Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Transfinite product reduction in fundamental groupoids

Jeremy Brazas

TL;DR

The paper develops a comprehensive reduction framework for transfinite product operations in fundamental groupoids, focusing on when reductions of infinite factorizations preserve path-homotopy. It introduces homotopy cut-sets and distinguishes between transfinite $\\pi_1$- and $\\Pi_1$-product well-definedness, proving that under the assumption that the algebraic $1$-wild set $\\mathbf{aw}(X)$ is scattered, well-defined transfinite $\\pi_1$-products imply well-defined transfinite $\\Pi_1$-products. The results hinge on a sequence of technical lemmas that allow reductions at non-wild points, on perfect Cantor-type cut-sets, at isolated image points, and via transfinite iterative reductions, culminating in a proof of the main theorem. This work bridges infinitary group and groupoid operations, with implications for generalized universal coverings in metrizable spaces and connections to homotopically Hausdorff properties, and it is shown to be optimally extended beyond previous first-countable results.

Abstract

Infinite products, indexed by countably infinite linear orders, arise naturally in the context of fundamental groupoids. Such products are called "transfinite" if the index orders are permitted to contain a dense suborder and are called "scattered" otherwise. In this paper, we prove several technical lemmas related to the reduction (i.e. combining of factors) of transfinite products in fundamental groupoids. Applying these results, we show that if the transfinite fundamental group operations are well-defined in a space $X$ with a scattered algebraic $1$-wild set $\mathbf{aw}(X)$, then all transfinite fundamental groupoid operations are also well-defined.

Transfinite product reduction in fundamental groupoids

TL;DR

The paper develops a comprehensive reduction framework for transfinite product operations in fundamental groupoids, focusing on when reductions of infinite factorizations preserve path-homotopy. It introduces homotopy cut-sets and distinguishes between transfinite - and -product well-definedness, proving that under the assumption that the algebraic -wild set is scattered, well-defined transfinite -products imply well-defined transfinite -products. The results hinge on a sequence of technical lemmas that allow reductions at non-wild points, on perfect Cantor-type cut-sets, at isolated image points, and via transfinite iterative reductions, culminating in a proof of the main theorem. This work bridges infinitary group and groupoid operations, with implications for generalized universal coverings in metrizable spaces and connections to homotopically Hausdorff properties, and it is shown to be optimally extended beyond previous first-countable results.

Abstract

Infinite products, indexed by countably infinite linear orders, arise naturally in the context of fundamental groupoids. Such products are called "transfinite" if the index orders are permitted to contain a dense suborder and are called "scattered" otherwise. In this paper, we prove several technical lemmas related to the reduction (i.e. combining of factors) of transfinite products in fundamental groupoids. Applying these results, we show that if the transfinite fundamental group operations are well-defined in a space with a scattered algebraic -wild set , then all transfinite fundamental groupoid operations are also well-defined.

Paper Structure

This paper contains 9 sections, 18 theorems, 3 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Sets of 1-dimensional wildness
  3. Homotopy Cut-Sets
  4. Reductions of Homotopy Cut-sets
  5. Reduction at non-wild cut points
  6. Reduction to perfect cut-sets
  7. Reduction at isolated image points
  8. Transfinite sequences of reductions
  9. A Proof of Theorem \ref{['mainthm']}

Key Result

Theorem 1.1

Suppose $X$ is a space in which the subspace of points $\mathbf{aw}(X)$ at which there exists an infinite concatenation of non-contractible loops is scattered. Then $X$ has well-defined transfinite $\pi_1$-products if and only if $X$ has well-defined transfinite $\Pi_1$-products.

Figures (2)

  • Figure 1: The subspace of $X$ in Example \ref{['example1']} including the circles corresponding to $\alpha<\omega+\omega+\omega$.
  • Figure 2: The space $X$ in Example \ref{['example2']}

Theorems & Definitions (46)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Remark 2.7
  • Proposition 2.8
  • ...and 36 more