Floer introduced an action of H∗(M) on HF∗(H) in (Fl89b) in the context of Hamiltonian fixed points and used it to estimate the number of fixed points for general Hamiltonian diffeomorphisms for the case of CPn. He conjectured the presence of such an action under the assumption that a certain transversality is available for a generic choice of almost-complex structures, and, as a consequence, he implicitly introduced a ring structure on HF∗(H). Piunikhin (Pi94) used the interpretation of the quantum cohomology QH∗(M) as a Bott– Morse generalization of the Floer cohomology for the zero Hamiltonian H and proposed the isomorphism between QH∗(M) and HF∗(H) by interpolating the given H and the ‘zero’ Hamiltonian in the level of rings. (See also (RT95b) for further elaboration.) This would give rise to an isomorphism between QH∗(M) and HF∗(H). Piunikhin, Salamon and Schwarz (PSS96) later made a proposal for the construction of the ring isomorphism that is somewhat different from (Pi94, RT95b), being based on the construction of the so-called PSS map between HF∗(H) and QH∗(M). The isomorphism property, especially at the level of a ring, plays an important role in applications of Floer homology to problems in symplectic topology (Se97), (Schw00), (Oh05c, Oh05d). However, the proposals both of Piunikhin (Pi94) and of Piunikhin, Salamon and Schwarz (PSS96) lacked some non-trivial gluing analysis, which was not available at the time of the proposals but has been developed later. For the construction in (Pi94), one needs the kind of gluing analysis developed in Chapter 7 (FOOO09), whereas for the construction in (PSS96), one needs the analysis presented in (OhZ11a) or (OhZ11b).

In this chapter, we explain the structure of a quantum cohomology ring and its chain-level description via the Floer complex of small Morse functions. Then a complete proof of the PSS isomorphism property will be given modulo the key gluing analysis entering into the proof of its isomorphism property, which we leave to (OhZ11b).