Classifying spaces for 1–truncated compact Lie groups

Rezk, Charles

doi:10.2140/agt.2018.18.525

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Classifying spaces for

$1$ –truncated compact Lie groups

Charles Rezk

Algebraic & Geometric Topology 18 (2018) 525–546

DOI: 10.2140/agt.2018.18.525

Abstract

A $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1</mn><mspace width="0.3em" class="thinspace"/></math>$ –truncated compact Lie group is any extension of a finite group by a torus. In this note we compute the homotopy types of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mo class="qopname"> Map</mo></mrow><mrow><mo class="MathClass-bin">∗</mo></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>B</mi><mi>G</mi><mo class="MathClass-punc">,</mo><mi>B</mi><mi>H</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ , $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="qopname"> Map</mo><mrow><mo class="MathClass-open">(</mo><mrow><mi>B</mi><mi>G</mi><mo class="MathClass-punc">,</mo><mi>B</mi><mi>H</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ , and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="qopname"> Map</mo><mrow><mo class="MathClass-open">(</mo><mrow><mi>E</mi><mi>G</mi><mo class="MathClass-punc">,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>G</mi></mrow></msub> <mi>H</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ for compact Lie groups $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>G</mi></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>H</mi></math>$ with $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>H</mi></math>$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1</mn><mspace width="0.3em" class="thinspace"/></math>$ –truncated, showing that they are computed entirely in terms of spaces of homomorphisms from $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>G</mi></math>$ to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>H</mi></math>$ . These results generalize the well-known case when $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>H</mi></math>$ is finite, and the case when $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>H</mi></math>$ is compact abelian due to Lashof, May, and Segal.

Keywords

classifying spaces, equivariant

Mathematical Subject Classification 2010

Primary: 55R91

Secondary: 55P92, 55R35, 55R37

References

Publication

Received: 2 February 2017

Revised: 30 June 2017

Accepted: 18 July 2017

Published: 10 January 2018

Authors

Charles Rezk
Department of Mathematics University of Illinois Urbana, IL United States
http://www.math.illinois.edu/~rezk

Volume 18, issue 1 (2018)