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On equivariant Kasparov theory and Spanier-Whitehead duality. (English) Zbl 0811.46075

Summary: Suppose that \(G\) is a second countable compact Lie group and that \(A\) and \(B\) are commutative \(G\)-\(C^*\)-algebras. Then the Kasparov group \(KK_ *^ G (A,B)\) is a bifunctor on \(G\)-spaces. It is computed here in terms of equivariant stable homotopy theory. This result is a consequence of a more geneal study of equivariant Spanier-Whitehead duality and uses in an essential way the extension of the Kasparov machinery to the setting of \(\sigma\)-\(G\)-\(C^*\)-algebras. As a consequence, we show that if \((X,x_ 0)\) is a based separable compact metric \(G\)-ENR (such as a smooth compact \(G\)-manifold) and \((Y,y_ 0)\) is a based countable \(G\)-CW-complex then there is a natural isomorphism \[ KK_ *^ G (C(X,x_ 0), C(Y,y_ 0)) \cong K_ G^*(Y\wedge {\mathbf F}X) \] where \({\mathbf F}X\) is the functional equivariant Spanier-Whitehead dual of \(X\). This specializes when \(Y\) is trivial to yield a natural isomorphism \[ KK_ *^ G (C(X,x_ 0), \mathbb{F}) \cong {}^ s K_ *^ G(X) \] where \({}^ s K_ *^ G(-)\) denotes equivariant Steenrod \(K\)-homology theory. This result is new even for \(X\) a finite \(G\)-CW-complex, in which case Steenrod \(K\)- homology coincides with the usual topological equivariant \(K\)-homology \(K_ *^ G(X)\).

MSC:

46L80 \(K\)-theory and operator algebras (including cyclic theory)
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
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