Dieter Kotschick: Ordering manifolds by maps of non-zero degree

The existence of a map of non-zero degree defines an interesting transitive relation, called the domination relation, between homotopy types of closed oriented manifolds. In dimension two this relation coincides with the ordering given by the genus. We study the domination relation in higher dimensions, with special emphasis on the case where the domain is a non-trivial product and the target has a large universal covering in a suitable sense, e.g. the target could be non-positively curved. In many such situations we prove that there are no maps of non-zero degree. Gromov predicted this outcome for the situation where the target is an irreducible locally symmetric space of non-compact type. Our discussion leads naturally to the algebraic concept of groups not presentable by products, which is of independent interest. (The talk is based on some joint papers with C. Löh, respectively with C. Neofytidis.)