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Enlargeability and index theory: infinite covers. (English) Zbl 1128.58012

Summary: In [B. Hanke and T. Schick, J. Differ. Geom. 74, No. 2, 293–320 (2006; Zbl 1122.58011)] we showed the non-vanishing of the universal index elements in the \(K\)-theory of maximal \(C^*\)-algebras of fundamental groups of enlargeable spin manifolds. The underlying notion of enlargeability was the one from M. Gromov and H. B. Lawson [Ann. Math. (2) 111, No. 2, 209–230 (1980; Zbl 0445.53025)], involving contracting maps defined on finite covers of the given manifolds.
In the paper at hand, we weaken this assumption to the one in [M. Gromov and H. B. Lawson, Publ. Math., Inst. Hautes Étud. Sci. 58, 83–196 (1983; Zbl 0538.53047)] where infinite covers are allowed. The new idea is the construction of a geometrically given \(C^*\)-algebra with trace which encodes the information given by these infinite covers. Along the way we obtain an easy proof of a relative index theorem relevant in this context.

MSC:

58J22 Exotic index theories on manifolds
46L80 \(K\)-theory and operator algebras (including cyclic theory)
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References:

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