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Concordance of decompositions given by defining sequences

Boldizsar Kalmar

TL;DR

The article develops a framework for the concordance and cobordism of decompositions of $S^3$ defined by toroidal defining sequences, linking these decompositions to homology-manifold cobordism. It introduces invariants for the concordance group, notably a mod-2 component-linking sequence, and demonstrates that the concordance group $ Delta_3^a$ is uncountable, with Antoine decompositions providing a rich source of examples. It further defines a natural map $eta_n$ from the cobordism of decompositions to the cobordism group of ENR homology manifolds, proving low-dimensional surjectivity and clarifying how decomposition spaces relate to topological manifolds and resolvable homology manifolds. Overall, the work situates wild Cantor-set decompositions within manifold topology, connecting knot-concordance ideas, dimension theory, and homology-manifold cobordism to illuminate their structure and classification.

Abstract

We study the concordance and bordism of decompositions associated with defining sequences and we relate them to some invariants of toroidal decompositions and to the cobordism of homology manifolds. These decompositions are often wild Cantor sets and they arise as nested intersections of knotted solid tori. We show that there are at least uncountably many concordance classes of such decompositions in the 3-sphere.

Concordance of decompositions given by defining sequences

TL;DR

The article develops a framework for the concordance and cobordism of decompositions of defined by toroidal defining sequences, linking these decompositions to homology-manifold cobordism. It introduces invariants for the concordance group, notably a mod-2 component-linking sequence, and demonstrates that the concordance group is uncountable, with Antoine decompositions providing a rich source of examples. It further defines a natural map from the cobordism of decompositions to the cobordism group of ENR homology manifolds, proving low-dimensional surjectivity and clarifying how decomposition spaces relate to topological manifolds and resolvable homology manifolds. Overall, the work situates wild Cantor-set decompositions within manifold topology, connecting knot-concordance ideas, dimension theory, and homology-manifold cobordism to illuminate their structure and classification.

Abstract

We study the concordance and bordism of decompositions associated with defining sequences and we relate them to some invariants of toroidal decompositions and to the cobordism of homology manifolds. These decompositions are often wild Cantor sets and they arise as nested intersections of knotted solid tori. We show that there are at least uncountably many concordance classes of such decompositions in the 3-sphere.

Paper Structure

This paper contains 8 sections, 10 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cell-like decompositions
  4. Homology manifolds
  5. Concordance and cobordism of decompositions
  6. Results
  7. Computations in the concordance groups
  8. Computations in the cobordism group

Key Result

Lemma 2.3

Let $\mathcal{D}$ be a decomposition of a manifold $X$ possibly with non-empty boundary given by a defining sequence. Then the decomposition space $X / \mathcal{D}$ is finite dimensional.

Theorems & Definitions (33)

  • Definition 2.1: Defining sequence for a subset
  • Definition 2.2: Cell-like set
  • Lemma 2.3
  • proof
  • Definition 2.4: Homology manifold
  • Lemma 2.5
  • proof
  • Definition 2.6: Cobordism of homology manifolds
  • Theorem 2.7: Qu82Qu83Qu87
  • Lemma 2.8
  • ...and 23 more