Transfinite product reduction in fundamental groupoids

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.

17 pages, 2 figures

References in corpus (5)

• On small homotopies of loops
• Test map characterizations of local properties of fundamental groups
• On the failure of the first Čech homotopy group to register geometrically relevant fundamental group elements
• Scattered products in fundamental groupoids
• Dense products in fundamental groupoids