Petrie conjecture

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This page has not been refereed. The information given here might be incomplete or provisional.

Contents

1 Problem

If a compact Lie group G acts smoothly and non-trivially 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?

In the case where M is homotopy equivalent to \CP^n, Petrie [Petrie1972] restricted his attention to smooth actions of the Lie group S^1 = \{z \in \mathbb{C} \ \colon \ |z| = 1\}, and posed the following conjecture.

Conjecture 1.1 [Petrie1972]. Suppose that M is a closed smooth 2n-manifold homotopy equivalent to \CP^n and that S^1 acts smoothly and non-trivially on M. Then the total Pontrjagin class p(M) of M agrees with that of \CP^n,

i.e., p(M) = (1+x^2)^{n+1} for a generator x \in H^2(M; \mathbb{Z}).

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 action of S^1 on M extends to a smooth action of the torus T^n.
  • If \dim M \leq 8, the statement is true by the work of [Dejter1976] and [James1985].
  • According to [Hattori1978], the conjecture is also true if M admits an invariant almost complex structure with the first Chern class c_1(M) = (n+1)x.
  • Other special cases where the Petrie conjecture holds are described by [Wang1975], [Yoshida1975/76], [Iberkleid1978], [Muslin1978], [Tsukada&Washiyama1979], and [Masuda1981].
  • By the work of [Dessai2002], the total Pontrjagin class of M agrees with that of \CP^n if \dim M \leq 22 and the action of S^1 on M extends to an appropriate action of Pin(2).
  • Similarly, the work of [Dessai&Wilking2004] confirms the Petrie conjecture under the assumption that \dim M \leq 8k-4 and the action of S^1 on M extends to a smooth action of T^k for k \geq 1.

3 Further discussion

A symplectic version of the Petrie conjecture is discussed in the article of [Tolman2010].

4 References

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