Comparing seminorms on homology

Abstract

We compare the l 1 -seminorm · 1 and the manifold seminorm · man on n-dimensional integral homology classes. Crowley and Löh showed that for any topological space X and any α ∈ H n (X; ‫,)ޚ‬ with n = 3, the equality α man = α 1 holds. We compute the simplicial volume of the 3-dimensional Tomei manifold and apply Gaifullin's desingularization to establish the existence of a constant δ 3 ≈ 0.0115416, with the property that for any X and any α ∈ H 3 (X; ‫,)ޚ‬ one has the inequality δ 3 α man ≤ α 1 ≤ α man .

Volume entropy semi-norm

“…Since the boundary of every (m + 1)-simplex is formed of m + 2 simplices of dimension m, we have Proof. The inequality a Z ∆ ≤ κ(a) is obvious and the reverse inequality κ(a) ≤ a Z ∆ can be found in [30,Proposition 2.1] (and also follows from [25, p. 108-109]). Hence the relation (4.2).…”