Spanning via Cech Cohomology

TL;DR

This paper introduces a new approach to Plateau's problem by using ch cohomology to define spanning surfaces across all dimensions and codimensions, overcoming previous limitations of ch homology.

Contribution

The authors develop a ch cohomology-based framework for defining spanning sets, extending previous methods to all dimensions and codimensions.

Findings

Provides a ch cohomology-based definition for spanning sets.

Overcomes limitations of ch homology in multi-component boundaries.

Extends the theory of spanning surfaces to all dimensions and codimensions.

Abstract

Plateau's problem is to find a surface with minimal area spanning a given boundary. In 1960, Reifenberg and Adams developed a definition for "span" using \v{C}ech homology, and variants of this definition have been used ever sense. However, limitations of \v{C}ech homology resulted in the lack of a natural definition for a boundary consisting of more than one component. The authors avoided this problem in an earlier paper for codimension one surfaces using linking numbers to define spanning sets. In this paper, we show how to use \v{C}ech cohomology to provide a similar definition for all dimensions and codimensions.