Symplectic quasi-states and semi-simplicity of quantum homology

Abstract

We review and streamline our previous results and the results of Y. Ostrover on the existence of Calabi quasi-morphisms and symplectic quasi-states on symplectic manifolds with semi-simple quantum homology. As an illustration, we discuss the case of symplectic toric Fano 4-manifolds. We present also new results due to D. McDuff: she observed that for the existence of quasi-morphisms/quasi-states it suffices to assume that the quantum homology contains a field as a direct summand, and she showed that this weaker condition holds true for one point blow-ups of non-uniruled symplectic manifolds.

Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms

“…3 This effectively answers a question raised by Entov and Polterovich [14,Remark 3.2] as to whether semisimplicity in the sense studied there (which in our language amounts to the semisimplicity of QH.M; !/ 0 ) can be deduced from generic semisimplicity in the algebraic geometry sense. As stated, the answer is evidently negative, since the Ostrover-Tyomkin example has generically semisimple quantum homology but fails to have QH.M; !/ 0 semisimple.…”

“…/; Á / will be graded by Z=2Z, not Z. Correspondingly, there is no "degree-shifting element" such as that denoted q in Entov and Polterovich [14].…”

“…and the field denoted by K in Entov and Polterovich [14], and the algebra ƒ ! -algebra QH.M; !/ 0 is straightforwardly seen to be isomorphic to the K -algebra QH 2n .M; !/ from [14] (in [14] a degree-shifting parameter q , which does not belong to K , is used to move all of the even-degree homology into degree 2n; note that the ring denoted by ƒ in [14] plays a different role than our ƒ ! ).…”

“…). In particular the notion of semisimple quantum homology from [14] is equivalent to, in our notation, the property that QH.M; !/ 0 is semisimple. In turn, in the case that .M; !/ is monotone, this notion can be identified with that in Entov and Polterovich [12] by the argument in [14,Section 5].…”

“…At the other extreme, if M is not uniruled, then the undeformed quantum homology of the blowup has a field direct summand; this fact is proven based on results of McDuff [43] in Entov and Polterovich [14,Section 3], where its discovery is attributed to McDuff.…”