# Cohomology of Lie $2$-groups

### Grégory Ginot

Université Pierre et Marie Curie, Paris, France### Ping Xu

University of Luxembourg, Luxembourg

## Abstract

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 $G\to \mathrm {Aut}(G)$ and $G\to \mathrm {Aut}^+(G)$ is the abutment of a spectral sequence involving the cohomology of $GL(n,\mathbb Z)$ and $SL(n,\mathbb Z)$. When the dimension of the center of $G$ is less than 3, we compute these cohomology groups explicitly. We also compute the cohomology of the Lie 2-group corresponding to a crossed module G{\xrightarrow[i]} H for which $\ker(i)$ is compact and Coker$(i)$ is connected, simply connected and compact, and we apply the result to the {\it string} 2-group.

## Cite this article

Grégory Ginot, Ping Xu, Cohomology of Lie $2$-groups. Enseign. Math. 55 (2009), no. 3, pp. 373–396

DOI 10.4171/LEM/55-3-8