Cohomology of Lie 2-groups

TL;DR

This paper investigates the cohomology of strict Lie 2-groups, providing explicit maps, spectral sequence analyses, and computations for specific cases including the string 2-group, advancing understanding of their algebraic topology.

Contribution

It introduces explicit cohomology maps for certain Lie 2-groups and computes their cohomology groups in specific cases, including the string 2-group, using spectral sequences.

Findings

Explicit Bott-Shulman type map for Lie 2-groups from crossed modules.

Cohomology of universal crossed module Lie 2-groups linked to spectral sequences.

Explicit cohomology computations for Lie 2-groups with small center and for the string 2-group.

Abstract

In this paper we study the cohomology of (strict) Lie 2-groups. We obtain an explicit Bott-Shulman type map in the case of a Lie 2-group corresponding to the crossed module $A\to 1$. The cohomology of the Lie 2-groups corresponding to the universal crossed modules $G\to \Aut(G)$ and $G\to \Aut^+(G)$ is the abutment of a spectral sequence involving the cohomology of $GL(n,\Z)$ and $SL(n,\Z)$. When the dimension of the center of $G$ is less than 3, we compute explicitly these cohomology groups. We also compute the cohomology of the Lie 2-group corresponding to a crossed module $G\to H$ whose kernel is compact and cokernel is connected, simply connected and compact and apply the result to the string 2-group.