Simply connected manifolds of positive scalar curvature. (English) Zbl 0784.53029

Stolz, Stephan

Ann. Math. (2) 136, No. 3, 511-540 (1992).

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 \(\alpha(M)=0\).

Reviewer: K.Sekigawa (Niigata)

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

Keywords:

Dirac operator; Fredholm operators; KO-characteristic number; spin manifold

Citations:

Zbl 0463.53025; Zbl 0136.184; Zbl 0284.58016