\(C^ *\)-algebras, positive scalar curvature, and the Novikov conjecture. III
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.
scientific article; zbMATH DE number 3977858