K-homology

Summary

K-homology is the homology theory dual to topological K-theory. In its analytic form it assigns to a separable C*-algebra $A$ the groups $K^0(A)$ and $K^1(A)$ of classes of Fredholm modules over $A$. For a compact metric space $X$, it gives the groups $K_i(X) = K^i(C(X))$. Elliptic operators on a manifold give K-homology classes, and pairing them with K-theory classes gives Fredholm indices. It is the special case $KK(A, \C)$ of KK-theory, and the left-hand side of the Baum–Connes conjecture is an equivariant K-homology group.