Semi-coarse Spaces: Fundamental Groupoid and the van Kampen Theorem

TL;DR

This paper introduces a semi-coarse fundamental groupoid in algebraic topology, extending classical concepts to semi-coarse spaces and establishing a Van Kampen theorem for it.

Contribution

It constructs a semi-coarse fundamental groupoid, which is well-defined for general semi-coarse spaces and admits a Van Kampen theorem, unlike previous coarse homotopy invariants.

Findings

The semi-coarse fundamental groupoid is not necessarily trivial for coarse spaces.

It is well-defined for all semi-coarse spaces.

A version of the Van Kampen Theorem is established for this groupoid.

Abstract

In algebraic topology, the fundamental groupoid is a classical homotopy invariant which is defined using continuous maps from the closed interval to a topological space. In this paper, we construct a semi-coarse version of this invariant, using as paths a finite sequences of maps from $\mathbb{Z}_1$ to a semi-coarse space, connecting their tails through semi-coarse homotopy. In contrast to semi-coarse homotopy groups, this groupoid is not necessarily trivial for coarse spaces, and, unlike coarse homotopy, it is well-defined for general semi-coarse spaces. In addition, we show that the semi-coarse fundamental groupoid which we introduce admits a version of the Van Kampen Theorem.