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INFINITARY COMMUTATIVITY AND ABELIANIZATION IN FUNDAMENTAL GROUPS

Part of: Homotopy groups Low-dimensional topology Algebraic structures

Published online by Cambridge University Press:  23 November 2020

JEREMY BRAZAS Open the ORCID record for JEREMY BRAZAS [Opens in a new window]  and
PATRICK GILLESPIE
JEREMY BRAZAS
Affiliation:
West Chester University, Department of Mathematics, West Chester, PA 19383, USA
PATRICK GILLESPIE
Affiliation:
West Chester University, Department of Mathematics, West Chester, PA 19383, USA
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Abstract

Infinite product operations are at the forefront of the study of homotopy groups of Peano continua and other locally path-connected spaces. In this paper, we define what it means for a space X to have infinitely commutative $\pi _1$-operations at a point $x\in X$. Using a characterization in terms of the Specker group, we identify several natural situations in which this property arises. Maintaining a topological viewpoint, we define the transfinite abelianization of a fundamental group at any set of points $A\subseteq X$ in a way that refines and extends previous work on the subject.

Keywords

fundamental groupinfinitary operationtransfinite abelianizationtransfinite commutativity

MSC classification

Primary: 57M05: Fundamental group, presentations, free differential calculus
Secondary: 08A65: Infinitary algebras 55Q52: Homotopy groups of special spaces

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 112 , Issue 3 , June 2022 , pp. 289 - 311
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by George Willis

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