Cohomology of Lie 22-groups

  • Grégory Ginot

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

    University of Luxembourg, Luxembourg


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 A1A \to 1. The cohomology of the Lie 2-groups corresponding to the universal crossed modules GAut(G)G\to \mathrm {Aut}(G) and GAut+(G)G\to \mathrm {Aut}^+(G) is the abutment of a spectral sequence involving the cohomology of GL(n,Z)GL(n,\mathbb Z) and SL(n,Z)SL(n,\mathbb Z). When the dimension of the center of GG 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)\ker(i) is compact and Coker(i)(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 22-groups. Enseign. Math. 55 (2009), no. 3/4, pp. 373–396

DOI 10.4171/LEM/55-3-8