On Wagoner complexes

TL;DR

This paper introduces Wagoner complexes linked to Kac-Moody groups, explores their properties, computes low-dimensional homotopy groups, and extends the concept to affine Wagoner complexes for groups with valuation-based root data.

Contribution

It provides a general definition of Wagoner complexes, analyzes their homotopy groups, and introduces affine Wagoner complexes for groups with valuation-based root data.

Findings

Wagoner complexes have interesting homotopy groups related to group homology.

Low-dimensional homotopy groups of Wagoner complexes are explicitly calculated.

Affine Wagoner complexes are defined for groups with valuation-based root data.

Abstract

Wagoner complexes are simplicial complexes associated to groups of Kac-Moody type. They admit interesting homotopy groups which are related to integral group homology if the root datum is of 2-spherical type. We give a general definition of Wagoner complexes, exhibit some simple properties and calculate low dimensional homotopy groups. In addition, we give a definition of affine Wagoner complexes related to groups admitting a root datum with valuation.