Cohomology of Lie 2-groups (Q845074)

A Lie \(2\)-group \(\Gamma\) is a Lie groupoid such that the spaces of objects and morphisms are both Lie groups, and the structure maps are Lie group homomorphisms. Lie \(2\)-groups are equivalent to crossed modules, Lie group morphisms \(G\to H\) with a right action of \(H\) on \(G\) satisfying compatibility conditions. As in the case of groups, the cohomology of a Lie \(2\)-group \(\Gamma\) with real coefficients is defined to be the cohomology of a simplicial manifold \(N.\Gamma\) associated to \(\Gamma\) called the nerve of \(\Gamma\); the cohomology of \(N.\Gamma\) can be computed using a double de Rham cochain complex. This paper explores the question of whether there is a Bott-Shulman map such as that developed in \textit{R. Bott} [Adv. Math. 11, 289--303 (1973; Zbl 0276.55011)] and \textit{R. Bott, H. Shulman} and \textit{J. Stasheff} [Adv. Math. 20, 43--56 (1976; Zbl 0342.57016)] for the Lie 2-group \([i:G\to H]\) associated to the crossed module \(i:G\to H\) expressing the cohomology of the Lie 2-group in terms of invariant polynomials on \(\mathfrak{g}^\ast\). The authors use a Leray-Serre spectral sequence associated to the fibration \([G\to H] \rightarrow [1 \to H/i(G)]\) to describe a class of cocycles in \(\Omega^{3r}([G\to H])\) generated by elements in the symmetric algebra on the vector space \((\mathfrak{g}^\ast)^{\mathfrak{g},H}\) with degree \(3\). This is used to obtain a Bott-Shulman map in the case where the cokernel \(H/i(G)\) is finite as well as in the case of the Lie \(2\)-group \([A \to 1]\) where \(A\) is compact and abelian. In addition, the authors compute the cohomology of a Lie \(2\)-group with connected and simply connected cokernel as well as the cohomology of the Lie \(2\)-groups \([G\to \Aut(G)]\) and \([G\to \Aut^+(G)]\), where \(\Aut^+(G)\) denotes the orientation-preserving automorphisms, when the dimension of the center of \(G\) is at most \(3\).