Classifying covering types in homotopy type theory

TL;DR

This paper formalizes the classification of covering spaces within homotopy type theory, extending to n-dimensional cases and applying to lens spaces and the Poincaré homology sphere.

Contribution

It introduces a formal framework for classifying covering spaces in homotopy type theory, including higher-dimensional generalizations and specific applications.

Findings

Formalized Galois correspondence in homotopy type theory.

Developed an n-dimensional generalization of covering spaces.

Classified coverings of lens spaces and constructed the Poincaré homology sphere.

Abstract

Covering spaces are a fundamental tool in algebraic topology because of the close relationship they bear with the fundamental groups of spaces. Indeed, they are in correspondence with the subgroups of the fundamental group: this is known as the Galois correspondence. In particular, the covering space corresponding to the trivial group is the universal covering, which is a "1-connected" variant of the original space, in the sense that it has the same homotopy groups, except for the first one which is trivial. In this article, we formalize this correspondence in homotopy type theory, a variant of Martin-L\"of type theory in which types can be interpreted as spaces (up to homotopy). Along the way, we develop an n-dimensional generalization of covering spaces. Moreover, in order to demonstrate the applicability of our approach, we formally classify the covering of lens spaces and explain how to construct the Poincar\'e homology sphere.