Petrie conjecture
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− | 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 $x \in H^2(M; \mathbb{Z})$ | + | 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 $x \in H^2(M; \mathbb{Z})$, $$p(M) = (1+x^2)^{n+1}.$$ |
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Revision as of 03:44, 30 November 2010
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1 Introduction
The Petrie conjecture was formulated in the following context: suppose that a Lie group acts smoothly on a closed smooth manifold , what constraints does this place on the topology of in general and on the Pontrjagin classes of in particular.
Petrie restricted his attention to smooth actions of the Lie group [Petrie1972] (or more generally, the torus for [Petrie1973]) on closed smooth manifolds which are homotopy equivalent to . He has formulated the following conjecture.
Conjecture 0.1 [Petrie1972].
Suppose that is a closed smooth manifold homotopy equivalent to and that acts smoothly and non-trivially on . Then the total Pontrjagin class of agrees with that of , i.e., for a generator ,2 References
- [Petrie1972] T. Petrie, Smooth -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