Enlargeability and index theory (Q2434080)

This paper relates two important concepts. The first one is (a complex version of) certain \(K\)-theory index element, \(\alpha_{\max}(M)\in K_n(C^*_{\max}(\pi_1(M)))\), constructed by \textit{J.~Rosenberg} for any closed spin manifold \(M^n\) [Topology 25, 319--336 (1986; Zbl 0605.53020)]. The Lichnerowicz-Schrödinger-Weitzenböck formula shows that this index is zero if \(M\) admits a metric of positive scalar curvature. The converse was proved to be false by the second author [Topology 37, 1165--1168 (1998; Zbl 0976.53052)]. The second fundamental concept of the paper is the following: a closed oriented manifold \(M^n\) is called enlargeable if, for some metric and all \(\varepsilon>0\), there is a finite connected cover \(\overline M\) of \(M\) and an \(\varepsilon\)-contracting map \(\overline M\to\mathbb{S}^n\). An area-enlargeable version can be given, too. This notion was introduced by Gromov and Lawson, who proved that enlageable spin manifolds do not admit metrics of positive scalar curvature [\textit{M. Gromov} and \textit{H. B. Lawson jun.}, Ann. Math. (2) 111, 209--230 (1980; Zbl 0445.53025)]. The first main result of the paper shows that, for the problem of existence of metrics with positive scalar curvature, the index obstruction \(\alpha_{\max}(M)\) subsumes the enlargeability obstruction; i.e., if a spin manifold \(M\) is enlargeable or area-enlargeable, then \(\alpha_{\max}(M)\neq0\). As another application of their methods, the authors also give an affirmative answer to a question of Burghelea: if \(M\) is an enlargeable or area-enlargeable manifold, \(f:M\to B\pi_1(M)\) a classifying map for its universal cover and \([M]\in H_n(M;\mathbb{Q})\) its fundamental class, then \(f_*([M])\neq0\) in \(H_n(B\pi_1(M);\mathbb{Q})\). It is remarkable that no spin assumption is required here.