Petrie conjecture

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{{beginthm|Conjecture|(Petrie conjecture)}}
{{beginthm|Conjecture|(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 $M$ agrees with that of $\CP^n$, i.e.,
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.,
$p(M) = (1+x^2)^{n+1}$ for a generator $x$ of $H^2(M; \mathbb{Z})$.
+
$p(M) = (1+x^2)^{n}$ for a generator $x$ of $H^2(M; \mathbb{Z})$.
{{endthm}}
{{endthm}}
</wikitex>
</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
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, what constraints does this place on the topology of
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in general and on the Pontrjagin classes of
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in particular? In the case where
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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
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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
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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
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    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
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    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
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    admits an appropriate smooth action of Pin(2) and \dim M \leq 22.
  • It follows from [Dessai&Wilking2004] that the Petrie conjecture holds if
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    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
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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
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determined by the cohomology ring H^*(M;\Zz)?

[edit] 4 References

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