On a symplectic generalization of Petrie’s conjecture

Tolman, Susan

doi:10.1090/S0002-9947-10-04985-8

On a symplectic generalization of Petrie’s conjecture
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Trans. Amer. Math. Soc. 362 (2010), 3963-3996 Request permission

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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