Appears in collection : Not Only Scalar Curvature Seminar

In this talk I want to study the (conformal) Yamabe constant of a closed Riemannian (resp. conformal) manifold and how it is affected by Gromov-Lawson type surgeries. This yields information about Yamabe invariants and their bordism invariance. So far the talk gives an overview over older results of mine in joint work with M. Dahl, N. Große, E. Humbert, and N. Otoba. A further consequence is that many results about the space of metrics with positive scalar curvature may be generalized to spaces of metrics with Yamabe constant above $t>0$. In particular we will present the following Chernysh-Walsh type result which is work in progress: if $N^n$ arises from $M^n$ by a surgery of dimension $k\in{2,3,\ldots,n-3}$, then a Gromov-Lawson type surgery construction defines a homotopy equivalence from the space of metrics on $M$ with Yamabe constant above $t\in (0,\Lambda_{n,k})$ to the corresponding space on $N$.