A Thom isomorphism for infinite rank Euclidean bundles (Q1397811)

In the paper under reviewing the author proves an equivariant version of Thom's isomorphism theorem for infinite-rank Euclidean vector bundles over finite-dimensional Riemannian manifolds with isoreductive actions (Definition 4.1) of non-compact topological groups, which is a generalization of proper actions in the sense of Baum-Connes-Higson [\textit{P. Baum, A. Connes} and \textit{N. Higson}, Contemp. Math. 167, 241-291 (1994; Zbl 0830.46061)]. Let \(G\) be a smooth, second countable, locally compact group and \(M\) a smooth Riemannian \(G\)-manifold. Consider a smooth Euclidean \(G\)-bundle \(p: \mathfrak E \to M\) modeled on infinite-dimensional Euclidean space \(\mathcal E\) of countable infinite dimension and the structural group \(O(\mathcal E)\) in the norm (or strong) operator topology. The group \(G\) is supposed to act isoreductively on the bundle. For a finite-rank Euclidean subbundle \(E^\alpha\) the C*-algebra \(\mathcal C(M)\) is defined as the C*-algebra of continuous sections of the complexified Clifford algebra bundle \(\text{Cliff}(TM)\) of the tangent bundle TM vanishing at infinity and \(\mathcal A(M) = C_0(M) \widehat{\otimes} \mathcal C(M)\) is the \(\mathbb Z_2\)-graded suspension of \(\mathcal C(M)\). There is a Morita equivalence given by the spinor bundle between \(\mathcal C(M)\) and \(C_0(M)\). Equip \(\mathfrak E\) with a compatible \(G\)-invariant connection \(\nabla\) which controls splittings used in defining the connecting maps of the direct limit \(\mathcal A(\mathfrak E) = \varinjlim_{E^\alpha \subset \mathfrak E} \mathcal A(E^\alpha)\), taken over the directed system of all smooth finite rank Euclidean subbundles, ordered by inclusion of subbundles. The equivariant inclusion \(M \hookrightarrow \mathfrak E\) as the zero section, canonically induces an equivariant Thom *-homomorphism \(\Psi_p: \mathcal A(M) \to \mathcal A(\mathfrak E)\). It induces an equivariant Thom homomorphism and the main result of the paper is Theorem 1.1 asserts an isomorphism \(\Psi^G : K^G_*(\mathcal A(M)) \to K^G_*(\mathcal A(\mathfrak E))\).