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Higher homotopy groups of topological Kac-Moody groups, Whitehead tower, and string groups

Ralf Köhl

Abstract

This note establishes that homotopy groups of topological split real Kac-Moody groups are countable and, hence, concludes the existence of Whitehead towers consisting of topological groups for these groups and their maximal compact subgroups. Moreover, this note proposes a construction for string groups of the $E_n$-series.

Higher homotopy groups of topological Kac-Moody groups, Whitehead tower, and string groups

Abstract

This note establishes that homotopy groups of topological split real Kac-Moody groups are countable and, hence, concludes the existence of Whitehead towers consisting of topological groups for these groups and their maximal compact subgroups. Moreover, this note proposes a construction for string groups of the -series.
Paper Structure (5 sections, 10 theorems, 19 equations)

This paper contains 5 sections, 10 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Homogeneous spaces of $k_\omega$-Kac--Moody groups
  3. Homotopy groups of Kac--Moody groups
  4. String groups
  5. Whitehead towers

Key Result

Proposition 2.1

Let $G$ be a split real Kac--Moody group endowed with its $k_\omega$-group topology and let $H$ be a Zariski-closed subgroup of $G$. Then $H \to G \to G/H$ is a Serre fibration and, in fact, a Hurewicz fibration.

Theorems & Definitions (20)

  • Proposition 2.1
  • proof
  • Proposition 2.2: Harring/Koehl:2023
  • Corollary 2.3
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • ...and 10 more