[Paper Review] Isomorphisms in l^1-homology

Clara Loeh|ArXiv.org|Dec 20, 2006

Homotopy and Cohomology in Algebraic Topology22 references3 citations

TL;DR

This paper establishes a duality principle between ℓ¹-homology and bounded cohomology via dual Banach chain complexes, enabling transfer of structural results from bounded cohomology to ℓ¹-homology. It proves that ℓ¹-homology depends only on the fundamental group and provides a homological algebra description using projective resolutions, resolving gaps in prior work and offering new insights into simplicial volume of non-compact manifolds.

ABSTRACT

Taking the l^1-completion and the topological dual of the singular chain complex gives rise to l^1-homology and bounded cohomology respectively. In contrast to l^1-homology, major structural properties of bounded cohomology are well understood by the work of Gromov and Ivanov. Based on an observation by Matsumoto and Morita, we derive a mechanism linking isomorphisms on the level of homology of Banach chain complexes to isomorphisms on the level of cohomology of the dual Banach cochain complexes and vice versa. Therefore, certain results on bounded cohomology can be transferred to l^1-homology. For example, we obtain a new proof of the fact that l^1-homology depends only on the fundamental group and that l^1-homology with twisted coefficients admits a description in terms of projective resolutions. The latter one in particular fills a gap in Park's approach. In the second part, we demonstrate how l^1-homology can be used to get a better understanding of simplicial volume of non-compact manifolds.

Motivation & Objective

• To establish a duality mechanism linking isomorphisms in ℓ¹-homology and bounded cohomology via dual Banach chain complexes.
• To resolve gaps in prior approaches—particularly Park’s incomplete proof—on the dependence of ℓ¹-homology on the fundamental group.
• To provide a homological algebra framework for ℓ¹-homology using strong relatively projective resolutions.
• To apply ℓ¹-homology to understand the simplicial volume of non-compact manifolds, especially when it is finite or non-zero.

Proposed method

• Develops a translation principle linking isomorphisms in homology of Banach chain complexes to isomorphisms in cohomology of their duals.
• Applies this principle to ℓ¹-chain complexes and bounded cochain complexes of topological spaces and discrete groups.
• Uses the duality to transfer known results from bounded cohomology—such as fundamental group dependence and amenability criteria—to ℓ¹-homology.
• Applies the finiteness criterion for simplicial volume based on ℓ¹-invisibility of boundaries to analyze non-compact manifolds.
• Constructs locally finite cycles with decaying ℓ¹-norm to prove vanishing of simplicial volume in certain product cases.
• Employs the uniform boundary condition and relative fundamental cycles to bound simplicial volume from below.

Experimental results

Research questions

• RQ1 Does ℓ¹-homology depend only on the fundamental group, as bounded cohomology does?
• RQ2 Can the duality between ℓ¹-homology and bounded cohomology be formalized to transfer structural results?
• RQ3 How can ℓ¹-homology be described using homological algebra, particularly via projective resolutions?
• RQ4 What determines the finiteness or non-vanishing of simplicial volume in non-compact manifolds?
• RQ5 Can ℓ¹-invisibility of boundaries be used to characterize when the simplicial volume of a product manifold is finite or infinite?

Key findings

• ℓ¹-homology of countable, connected CW-complexes depends only on the fundamental group, confirming a conjecture with a new proof.
• Amenability of a discrete group is characterized by the vanishing of its ℓ¹-homology in degree one.
• ℓ¹-homology of connected, countable CW-complexes coincides with ℓ¹-homology of the fundamental group and can be computed via strong relatively projective resolutions.
• Simplicial volume of a non-compact manifold N obtained by removing points from a closed manifold M is finite and positive if M has non-zero simplicial volume.
• The product M×ℝ has simplicial volume zero if M is ℓ¹-invisible, and infinite otherwise, providing a sharp criterion for such products.
• For a compact surface M with boundary of genus ≥1, the simplicial volume of M∘×ℝ is infinite, as its boundary is not ℓ¹-invisible.