Petrie conjecture

Revision as of 21:02, 30 November 2010

1 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> Results confirming that in some cases the Petrie conjecture is true go back to {{cite|Dejter1976}}, {{cite|Iberkleid1978}}, {{cite|Tsukada&Washiyama1979}}, {{cite|Dessai2002}},{{cite|Dessai&Wilking2004}}, and {{cite|Tolman2009}}. == 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 $x \in H^2(M; \mathbb{Z})$,

$$\displaystyle p(M) = (1+x^2)^{n+1}.$$

Results confirming that in some cases the Petrie conjecture is true go back to [Dejter1976], [Iberkleid1978], [Tsukada&Washiyama1979], [Dessai2002],[Dessai&Wilking2004], and [Tolman2010].

2 References

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• [Tolman2010] S. Tolman, On a symplectic generalization of Petrie's conjecture, Trans. Amer. Math. Soc. 362 (2010), no.8, 3963–3996. MR2638879 Zbl 1216.53074
• [Tsukada&Washiyama1979] E. Tsukada and R. Washiyama, $S^1$-actions on cohomology complex projective spaces with three components of the fixed point sets, Hiroshima Math. J. 9 (1979), no.1, 41–46. MR529325 (80j:57043) Zbl 0411.57037