We determine which three-manifolds are dominated by products. The result is that a closed, oriented, connected three-manifold is dominated by a product if and only if it is finitely covered either by a product or by a connected sum of copies of S 2 × S 1 . This characterization can also be formulated in terms of Thurston geometries, or in terms of purely algebraic properties of the fundamental group. We also determine which three-manifolds are dominated by non-trivial circle bundles, and which three-manifold groups are presentable by products.

We define a new class of irreducible groups, called groups not infinite-index presentable by products or not IIPP. We prove that certain aspherical manifolds with fundamental groups not IIPP do not admit maps of non-zero degree from direct products. This extends previous results of Kotschick and Löh, providing new classes of aspherical manifolds -beyond those non-positively curved ones which were predicted by Gromov -that do not admit maps of non-zero degree from direct products.A sample application is that an aspherical geometric 4-manifold admits a map of non-zero degree from a direct product if and only if it is a virtual product itself. This completes a characterization of the product geometries due to Hillman. Along the way we prove that for certain groups the property IIPP is a criterion for reducibility. This especially implies the vanishing of the simplicial volume of the corresponding aspherical manifolds. It is shown that aspherical manifolds with reducible fundamental groups do always admit maps of non-zero degree from direct products. FUNDAMENTAL GROUPS OF ASPHERICAL MANIFOLDS AND MAPS OF NON-ZERO DEGREE 3 manifolds using maps of non-zero degree [4,38] and the monotonicity of Kodaira dimensions with respect to the existence of maps of non-zero degree [39] will be presented in a subsequent paper [28].

We obtain an ordering of closed aspherical 4-manifolds that carry a non-hyperbolic Thurston geometry. As application, we derive that the Kodaira dimension of geometric 4-manifolds is monotone with respect to the existence of maps of non-zero degree.

Every closed oriented manifold M is associated with a set of integers D(M ), the set of self-mapping degrees of M . In this paper we investigate whether a product M × N admits a self-map of degree d, when neither D(M ) nor D(N ) contains d. We find sufficient conditions so that D(M × N ) contains exactly the products of the elements of D(M ) with the elements of D(N ). As a consequence, we obtain manifolds M × N that do not admit self-maps of degree −1 (strongly chiral), that have finite sets of self-mapping degrees (inflexible) and that do not admit any self-map of degree dp for a prime number p. Furthermore we obtain a characterization of odd-dimensional strongly chiral hyperbolic manifolds in terms of self-mapping degrees of their products.

We show that non-domination results for targets that are not dominated by products are stable under Cartesian products.