Cochain
A homogeneous element of an Abelian cochain group 
(or, in the general case, a module), i.e. a graded Abelian group equipped with an endomorphism 
of degree 
such that 
. The endomorphism 
is called the coboundary mapping or the coboundary.
A cochain group 
usually arises as a group 
or 
, where 
is an arbitrary Abelian group, called the coefficient group, and 
is a chain group, i.e. a graded Abelian group equipped with an endomorphism 
of degree 
(the boundary mapping or boundary) with 
. In this situation the mapping 
on the group 
is defined as the adjoint of 
, where 
, 
.
Given a topological space 
, one defines the group 
of singular chains as the Abelian group of formal finite sums 
, where 
and the 
are arbitrary singular simplices in 
, i.e. continuous mappings of the standard simplex into 
. A singular cochain in 
with coefficients in 
is defined as a homogeneous element of the group 
.
Similarly, a simplicial 
-cochain of a simplicial complex in 
with coefficients in an Abelian group 
is defined as a homomorphism 
, where 
is the group of 
-chains of 
, i.e. the group of formal finite sums 
, where 
and the 
are 
-simplices in 
. In particular, a cochain in the sense of Aleksandrov–Čech in an arbitrary topological space 
is a cochain of the nerve of an open covering of 
.
If 
is a 
-complex (and 
denotes the 
-skeleton of 
), then the Abelian group 
is called the group of 
-dimensional cellular cochains of the complex 
. The coboundary homomorphism 
is put equal to the connecting mappings of the triple 
.
In practice, the group 
is frequently provided with an additional multiplicative structure, i.e. it is a graded algebra. In these cases the coboundary mapping 
possesses the Leibniz property: 
, where the element 
is assumed to be homogeneous of degree 
. An example of such a graded cochain algebra is the algebra of differential forms on a smooth manifold, in which the exterior differential acts as coboundary.
References
| [1] | P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1960) |
| [2] | S. MacLane, "Homology" , Springer (1963) |
Comments
References
| [a1] | N.E. Steenrod, S. Eilenberg, "Foundations of algebraic topology" , Princeton Univ. Press (1966) |
| [a2] | E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) |
| [a3] | C.R.F. Maunder, "Algebraic topology" , v. Nostrand-Reinhold (1970) |
Cochain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cochain&oldid=31316