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),π(A)). In this paper H2(G,A) is determined as a direct sum
of subgroups of H2(UG,UA) and H2(π(G),π(A)), and of a third group which is
computed.
