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Simple connectedness of the Ran space

Jānis Lazovskis

TL;DR

This work analyzes the fundamental group of finite-subset Ran spaces $\text{Ran}_{\le n}(X)$ for path-connected $X$ by providing explicit loop-homotopy constructions. The author shows that the inclusion-induced maps $\pi_1(\text{Ran}_{\le n}(X)) \to \pi_1(\text{Ran}_{\le n+2}(X))$ are trivial for all $n$, and proves $\pi_1(\text{Ran}_{\le n}(X)) = 0$ whenever $n \ge 4$, via a constructive strategy that decomposes loops through $X^n$, reparametrizes to $\text{Ran}_{\le 2}(X)$, and contracts components with explicit homotopies. This extends prior results on the triviality of homotopy groups for the full Ran space by delivering explicit contractions and broadening the class of spaces for which simple connectedness holds. The work thus clarifies the fundamental group behavior of finite-subset configuration spaces and has potential implications for related topological constructions such as factorization homology and braid-like phenomena in restricted Ran spaces.

Abstract

The space of all finite non-empty subsets of a topological space $X$, also known as the Ran space of $X$, is weakly contractible for $X$ path connected. We consider subspaces $\mathrm{Ran}_{\leqslant n}(X)$ of the Ran space given by all subsets of $X$ of size at most $n$, and present results on their first homotopy groups. In particular, we show that the induced map $π_1(\mathrm{Ran}_{\leqslant n}(X)) \to π_1(\mathrm{Ran}_{\leqslant n+2}(X))$ is trivial for all positive integers $n$, and even more, show that $π_1(\mathrm{Ran}_{\leqslant n}(X)) = 0$ for all $n\geqslant 4$, by explicitly drawing the path homotopies that contract any loop to a point.

Simple connectedness of the Ran space

TL;DR

This work analyzes the fundamental group of finite-subset Ran spaces for path-connected by providing explicit loop-homotopy constructions. The author shows that the inclusion-induced maps are trivial for all , and proves whenever , via a constructive strategy that decomposes loops through , reparametrizes to , and contracts components with explicit homotopies. This extends prior results on the triviality of homotopy groups for the full Ran space by delivering explicit contractions and broadening the class of spaces for which simple connectedness holds. The work thus clarifies the fundamental group behavior of finite-subset configuration spaces and has potential implications for related topological constructions such as factorization homology and braid-like phenomena in restricted Ran spaces.

Abstract

The space of all finite non-empty subsets of a topological space , also known as the Ran space of , is weakly contractible for path connected. We consider subspaces of the Ran space given by all subsets of of size at most , and present results on their first homotopy groups. In particular, we show that the induced map is trivial for all positive integers , and even more, show that for all , by explicitly drawing the path homotopies that contract any loop to a point.
Paper Structure (5 sections, 6 theorems, 3 equations, 3 figures)

This paper contains 5 sections, 6 theorems, 3 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Contribution
  4. Preliminary constructions
  5. Main results

Key Result

Lemma 3

For every continous $\sigma \colon S^1 \to \textup{Ran}_{\leqslant n}(X)$, there exists a continuous $\hat{\sigma} \colon S^1 \to \textup{Ran}_{\leqslant n}(X)$ which factors through $X^n$, satsfies $|\hat{\sigma}(0)| = 1$, and is homotopic to $\sigma$.

Figures (3)

  • Figure 1: Two nearby configurations on a topological space $X$ (left), one in a darker color the other in white, with a neighborhood emphasized around the element in a darker color. A loop $S^1\to \textup{Conf}_2(X)$ composed of two nearby loops $S^1\to X$ (center left). A loop $S^1\to \textup{Ran}_{\leqslant 2}(X)$ composed of two loops $S^1\to X$ going in opposite directions (center right). A continuous image $[0,1]\to X$ is drawn by a path going from a darker shade $(t=0)$ to a lighter shade $(t=1)$.
  • Figure 2: The steps of the proof of \ref{['lem_xlem']} presented visually. Given a continuous loop in $\textup{Ran}_{\leqslant n}(X)$, drawn as a subset of $X$, a new basepoint (circled) is added via homotopy (left). Branch points and merge points (dashed circles and dotted circles, respectively) are identified and pushed (following the arrow) to the basepoint via homotopy (right).
  • Figure 3: Path homotopies for the proof of \ref{['prop_s1prop1']}. A small separation between common endpoints of paths is used to visually distinguish the different paths, even though the endpoints coincide.

Theorems & Definitions (13)

  • Definition 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Proposition 5
  • proof
  • Theorem 6
  • proof
  • Theorem 7
  • ...and 3 more