Petrie conjecture

Revision as of 03:35, 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 {\cite|Petrie1972} restricted his attention to smooth actions of the Lie group $S^1$ (or more generally, the torus $T^k$ for $k \geq 1$) on 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 effectively on $M$. Then the total Pontrjagin class of $M$ agrees with that of $\CP^n$, i.e., $p(M) = (1+x^2)^{n+1}$ for a generato $x \in H^2(M; \mathbb{Z})$. {{endthm}} </wikitex> == 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 manifolds $M$ which are homotopy equivalent to $\CP^n$. He has formulated the following conjecture.

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