Delooping cyclic groups with lens spaces in homotopy type theory

TL;DR

This paper constructs cellular deloopings of cyclic groups within homotopy type theory, using lens spaces and iterative joins, enabling more practical manipulations and applications in algebraic topology.

Contribution

It introduces a novel cellular construction of cyclic group deloopings in homotopy type theory using lens spaces, overcoming previous limitations of elimination principles.

Findings

Constructed cellular deloopings of cyclic groups $\

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Abstract

In the setting of homotopy type theory, each type can be interpreted as a space. Moreover, given an element of a type, i.e. a point in the corresponding space, one can define another type which encodes the space of loops based at this point. In particular, when the type we started with is a groupoid, this loop space is always a group. Conversely, to every group we can associate a type (more precisely, a pointed connected groupoid) whose loop space is this group: this operation is called delooping. The generic procedures for constructing such deloopings of groups (based on torsors, or on descriptions of Eilenberg-MacLane spaces as higher inductive types) are unfortunately equipped with elimination principles which do not directly allow eliminating to untruncated types, and are thus difficult to work with in practice. Here, we construct deloopings of the cyclic groups $\mathbb{Z}_m$ which are cellular, and thus do not suffer from this shortcoming. In order to do so, we provide type-theoretic implementations of lens spaces, which constitute an important family of spaces in algebraic topology. Our definition is based on the computation of an iterative join of suitable maps from the circle to an arbitrary delooping of $\mathbb{Z}_m$. In some sense, this work generalizes the construction of real projective spaces by Buchholtz and Rijke, which handles the case m=2, although the general setting requires more involved tools. Finally, we use this construction to also provide cellular descriptions of dihedral groups, and explain how we can hope to use those to compute the cohomology and higher actions of such groups.