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Open AccessArticle

Eilenberg–Mac Lane Spaces for Topological Groups

Department of Mathematics, University of the Basque Country, 48940 Leioa, Spain
Axioms 2019, 8(3), 90; https://doi.org/10.3390/axioms8030090
Received: 30 June 2019 / Revised: 22 July 2019 / Accepted: 23 July 2019 / Published: 27 July 2019
(This article belongs to the Collection Topological Groups)
In this paper, we establish a topological version of the notion of an Eilenberg–Mac Lane space. If X is a pointed topological space, π 1 ( X ) has a natural topology coming from the compact-open topology on the space of maps S 1 X . In general, the construction does not produce a topological group because it is possible to create examples where the group multiplication π 1 ( X ) × π 1 ( X ) π 1 ( X ) is discontinuous. This discontinuity has been noticed by others, for example Fabel. However, if we work in the category of compactly generated, weakly Hausdorff spaces, we may retopologise both the space of maps S 1 X and the product π 1 ( X ) × π 1 ( X ) with compactly generated topologies to see that π 1 ( X ) is a group object in this category. Such group objects are known as k-groups. Next we construct the Eilenberg–Mac Lane space K ( G , 1 ) for any totally path-disconnected k-group G. The main point of this paper is to show that, for such a G, π 1 ( K ( G , 1 ) ) is isomorphic to G in the category of k-groups. All totally disconnected locally compact groups are k-groups and so our results apply in particular to profinite groups, answering a question of Sauer’s. We also show that analogues of the Mayer–Vietoris sequence and Seifert–van Kampen theorem hold in this context. The theory requires a careful analysis using model structures and other homotopical structures on cartesian closed categories as we shall see that no theory can be comfortably developed in the classical world. View Full-Text
Keywords: Eilenberg–Mac Lane space; k-group; homotopical algebra Eilenberg–Mac Lane space; k-group; homotopical algebra
MDPI and ACS Style

Corob Cook, G. Eilenberg–Mac Lane Spaces for Topological Groups. Axioms 2019, 8, 90.

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