I will present an obstruction to positive scalar curvature (psc) on complete manifolds with at least two ends based on the existence of incompressible hypersurfaces that do not admit psc. This result mixes an analytic technique based on $\mu$-bubbles, an augmentation of the classical minimal hypersurface obstructions to psc, with a topological argument based on positive scalar curvature surgery. Due to the latter a surprising (but necessary!) spin condition enters our result even though our methods are not based on the Dirac operator. Concretely, let $M$ be an orientable connected $n$-dimensional manifold with $n\in\{6,7\}$ and $Y\subset M$ a two-sided closed connected incompressible hypersurface that does not admit a metric of psc. Suppose that the universal covers of $M$ and $Y$ are either both spin or both non-spin. Then $M$ does not admit a complete metric of psc. As a consequence, our result answers questions of Rosenberg-Stolz and Gromov up to dimension $7$. Joint work with Simone Cecchini and Daniel Räde.
Abstract

I will introduce a new perspective of K-homology of spaces. This work is motivated by a paper of Guoliang Yu, where he showed that the K-theory of the localization algebra is isomorphic to K-homology for finite simplicial complexes. The localization algebra consists of functions from [1,\infty) to Roe algebra whose propagations go to 0. "The reason" we get K-homology is that by focusing operators whose propagation is small, we can recover some local information on spaces we lost by taking Roe algebras. Here we discuss how we can recover K-homology by focusing on operators whose propagation is smaller than a certain threshold r instead of thinking of operator valued functions. I will report what we can prove and what should be true.