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The only difference between homology and cohomology is that in cohomology the chain complexes depend in a contravariant manner on X, and that therefore the homology groups (which are called cohomology groups in this context and denoted by H n) form contravariant functors from the category that X belongs to into the category of abelian groups or ...
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology ) and abstract algebra (theory of modules and syzygies ) at the end of the 19th century, chiefly by ...
A continuous map f between topological spaces X and Y induces a chain map between the singular chain complexes of X and Y, and hence induces a map f * between the singular homology of X and Y as well. When X and Y are both equal to the n-sphere, the map induced on homology defines the degree of the map f.
Orthology (biology) - homologous sequences originate from the same ancestors (homolog e.g. all globin protein), which are separated from each other after a speciation event, e.g. human beta and chimp beta globin. An orthologous gene is a gene in different species that evolved from a common ancestor by speciation.
In noncommutative geometry and related branches of mathematics, cyclic homology and cyclic cohomology are certain (co)homology theories for associative algebras which generalize the de Rham (co)homology of manifolds. These notions were independently introduced by Boris Tsygan (homology) [1] and Alain Connes (cohomology) [2] in the 1980s.
In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolute homology group comes from which subspace.
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Example of singular 1-chains: The violet and orange 1-chains cannot be realized as a boundary of a 2-chain. The usual construction of singular homology proceeds by defining formal sums of simplices, which may be understood to be elements of a free abelian group, and then showing that we can define a certain group, the homology group of the topological space, involving the boundary operator.