 01:12, 25 October 2012

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

Contents

[edit] 1 Problem

If a compact Lie group G acts smoothly and non-trivially on a closed smooth manifold 
Tex syntax error
, what constraints does this place on the topology of 
Tex syntax error
 in general and on the Pontrjagin classes of 
Tex syntax error
 in particular? In the case where 
Tex syntax error
 is homotopy equivalent to \CP^n, M \simeq \CP^n, Petrie [Petrie1972] restricted his attention to actions of the Lie group S^1 = \{z \in \mathbb{C} \ \colon \ |z| = 1\} and proved that if the fixed point set of the action consists only of isolated fixed points, then the Pontrjagin classes of 
Tex syntax error
 are determined by the representations of S^1 at the fixed points. Motivated by this result, Petrie [Petrie1972] posed the following conjecture.

Conjecture 1.1 (Petrie conjecture).

Suppose that S^1 acts smoothly and non-trivially on a closed smooth 2n-manifold M \simeq \CP^n. Then the total Pontrjagin class p(M) of 
Tex syntax error
 agrees with that of \CP^n, i.e.,

p(M) = (1+x^2)^{n} for a generator x of H^2(M; \mathbb{Z}).

[edit] 2 Progress to date

As of December 21, 2010, the Petrie conjecture has not been confirmed in general. However, it has been proven in the following special cases.

  • Petrie [Petrie1973] has verified his conjecture under the assumption that the manifold M \simeq \CP^n admits a smooth action of the torus T^n.
  • By the work of [Dejter1976], the Petrie conjecture is true if \dim M = 6, i.e., M \simeq \CP^3 and hence, if \dim M \leq 6.
  • Related results go back to [Musin1978] and [Musin1980], in particular, the latter work shows that the Petrie conjecture holds if \dim M = 8, i.e., M \simeq \CP^4.
  • According to [Hattori1978], the Petrie conjecture holds if 
    Tex syntax error
     admits an invariant almost complex structure with the first Chern class c_1(M) = (n+1)x.
  • Other special cases where the Petrie conjecture holds are described by [Wang1975], [Yoshida1975/76], [Iberkleid1978].
  • By [Tsukada&Washiyama1979] and [Masuda1981], the Petrie conjecture is true if the fixed point set consists of three or four connected components.
  • Masuda [Masuda1983] proved the Petrie conjecture in the case where 
    Tex syntax error
     admits a specific smooth action of T^k for k \geq 2.
  • The work of [James1985] confirms the result of [Musin1980] that the Petrie conjecture is true if \dim M = 8, i.e., M \simeq \CP^4.
  • According to [Dessai2002], the Petrie conjecture holds if 
    Tex syntax error
     admits an appropriate smooth action of Pin(2) and \dim M \leq 22.
  • It follows from [Dessai&Wilking2004] that the Petrie conjecture holds if 
    Tex syntax error
     admits a smooth action of T^k and \dim M \leq 8k-4.

[edit] 3 Further discussion

Masuda and Suh [Masuda&Suh2008] posed the following question about the invariance of Pontrjagin classes for toric 2n-manifolds.

Question 3.1. For two toric 2n-manifolds with isomorphic cohomology rings, is it true that any isomorphism between the cohomology rings preserves the Pontrjagin classes of the two manifolds?

A symplectic version of the Petrie conjecture is discussed by Tolman [Tolman2010]. In particular, the following question has been posed.

Question 3.2.

If the circle S^1 acts in a Hamiltonian way on a compact symplectic manifold 
Tex syntax error
 with H^{2i}(M;\Rr) \cong H^{2i}(\CP^n; \Rr) for all i \geq 0, is it true that H^{j}(M;\Zz) \cong H^{j}(\CP^n; \Zz) for all j \geq 0? Is the total Chern class of 
Tex syntax error
 determined by the cohomology ring H^*(M;\Zz)?

[edit] 4 References

Retrieved from "http://www.map.mpim-bonn.mpg.de/index.php?title=Petrie_conjecture&oldid=9504"
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