A conjecture concerning positive Ricci curvature and the Witten genus
\textit{J. Lohkamp} [Bull. Am. Math. Soc., New Ser. 27, No. 2, 288-291 (1992; Zbl 0753.53026)] has proven that every manifold of dimension \(n\geq 4\) admits a Riemannian metric of negative Ricci curvature. In contrast, apart from the classical work of S. B. Myers (1935) and A. Lichnerowicz (1963), relatively little is known about the corresponding case of positive Ricci curvature. The present paper is intended to remedy this situation by investigating the following Conjecture: Let \(M\) be a smooth, closed spin-manifold of dimension \(4k\) such that \(1/2 p_1(M)=0\) (which is the characteristic class for spin-manifolds twice of which is the usual Pontryagin class). Then if \(M\) admits a Riemannian metric with positive Ricci curvature, the Witten genus of \(M\) vanishes. It is shown that this conjecture is true for certain classes of manifolds which are known to possess Riemannian metrics of positive Ricci curvature, and in particular that it implies that there are simply-connected closed \(4k\)-dimensional manifolds, \(4k=24\) and \(4k\geq 32\) which admit metrics of positive scalar curvature but have nonpositive Ricci curvature. Contents include: an introduction; the index theorem and Witten genus; evidence for the conjecture; a heuristic ``proof''; a group whose classifying space is \(BO\langle 8\rangle\); questions and comments; and a 26 item bibliography.
scientific article; zbMATH DE number 892574
scientific article