Three approaches towards Floer homology of cotangent bundles
نویسنده
چکیده
Consider the cotangent bundle of a closed Riemannian manifold and an almost complex structure close to the one induced by the Riemannian metric. For Hamiltonians which grow for instance quadratically in the fibers outside of a compact set, one can define Floer homology and show that it is naturally isomorphic to singular homology of the free loop space. We review the three isomorphisms constructed by Viterbo [V96], Salamon-Weber [SW03] and Abbondandolo-Schwarz [AS04]. The theory is illustrated by calculating Morse and Floer homology in case of the euclidean n-torus. Applications include existence of noncontractible periodic orbits of compactly supported Hamiltonians on open unit disc cotangent bundles which are sufficiently large over the zero section. 1 Chain group and boundary operators Let M be a closed smooth manifold and fix a Riemannian metric. Let ∇ be the associated Levi-Civita connection. This endows the free loop space LM = C(S,M) with an L and a W 1,2 metric given by 〈ξ, η〉L2 = ∫ 1 0 〈ξ(t), η(t)〉 dt, 〈ξ, η〉W 1,2 = 〈ξ, η〉L2 + 〈∇tξ,∇tη〉L2 , where ξ and η are smooth vector fields along x ∈ LM . Here and throughout we identify S = R/Z and think of x ∈ LM as a smooth map x : R → M which satisfies x(t + 1) = x(t). Fix a time-dependent function V ∈ C(S ×M) and set Vt(q) := V (t, q). The classical action functional on LM is defined by SV (x) := ∫ 1
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