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