Cohomology

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory. In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in 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 method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. 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 ∘ f on X. The most important cohomology theories have a product, the cup product, which gives them a ring structure. Because of this feature, cohomology is usually a stronger invariant than homology. Singular cohomology is a powerful invariant in topology, associating a graded-commutative ring to any topological space. Every continuous map f: X → Y determines a homomorphism from the cohomology ring of Y to that of X; this puts strong restrictions on the possible maps from X to Y. Unlike more subtle invariants such as homotopy groups, the cohomology ring tends to be computable in practice for spaces of interest. For a topological space X, the definition of singular cohomology starts with the singular chain complex: By definition, the singular homology of X is the homology of this chain complex (the kernel of one homomorphism modulo the image of the previous one). In more detail, Ci is the free abelian group on the set of continuous maps from the standard i-simplex to X (called 'singular i-simplices in X'), and ∂i is the ith boundary homomorphism. The groups Ci are zero for i negative. Now fix an abelian group A, and replace each group Ci by its dual group C i ∗ := H o m ( C i , A ) , {displaystyle C_{i}^{*}:=mathrm {Hom} (C_{i},A),} and ∂ i {displaystyle partial _{i}} by its dual homomorphism This has the effect of 'reversing all the arrows' of the original complex, leaving a cochain complex For an integer i, the ith cohomology group of X with coefficients in A is defined to be ker(di)/im(di−1) and denoted by Hi(X, A). The group Hi(X, A) is zero for i negative. The elements of C i ∗ {displaystyle C_{i}^{*}} are called singular i-cochains with coefficients in A. (Equivalently, an i-cochain on X can be identified with a function from the set of singular i-simplices in X to A.) Elements of ker(d) and im(d) are called cocycles and coboundaries, respectively, while elements of ker(d)/im(d) = Hi(X, A) are called cohomology classes (because they are equivalence classes of cocycles). In what follows, the coefficient group A is sometimes not written. It is common to take A to be a commutative ring R; then the cohomology groups are R-modules. A standard choice is the ring Z of integers.