Recognizing nullhomotopic maps into the classifying space of a Kac-Moody group

TL;DR

This paper investigates when maps into the classifying space of Kac-Moody groups are nullhomotopic, extending known results from p-compact groups and using algebraic and topological tools to identify detection methods.

Contribution

It generalizes nullhomotopy characterizations to Kac-Moody groups and introduces new detection techniques involving maximal tori and algebraic Morse theory.

Findings

Null maps from compact Lie groups to Kac-Moody classifying spaces can be detected by maximal tori.

Obstructions to detection are characterized by an explicit abelian group.

Partial detection results are obtained using algebraic discrete Morse theory.

Abstract

This paper extends certain characterizations of nullhomotopic maps between p-compact groups to maps with target the p-completed classifying space of a connected Kac-Moody group and source the classifying space of either a p-compact group or a connected Kac-Moody group. A well known inductive principle for p-compact groups is applied to obtain general, mapping space level results. An arithmetic fiber square computation shows that a null map from the classifying space of a connected compact Lie group to the classifying space of a connected topological Kac-Moody group can be detected by restricting to the maximal torus. Null maps between the classifying spaces of connected topological Kac-Moody groups cannot, in general, be detected by restricting to the maximal torus due to the nonvanishing of an explicit abelian group of obstructions described here. Nevertheless, partial results are obtained via the application of algebraic discrete Morse theory to higher derived limit calculations. These partial results show that null maps are detected by restricting to the maximal torus in many cases of interest.