Index theory for manifolds with Baas-Sullivan singularities (Q1747030)

This article generalises the geometric model for \(K\)-homology by \textit{P. Baum} and \textit{R. G. Douglas} [Proc. Symp. Pure Math. 38, 117--173 (1982; Zbl 0532.55004); Contemp. Math. 10, 1--31 (1982; Zbl 0507.55004)] to manifolds with Baas-Sullivan singularities. The latter means a manifold \(Q\) with boundary \(\partial Q\) and a diffeomorphism \(\partial Q \cong \beta Q \times P\) for a fixed closed manifold \(P\). Important examples for \(P\) that are studied in depth in the article are the discrete set with \(k\) points, which leads to \(\mathbb{Z}/k\)-manifolds, and the sphere \(S^1\). It is assumed that the stable normal bundle of \(P\) is trivial because this is needed to prove key properties of the theory. The definition of the geometric \(K\)-homology for this situation and the proofs of its basic properties are similar in many respects to the previous work for \(\mathbb{Z}/k\)-manifolds. A key property of the \(P\)-manifold \(K\)-homology defined here is an analogue of the Böckstein sequence and a uniqueness theorem that says when two manifolds \(P\) and \(P'\) give the same \(K\)-homology theory. There is also an analogue of the Baum-Connes assembly map for the \(P\)-manifold \(K\)-homology theory. The variant of geometric \(K\)-homology defined using \(P\)-manifolds has an analytic counterpart, and the comparison of the geometric and analytic versions of the \(K\)-homology theory encodes important index theoretic constructions. For instance, consider the classical Baum-Douglas model for \(K\)-homology. A cycle for it is a triple \((M,E,\varphi)\) consisting of a \(\mathrm{Spin}^c\)-manifold \(M\), a vector bundle \(E\) over \(M\) and a continuous map \(\varphi: M\to X\). The map to Kasparov's analytic \(K\)-homology takes this cycle to the pushforward of the Dirac operator on \(M\) with coefficients in \(E\) along the map \(\varphi\). Explicit computations of this map therefore involve the Atiyah-Singer Index Theorem. When dealing with \(\mathbb{Z}/k\)-manifolds, this is replaced by an index theorem by \textit{D. S. Freed} and \textit{R. B. Melrose} [Invent. Math. 107, No. 2, 283--299 (1992; Zbl 0760.58039)]. It is explained how this fits into the more general framework considered in this article, and an analogous comparison map from the geometric to an analytic theory for \(S^1\)-manifolds is computed in the case where \(X\) is a point.