×

Topological monoids are transfinitely \(\Pi_1\)-commutative at the identity element. (English) Zbl 1478.57024

Inspired by the behaviour of loops in the fundamental group of the Hawaiian earring, the authors of the paper under review define the infinite concatenation \(\prod_{n=1}^{\infty}\alpha_n\) and the transfinite concatenation \(\prod_{\tau}\alpha_n\) of a null-sequence of loops \( \{\alpha_n\}\) in a topological space \(X\) at a point \(x\). A sequence of loops \(\{\alpha_n\}\) is called a null-sequence if it convergences to the null loop in the compact-open topology. A topological space \(M\) with a monoid operation \(*\) is called a pre-\(\Delta\)-monoid if for any paths \(\alpha\) and \(\beta\) in \(M\) the product path \(\alpha * \beta:I\rightarrow M\) defined by \(\alpha * \beta(t)=\alpha(t)*\beta(t)\) is continuous. The main result of the paper under review is as follows:
Theorem. Every pre-\(\Delta\)-monoid is transfinitely \(\pi_1\)-commutative at its identity element.
A topological space \(X\) is called transfinitely \(\pi_1\)-commutative at \(x\) if for every null-sequence of loops \(\{\alpha_n\}\) in \(X\) at \(x\) and any bijection \(\phi :\mathbb{N}\rightarrow \mathbb{N}\) the following equality holds in \(\pi_1(X,x)\): \[[\prod_{\tau}\alpha_n]=[\prod_{\tau}\alpha_{\phi(n)}].\]

MSC:

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

References:

[1] J. Brazas, P. Gillespie, Infinitary commutativity and abelianization in fundamen-tal groups, Preprint. 2020.
[2] J.D. Christensen, G. Sinnamon, E. Wu, The D-topology for diffeological spaces, Pacific J. Math. 272 (2014), no. 1, 87-110. · Zbl 1311.57031
[3] A. Dold, R. Thom, Quasifaserungen und Unendliche Symmetrische Produkte, Annals of Math. 67 (1958), 239-281. · Zbl 0091.37102
[4] B. Eckmann, P.J. Hilton, Group-like structures in general categories. I. Multipli-cations and comultiplications, Mathematische Annalen 145 (1962), no. 3, 227-255. · Zbl 0099.02101
[5] K. Eda, Free σ-products and noncommutatively slender groups, J. of Algebra 148 (1992), 243-263. · Zbl 0779.20012
[6] K. Eda, K. Kawamura, Homotopy and homology groups of the n-dimensional Hawaiian earring, Fundamenta Mathematicae 165 (2000), 17-28. · Zbl 0959.55010
[7] T. Fay, E. Ordman, B.V.S. Thomas, The free topological group over the rationals. Gen. Topol. Appl. 10 (1979), no. 1, 33-47. · Zbl 0403.22003
[8] L. Fajstrup, J. Rosický, A convenient category for directed homotopy, Theory Appl. Categ. 21 (2008), no. 1, 7-20. · Zbl 1157.18003
[9] M.I. Graev, Free topological groups, Amer. Math. Soc. Transl. 8 (1962), 305-365.
[10] J.P.L. Hardy, S.A. Morris, The Free Topological Group on a Simply Connected Space, Proc. Amer. Math. Soc. 55 (1976), no. 1, 155-159. · Zbl 0322.57028
[11] I.M. James, Reduced product spaces, Annals of Math. 62 (1955), 170-197. · Zbl 0064.41505
[12] K. Kawamura, Low dimensional homotopy groups of suspensions of the Hawaiian earring, Colloq. Math. 96 (2003), no. 1, 27-39. · Zbl 1037.55003
[13] J.W. Morgan, I.A. Morrison, A van Kampen theorem for weak joins. Proc. London Math. Soc. 53 (1986), no. 3, 562-576. · Zbl 0609.57002
[14] G.W. Whitehead, Elements of Homotopy Theory, Springer Verlag, Graduate Texts in Mathematics. 1978. · Zbl 0406.55001
[15] West Chester University, Department of Mathematics, West Chester, PA 19383, USA Email address, Brazas: jbrazas@wcupa.edu Email address, Gillespie: pg915111@wcupa.edu
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.