Index theory in spaces of manifolds (Q2423433)

The author proves the families index theorem by realizing analytic and topological indices as homotopic weak maps of spectra, with the traditional families indices corresponding to the maps of the \(n=0\) spaces. Because the results extend to apply to operators of Dirac type over \(C^*\)-algebras, they generalize results of \textit{A. S. Mishchenko} and \textit{A. T. Fomenko} [Izv. Akad. Nauk SSSR, Ser. Mat. 43, 831--859 (1979; Zbl 0416.46052)]. For a single operator, the map between the \(n=1\) spaces recovers the index theorem for partitioned manifolds of \textit{J. Roe} [Lond. Math. Soc. Lect. Note Ser. 135, 187--228 (1988; Zbl 0677.58042)]. \par The target spectrum is the (appropriate (de)suspension of the) \(K\)-theory spectrum. The domain spectrum for the index map is the space of manifolds described in [\textit{S. Galatius} and \textit{O. Randal-Williams}, Geom. Topol. 14, No. 3, 1243--1302 (2010; Zbl 1205.55007)]. The \(n^{th}\) space in the spectrum is the space of not necessarily compact manifolds (having specified structure) with proper maps to \(\mathbb R ^n\). Galatius and Randal-Williams [loc. cit.] showed that this spectrum is stably equivalent to the corresponding spectrum defined by affine subspaces of \(\mathbb R ^n\). The author uses this observation to define the topological index and prove that it is homotopic to the analytic index.