Topological homotopy groups. (English) Zbl 1155.55004

Mashayekhy, B.; Ghane, H.; Hamed, Z.; Mirebrahimi, H.

Bull. Belg. Math. Soc. - Simon Stevin 15, No. 3, 455-464 (2008).

Starting point is the paper by J. Dugundji [Proc. Natl. Acad. Sci. USA 36, 141–143 (1950; Zbl 0037.10202)], who put a topology on \(\pi_1\) and obtained new results related to covering spaces. This has been recently revitalized by several authors, e.g. D. K. Bliss [Topology Appl. 124, No. 3, 355–371 (2002; Zbl 1016.57002)] and P. Fabel [Topol. Proc. 30, No. 1, 187–195 (2006; Zbl 1125.55007)]. If \(\text{Hom}((S^1,1),(X,x))\) denotes the space of pairs of maps with the compact-open topology, then the topological fundamental group \(\pi^{\text{top}}_1\) is the set \(\pi_1\) equipped with the quotient topology. The authors transfer this construction to higher homotopy groups, i.e. \(\pi^{\text{top}}_n\) is \(\pi_n\) with the quotient topology from the space of maps \(((I^n,\partial I^n),(X,x))\) with the compact-open topology. First they show that \(\pi^{\text{top}}_n\) is a topological group for \(n\geq 1\). Then they examine properties of \(\pi^{\text{top}}_n\), like independency of the base point and behaviour with respect to the product of spaces. Furthermore, they study conditions for \(\pi^{\text{top}}_n\) to be discrete. This happens if \(X\) is an \(n\)-connected metric space independently of the base point.

Reviewer: Ivan Ivanšić (Zagreb)

MSC:

55Q05	Homotopy groups, general; sets of homotopy classes
55U40	Topological categories, foundations of homotopy theory
54H11	Topological groups (topological aspects)
55Q70	Homotopy groups of special types

Keywords:

Topological homotopy groups; topological groups; homotopy groups; semi locally simply connected

Citations:

Zbl 0037.10202; Zbl 1016.57002; Zbl 1125.55007