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In mathematics, Floer homology is a tool mathematicians use to study symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analog of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called Hamiltonian Floer homology, in his proof of the Arnold conjecture in symplectic geometry. Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold. A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the Yang–Mills functional. These constructions and their descendants play a fundamental role in current investigations into the topology of symplectic and contact manifolds as well as (smooth) three- and four-dimensional manifolds. Floer homology is typically defined by associating to the object of interest an infinite-dimensional manifold and a real valued function on it. In the symplectic version, this is the free loop space of a symplectic manifold with the symplectic action functional. For the (instanton) version for three-manifolds, it is the space of SU(2)-connections on a three-dimensional manifold with the Chern–Simons functional. Loosely speaking, Floer homology is the Morse homology of the function on the infinite-dimensional manifold. A Floer chain complex is formed from the abelian group spanned by the critical points of the function (or possibly certain collections of critical points). The differential of the chain complex is defined by counting the function's gradient flow lines connecting certain pairs of critical points (or collections thereof). Floer homology is the homology of this chain complex. The gradient flow line equation, in a situation where Floer's ideas can be successfully applied, is typically a geometrically meaningful and analytically tractable equation. For symplectic Floer homology, the gradient flow equation for a path in the loopspace is (a perturbed version of) the Cauchy–Riemann equation for a map of a cylinder (the total space of the path of loops) to the symplectic manifold of interest; solutions are known as pseudoholomorphic curves. The Gromov compactness theorem is then used to show that the differential is well-defined and squares to zero, so that the Floer homology is defined. For instanton Floer homology, the gradient flow equations is exactly the Yang-Mills equation on the three-manifold crossed with the real line. ==Symplectic Floer homology== Symplectic Floer Homology (SFH) is a homology theory associated to a symplectic manifold and a nondegenerate symplectomorphism of it. If the symplectomorphism is Hamiltonian, the homology arises from studying the symplectic action functional on the (universal cover of the) free loop space of a symplectic manifold. SFH is invariant under Hamiltonian isotopy of the symplectomorphism. Here, nondegeneracy means that 1 is not an eigenvalue of the derivative of the symplectomorphism at any of its fixed points. This condition implies that the fixed points are isolated. SFH is the homology of the chain complex generated by the fixed points of such a symplectomorphism, where the differential counts certain pseudoholomorphic curves in the product of the real line and the mapping torus of the symplectomorphism. This itself is a symplectic manifold of dimension two greater than the original manifold. For an appropriate choice of almost complex structure, punctured holomorphic curves (of finite energy) in it have cylindrical ends asymptotic to the loops in the mapping torus corresponding to fixed points of the symplectomorphism. A relative index may be defined between pairs of fixed points, and the differential counts the number of holomorphic cylinders with relative index 1. The symplectic Floer homology of a Hamiltonian symplectomorphism of a compact manifold is isomorphic to the singular homology of the underlying manifold. Thus, the sum of the Betti numbers of that manifold yields the lower bound predicted by one version of the Arnold conjecture for the number of fixed points for a nondegenerate symplectomorphism. The SFH of a Hamiltonian symplectomorphism also has a pair of pants product that is a deformed cup product equivalent to quantum cohomology. A version of the product also exists for non-exact symplectomorphisms. For the cotangent bundle of a manifold M, the Floer homology depends on the choice of Hamiltonian due to its noncompactness. For Hamiltonians that are quadratic at infinity, the Floer homology is the singular homology of the free loop space of M (proofs of various versions of this statement are due to Viterbo, Salamon–Weber, Abbondandolo–Schwarz, and Cohen). There are more complicated operations on the Floer homology of a cotangent bundle that correspond to the string topology operations on the homology of the loop space of the underlying manifold. The symplectic version of Floer homology figures in a crucial way in the formulation of the homological mirror symmetry conjecture. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Floer homology」の詳細全文を読む スポンサード リンク
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