Petrie conjecture

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* By the work of {{cite|Dejter1976}} and {{cite|James1985}}, the Petrie conjecture is true if $\dim M \leq 6$ and $8$, respectively.
* By the work of {{cite|Dejter1976}} and {{cite|James1985}}, the Petrie conjecture is true if $\dim M \leq 6$ and $8$, respectively.
* According to {{cite|Hattori1978}}, the Petrie conjecture holds if $M$ admits an invariant almost complex structure with the first Chern class $c_1(M) = (n+1)x$. Remember $\dim M = 2n$.
* According to {{cite|Hattori1978}}, the Petrie conjecture holds if $M$ admits an invariant almost complex structure with the first Chern class $c_1(M) = (n+1)x$. Remember $\dim M = 2n$.
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* Other special cases where the Petrie conjecture holds are described by {{cite|Wang1975}}, {{cite|Yoshida1975/76}}, {{cite|Iberkleid1978}}, {{cite|Muslin1978}}, {{cite|Tsukada&Washiyama1979}}, and {{cite|Masuda1981}}.
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* Other special cases where the Petrie conjecture holds are described by {{cite|Wang1975}}, {{cite|Yoshida1975/76}}, {{cite|Iberkleid1978}}, and {{cite|Muslin1978}}.
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* It follows from {{cite|Tsukada&Washiyama1979}}, {{cite|Masuda1981}}, and {{cite|Masuda1983}}, that the Petrie conjecture holds
* The work of {{cite|Dessai2002}} confirms the Petrie conjecture under the assumption that the action of $S^1$ on $M$ extends to an appropriate action of $Pin(2)$ and $\dim M \leq 22$.
* The work of {{cite|Dessai2002}} confirms the Petrie conjecture under the assumption that the action of $S^1$ on $M$ extends to an appropriate action of $Pin(2)$ and $\dim M \leq 22$.
* According to {{cite|Dessai&Wilking2004}}, the Petrie conjecture holds if the action of $S^1$ on $M$ extends to a smooth action of the torus $T^k$ and $\dim M \leq 8k-4$.
* According to {{cite|Dessai&Wilking2004}}, the Petrie conjecture holds if the action of $S^1$ on $M$ extends to a smooth action of the torus $T^k$ and $\dim M \leq 8k-4$.

Revision as of 02:50, 2 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 M, what constraints does this place on the topology of M in general and on the Pontrjagin classes of M in particular?

In the case where M is homotopy equivalent to \CP^n, M \simeq \CP^n, Petrie [Petrie1972] restricted his attention to smooth actions of the Lie group S^1 = \{z \in \mathbb{C} \ \colon \ |z| = 1\}, and 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 M agrees with that of \CP^n, i.e.,

p(M) = (1+x^2)^{n+1} for a generator x \in 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 action of S^1 on M \simeq \mathbb{C}P^n extends to a smooth action of the torus T^n.
  • By the work of [Dejter1976] and [James1985], the Petrie conjecture is true if \dim M \leq 6 and 8, respectively.
  • According to [Hattori1978], the Petrie conjecture holds if M admits an invariant almost complex structure with the first Chern class c_1(M) = (n+1)x. Remember \dim M = 2n.
  • Other special cases where the Petrie conjecture holds are described by [Wang1975], [Yoshida1975/76], [Iberkleid1978], and [Muslin1978].
  • It follows from [Tsukada&Washiyama1979], [Masuda1981], and [Masuda1983], that the Petrie conjecture holds
  • The work of [Dessai2002] confirms the Petrie conjecture under the assumption that the action of S^1 on M extends to an appropriate action of Pin(2) and \dim M \leq 22.
  • According to [Dessai&Wilking2004], the Petrie conjecture holds if the action of S^1 on M extends to a smooth action of the torus T^k and \dim M \leq 8k-4.

3 Further discussion

A symplectic version of the Petrie conjecture is discussed in the article of [Tolman2010].

4 References

Retrieved from "http://www.map.mpim-bonn.mpg.de/index.php?title=Petrie_conjecture&oldid=5935"
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