Petrie conjecture

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1 Introduction

The Petrie conjecture was formulated in the following context: suppose that a Lie group $G$${{Stub}}== Introduction == <wikitex>; The Petrie conjecture was formulated in the following context: suppose that a Lie group $G$ acts on a closed smooth manifold $M$, what constraints does this place on the topology of $M$ in general and on the Pontrjagin classes of $M$ in particular. Petrie restricted his attention to actions of the Lie group $S^1$ on manifolds $M$ which are [[Fake complex projective spaces|homotopy equivalent to $\CP^n$]]. He formulated the following {{beginthm|Conjecture|{{cite|Petrie1972}}}} Suppose that $M$ is a closed smooth manifold homotopy equivalent to $\CP^n$ and that $S^1$ acts effectively on $M$. Then the total Pontrjagin class of $M$ agrees with that of $\CP^n$. {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Problems]] [[Category:Group actions on manifolds]]G$ acts on a closed smooth manifold $M$$M$, what constraints does this place on the topology of $M$$M$ in general and on the Pontrjagin classes of $M$$M$ in particular.

Petrie restricted his attention to actions of the Lie group $S^1$$S^1$ on manifolds $M$$M$ which are homotopy equivalent to $\CP^n$$\CP^n$. He formulated the following

Conjecture 0.1 [Petrie1972]. Suppose that $M$$M$ is a closed smooth manifold homotopy equivalent to $\CP^n$$\CP^n$ and that $S^1$$S^1$ acts effectively on $M$$M$. Then the total Pontrjagin class of $M$$M$ agrees with that of $\CP^n$$\CP^n$.

2 References

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