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 extend this construction for ($p$,$n$)-intermediate scalar curvature for 0≤$p$≤$n$−2 for surgeries in codimension at least $p$+3. We then use it to generalize a well known theorem of Carr. Letting $R^{sp,n>0}$($M$) denote the space of positive ($p$,$n$)-intermediate scalar curvature metrics on an n-manifold $M$, we show for 0 ≤ p ≤ 2$n$−3 and $n$ ≥ 2, that for a closed, spin, (4$n$−1)-manifold M admitting a metric of positive ($p$, 4$n$−1)-intermediate scalar curvature, $R^{sp,n>0$,4n−1^>0}($$M$) has infinitely many path components.