Petrie conjecture

Revision as of 23:43, 30 November 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 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 $S^1$ {{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. {{beginthm|Conjecture|{{cite|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 $x \in H^2(M; \mathbb{Z})$, $$p(M) = (1+x^2)^{n+1}.$$ {{endthm}} </wikitex> The Petrie conjecture has not been confirmed in general (no counterexample is know at the moment). However, if $\dim M \leq 8$, the total Pontrjagin class of $M$ agrees with that of $\CP^n$ by the work of {{cite|Dejter1976}} and {{cite|James1978}}. {{cite|Iberkleid1978}}, {{cite|Tsukada&Washiyama1979}}, {{cite|Dessai2002}},{{cite|Dessai&Wilking2004}}, and {{cite|Tolman2010}}. == References == {{#RefList:}} [[Category:Problems]] [[Category:Group actions on manifolds]]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 $S^1$ [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 (no counterexample is know at the moment). However, if $\dim M \leq 8$, the total Pontrjagin class of $M$ agrees with that of $\CP^n$ by the work of [Dejter1976] and [James1978].

[Iberkleid1978], [Tsukada&Washiyama1979], [Dessai2002],[Dessai&Wilking2004], and [Tolman2010].

References

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