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The metric fundamental class of non-orientable manifolds and manifolds with boundary

Denis Marti

Abstract

We introduce the metric fundamental class for metric spaces that are homeomorphic to compact, non-orientable, smooth manifolds with (possibly empty) boundary. This is an integer rectifiable current that provides an analytic representation of the topological fundamental class of the space. Under certain weak geometric conditions, we show the existence of such a current, extending earlier results for orientable, closed manifolds obtained in collaboration with Basso and Wenger. As an application, we present new rectifiability results.

The metric fundamental class of non-orientable manifolds and manifolds with boundary

Abstract

We introduce the metric fundamental class for metric spaces that are homeomorphic to compact, non-orientable, smooth manifolds with (possibly empty) boundary. This is an integer rectifiable current that provides an analytic representation of the topological fundamental class of the space. Under certain weak geometric conditions, we show the existence of such a current, extending earlier results for orientable, closed manifolds obtained in collaboration with Basso and Wenger. As an application, we present new rectifiability results.
Paper Structure (25 sections, 30 theorems, 142 equations)

This paper contains 25 sections, 30 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Structure of the article
  4. Preliminaries
  5. Metric notions
  6. Orientation and degree
  7. Rectifiability
  8. Nagata dimension
  9. Metric currents
  10. Basic definitions
  11. Integer rectifiable and integral currents
  12. Isoperimetric inequality
  13. Flat currents modulo $p$
  14. Slicing
  15. Product currents and cone constructions
  16. ...and 10 more sections

Key Result

Theorem 1.2

Let $X$ be a compact, orientable metric $n$-manifold with finite Hausdorff $n$-measure and (possibly empty) boundary. Suppose that $X$ is linearly locally contractible and $\mathcal{H}^n(\partial X) = 0$. Then, $X$ has a metric fundamental class $T \in \mathcal{I}_n(X)$ with and whenever $S \in \mathop{\mathrm{\mathbf{I}}}\nolimits_n(X)$ satisfies $\operatorname{spt} (\partial S) \subset \partial

Theorems & Definitions (60)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.1
  • Lemma 2.2
  • ...and 50 more