Cohomology of Lie $2$-groups

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.