Petrie conjecture
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1 Problem
If a compact Lie group acts smoothly and non-trivially on a closed smooth manifold , what constraints does this place on the topology of in general and on the Pontrjagin classes of in particular? In the case where is homotopy equivalent to , , Petrie [Petrie1972] restricted his attention to actions of the Lie group . He proved that if acts smoothly on with isolated fixed points, then the Pontrjagin classes of are determined by the representations of at the fixed points. Having this as well as other results in mind, Petrie posed the following conjecture.
Conjecture 1.1 [Petrie1972]. Suppose that acts smoothly and non-trivially on a closed smooth -manifold . Then the total Pontrjagin class of agrees with that of , i.e., for a generator of .
2 Progress to date
As of November 30, 2010, the Petrie conjecture has not been confirmed in general. However, it has been proven in the following special cases.
- Petrie [Petrie1973] has verified his conjecture under the assumption that the manifold admits a smooth action of the torus .
- By the work of [Dejter1976], the Petrie conjecture is true if , i.e., , and more generaly, if .
- According to [Hattori1978], the Petrie conjecture holds if admits an invariant almost complex structure with the first Chern class .
- Other special cases where the Petrie conjecture holds are described by [Wang1975], [Yoshida1975/76], [Iberkleid1978], and [Musin1978] and [Musin1980]}.
- By [Tsukada&Washiyama1979] and [Masuda1981], the Petrie conjecture is true if the fixed point set consists of three or four connected components.
- Masuda [Masuda1983] proved the Petrie conjecture in the case where admits a specific smooth action of for .
- It follows from the work [James1985] that the Petrie conjecture is true if , i.e., .
- The work of [Dessai2002] confirms the Petrie conjecture if admits an appropriate smooth action of and .
- According to [Dessai&Wilking2004], the Petrie conjecture holds if admits to a smooth action of and .
3 Further discussion
A related problem posed by Masuda and Suh [Masuda&Suh2008] reads as follows. For two toric -manifolds with isomorphic cohomology rings, is there an isomorphism between the cohomology rings which preserves the Pontrjagin classes of the two manifolds?
A symplectic version of the Petrie conjecture is discussed by Tolman [Tolman2010]. In particular, in the case where acts in a Hamiltonian way on a compact symplectic manifold , is the total Chern class of determined by the cohomology ring ?
4 References
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