Vol. 36, No. 2, 1971

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A decomposition theorem for topological group extensions

Arnold Joseph Insel

Vol. 36 (1971), No. 2, 357–378

G. W. Mackey has developed a characterization of the group of equivalence classes of extensions of a fixed group G by a fixed abelian group A restricting all groups to be locally compact second countable Hausdorff spaces. Calvin C. Moore incorporated his results into a cohomology theory of group extensions such that the second cohomology group, H2(G,A), coincides with Mackey’s group of extension classes. The purpose of this paper is to consider the special case in which G and A are connected, A is a Lie group, G is locally arcwise connected and locally simply connected. Under these conditions G and A admit universal covering groups UG and UA. Allowing π(G) and π(A) to denote the fundamental groups of G and A respectively (with basepoint the identity) any extension of G by A determines an extension of UG by UA and an extension of π(G) by π(A) uniquely up to equivalence. Hence there is a map Φ, in fact a homomorphism, constructed from H2(G,A) to H2(UG,UA) H2(π(G)(A)). In this paper H2(G,A) is determined as a direct sum of subgroups of H2(UG,UA) and H2(π(G)(A)), and of a third group which is computed.

Mathematical Subject Classification
Primary: 22.20
Received: 12 January 1970
Published: 1 February 1971
Arnold Joseph Insel