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One-dimensional and codimension one homology of metric manifolds

Denis Marti

Abstract

We compare singular homology and homology via integral currents in metric spaces that are homeomorphic to smooth manifolds. For such spaces, we provide sufficient conditions that guarantee the existence of a surjective homomorphism from the codimension one homology group via integral currents to the codimension one singular homology group. Moreover, we show that a one-dimensional isoperimetric inequality for integral currents implies that the one-dimensional homology groups coincide.

One-dimensional and codimension one homology of metric manifolds

Abstract

We compare singular homology and homology via integral currents in metric spaces that are homeomorphic to smooth manifolds. For such spaces, we provide sufficient conditions that guarantee the existence of a surjective homomorphism from the codimension one homology group via integral currents to the codimension one singular homology group. Moreover, we show that a one-dimensional isoperimetric inequality for integral currents implies that the one-dimensional homology groups coincide.
Paper Structure (11 sections, 15 theorems, 78 equations, 2 figures)

This paper contains 11 sections, 15 theorems, 78 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Metric notions
  4. Metric currents
  5. Homology
  6. Isoperimetric inequalities
  7. Codimension one homology
  8. One-dimensional homology
  9. Examples
  10. The first example
  11. The second example

Key Result

Theorem 1.2

Let $X$ be a metric space with finite Hausdorff $n$-measure that is homeomorphic to a closed, oriented, connected smooth $n$-manifold. Suppose that $X$ has a metric fundamental class. Then, there exists a surjective homomorphism from $H_{n-1}^\textup{IC}(X)$ to $H_{n-1}(X)$.

Figures (2)

  • Figure 1: Two subsequent mushrooms $M_k$ and $M_{k+1}$
  • Figure 2: The generator $G$ and the first iteration $S_1$.

Theorems & Definitions (28)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • ...and 18 more