On stability of non-domination under taking products

Abstract

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

Exotic Finite Functorial Semi-Norms on Singular Homology

“…The proof is based on Thom's representation of homology classes by manifolds and straightforward calculations in singular homology and cohomology; similar arguments appear in related work on domination of/by product manifolds [6,7,12].…”

“…The classical example of a finite functorial semi-norm on H d ( · ; R) is the ℓ 1 -semi-norm (see Section 2.2 below). Other examples of functorial semi-norms can be constructed by means of manifold topology, e.g., the products-of-surfaces semi-norm [1,Sections 2,7] or infinite functorial seminorms that exhibit exotic behaviour on certain classes of simply connected spaces of high dimension [1, Theorem 1.2]; such a construction principle via manifold topology will be recalled in Section 4.1.…”

Ordering Thurston’s geometries by maps of nonzero degree

“…The following result gives a sufficient condition for non-domination stability under taking direct products: The proof of the above statement is based on the celebrated realization theorem of Thom [20]; see [10,17] for details. In the same spirit, we obtain the following:…”

“…[10,17]). Let M, W and N be closed oriented manifolds of dimensions m, k and n respectively such that m, k < n < m + k. If N is not dominated by products, then M × W ≥ N × V for no closed oriented manifold V of dimension m + k − n.…”

Domination between different products and finiteness of associated semi-norms

“…, Σ are closed oriented (hyperbolic) surfaces. By Theorem 1.4 (or by [5,Theorem 2.3]) we conclude that each X i is a surface, and since the X i are not dominated by products, we deduce that each X i is a hyperbolic surface (and also m = ).…”