By Seidel P.
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Extra resources for Pi-1 of Symplectyc Autimorphism Groups and Invertibles in Quantum Homology Rings
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Proof. In this case, I¯ is Z-valued because I(g, g˜) ∈ Z is independent of the choice of g˜. Since I¯ is a homomorphism, it is sufficient to show that ¯ ≤ 0 for all g. The assumption on c1 also implies that the grading of the I(g) Novikov ring Λ is trivial. 11 that HF0 (M, ω) ∼ = H0 (M ; Z/2) ⊗ Λ ∼ = Λ and HFk (M, ω) = 0 for k < 0. 4, HF∗ (g, g˜) maps HF0 (M, ω) isomorphically to HF−2I(g,˜g) (M, ω), which is clearly impossible if I(g, g˜) > 0. 1088 P. SEIDEL GAFA To obtain more general results, it is necessary to use the multiplicative structure.
6. It is of deˆ γ + S) is d + 2c1 (γ) and γ has gree d because the dimension of S(j, J, degree −2c1 (γ). 10. Q(E, Ω, S) is independent of the choice of j, J and J. We omit the proof. 1]. 1078 P. SEIDEL GAFA Sometimes it is convenient to define QH∗ (M, ω) in terms of Morse homology as the homology of the graded tensor product (CM∗ (f )⊗Λ,∂(f ,h)⊗Id). An element of CM∗ (f ) ⊗ Λ is a (possibly infinite) linear combination of y ⊗ γ for y ∈ Crit(f ), γ ∈ Γ. 8 with respect to the pseudo-cycles evz0 : S(j, J, all S , and define ˆ S , y) = s ∈ S(j, J, ˆ S ) | evz (s) ∈ W s (y; f, h) S(j, J, 0 for y ∈ Crit(f ).
Proof. In [M1] McDuff showed that (M, ωλ ) is symplectically isomorphic to CP 1 × CP 1 with the product structure λ(τ1 × 1) + 1 × τ1 . Such an isomorphism maps x± to a ± b, where a = [CP 1 × pt] and b = [pt × CP 1 ]. Let a ¯, ¯b the mod 2 reductions of these classes. g. 5]); it satisfies 2 2 = [CP 1 × CP 1 ] ⊗ b , ¯b ⊗ 0 = [CP 1 × CP 1 ] ⊗ a . a ¯⊗ 0 Because of the Z/2-coefficients, this implies the relation stated above. This can be used to give a proof of the following result of McDuff. 4. For all λ > 1, [g] ∈ π1 (Ham(M, ωλ )) has infinite order.