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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On a symplectic generalization of Petrie's conjecture

Author: Susan Tolman
Journal: Trans. Amer. Math. Soc. 362 (2010), 3963-3996
MSC (2000): Primary 53D20
Published electronically: March 17, 2010
MathSciNet review: 2638879
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Abstract: Motivated by the Petrie conjecture, we consider the following questions: Let a circle act in a Hamiltonian fashion on a compact symplectic manifold $ (M,\omega)$ which satisfies $ H^{2i}(M;\mathbb{R}) = H^{2i}({\mathbb{C}}{\mathbb{P}}^n,\mathbb{R})$ for all $ i$. Is $ H^j(M;\mathbb{Z}) = H^j({\mathbb{C}}{\mathbb{P}}^n;\mathbb{Z})$ for all $ j$? Is the total Chern class of $ M$ determined by the cohomology ring $ H^*(M;\mathbb{Z})$? We answer these questions in the six-dimensional case by showing that $ H^j(M;\mathbb{Z})$ is equal to $ H^j({\mathbb{C}}{\mathbb{P}}^3;\mathbb{Z})$ for all $ j$, 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 $ H^*(M;\mathbb{Z})$ is isomorphic to $ H^*({\mathbb{C}}{\mathbb{P}}^3;\mathbb{Z})$ or $ H^*(\widetilde{G}_2(\mathbb{R}^5);\mathbb{Z})$, 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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Susan Tolman
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801

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

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