What is a Cochain?
Definition and properties A cochain group C⋆ is a graded Abelian group, which means that C⋆ is decomposed as the direct sum of subgroups Ak, indexed with k∈Z, some of which might be trivial; an homogenenous element f is an element belonging to some Ak, where k is called the degree of the element and denoted by deg(f).
What does group cohomology measure?
. The cohomology groups in turn provide insight into the structure of the group G and G-module M themselves. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper.
What is the difference between homology and cohomology?
In homology, you look at sums of simplices in the topological space, upto boundaries. In cohomology, you have the dual scenario, ie you attach an integer to every simplex in the topological space, and make identifications upto coboundaries.
Why is cohomology important?
Cohomology is important because you can take cup products. In particular, you get multiplication maps H^k(X) x H^m(X) —-> H^{k+m}(X) given by the cup product (again, you define this on the chain complex, and then show that it induces a map on its homology, i.e., on cohomology).
What is the purpose of homological algebra?
Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other ‘tangible’ mathematical objects. A powerful tool for doing this is provided by spectral sequences.
Why is cohomology useful?
Cohomology is used in physics to compute topological structure of gauge fields, like the electromagnetic field in the AB effect. Here, the electron encircles a magnetic flux, which you can measure in the self interference pattern of the electron.
What is a cochain?
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.
What is cohomology in topology?
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with 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.
What is a cochain of a CW complex?
In particular, a cochain in the sense of Aleksandrov–Čech in an arbitrary topological space X is a cochain of the nerve of an open covering of X (see Simplicial complex ). If X is a CW-complex (and X n denotes the n -skeleton of X ), then the Abelian group H n ( X n, X n − 1) is called the group of n -dimensional cellular cochains of the complex X.
How do you define cosimplices and cochains?
Cosimplices and cochains are defined by passing to dual spaces Ck ( X, G) of cochains on X with values in G (ℝ or Z ). Ck ( X, G) is defined as the set of maps f: Ck ( X, G) → G such that for any finite formal sum one has f(∑g iσ i) = ∑g if(σ i), g i ∈ ℕ[resp. Z], f(σ i) ∈ ℕ[resp. Z].