Positive $(p, n)$-intermediate scalar curvature and cobordism

Abstract

In this paper we consider a well-known construction due to Gromov and Lawson, Schoen and Yau, Gajer, and Walsh which allows for the extension of a metric of positive scalar curvature over the trace of a surgery in codimension at least 3 to a metric of positive scalar curvature which is a product near the boundary. We generalize this construction to work for pp, nq-intermediate scalar curvature for 0 ď p ď n ´2 for surgeries in codimension at least p`3. We then use it to generalize a well known theorem of Carr. Letting R sp,ną0 pM q denote the space of positive pp, nq-intermediate scalar curvature metrics on an n-manifold M , we show for 0 ď p ď 2n ´3 and n ě 2, that for a closed, spin, p4n ´1q-manifold M admitting a metric of positive pp, 4n ´1q-intermediate scalar curvature, R s p,4n´1 ą0 pM q has infinitely many path components.