Thin homotopy and the signature of piecewise linear surfaces

TL;DR

This paper introduces a new algebraic framework for piecewise linear surfaces, demonstrating that the surface signature uniquely characterizes surfaces up to translation and thin homotopy, with efficient computation methods.

Contribution

It generalizes Chen's result to surfaces, introduces a crossed module structure, and develops explicit methods for computing the surface signature.

Findings

Surface signature uniquely characterizes surfaces up to translation and thin homotopy.

Provides multiple equivalent definitions of thin homotopy for surfaces.

Develops efficient algorithms for computing the surface signature.

Abstract

We introduce a crossed module of piecewise linear surfaces and study the signature homomorphism, defined as the surface holonomy of a universal translation invariant $2$-connection. This provides a transform whereby surfaces are represented by formal series of tensors. Our main result is that the signature uniquely characterizes a surface up to translation and thin homotopy, also known as tree-like equivalence in the case of paths. This generalizes a result of Chen and positively answers a question of Kapranov in the setting of piecewise linear surfaces. As part of this work, we provide several equivalent definitions of thin homotopy, generalizing the plethora of definitions which exist in the case of paths. Furthermore, we develop methods for explicitly and efficiently computing the surface signature.