Petrie conjecture

Revision as of 02:56, 1 December 2010

Introduction

The Petrie conjecture was formulated in the following context: suppose that a Lie group ${{Stub}}== Introduction == <wikitex>; The Petrie conjecture was formulated in the following context: suppose that a 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. Petrie restricted his attention to smooth actions of the Lie group $S^1$, 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 $\CP^n$]]. {{beginthm|Conjecture|{{cite|Petrie1972}}}} Suppose that $M$ is a closed smooth n$-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 $x \in H^2(M; \mathbb{Z})$, $$p(M) = (1+x^2)^{n+1}.$$ {{endthm}} Petrie {{cite|Petrie1973}} has shown that the total Pontrjagin class of $M$ agrees with that of $\CP^n$ if the action of $S^1$ on $M$ extends to a smooth action of $T^n$. 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}}. According to {{cite|Hattori1978}}, the conjecture is also true if $M$ admits an invarint almost complex structure with appropriate first Chern class. Another special cases where the Petrie conjecture holds are described by {{cite|Wang1975}}, {{cite|Yoshida1976}}, {{cite|Iberkleid1978}}, {{cite|Muslin1978}}, {{cite|Tsukada&Washiyama1979}}, and {{cite|Masuda1981}}. Moreover, by the work of {{cite|Dessai2002}}, the Petrie conjecture holds 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 {{cite|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 Tolman {{cite|Tolman2010}}. == References == {{#RefList:}} [[Category:Problems]] [[Category:Group actions on manifolds]] </wikitex>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.

Petrie restricted his attention to smooth actions of the Lie group $S^1$, 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 $\CP^n$.

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 shown that the total Pontrjagin class of $M$ agrees with that of $\CP^n$ if the action of $S^1$ on $M$ extends to a smooth action of $T^n$. 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 appropriate first Chern class. Another special cases where the Petrie conjecture holds are described by [Wang1975], [Yoshida1976], [Iberkleid1978], [Muslin1978], [Tsukada&Washiyama1979], and [Masuda1981]. Moreover, by the work of [Dessai2002], the Petrie conjecture holds 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 [Tolman2010].

References

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