Erratum to “π1 of Symplectic Automorphism Groups and Invertibles in Quantum Homology Rings”

We note an error in [2]. This Erratum will not be published. The paper defines Ham(M,ω) to be the group of Hamiltonian automorphisms, equipped with the C∞-topology, and G as “the group of smooth based loops in Ham(M,ω)”. This is a misleading formulation, since what the paper really means is that elements of G are Hamiltonian loops. If one understands it in that way, then the proof of [2, Lemma 2.1] as given is incorrect. However, the distinction between “smooth loops in the symplectic automorphism group which remain inside Ham(M,ω)” and “Hamiltonian loops” is ultimately irrelevant, because of the following: Lemma. Let (φt)0≤t≤1 be a smooth isotopy of symplectic automorphisms of M , such that each φt is Hamiltonian. Then, the isotopy itself is a Hamiltonian isotopy. Proof. Let at ∈ H(M ;R) be the infinitesimal flux of the isotopy. This depends smoothly on t. If at is nonzero at some point t ∈ (0, 1), one can find arbitrarily small such that (1) ∫ t+ t− at dt 6= 0. By assumption, φt− and φt+ are both Hamiltonian. By connecting them to the identity, one forms a loop in the symplectic automorphism group whose flux is (1). But this flux can be made arbitrarily small, contradicting [1]. Hence, at is necessarily identically zero. After appealing to that, the proof of [2, Lemma 2.1] goes through as stated in thepaper. References[1] K. Ono. Floer-Novikov cohomology and the flux conjecture. Geom. Funct. Anal., 16:981–1020,2006.[2] P. Seidel. π1 of symplectic automorphism groups and invertibles in quantum homology rings.Geom. Funct. Anal., 7:1046–1095, 1997.