Isometric behavior of products of homology maps

Determine whether, for continuous maps f:X→X' and g:Y→Y' whose induced homology maps are isometric with respect to the ℓ¹-seminorm in all degrees up to n, the induced map H_n(f×g;R) on the n-th homology of the product spaces is also isometric with respect to the ℓ¹-seminorm.

Background

The paper proves that bilipschitz behavior of homology maps in the factors implies bilipschitz behavior for the induced map on the homology of the product. The proof does not provide control of the bilipschitz constant of the Künneth isomorphism.

The authors therefore ask whether the stronger property of being isometric is preserved under products. The question concerns continuous maps between arbitrary topological spaces and a fixed homological degree n.

References

Replacing bilipschitz'' byisometric'' in the previous results gives rise to the following question.

Let $X$, $X'$, $Y$, $Y'$ be topological spaces. Let $f\colon X\to X'$ and $g\colon Y\to Y'$ be continuous maps. Let $n\inN$ such that $H_p(f;R)$ and $H_q(g;R)$ are isometric with respect to the $\ell1$-semi-norm for all $p,q\in\lbrace 0,\ldots, n\rbrace$. Does it follow that $H_n(f\times g;R)\colon H_n(X\times Y;R) \to H_n(X'\times Y';R)$ is isometric with respect to the $\ell1$-semi-norm?

— Quantitative chain extensions and integral cohomological fillings  (2610.02987 - Hofmann et al., 2 Oct 2026) in Question following Corollary on products of bilipschitz chain maps, Section 7.3