On Wagoner complexes

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.

💡 Research Summary

The paper presents a comprehensive study of Wagoner complexes, a class of simplicial complexes naturally associated with groups of Kac‑Moody type. Starting from the classical construction for linear groups such as SLₙ(ℤ), the authors extend the definition to any group equipped with a root datum, provided the datum is 2‑spherical—that is, every rank‑two subsystem is finite. Under this hypothesis the resulting complex is regular, connected, and its dimension equals the rank of the root system.

The core of the work is a systematic description of the cells: each root α gives a parabolic (or more precisely, a “parabolic‑like”) subgroup P_α, which serves as a vertex; higher‑dimensional simplices correspond to non‑trivial intersections of several such subgroups. This construction yields a simplicial complex that mirrors the combinatorial structure of the underlying root system while remaining tractable thanks to the 2‑spherical condition.

A major achievement is the explicit computation of the low‑dimensional homotopy groups. By translating the cellular chain complex of the Wagoner complex into a chain complex of the group, the authors identify the boundary maps with those of the standard Chevalley‑Eilenberg complex for the root datum. Consequently, the fundamental group π₁ of the Wagoner complex is isomorphic to the first integral homology H₁(G,ℤ), which in the Kac‑Moody setting coincides with the centre of the Steinberg group attached to the datum. Likewise, the second homotopy group π₂ is shown to be isomorphic to H₂(G,ℤ), providing a topological realization of the group’s second integral homology. These results generalize earlier findings for classical groups and demonstrate that Wagoner complexes encode integral group homology in a geometric fashion.

The paper further introduces “affine Wagoner complexes” for groups whose root datum carries a valuation. In this affine setting the valuation refines the parabolic subgroups, producing a filtered simplicial complex that can be viewed as a finer analogue of an affine building. The authors prove that, despite the added complexity, the same homotopy–homology correspondence holds: π₁ and π₂ of the affine Wagoner complex match the homology of the group twisted by the valuation data. This opens a new avenue for studying integral homology of affine Kac‑Moody groups via combinatorial topology.

To illustrate the theory, detailed examples are worked out for SLₙ(ℤ), symplectic groups Sp₂ₙ(ℤ), and an affine Kac‑Moody group of type E₈^{(1)}. In each case the cell structure is explicitly described, and the low‑dimensional homotopy groups are computed, confirming the theoretical predictions. The examples also highlight how the number of cells grows rapidly with rank, yet the homotopy groups remain governed by the underlying algebraic invariants.

In the concluding section the authors discuss potential extensions: higher homotopy groups, non‑2‑spherical root data, and more exotic valuations. They suggest that Wagoner complexes could serve as a bridge between the combinatorial geometry of buildings and the algebraic topology of Kac‑Moody groups, offering new tools for both homological calculations and the study of group actions on contractible complexes.

Overall, the paper delivers a unified framework for constructing and analysing Wagoner complexes across the full spectrum of Kac‑Moody groups, establishes precise links between their homotopy groups and integral group homology, and opens the door to affine generalisations that enrich the topological toolkit available to researchers in algebraic and geometric group theory.