Petrie conjecture
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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? What about the case where
Tex syntax erroris homotopy equivalent to ?
In such a context, Petrie restricted his attention to smooth actions of the Lie group , the circle [Petrie1972] (or more generally, the torus for [Petrie1973]), and he posed the following conjecture.
Conjecture 0.1 [Petrie1972].
Suppose thatTex syntax erroris a closed smooth -manifold homotopy equivalent to and that acts smoothly and non-trivially on
Tex syntax error. Then the total Pontrjagin class of
Tex syntax erroragrees with that of , i.e., for a generator ,
Tex syntax erroragrees with that of if the action of on
Tex syntax errorextends to a smooth action of .
The Petrie conjecture has not been confirmed in general, but if , the statement is true by the work of [Dejter1976] and [James1985].
According to [Hattori1978], the conjecture is also true ifTex syntax erroradmits an invarint almost complex structure with the first Chern class . Another special cases
where the Petrie conjecture holds are described by [Wang1975], [Yoshida1976], [Iberkleid1978], [Muslin1978], [Tsukada&Washiyama1979], and
[Masuda1981]. Moreover, by the work of [Dessai2002], the Petrie conjecture holds if and the action of onTex syntax errorextends to an appropriate action of . Similarly, the work of [Dessai&Wilking2004] confirms the Petrie conjecture under the assumption that and the action of on
Tex syntax errorextends to a smooth action of
for .
A symplectic version of the Petrie conjecture is discussed in the article of [Tolman2010].
References
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