Geometry and Topology Seminar

The existence of a map of non-zero degree defines a transitive relation on the homotopy types of closed oriented manifolds of the same dimension, called domination relation. Gromov suggested investigating the domination relation as defining an ordering of manifolds and formulated numerous conjectures regarding classes that might (not) be comparable under this relation. A particular case of the domination question, which specifies Steenrod's classical problem on the realization of homology classes by manifolds, and is motivated by Milnor-Thurston and Gromov's theory of functorial semi-norms on homology (such as the simplicial volume), is when the domain is a non-trivial direct product. In this talk, I will discuss the (non-)existence of maps of non-zero degree from direct products to aspherical manifolds with fundamental groups with non-trivial center. As an application, I will describe an ordering of all non-hyperbolic aspherical 4-manifolds possessing a Thurston geometry.