Petrie conjecture

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As of November 30, 2010, the Petrie conjecture has not been confirmed in general. However, it has been proven in the following special cases.
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 {{cite|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 $T^n$.
* Petrie {{cite|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 $T^n$.
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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.
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* By the work of {{cite|Dejter1976}}, the Petrie conjecture is true if $\dim M \leq 6$.
* 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 $\dim M = 2n$.
* Other special cases where the Petrie conjecture holds are described by {{cite|Wang1975}}, {{cite|Yoshida1975/76}}, {{cite|Iberkleid1978}}, and {{cite|Muslin1978}}.
* Other special cases where the Petrie conjecture holds are described by {{cite|Wang1975}}, {{cite|Yoshida1975/76}}, {{cite|Iberkleid1978}}, and {{cite|Muslin1978}}.
−
* It follows from {{cite|Tsukada&Washiyama1979}}, {{cite|Masuda1981}}, and {{cite|Masuda1983}}, that the Petrie conjecture holds
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* By {{cite|Tsukada&Washiyama1979}}, {{cite|Masuda1981}}, and {{cite|Masuda1983}}, the Petrie conjecture is true if $M$ has three or four connected components.
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* It follows from {{cite|James1985}} that the Petrie conjecture is also true if $\dim M = 8$.
* 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 02:59, 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], the Petrie conjecture is true if \dim M \leq 6.
  • 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].
  • By [Tsukada&Washiyama1979], [Masuda1981], and [Masuda1983], the Petrie conjecture is true if M has three or four connected components.
  • It follows from [James1985] that the Petrie conjecture is also true if \dim M = 8.
  • 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=5936"
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