Petrie conjecture

Revision as of 16:06, 26 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 formulate in the following context: suppose that a Lie group $G$ acts 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 actions of the Lie group $S^1$ on manifolds $M$ which are [[Fake complex projective spaces|homotopy equivalent to $\CP^n$]]. He formulate the following {{beginthm|Conjecture|{{cite|Petrie1972}}}} Suppose that $M$ is a closed smooth manifold homotopy equivalent to $\CP^n$ and that $S^1$ acts effectively on $M$. Then the total Pontrjagin class of $M$ agrees with that of $\CP^n$. {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Problems]] [[Category:Symmetries of manifolds]]G$ acts 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 actions of the Lie group $S^1$ on manifolds $M$ which are homotopy equivalent to $\CP^n$. He formulated the following

2 References

• [Petrie1972] T. Petrie, Smooth $S^1$-actions on homotopy complex projective spaces and related topics, Bull. Amer. Math. Soc. 78 (1972), 105–153. MR0296970 Zbl 0247.57010