Petrie conjecture
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{{Stub}}== Introduction == | {{Stub}}== Introduction == | ||
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− | 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. | + | 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 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 | + | Petrie {\cite|Petrie1972} restricted his attention to smooth actions of the Lie group $S^1$, or more general $T^k$ for $k \geq 1$, on manifolds $M$ which are [[Fake complex projective spaces|homotopy equivalent to $\CP^n$]]. He formulated the following conjecture. |
{{beginthm|Conjecture|{{cite|Petrie1972}}}} | {{beginthm|Conjecture|{{cite|Petrie1972}}}} | ||
− | Suppose that $M$ is a closed smooth manifold homotopy equivalent to $\CP^n$ and that $S^1$ acts effectively on $M$. | + | 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 $x \in H^(M; \mathbb{Z})$. |
{{endthm}} | {{endthm}} | ||
</wikitex> | </wikitex> |
Revision as of 03:29, 30 November 2010
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 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 {\cite|Petrie1972} restricted his attention to smooth actions of the Lie group , or more general for , on manifolds which are homotopy equivalent to . He formulated the following conjecture.
Conjecture 0.1 [Petrie1972]. Suppose that is a closed smooth manifold homotopy equivalent to and that acts effectively on . Then the total Pontrjagin class of agrees with that of , i.e., for a generato .
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