Central extensions of a Lie groupoid
The author generalizes central extensions of a Lie group by an Abelian Lie group to obtain central extensions of a Lie groupoid by an Abelian Lie group. Then, he establishes the following results: let \(M\) be a connected smooth manifold such that any nontrivial element of \(H^2(M,Z)\) is represented by a symplectic form. Then:NEWLINENEWLINENEWLINE(i) The group \(E(M\times M, S^1)\) of equivalence classes of central \(S^1\)-extension of the pair groupoid \(M\times M\to M\) is isomorphic to \(H^2(M;Z)\). This group is isomorphic to \(Z\) if \(M\) is a compact orientable surface.NEWLINENEWLINENEWLINE(ii) The group \(E(\Pi(M), S^1)\) of equivalence classes of central \(S^1\)-extension of the fundamental groupoid of \(M\) is isomorphic to \(p^*_M(H^2 (M;Z))\), where \(p_M\) is the universal covering projection mapping. This group is zero, if \(M\) is a compact orientable surface with positive genus.
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