On the Moduli of Lipschitz Homology Classes

TL;DR

This paper introduces a new Lipschitz homology class modulus based on $L^p$ forms, establishing a duality theorem and generalizing classical surface modulus estimates within Lipschitz Riemannian manifolds.

Contribution

It defines a novel modulus for Lipschitz homology classes, proves a homological duality theorem, and extends classical modulus estimates to a broader Lipschitz manifold setting.

Findings

Established a duality between Lipschitz homology classes using the new modulus.

Generalized classical surface modulus inequalities to Lipschitz Riemannian manifolds.

Provided characterizations of Sobolev forms via Lipschitz chains.

Abstract

We define a type of modulus $\operatorname{dMod}_p$ for Lipschitz surfaces based on $L^p$-integrable measurable differential forms, generalizing the vector modulus of Aikawa and Ohtsuka. We show that this modulus satisfies a homological duality theorem, where for H\"older conjugate exponents $p, q \in (1, \infty)$, every relative Lipschitz $k$-homology class $c$ has a unique dual Lipschitz $(n-k)$-homology class $c'$ such that $\operatorname{dMod}_p^{1/p}(c) \operatorname{dMod}_q^{1/q}(c') = 1$ and the Poincar\'e dual of $c$ maps $c'$ to 1. As $\operatorname{dMod}_p$ is larger than the classical surface modulus $\operatorname{Mod}_p$, we immediately recover a more general version of the estimate $\operatorname{Mod}_p^{1/p}(c) \operatorname{Mod}_q^{1/q}(c') \leq 1$, which appears in works by Freedman and He and by Lohvansuu. Our theory is formulated in the general setting of Lipschitz Riemannian manifolds, though our results appear new in the smooth setting as well. We also provide a characterization of closed and exact Sobolev forms on Lipschitz manifolds based on integration over Lipschitz $k$-chains.