K-homology is the homology theory dual to topological K-theory, with an analytic model using operators
that are invertible modulo compact operators
(Higson and Roe 2000). For a compact Hausdorff space
, an odd analytic cycle consists of a
complex Hilbert space 
, a C*-algebra
representation 
,
and a bounded self-adjoint operator 
for which 
and 
are compact for all 
, taking a unital representation. Cycles are identified
by unitary equivalence, homotopy, and addition of cycles
whose compact error terms are zero. Direct
sum gives the odd group 
. The even version uses a decomposition 
preserved by 
and exchanged by 
, and gives 
.
The support convention matters for noncompact spaces. K-homology with compact support is obtained from
compact subsets, or finite
subcomplexes of a CW-complex. Locally finite analytic
K-homology instead uses 
and permits cycles extending through the noncompact space. Assembly
maps compare these topological classes with analytic index classes.
