Cohomology of classifying spaces of rank 3 Kac-Moody groups

TL;DR

The paper computes the rational and mod $p$ cohomology of classifying spaces $BG(A)$ for simply connected rank $3$ Kac–Moody groups by decomposing $BK(A)$ into a homotopy colimit of finite-type parabolic subgroups and then applying Mayer–Vietoris alongside Weyl-invariant invariants. It shows that these cohomology groups decompose as direct sums of invariants of Weyl groups and their quotients, and it proves the presence of $p$-torsion for every prime $p$ in the integral cohomology. Mod $p$ cohomology ($p>2$) mirrors the rational case, while mod $2$ requires a refined ten-class classification due to the appearance of $G_2$ as a rank-2 parabolic. The work also derives the ring structure of the rational cohomology for the main cases, with a complete description hindered only by a conjecture for one case; it emphasizes the implications for extending Borel-type results to indefinite Kac–Moody groups and highlights the significant p-torsion phenomena absent in finite-dimensional Lie groups.

Abstract

We represent the rational and mod $p$ cohomology groups of classifying spaces of rank 3 Kac-Moody groups by a direct sum of the invariants of Weyl groups and their quotients. As an application, the authors conclude that there is a $p$-torsion for each prime $p$ in the integral cohomology groups of classifying spaces of rank 3 Kac-Moody groups. We also determine the ring structure of the rational cohomology with one exception case.

Theorems & Definitions (44)

• Theorem 1.1: Theorem \ref{['s3t1']}
• Theorem 1.2: Theorem \ref{['s6t1']}
• Theorem 2.1
• Example 2.2
• Proposition 2.3
• Lemma 2.4
• proof
• Theorem 2.5: Bor53
• Remark 2.6
• Lemma 3.1