Petrie conjecture
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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 {\cite|Petrie1972} restricted his attention to smooth actions of the Lie group (or more generally, the torus for ) on 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 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