6 The topological side
6.1 K-homology of a cocompact piece
Let \(\Gamma \) be countable and discrete. Use locally compact, second-countable Hausdorff proper \(\Gamma \)-spaces with continuous actions. The concrete universal model is the geometric telescope from Chapter 4. Choose a common universe containing the space and module carriers.
For each closed invariant cocompact piece \(S\), define its K-homology to be the genuine full Chapter 5 group \(KK_i^\Gamma (C_0(S,\mathbb C),\mathbb C)\) for \(i=0,1\). The source action is \((g\cdot a)(x)=a(g^{-1}x)\) and the scalar coefficient action is trivial. Separability of the actual supremum-norm \(C_0\) algebra follows by isometric extension by zero to the one-point compactification.
6.2 Transition maps
For \(S\subseteq T\), actual restriction \(C_0(T)\to C_0(S)\) is an equivariant nonunital complex star homomorphism. It induces the forward additive map on K-homology by contravariance in the first KK-variable. Identity inclusions act as identities and successive inclusions compose.
Closed inclusions are proper and therefore preserve vanishing at infinity under pullback. Apply the proved Chapter 5 source-pullback laws to these restriction homomorphisms; the two successive restrictions are their actual composite on every function.
6.3 The filtered colimit
Construct
Here \(S\) ranges over all closed invariant cocompact pieces of the chosen model. The constructed additive group satisfies the colimit universal property; two representatives agree exactly when they agree in a larger piece. Replacing this system by any monotone cofinal directed family induces a canonical additive equivalence. The cofinality condition is \(\forall S\, \exists j\colon S\subseteq S_j\).
The group is independent of the locally compact second-countable genuine universal proper CW model satisfying the Chapter 4 fixed-point criterion. Every equivariant comparison map induces the same additive equivalence. The canonical implementation uses the actual geometric telescope.
The colimit is the quotient of the disjoint union of the actual piece groups by eventual equality. Its group operations and universal property are proved. Model comparison is genuinely piecewise: an equivariant map restricted to a cocompact source piece is proper and has a closed cocompact image. Compact orbit representatives and finite transporters prove this fact. The same argument bounds the entire interval of a homotopy inside one target piece. Proper homotopies then give point-norm \(C_0\) homotopies and equal maps on the full even and odd KK groups. Chapter 4’s proved CW uniqueness supplies inverse comparisons up to equivariant homotopy.
Cofinality is necessary: directed families that merely cover every point can have different colimits. No global properness of a universal comparison map is assumed. This chapter constructs the topological side and does not assert that the assembly map is an isomorphism.