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On the Moduli of Lipschitz Homology Classes

Ilmari Kangasniemi, Eden Prywes

TL;DR

This work defines a homology-friendly modulus, $\operatorname{dMod}_p$, built from $L^p$-differential forms on Lipschitz manifolds to measure Lipschitz homology classes. It establishes a sharp duality: for a non-torsion class $c$, there exists a unique dual $c'$ with $\operatorname{dMod}_p(c)^{1/p}\operatorname{dMod}_q(c')^{1/q}=1$ and Poincaré-dual coupling, extending to manifolds with boundary. The authors prove existence and regularity of modulus minimizers, show these minimizers are $p$-harmonic, and connect the theory to Sobolev de Rham cohomology and a Lipschitz de Rham theorem, thereby generalizing classical surface modulus results to a robust form-based framework. The results yield a generalization of duality inequalities (e.g., $\operatorname{Mod}_p^{1/p}(c)\operatorname{Mod}_q^{1/q}(c')\le 1$) and provide a principled link between capacity problems, Poincaré duality, and Sobolev form theory on Lipschitz manifolds. Overall, the paper advances a comprehensive analytic toolkit to study Lipschitz homology via differential forms, enabling precise duality statements and variational characterizations with potential applications to geometric analysis and quasiconformal-type questions on rough spaces.

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 Hölder 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 Poincaré 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.

On the Moduli of Lipschitz Homology Classes

TL;DR

This work defines a homology-friendly modulus, , built from -differential forms on Lipschitz manifolds to measure Lipschitz homology classes. It establishes a sharp duality: for a non-torsion class , there exists a unique dual with and Poincaré-dual coupling, extending to manifolds with boundary. The authors prove existence and regularity of modulus minimizers, show these minimizers are -harmonic, and connect the theory to Sobolev de Rham cohomology and a Lipschitz de Rham theorem, thereby generalizing classical surface modulus results to a robust form-based framework. The results yield a generalization of duality inequalities (e.g., ) and provide a principled link between capacity problems, Poincaré duality, and Sobolev form theory on Lipschitz manifolds. Overall, the paper advances a comprehensive analytic toolkit to study Lipschitz homology via differential forms, enabling precise duality statements and variational characterizations with potential applications to geometric analysis and quasiconformal-type questions on rough spaces.

Abstract

We define a type of modulus for Lipschitz surfaces based on -integrable measurable differential forms, generalizing the vector modulus of Aikawa and Ohtsuka. We show that this modulus satisfies a homological duality theorem, where for Hölder conjugate exponents , every relative Lipschitz -homology class has a unique dual Lipschitz -homology class such that and the Poincaré dual of maps to 1. As is larger than the classical surface modulus , we immediately recover a more general version of the estimate , 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 -chains.
Paper Structure (30 sections, 48 theorems, 131 equations, 1 figure)

This paper contains 30 sections, 48 theorems, 131 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Modulus and duality
  3. Modulus with differential forms
  4. Ideas of the proofs
  5. Modulus of measures
  6. Lipschitz manifolds
  7. Lipschitz manifolds without boundary
  8. Sobolev spaces on Lipschitz manifolds
  9. Lipschitz domains and submanifolds
  10. Lipschitz chains and integration
  11. Lipschitz chains in Euclidean domains
  12. Wolfe's theorem
  13. Integration of $W^{d, \infty}$-forms in Lipschitz manifolds
  14. Integration of $L^p$-forms in Lipschitz manifolds
  15. Sobolev cohomology theories
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $M$ be a closed Lipschitz Riemannian manifold. Let $c \in H_k^{\mathop{\mathrm{lip}}\nolimits}(M;{\mathbb Z})$ be a non-torsion element of the rank $k$ Lipschitz homology group. For all $p \in (1,\infty)$, there exists a unique element $c' \in H^{\mathop{\mathrm{lip}}\nolimits}_{n-k}(M;{\mathbb where $p^{-1}+q^{-1} = 1$.

Figures (1)

  • Figure 1: The dumbbell-shaped twisted solid torus of Freedman and He that acts as a counterexample to a version of Theorem \ref{['thm:mainthmZ']} for the classical modulus.

Theorems & Definitions (90)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3: Fuglede_surface-modulus
  • Corollary 2.4
  • ...and 80 more