Metrics of positive scalar curvature on spheres and the Gromov-Lawson conjecture

Abstract

Boguslaw Hajduk Institute of Mathematics, Wrociaw University, pl. Grunwaldzki 2/4, PL-50-384 Wrodaw, Poland 1. Introduction In [2] and [9] an ingenious procedure is given to construct a Riemannian metric of positive scalar curvature on a manifold obtained by surgery from one which already has such a metric. With some improvements of [1] this may be summarized as follows. 1.1. Theorem. Let (M, g) be a compact n-dimensional Riemannian manifold with positive scalar curvature and let W be a cobordism from M to M' such that W admits a handle decomposition on M with no handles of index greater than n- 2 (i.e. there exists a Morse function on W which is minimal on M and critical points have indices <-_ n- 2). Then there exists a metric of positive scalar curvature on W which extends g and is product on a collar of MUM'. This construction applied to the cobordism with one handle of index 1 between the disjoint sum MuN and the connected sum M4#N shows imme- diately that the connected sum is well defined for manifolds of positive scalar curvature. Furthermore, this operation gives an abelian group structure in the set n~ of concordance classes of positive scalar curvature metrics on S n, with the zero class represented by the standard metric gcan [1]. We say that two metrics go, gl of positive scalar curvature on M are concordant if there exists a metric g of positive scalar curvature on M x [0,1] such that glM x {i} =gi, i=0, 1, and g is product near M x d[0,1]. Our aim is to show how this group, or its subgroup ~ of classes of those metrics which are boundary restrictions of metrics of positive scalar curvature on compact spin manifolds, is related to some questions concerning positive scalar curvature. IfX n is a 2-connected dosed manifold, B a smooth n-baU in X, then by Morse- Srnale theory and Theorem 1.1 there is a metric of positive scalar curvature on X- Int B which is product near S n- i = O(X - Int B). This metric induces a metric of positive scalar curvatures on S "- 1. Our basic observation is that the concordance class 6(X) of the induced metric depends only on the spin cobordism class of X and 6(X)--. 0 if and only if any l-connected manifold spin cobordant to X admits a