Examples of "cocycle"
Here is a 2-cocycle on . This 2-cocycle can be obtained from albeit in a highly nontrivial way.
for some scalar in . It follows that the 2-cocycle or Schur multiplier satisfies the cocycle equation
Thus a (q−1)-cochain "f" is a cocycle if for all "q"-simplices σ the cocycle condition formula_27 holds. In particular, a 1-cochain "f" is a 1-cocycle if
The trivial 2-cocycle gives the trivial extension, and since a 2-coboundary is cohomologous with the trivial 2-cocycle, one has
Since every central extension comes from a 2-cocycle , it suffices to show that every 2-cocycle is a coboundary. Suppose is a 2-cocycle on . The task is to use this 2-cocycle to manufacture a 1-cochain such that .
In particular, note that this proof demonstrates that the [[Cocycle (algebraic topology)|cocycle]] condition "d" = 0 is in a sense dual to the Jacobi identity.
for almost all ("g", "h", "x"). A unitary cocycle is strict if and only if the above relations hold for all ("g", "h", "x"). It can be shown that for any unitary cocycle there is a strict unitary cocycle which is equal almost everywhere to it (Varadarajan, 1985).
The group elements must in addition satisfy the cocycle condition
defined on each nonempty overlap, such that the "cocycle condition"
accumulation of multiplied cocycle values (and limits thereof) according to
that satisfies the cocycle condition: writing "m" for multiplication,
as well as the "cocycle relation" (guaranteeing associativity)
A Haefliger structure on a space "X" is determined by a Haefliger cocycle. A codimension-"q" Haefliger cocycle consists of a covering of "X" by open sets "U", together with continuous maps Ψ from
The cup product of two cocycles is again a cocycle, and the product of a coboundary with a cocycle (in either order) is a coboundary. The cup product operation induces a bilinear operation on cohomology,
The third condition applies on triple overlaps "U" ∩ "U" ∩ "U" and is called the cocycle condition (see Čech cohomology). The importance of this is that the transition functions determine the fiber bundle (if one assumes the Čech cocycle condition).
That this product is well-defined follows from the cocycle relation formula_7. The map
so with this 2-cocycle, equivalent to the previous one, one has
The set of transition functions forms a Čech cocycle in the sense that
and then we have "u" = "a" in "K". This element "a" specifies a cocycle "c" by
whenever "v" is close enough to "u". The Haefliger cocycle is defined by