Petrie conjecture

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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.
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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Petrie restricted his attention to smooth actions of the Lie group $S^1$, i.e., the circle {{cite|Petrie1972}} (or more generally, the torus $T^n$ for $n \geq 1$ {{cite|Petrie1973}})
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Petrie restricted his attention to smooth actions of the Lie group $S^1$, i.e., the circle {{cite|Petrie1972}} (or more generally, the torus $T^k$ for $k \geq 1$ {{cite|Petrie1973}})
on closed smooth manifolds $M$ which are [[Fake complex projective spaces|homotopy equivalent to $\CP^n$]]. He has formulated the following conjecture.
on closed smooth manifolds $M$ which are [[Fake complex projective spaces|homotopy equivalent to $\CP^n$]]. He has formulated the following conjecture.
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by the work of {{cite|Dejter1976}} and {{cite|James1985}}.
by the work of {{cite|Dejter1976}} and {{cite|James1985}}.
According to {{cite|Hattori1978}}, the same is true if $M$ admits an invarint almost
According to {{cite|Hattori1978}}, the same is true if $M$ admits an invarint almost
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complex structure with the appropriate first Chern class. Another special cases where the Petrie conjecture holds are described by {{cite|Petrie1973}},
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complex structure with the appropriate first Chern class. Another special cases where the Petrie conjecture holds are described by {{cite|Wang1975}},
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{{cite|Wang1975}}, {{cite|Yoshida1976}}, {{cite|Iberkleid1978}}, {{cite|Muslin1978}}, {{cite|Tsukada&Washiyama1979}}, {{cite|Masuda1981}}, {{cite|Dessai2002}}
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{{cite|Yoshida1976}}, {{cite|Iberkleid1978}}, {{cite|Muslin1978}}, {{cite|Tsukada&Washiyama1979}}, {{cite|Masuda1981}}, {{cite|Dessai2002}}, and {{cite|Dessai&Wilking2004}}.
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and {{cite|Dessai&Wilking2004}}.
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== References ==
== References ==

Revision as of 01:37, 1 December 2010

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

Introduction

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.

Petrie restricted his attention to smooth actions of the Lie group S^1, i.e., the circle [Petrie1972] (or more generally, the torus T^k for k \geq 1 [Petrie1973]) on closed smooth manifolds M which are homotopy equivalent to \CP^n. He has formulated the following conjecture.

Conjecture 0.1 [Petrie1972].

Suppose that M is a closed smooth 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}),
\displaystyle p(M) = (1+x^2)^{n+1}.

The Petrie conjecture has not been confirmed in general, but if \dim M \leq 8, the total Pontrjagin class of M agrees with that of \CP^n by the work of [Dejter1976] and [James1985]. According to [Hattori1978], the same is true if M admits an invarint almost complex structure with the appropriate first Chern class. Another special cases where the Petrie conjecture holds are described by [Wang1975], [Yoshida1976], [Iberkleid1978], [Muslin1978], [Tsukada&Washiyama1979], [Masuda1981], [Dessai2002], and [Dessai&Wilking2004].

References

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