Examples of "cochains"
The coboundary of the cup product of cochains c and d is given by
The construction starts with a product of cochains: if "c" is a "p"-cochain and
Some 2-cocycles can be obtained from 1-cochains. A 1-cochain on is simply a linear map,
Technically speaking, "G" may be a profinite group, in which case the definitions need to be adjusted to allow only continuous cochains.
The relation to the coboundary operator "d" that was defined in the previous section, and which acts on the "inhomogeneous" cochains formula_20, is given by reparameterizing so that
This construction initially leads to a coboundary operator that acts on the "homogeneous" cochains. These are the elements of Hom("F", "M") i.e. functions φ: "G" → "M" that obey
defined by an explicit formula on singular cochains. The product of cohomology classes "u" and "v" is written as "u" ∪ "v" or simply as "uv". This product makes the direct sum
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.
Differential -forms can be integrated over a -simplex in a natural way, by pulling back to . Extending by linearity allows one to integrate over chains. This gives a linear map from the space of -forms to the th group of singular cochains, , the linear functionals on . In other words, a -form defines a functional
The Lie algebra cohomology of the Lie algebra formula_10 over the field formula_11, with values in the left formula_10-module formula_13 can be computed using the "Chevalley-Eilenberg complex" formula_14. The formula_15-cochains in this complex are the alternating formula_11-multilinear functions formula_17 of formula_15 variables with values in formula_13. The coboundary of an formula_15-cochain is the formula_21-cochain formula_22 given by
The definition using derived functors is conceptually very clear, but for concrete applications, the following computations, which some authors also use as a definition, are often helpful. For "n" ≥ 0, let "C"("G", "M") be the group of all functions from "G" to "M". This is an abelian group; its elements are called the (inhomogeneous) "n"-cochains. The coboundary homomorphisms
Using the "G"-invariants and the 1-cochains, one can construct the zeroth and first group cohomology for a group "G" with coefficients in a non-abelian group. Specifically, a "G"-group is a (not necessarily abelian) group "A" together with an action by "G".
In mathematics, Banach algebra cohomology of a Banach algebra with coefficients in a bimodule is a cohomology theory defined in a similar way to Hochschild cohomology of an abstract algebra, except that one takes the topology into account so that all cochains and so on are continuous.
This is also naturally isomorphic to the cohomology of the sub–chain complex formula_3 consisting of all singular cochains formula_4 that have compact support in the sense that there exists some compact formula_5 such that formula_6 vanishes on all chains in formula_7.
The first cohomology group is the quotient of the so-called "crossed homomorphisms", i.e. maps (of sets) "f" : "G" → "M" satisfying "f"("ab") = "f"("a") + "af"("b") for all "a", "b" in "G", modulo the so-called "principal crossed homomorphisms", i.e. maps "f" : "G" → "M" given by "f"("a") = "am"−"m" for some fixed "m" ∈ "M". This follows from the definition of cochains above.
A "q"-cochain of formula_1 with coefficients in formula_5 is a map which associates with each "q"-simplex σ an element of formula_14 and we denote the set of all "q"-cochains of formula_1 with coefficients in formula_5 by formula_17. formula_17 is an abelian group by pointwise addition.
If "u", "v", and "w" are 1-cochains dual to the 3 rings, then the product of any two is a multiple of the corresponding linking number and is therefore zero, while the Massey product of all three elements is non-zero, showing that the Borromean rings are linked. The algebra reflects the geometry: the rings are pairwise unlinked, corresponding to the pairwise (2-fold) products vanishing, but are overall linked, corresponding to the 3-fold product not vanishing.
Let ("A","ρ",[..]) be a Lie algebroid over a smooth manifold "M" and let Ω("A") denote its Lie algebroid complex. Let further "E" be a ℤ-graded vector bundle over "M" and Ω("A","E") = Ω("A") ⊗ Γ("E") be its ℤ-graded "A"-cochains with values in "E". A representation up to homotopy of "A" on "E" is a differential operator "D" that maps
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.
is induced. The so-called connecting homomorphisms δ : "H"("G", "N") → "H"("G", "L") can be described in terms of inhomogeneous cochains as follows. If "c" is an element of "H"("G", "N") represented by an "n"-cocycle φ : "G" → N, then δ("c") is represented by "d"(ψ), where ψ is an "n"-cochain "G" → M "lifting" φ (i.e. such that φ is the composition of ψ with the surjective map "M" → "N").