Group extensions and cohomology for locally compact groups. III

Abstract

We shall define and develop the properties of cohomology groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript n Baseline left-parenthesis upper G comma upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^n}(G,A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which can be associated to a pair (<italic>G, A</italic>) where <italic>G</italic> is a separable locally compact group operating as a topological transformation group of automorphisms on the polonais abelian group <italic>A</italic>. This work extends the results in [29] and [30], and these groups are to be viewed as analogues of the Eilenberg-Mac Lane groups for discrete <italic>G</italic> and <italic>A</italic>. Our cohomology groups in dimension one are classes of continuous crossed homomorphisms, and in dimension two classify topological group extensions of <italic>G</italic> by <italic>A</italic>. We characterize our cohomology groups in all dimensions axiomatically, and show that two different cochain complexes can be used to construct them. We define induced modules and prove a version of Shapiro’s lemma which includes as a special case the Mackey imprimitivity theorem. We show that the abelian groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript n Baseline left-parenthesis upper G comma upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^n}(G,A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are themselves topological groups in a natural way and we investigate this additional structure.