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Simply connected manifolds of positive scalar curvature. (English) Zbl 0784.53029
Concerning the question, “Which manifolds admit metrics of positive scalar curvature?”, M. Gromov and H. B. Lawson jun. have proved that a simply connected closed manifold \(M\) of dimension \(\geq 5\), which does not admit a spin structure, carries a metric of positive scalar curvature [Ann. Math., II. Ser. 111, 423-434 (1980; Zbl 0463.53025)]. The proof of this result is based on the “surgery lemma” - - if a manifold is obtained from a manifold \(N\) by surgery of codimension \(\geq 3\), and \(N\) admits a metric of positive scalar curvature, then so does \(M\). On one hand, Lichnerowicz proved a strong vanishing theorem – if an \(n\)-dimensional spin manifold has positive scalar curvature, the kernel and cokernel of the Dirac operator are trivial, in particular, \(n=0\) mod 4, the characteristic number \(\hat A(M)\) (the index of the Dirac operator) vanishes [C. R. Acad. Sci., Paris, Ser. A-B 257, 7-9 (1963; Zbl 0136.184)]. The above result was generalized by N. Hitchin, who constructed a family of Fredholm operators closely related to the Dirac operator, whose index is a KO-characteristic number \(\alpha(M)\in KO(S^ n)\) [Adv. Math. 14, 1-55 (1974; Zbl 0284.58016)]. He proved that \(\alpha(M)=0\) if a spin manifold \(M\) has a metric of positive scalar curvature. In the present paper, the author gives a positive answer to the conjecture by Gromov and Lawson in the same paper as above. Namely, the author proves that a simply connected, closed, spin manifold \(M\) of dimension \(\geq 5\) carries a metric with positive scalar curvature if and only if \(\alpha(M)=0\).

MSC:
53C20 Global Riemannian geometry, including pinching
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
57R20 Characteristic classes and numbers in differential topology
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