\(C^ *\)-algebras, positive scalar curvature, and the Novikov conjecture. III
type: Item
instance of: scholarly article
title: \(C^ *\)-algebras, positive scalar curvature, and the Novikov conjecture. III
zbMATH Open document ID: 0605.53020
DOI: 10.1016/0040-9383(86)90047-9
published in: Topology
publication date: 1986-01-01T00:00:00Z
review text: This paper is the third, by the author, in a series of papers on the subject matter of the title [Part I, cf. Publ. Math., Inst. Hautes Etud. Sci. 58, 409-424 (1983; Zbl 0526.53044), Part II, cf. Proc. U.S.-Japan Semin. on Geom. Meth. in Operator Algebras, Kyoto (1983)]. The question addressed by the author concerns the existence of Riemannian metrics of positive scalar curvature on closed, connected smooth manifolds. Among the results obtained is a proof that the Gromov-Lawson Conjecture is true for simply connected manifolds \(M^ n\) with \(5\leq n\leq 23\). The author then proposes a conjecture that if \(M^ n\) is a connected closed manifold and if the fundamental group \(\Pi =\Pi_ 1(M)\) is finite of odd order, then M admits a metric of positive scalar curvature if and only if its universal cover does so. The conjecture turns out to be true when \(n\geq 5\) and \(| \Pi_ 1(M)|\) is an odd prime. The concluding sections of the paper concern the Gromov-Lawson Conjecture in the case where \(\Pi_ 1(M)\) is infinite and torsion-free. The main theorem is essentially the result that the condition of the Gromov-Lawson Conjecture is necessary.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J22
Mathematics Subject Classification ID: 57R15
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46M20
zbMATH DE Number: 3977858
zbMATH Keywords: spin manifold
zbMATH Keywords: bordism class
zbMATH Keywords: Stiefel-Whitney class
zbMATH Keywords: positive scalar curvature
zbMATH Keywords: Gromov-Lawson Conjecture
zbMATH Keywords: fundamental group
Wikidata QID: Q123280663
author: Jonathan M. Rosenberg
reviewed by: A. P. Stone
MaRDI profile type: Publication
full work available at URL: 0040-9383(86)90047-9
OpenAlex ID: W1999535754
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description: scientific article; zbMATH DE number 3977858