Mapping analytic surgery to homology, higher rho numbers and metrics of positive scalar curvature

Abstract

Let $\Gamma$ be a f.g. discrete group and let $\tilde{M}$ be a Galois $\Gamma$-covering of a smooth closed manifold $M$. Let $S_{\ast }^{\Gamma }(\tilde{M})$ be the analytic structure group, appearing in the Higson-Roe analytic surgery sequence $\to S_{\ast }^{\Gamma }(\tilde{M})\to K_{\ast }(M)\to K_{\ast }(C_r^{\ast }\Gamma )\to$. We prove that for an arbitrary discrete group $\Gamma$ it is possible to map the whole Higson-Roe sequence to the long exact sequence of even/odd-graded noncommutative de Rham homology $\to H_{[\ast -1]}(\mathcal{A}\Gamma )\to H_{[\ast -1]}^{del}(\mathcal{A}\Gamma )\to H_{[\ast ]}^e(\mathcal{A}\Gamma )\to$, with $\mathcal{A}\Gamma$ a dense homomorphically closed subalgebra of $C_r^{\ast }\Gamma$. Here, $H_{\ast }^{del}(\mathcal{A}\Gamma )$ is the delocalized homology and $H_{\ast }^e(\mathcal{A}\Gamma )$ is the homology localized at the identity element. Then, under additional assumptions on $\Gamma$, we prove the existence of a pairing between $H{C}_{del}^{\ast }(\mathbb{C}\Gamma )$, the delocalized part of the cyclic cohomology of $\mathbb{C}\Gamma$, and $H_{\ast -1}^{del}(\mathcal{A}\Gamma )$. This, in particular, gives a pairing between $S_{\ast }^{\Gamma }(\tilde{M})$ and $H{C}_{del}^{\ast -1}(\mathbb{C}\Gamma )$. We also prove the existence of a pairing between $S_{\ast }^{\Gamma }(\tilde{M})$ and the relative cohomology $H^{[\ast -1]}(M\to B\Gamma )$. Both these parings are compatible with known pairings associated with the other terms in the Higson-Roe sequence. In particular, we define higher rho numbers associated to the rho class $\rho (\tilde{D})\in S_{\ast }^{\Gamma }(\tilde{M})$ of an invertible $\Gamma$-equivariant Dirac type operator on $\tilde{M}$. Finally, we provide a precise study for the behavior of all previous K-theoretic and homological objects and of the higher rho numbers under the action of the diffeomorphism group of $M$. Then, we establish new results on the moduli space of metrics of positive scalar curvature when $M$ is spin.

144 pages. Changes from the first version: the title has been modified; imprecisions have been corrected; more details are given; several new sections with many geometric applications have been added. v6: correction of typos and minor mistakes, typesetting changed. Close to final version to be published in Memoir AMS