On a symplectic generalization of Petrie’s conjecture

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 , ω ) (M,\omega ) which satisfies H 2 i ( M ; R ) = H 2 i ( C P n , R ) H^{2i}(M;\mathbb {R}) = H^{2i}({\mathbb C}{\mathbb P}^n,\mathbb {R}) for all i i . Is H j ( M ; Z ) = H j ( C P n ; Z ) H^j(M;\mathbb {Z}) = H^j({\mathbb C}{\mathbb P}^n;\mathbb {Z}) for all j j ? Is the total Chern class of M M determined by the cohomology ring H ∗ ( M ; Z ) H^*(M;\mathbb {Z}) ? We answer these questions in the six-dimensional case by showing that H j ( M ; Z ) H^j(M;\mathbb {Z}) is equal to H j ( C P 3 ; Z ) H^j({\mathbb C}{\mathbb P}^3;\mathbb {Z}) for all j 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 ; Z ) H^*(M;\mathbb {Z}) is isomorphic to H ∗ ( C P 3 ; Z ) H^*({\mathbb C}{\mathbb P}^3;\mathbb {Z}) or H ∗ ( G ~ 2 ( R 5 ) ; Z ) 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.