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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク singular homology ： ウィキペディア英語版 singular homology In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups $H\_n(X)$. Intuitively spoken, singular homology counts, for each dimension ''n'', the ''n''-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions. In brief, singular homology is constructed by taking maps of the standard ''n''-simplex to a topological space, and composing them into formal sums, called singular chains. The boundary operation on a simplex induces a singular chain complex. The singular homology is then the homology of the chain complex. The resulting homology groups are the same for all homotopically equivalent spaces, which is the reason for their study. These constructions can be applied to all topological spaces, and so singular homology can be expressed in terms of category theory, where the homology group becomes a functor from the category of topological spaces to the category of graded abelian groups. These ideas are developed in greater detail below. == Singular simplices == A singular ''n''-simplex is a continuous mapping $\backslash sigma\_n$ from the standard ''n''-simplex $\backslash Delta^n$ to a topological space ''X''. Notationally, one writes $\backslash sigma\_n:\backslash Delta^n\backslash to\; X$. This mapping need not be injective, so there can be non-equivalent singular simplices with the same image in ''X''. The boundary of $\backslash sigma\_n(\backslash Delta^n)$, denoted as $\backslash partial\_n\backslash sigma\_n(\backslash Delta^n)$, is defined to be the formal sum of the singular (''n'' − 1)-simplices represented by the restriction of $\backslash sigma$ to the faces of the standard ''n''-simplex, with an alternating sign to take orientation into account. (A formal sum is an element of the free abelian group on the simplices. The basis for the group is the infinite set of all possible images of standard simplices. The group operation is "addition" and the sum of image ''a'' with image ''b'' is usually simply designated ''a'' + ''b'', but ''a'' + ''a'' = 2''a'' and so on. Every image ''a'' has a negative −''a''.) Thus, if we designate the range of $\backslash sigma\_n$ by its vertices :$()=()$ corresponding to the vertices $e\_k$ of the standard ''n''-simplex $\backslash Delta^n$ (which of course does not fully specify the standard simplex image produced by $\backslash sigma\_n$), then :$\backslash partial\_n\backslash sigma\_n(\backslash Delta^n)=\backslash sum\_^n(-1)^k\; (p\_n)$ is a formal sum of the faces of the simplex image designated in a specific way. (That is, a particular face has to be the image of $\backslash sigma\_n$ applied to a designation of a face of $\backslash Delta^n$ which depends on the order that its vertices are listed.) Thus, for example, the boundary of $\backslash sigma=()$ (a curve going from $p\_0$ to $p\_1$) is the formal sum (or "formal difference") $()\; -\; ()$. 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「singular homology」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.