Construction of manifolds of positive scalar curvature

TLDR: In this article, it was shown that a regular neighborhood of a codimension > 3 subcomplex of a manifold can be chosen so that the induced metric on its boundary has positive scalar curvature.

Abstract: We prove that a regular neighborhood of a codimension > 3 subcomplex of a manifold can be chosen so that the induced metric on its boundary has positive scalar curvature. A number of useful facts concerning manifolds of positive scalar curvature follow from this construction. For example, we see that any finitely presented group can appear as the fundamental group of a compact 4-manifold with such a metric. 0. Outline of results. We give a new method for constructing complete Riemannian manifolds of positive scalar curvature and use it to continue the investigation of properties of positive scalar curvature. Our construction uses the idea that manifolds having spheres of dimension > 2 as "factors" will admit metrics of positive scalar curvature if the spheres can be made to carry sufficient positive curvature to dominate any negative curvature. Most of the known methods for constructing manifolds of positive scalar curvature employ this same idea. For example, any manifold of the form M X S2 can be given a warped-product metric of positive scalar curvature by suitably adjusting the radius of the S2-factor. Similarly, by deforming the standard metric on S3-{point} in a small neighborhood of the point and using the S2-factor to carry positive curvature around the corner we can construct a complete metric of positive scalar curvature on R3. This same idea was used by Gromov and Lawson [GL] and Schoen and Yau [SY] in proving that codimension > 3 surgeries on a manifold of positive scalar curvature yields a manifold which also carries positive scalar curvature. In this paper we generalize the above techniques to cover any manifold formed as the boundary of a regular neighborhood of a subcomplex K of a manifold M. If the codimension of K > 3 this boundary looks locally like K X S2, and so should carry positive scalar curvature. THEOREM 1. Let M be an n-dimensional Riemannian manifold with a fixed smooth cell decomposition and K a codimension q > 3 subcomplex of M. Then there is a regular neighborhood U of K in M so that the induced metric on the boundary dU has positive scalar curvature. An easy consequence of this theorem is the following. COROLLARY 2. Let 7r be a finitely presented group. Then there exists a compact 4-manifold M of positive scalar curvature with 7rl (M) = 7r. This fact is interesting since it is generally believed that manifolds that are "large" in some sense should not adrrlit metrics of positive curvature. Corollary 2 Receiv?d by the editors Septem})er 25, 1985 and, ill revised form, July 17, 1986. 1980 M(lthf'rB(lti('.s.S?l{Jjf'('t (l(l.N'.N'iJl('(ltiOn (1985 RfviSion). Primary 53C20. (r)1988 Americatl Mathematic.al Society 0002-9947/88 $1.00 + $.25 per page