On a symplectic generalization of Petrie's conjecture

Author:
Susan Tolman

Journal:
Trans. Amer. Math. Soc. **362** (2010), 3963-3996

MSC (2000):
Primary 53D20

DOI:
https://doi.org/10.1090/S0002-9947-10-04985-8

Published electronically:
March 17, 2010

MathSciNet review:
2638879

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Abstract | References | Similar Articles | Additional Information

Abstract: Motivated by the Petrie conjecture, we consider the following questions: Let a circle act in a Hamiltonian fashion on a compact symplectic manifold which satisfies for all . Is for all ? Is the total Chern class of determined by the cohomology ring ? We answer these questions in the six-dimensional case by showing that is equal to for all , by proving that only four cohomology rings can arise, and by computing the total Chern class in each case. We also prove that there are no exotic actions. More precisely, if is isomorphic to or , then the representations at the fixed components are compatible with one of the standard actions; in the remaining two cases, the representation is strictly determined by the cohomology ring. Finally, our results suggest a natural question: Do the remaining two cohomology rings actually arise? This question is closely related to some interesting problems in symplectic topology, such as embeddings of ellipsoids.

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Additional Information

**Susan Tolman**

Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801

Email:
stolman@math.uiuc.edu

DOI:
https://doi.org/10.1090/S0002-9947-10-04985-8

Received by editor(s):
September 19, 2007

Published electronically:
March 17, 2010

Additional Notes:
The author was partially supported by National Science foundation grant DMS #07-07122.

Article copyright:
© Copyright 2010
American Mathematical Society