Eilenberg–Mac Lane Spaces for Topological Groups

• Main Author: Ged Corob Cook
• Format: Article
• Language: English
• Published: MDPI AG 2019-07-01
• Series: Axioms
• Subjects: Eilenberg–Mac Lane space; k-group; homotopical algebra
• Online Access: https://www.mdpi.com/2075-1680/8/3/90

In this paper, we establish a topological version of the notion of an Eilenberg&#8722;Mac Lane space. If <i>X</i> is a pointed topological space, <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>&#960;</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> has a natural topology coming from the compact-open topology on the space of maps <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>S</mi> <mn>1</mn> </msup> <mo>&#8594;</mo> <mi>X</mi> </mrow> </semantics> </math> </inline-formula>. In general, the construction does not produce a topological group because it is possible to create examples where the group multiplication <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>&#960;</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>&#215;</mo> <msub> <mi>&#960;</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>&#8594;</mo> <msub> <mi>&#960;</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> 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 <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>S</mi> <mn>1</mn> </msup> <mo>&#8594;</mo> <mi>X</mi> </mrow> </semantics> </math> </inline-formula> and the product <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>&#960;</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>&#215;</mo> <msub> <mi>&#960;</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> with compactly generated topologies to see that <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>&#960;</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is a group object in this category. Such group objects are known as <i>k</i>-groups. Next we construct the Eilenberg&#8722;Mac Lane space <inline-formula> <math display="inline"> <semantics> <mrow> <mi>K</mi> <mo>(</mo> <mi>G</mi> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> for any totally path-disconnected <i>k</i>-group <i>G</i>. The main point of this paper is to show that, for such a <i>G</i>, <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>&#960;</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>K</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is isomorphic to <i>G</i> in the category of <i>k</i>-groups. All totally disconnected locally compact groups are <i>k</i>-groups and so our results apply in particular to profinite groups, answering a question of Sauer&#8217;s. We also show that analogues of the Mayer&#8722;Vietoris sequence and Seifert&#8722;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.