On the Nielsen-Schreier Theorem in Homotopy Type Theory

Andrew W Swan

doi:10.46298/lmcs-18(1:18)2022

Abstract

We give a formulation of the Nielsen-Schreier theorem (subgroups of free groups are free) in homotopy type theory using the presentation of groups as pointed connected 1-truncated types. We show the special case of finite index subgroups holds constructively and the full theorem follows from the axiom of choice. We give an example of a boolean infinity topos where our formulation of the theorem does not hold and show a stronger "untruncated" version of the theorem is provably false in homotopy type theory.

Highlights

The statement of the Nielsen–Schreier theorem sounds very simple at first: subgroups of free groups are themselves free
We give a formulation of the Nielsen–Schreier theorem in homotopy type theory using the presentation of groups as pointed connected 1-truncated types
Any covering space is homotopic to the geometric realisation of a graph, so the problem is reduced to showing that the fundamental groups of graphs are free groups

Summary

Introduction

The statement of the Nielsen–Schreier theorem sounds very simple at first: subgroups of free groups are themselves free. Any covering space is homotopic to the geometric realisation of a graph, so the problem is reduced to showing that the fundamental groups of graphs are free groups This is proved by constructing a spanning tree of the graph, which is contracted down to point, leaving the remaining edges outside the spanning tree as edges from that point to itself, showing that the graph is homotopy equivalent to a bouquet of circles. This use of ideas from topology makes the Nielsen–Schreier theorem a natural candidate for formalisation in homotopy type theory [Uni13].

Group Theory and Higher Group Theory in HoTT

Coequalizers in HoTT

Spanning Trees in HoTT

The Nielsen–Schreier Theorem

The Untruncated Nielsen–Schreier Theorem is False

Conclusion