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 $G\stackrel{i}{\to }H$ for which $\ker (i)$ is compact and Coker$(i)$ is connected, simply connected and compact, and we apply the result to the string 2-group.