Petrie conjecture

This page has not been refereed. The information given here might be incomplete or provisional.

1 Introduction

The Petrie conjecture was formulated in the following context: suppose that a Lie group $G$${{Stub}}== Introduction == <wikitex>; The Petrie conjecture was formulated in the following context: suppose that a Lie group $G$ acts smoothly on a closed smooth manifold $M$, what constraints does this place on the topology of $M$ in general and on the Pontrjagin classes of $M$ in particular. Petrie restricted his attention to smooth actions of the Lie group $S^1$ {{cite|Petrie1972}} (or more generally, the torus $T^k$ for $k \geq 1$ {{cite|Petrie1973}}) on closed smooth manifolds $M$ which are [[Fake complex projective spaces|homotopy equivalent to $\CP^n$]]. He has formulated the following conjecture. {{beginthm|Conjecture|{{cite|Petrie1972}}}} Suppose that $M$ is a closed smooth manifold homotopy equivalent to $\CP^n$ and that $S^1$ acts smoothly and non-trivially on $M$. Then the total Pontrjagin class of $M$ agrees with that of $\CP^n$, i.e., for a generator $x \in H^2(M; \mathbb{Z})$: $$p(M) = (1+x^2)^{n+1}.$$ {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Problems]] [[Category:Group actions on manifolds]]G$ acts smoothly on a closed smooth manifold $M$$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 smooth actions of the Lie group $S^1$$S^1$ [Petrie1972] (or more generally, the torus $T^k$$T^k$ for $k \geq 1$$k \geq 1$ [Petrie1973]) on closed smooth manifolds $M$$M$ which are homotopy equivalent to $\CP^n$$\CP^n$. He has formulated the following conjecture.

Conjecture 0.1 [Petrie1972].

Suppose that $M$$M$ is a closed smooth manifold homotopy equivalent to $\CP^n$$\CP^n$ and that $S^1$$S^1$ acts smoothly and non-trivially on $M$$M$. Then the total Pontrjagin class of $M$$M$ agrees with that of $\CP^n$$\CP^n$, i.e., for a generator $x \in H^2(M; \mathbb{Z})$$x \in H^2(M; \mathbb{Z})$:

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

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
• [Petrie1973] T. Petrie, Torus actions on homotopy complex projective spaces, Invent. Math. 20 (1973), 139–146. MR0322893 (48 #1254) Zbl 0262.57021