Cohomology of Lie 2-groups

Details Cohomology of Lie 2-groups Cohomology of Lie 2-groups

2007-12-13

arxiv.org/abs/0712.2069v2

Enseign. Math. (2) 55 (2009), no. 3-4, 373--396

Mathematics

Algebraic Topology

21 pages; updated references; corrected typos; a few more examples

Scientific paper

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.

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