Petrie conjecture

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Suppose that $M$ is a closed smooth $2n$-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 $x \in H^2(M; \mathbb{Z})$, $$p(M) = (1+x^2)^{n+1}.$$
Suppose that $M$ is a closed smooth $2n$-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 $x \in H^2(M; \mathbb{Z})$, $$p(M) = (1+x^2)^{n+1}.$$
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Petrie {{cite|Petrie1973}} has proven that his conjecture holds true if the action of $S^1$ on $M$ extends to a smooth action of the torus $T^n$.
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Petrie {{cite|Petrie1973}} has verified his conjecture under the assumtion that the action of $S^1$ on $M$ extends to a smooth action of the torus $T^n$.
The Petrie conjecture has not been confirmed in general, but if $\dim M \leq 8$, the statement is true by the work of {{cite|Dejter1976}} and {{cite|James1985}}.
The Petrie conjecture has not been confirmed in general, but if $\dim M \leq 8$, the statement is true by the work of {{cite|Dejter1976}} and {{cite|James1985}}.
According to {{cite|Hattori1978}}, the conjecture is also true if $M$ admits an invarint almost complex structure with the first Chern class $c_1(M) = (n+1)x$. Another special cases
According to {{cite|Hattori1978}}, the conjecture is also true if $M$ admits an invarint almost complex structure with the first Chern class $c_1(M) = (n+1)x$. Another special cases

Revision as of 04:20, 1 December 2010

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

Introduction

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? What about the case where M is homotopy equivalent to \CP^n?

In such a context, Petrie restricted his attention to smooth actions of the Lie group S^1, the circle [Petrie1972] (more generally, the torus T^k for k \geq 1 [Petrie1973]), and he posed the following conjecture.

Conjecture 0.1 [Petrie1972].

Suppose that M is a closed smooth 2n-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}.

Petrie [Petrie1973] has verified his conjecture under the assumtion that the action of S^1 on M extends to a smooth action of the torus T^n. The Petrie conjecture has not been confirmed in general, but if \dim M \leq 8, the statement is true by the work of [Dejter1976] and [James1985]. According to [Hattori1978], the conjecture is also true if M admits an invarint almost complex structure with the first Chern class c_1(M) = (n+1)x. Another special cases where the Petrie conjecture holds are described by [Wang1975], [Yoshida1976], [Iberkleid1978], [Muslin1978], [Tsukada&Washiyama1979], and [Masuda1981]. By the work of [Dessai2002], the total Pontrjagin class of M agrees with that of \CP^n if \dim M \leq 22 and the action of S^1 on M extends to an appropriate action of Pin(2). Similarly, the work of [Dessai&Wilking2004] confirms the Petrie conjecture under the assumption that \dim M \leq 8k-4 and the action of S^1 on M extends to a smooth action of T^k for k \geq 1.

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

References

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