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In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some points onward (Betti numbers vanish above the dimension of a space), and they are all finite. The ''n''th Betti number represents the rank of the ''n''th homology group, denoted ''H''''n'', which tells us the maximum amount of cuts that must be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc. These numbers are used today in fields such as simplicial homology, computer science, digital images, etc. The term "Betti numbers" was coined by Henri Poincaré after Enrico Betti. ==Definition== Informally, the ''k''th Betti number refers to the number of ''k''-dimensional holes on a topological surface. The first few Betti numbers have the following definitions for 0-dimensional, 1-dimensional, and 2-dimensional simplicial complexes: * b0 is the number of connected components * b1 is the number of one-dimensional or "circular" holes * b2 is the number of two-dimensional "voids" or "cavities" The two-dimensional Betti numbers are easier to understand because we see the world in 0, 1, 2, and 3-dimensions, however. The following Betti numbers are higher-dimensional than apparent physical space. For a non-negative integer ''k'', the ''k''th Betti number ''b''''k''(''X'') of the space ''X'' is defined as the rank (number of linearly independent generators) of the abelian group ''H''''k''(''X''), the ''k''th homology group of ''X''. The ''k''th homology group is , the are the boundary maps of the simplicial complex and the rank of Hk is the ''k''th Betti number. Equivalently, one can define it as the vector space dimension of ''H''''k''(''X''; Q) since the homology group in this case is a vector space over Q. The universal coefficient theorem, in a very simple torsion-free case, shows that these definitions are the same. More generally, given a field ''F'' one can define ''b''''k''(''X'', ''F''), the ''k''th Betti number with coefficients in ''F'', as the vector space dimension of ''H''''k''(''X'', ''F''). 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「Betti number」の詳細全文を読む スポンサード リンク
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