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In mathematics, specifically in the field of differential topology, Morse homology is a homology theory defined for any smooth manifold. It is constructed using the smooth structure and an auxiliary metric on the manifold, but turns out to be topologically invariant, and is in fact isomorphic to singular homology. Morse homology also serves as a model for the various infinite-dimensional generalizations known as Floer homology theories. == Formal definition == Given any (compact) smooth manifold, let ''f'' be a Morse function and ''g'' a Riemannian metric on the manifold. (These are auxiliary; in the end, the Morse homology depends on neither.) The pair (''f'', ''g'') gives us a gradient vector field. We say that (''f'', ''g'') is Morse–Smale if the stable and unstable manifolds associated to all of the critical points of ''f'' intersect each other transversely. For any such (''f'', ''g''), it can be shown that the difference in index between any two critical points is equal to the dimension of the moduli space of gradient flows between those points. Thus there is a one-dimensional moduli space of flows between a critical point of index ''i'' and one of index ''i'' − 1. Each flow can be reparametrized by a one-dimensional translation in the domain. After modding out by these reparametrizations, the quotient space is zero-dimensional — that is, a collection of oriented points representing unparametrized flow lines. A chain complex may then be defined as follows. The set of chains is the Z-module generated by the critical points. The differential ''d'' of the complex sends a critical point ''p'' of index ''i'' to a sum of index-(''i'' − 1) critical points, with coefficients corresponding to the (signed) number of unparametrized flow lines from ''p'' to those index-(''i'' − 1) critical points. The fact that the number of such flow lines is finite follows from the compactness of the moduli space. The fact that this defines a complex (that is, that ''d''2 = 0) follows from an understanding of how the moduli spaces of gradient flows compactify. Namely, in ''d''2 ''p'' the coefficient of an index-(''i'' − 2) critical point ''q'' is the (signed) number of ''broken flows'' consisting of an index-1 flow from ''p'' to some critical point ''r'' of index ''i'' − 1 and another index-1 flow from ''r'' to ''q''. These broken flows exactly constitute the boundary of the moduli space of index-2 flows: The limit of any sequence of unbroken index-2 flows can be shown to be of this form, and all such broken flows arise as limits of unbroken index-2 flows. Unparametrized index-2 flows come in one-dimensional families, which compactify to compact one-manifolds. The fact that the boundary of a compact one-manifold is always zero proves that ''d''2 ''p'' = 0. 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「Morse homology」の詳細全文を読む スポンサード リンク
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