A brief summary of definition of Floer homology(not complete)

잠시 정리.

For given a Hamiltonian function H(which is usually time-dependent) on a symplectic manifold M, we can think of a natural (perturbed) action functional A_H which is defined on the contractible loop space of M(in fact, the action functional is derived from the weak symplectic structure \Omega on the whole loop space of M. Roughly, in proving the closedness of \Omega, we construct a local primitive A, and it is in fact the guy). By computation, we may prove that all critical points of A_H are periodic solutions of the Hamiltonian differential equation. On the other hand, there is an obvious action of the second homotopy group of M on L_0(notation for contractible loop space, from now on), and it gives a universal covering of L_0. Observe that a path between two critical points of A_H(hence they are loops) is a cylinder in M. With compatible metric g from given symplectic structure and complex structure, we find the gradient of A_H, and the equation of the gradient flow is seen to be the Floer's perturbed Cauchy-Riemann equation.

We collect all critical points of A_H according to the natural grading which is due to Conley-Zehnder. Consider moduli spaces of gradient flows(of finite energy) between periodic orbits of which gradings differ by 1, and denote them by M^1(of course, later we want to consider M^k in general). Then we can prove regularity theorems, e.g. for generic choices of Hamiltonians and complex structures, the moduli spaces are of compact and dimension 0. Then we can count them considering orientations, and it gives a boundary map between two Floer complexes(=modules of formal sums (but with some rules) of critical points) which differ by grading 1. Examining boundary elements of moduli spaces of M^2, we can prove that the boundary is indeed a differential, i.e. its square is zero.

To define chain maps between two different(which means 'different Hamiltonians') Floer complexes, we consider a homotopy connecting two Hamiltonians, and using nonautonomous Cauchy-Riemann equations, we repeat same arguments above, but by issue of dimensions, now the moduli spaces of dimension zero are those such that the periodic orbits have same Conley-Zehnder index. Again, by examining dimension 2 moduli spaces, we may check that the maps are indeed chain maps.

by 상욱 | 2009/02/25 12:39 | Math | 트랙백 | 덧글(0)
트랙백 주소 : http://leemky7.egloos.com/tb/2301954
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