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In algebraic topology, simplicial homology formalizes the idea of the number of holes of a given dimension in a simplicial complex. This generalizes the number of connected components (the case of dimension 0). Simplicial homology arose as a way to study topological spaces whose building blocks are ''n''-simplices, the ''n''-dimensional analogs of triangles. This includes a point (0-dimensional simplex), a line segment (1-dimensional simplex), a triangle (2-dimensional simplex) and a tetrahedron (3-dimensional simplex). By definition, such a space is homeomorphic to a simplicial complex (more precisely, the geometric realization of an abstract simplicial complex). Such a homeomorphism is referred to as a ''triangulation'' of the given space. Many topological spaces of interest can be triangulated, including every smooth manifold, by Cairns and Whitehead.〔V. V. Prasolov. Elements of combinatorial and differential topology. Section 5.3.2〕 Simplicial homology is defined by a simple recipe for any abstract simplicial complex. It is a remarkable fact that simplicial homology only depends on the associated topological space.〔 M. A. Armstrong. Basic topology. Section 8.6.〕 As a result, it gives a computable way to distinguish one space from another. Singular homology is a related theory which is more commonly used by mathematicians today. Singular homology is defined for all topological spaces, and it agrees with simplicial homology for spaces which can be triangulated.〔 A. Hatcher. Algebraic topology. Theorem 2.27〕 Nonetheless, because it is possible to compute the simplicial homology of a simplicial complex automatically and efficiently, simplicial homology has become important for application to real-life situations, such as image analysis, medical imaging, and data analysis in general. == Definition == A key concept in defining simplicial homology is the notion of an orientation of a simplex. By definition, an orientation of a ''k''-simplex is given by an ordering of the vertices, written as (''v''0,...,''v''''k''), with the rule that two orderings define the same orientation if and only if they differ by an even permutation. Thus every simplex has exactly two orientations, and switching the order of two vertices changes an orientation to the opposite orientation. For example, choosing an orientation of a 1-simplex amounts to choosing one of the two possible directions, and choosing an orientation of a 2-simplex amounts to choosing what "counterclockwise" should mean. Let ''S'' be a simplicial complex. A simplicial ''k''-chain is a finite formal sum :, where each ''c''''i'' is an integer and σ''i'' is an oriented ''k''-simplex. In this definition, we declare that each oriented simplex is equal to the negative of the simplex with the opposite orientation. For example, : The group of ''k''-chains on ''S'' is written ''Ck''. This is a free abelian group which has a basis in one-to-one correspondence with the set of ''k''-simplices in ''S''. To define a basis explicitly, one has to choose an orientation of each simplex. One standard way to do this is to choose an ordering of all the vertices and give each simplex the orientation corresponding to the induced ordering of its vertices. Let σ = (''v''0,...,''v''''k'') be an oriented ''k''-simplex, viewed as a basis element of ''Ck''. The boundary operator : is the homomorphism defined by: : where the oriented simplex : is the ''i''th face of ''σ'', obtained by deleting its ''i''th vertex. In ''Ck'', elements of the subgroup : are referred to as cycles, and the subgroup : is said to consist of boundaries. A direct computation shows that ∂2 = 0. In geometric terms, this says that the boundary of anything has no boundary. Equivalently, the abelian groups : form a chain complex. Another equivalent statement is that ''Bk'' is contained in ''Zk''. The ''k''th homology group ''Hk'' of ''S'' is defined to be the quotient abelian group : It follows that the homology group ''Hk''(''S'') is nonzero exactly when there are ''k''-cycles on ''S'' which are not boundaries. In a sense, this means that there are ''k''-dimensional holes in the complex. For example, consider the complex ''S'' obtained by gluing two triangles (with no interior) along one edge, shown in the image. The edges of each triangle can be oriented so as to form a cycle. These two cycles are by construction not boundaries (since every 2-chain is zero). One can compute that the homology group ''H''1(''S'') is isomorphic to Z2, with a basis given by the two cycles mentioned. This makes precise the informal idea that ''S'' has two "1-dimensional holes". Holes can be of different dimensions. The rank of the ''k''th homology group, the number : is called the ''k''th Betti number of ''S''. It gives a measure of the number of ''k''-dimensional holes in ''S''. 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「simplicial homology」の詳細全文を読む スポンサード リンク
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