Petrie conjecture
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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. | 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 | + | 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}}}} | {{beginthm|Conjecture|{{cite|Petrie1972}}}} |
Revision as of 03:38, 30 November 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
1 Introduction
Tex syntax error, what constraints does this place on the topology of
Tex syntax errorin general and on the Pontrjagin classes of
Tex syntax errorin particular. Petrie restricted his attention to smooth actions of the Lie group [Petrie1972] (or more generally, the torus for [Petrie1973]) on closed smooth manifolds
Tex syntax errorwhich are homotopy equivalent to . He has formulated the following conjecture.
Conjecture 0.1 [Petrie1972].
Suppose thatTex syntax erroris a closed smooth manifold homotopy equivalent to and that acts effectively on
Tex syntax error. Then the total Pontrjagin class of
Tex syntax erroragrees 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
- [Petrie1973] T. Petrie, Torus actions on homotopy complex projective spaces, Invent. Math. 20 (1973), 139–146. MR0322893 (48 #1254) Zbl 0262.57021