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In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign 'quantities' to the ''chains'' of homology theory. From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century; from the initial idea of ''homology'' as a topologically invariant relation on ''chains'', the range of applications of homology and cohomology theories has spread out over geometry and abstract algebra. The terminology tends to mask the fact that in many applications ''cohomology'', a contravariant theory, is more natural than ''homology''. At a basic level this has to do with functions and pullbacks in geometric situations: given spaces ''X'' and ''Y'', and some kind of function ''F'' on ''Y'', for any mapping ''f'' : ''X'' → ''Y'' composition with ''f'' gives rise to a function ''F'' o ''f'' on ''X''. Cohomology groups often also have a natural product, the cup product, which gives them a ring structure. Because of this feature, cohomology is a stronger invariant than homology, as it can differentiate between certain algebraic objects that homology cannot. ==Definition== In algebraic topology, the cohomology groups for spaces can be defined as follows (see Hatcher). Given a topological space ''X'', consider the chain complex : as in the definition of singular homology (or simplicial homology). Here, the ''Cn'' are the free abelian groups generated by formal linear combinations of the singular ''n''-simplices in ''X'' and ∂''n'' is the ''n''th boundary operator. Now replace each ''Cn'' by its dual space ''C *n−1'' = Hom(''Cn, G''), and ∂''n'' by its transpose : to obtain the cochain complex : Then the nth cohomology group with coefficients in G is defined to be Ker(δ''n''+1)/Im(δ''n'') and denoted by ''Hn''(''C''; ''G''). The elements of ''C *n'' are called singular ''n''-cochains with coefficients in ''G'' , and the δ''n'' are referred to as the coboundary operators. Elements of Ker(δ''n''+1), Im(δ''n'') are called cocycles and coboundaries, respectively. Note that the above definition can be adapted for general chain complexes, and not just the complexes used in singular homology. The study of general cohomology groups was a major motivation for the development of homological algebra, and has since found applications in a wide variety of settings (see below). Given an element φ of ''C *n-1'', it follows from the properties of the transpose that as elements of ''C *n''. We can use this fact to relate the cohomology and homology groups as follows. Every element φ of Ker(δ''n'') has a kernel containing the image of ∂''n''. So we can restrict φ to Ker(∂''n''−1) and take the quotient by the image of ∂''n'' to obtain an element ''h''(φ) in Hom(''Hn, G''). If φ is also contained in the image of δ''n''−1, then ''h''(φ) is zero. So we can take the quotient by Ker(δ''n''), and to obtain a homomorphism : It can be shown that this map h is surjective, and that we have a short split exact sequence : 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「Cohomology」の詳細全文を読む スポンサード リンク
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