The functoriality painfully collected through Sections 7 and 8 will now allow us to establish powerful cohomological techniques.
We will first see in Section 10 a couple of constructions which are very analogous to elementary procedures in Eilenberg-MacLane cohomology, except for some difficulties arising notably from the presence of a topology on the groups.
Then, in Section 11, we encounter a new phenomenon which pertains genuinely to the theory of [continuous] bounded cohomology : the existence of doubly ergodic amenable spaces. This phenomenon, which has been presented by M. Burger and the author in  as a generalization of Mautner’s property, has remarkable consequences for bounded cohomology and its applications.
Finally, the Section 12 attacks the issue of Hochschild-Serre spectral sequences for group extensions. There, if the strategy is again inspired by usual cohomology, the outcome is very much influenced by the phenomenon of double ergodicity. In particular, we obtain an exact sequence up to degree three, with a refined term in degree two.
KeywordsExact Sequence Compact Group Spectral Sequence Closed Subgroup Finite Index
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