Positive scalar curvature and product formulas for secondary index invariants (Q2826642)

Let \((M,g)\) be a Riemannian manifold of dimension \(n\) where \(g\) has uniformly positive scalar curvature. Assume that \(M\) is spin and that \(\Gamma\) is a countable discrete group acting freely and properly by spin isometries on \(M\). Let \(\mu:K_*^1(M)\rightarrow K_*(C^*(X)^\Gamma)\) be the coarse assembly map. The Dirac operator gives rise to an element \([D]\in K_n^\Gamma(X)\) and the associated coarse index \(\mu([D])\) vanishes by the Lichnerowicz formula. This fact gives rise to a secondary invariant \(\rho^\Gamma(t)\in S_n^\Gamma(X)\). Given two such metrics \(g_0\) and \(g_1\), there is a relative index \(\mathrm {ind}_{\mathrm {real}}^\Gamma(g_0,g_1)\) in \(K_{n+1}(C^*(X)^\Gamma)\) such that \(\partial \mathrm {ind}_{\mathrm {real}}^\Gamma(g_0,g_1)=\rho^\Gamma(g_0)-\rho^\Gamma(g_1)\). These secondary invariants have often been applied previously to the case that \(M\) is the universal cover of a closed spin manifold \(M_0\) and \(\Gamma\) is the fundamental group. However, one can also apply them in the non-(co)compact setting. The author uses this to give conceptual proofs for arbitrary \(m\) of the delocalized Atiyah-Patodi-Singer index theorem using appropriate product formulas to change the parity of \(m\). The author also derives a product formula for the relative index. The author combines the localization algebra of Yu with the description of K-theory for graded \(C^*\) algebras of Trout. The first section of the paper is an introduction that puts matters in historical context and summarizes the main results. The second section treats graded \(C^*\) algebras and K-theory, the external product, Bott periodicity, and suitable long exact sequences. The third section deals with Yu's localization algebras, the \(Cl_n\) localization algebras, the functoriality of the localization algebra, and the external product in this setting. The fourth section discusses local index classes, secondary invariants, and product formulas and the fifth section discusses the compatibility with boundary maps. The sixth section deals with geometric applications: the delocalized Atiyah-Patodi-Singer index theorem and the stability of higher secondary invariants. There is an appendix dealing with the complex setting which gives explicit descriptions in terms of projections and unitaries.