Petrie conjecture

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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.
* By {{cite|Tsukada&Washiyama1979}} and {{cite|Masuda1981}}, the Petrie conjecture is true if the fixed point set consists of three or four connected components.
* Masuda {{cite|Masuda1983}} proved the Petrie conjecture in the case where $M$ admits a specific smooth action of $T^k$ for $k \geq 2$.
* 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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* It follows from the work {{cite|James1985}} that the Petrie conjecture is true if $\dim M = 8$, i.e., $M \simeq \CP^4$.
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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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* The work of {{cite|Dessai2002}} confirms the Petrie conjecture if $M$ admits an appropriate smooth action of $Pin(2)$ and $\dim M \leq 22$.
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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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* According to {{cite|Dessai&Wilking2004}}, the Petrie conjecture holds if $M$ admits to a smooth action of $T^k$ and $\dim M \leq 8k-4$.
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* It follows from {{cite|Dessai&Wilking2004}} that the Petrie conjecture holds if $M$ admits to a smooth action of $T^k$ and $\dim M \leq 8k-4$.
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Revision as of 16:19, 3 December 2010

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

Contents

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\}. He proved that if S^1 acts smoothly on M \simeq \CP^n with isolated fixed points, then the Pontrjagin classes of 
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 are determined by the representations of S^1 at the fixed points. Having this as well as other results in mind, Petrie posed the following conjecture.

Conjecture 1.1 [Petrie1972].

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

2 Progress to date

As of November 30, 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 \mathbb{C}P^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 to a smooth action of T^k and \dim M \leq 8k-4.

3 Further discussion

A related problem posed by Masuda and Suh [Masuda&Suh2008] reads as follows. For two toric 2n-manifolds with isomorphic cohomology rings, is there an isomorphism between the cohomology rings which preserves the Pontrjagin classes of the two manifolds?

A symplectic version of the Petrie conjecture is discussed by Tolman [Tolman2010]. In particular, in the case where S^1 acts in a Hamiltonian way on a compact symplectic manifold 
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, is the total Chern class of 
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 determined by the cohomology ring H^*(M,\mathbb{Z})?

4 References

Retrieved from "http://www.map.mpim-bonn.mpg.de/index.php?title=Petrie_conjecture&oldid=6032"
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