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Acyclicity of certain homeomorphism groups. (English) Zbl 0711.57022
A group G is called pseudo-mitotic if for every finitely generated subgroup B there are homomorphisms $$\sigma$$ and $$\delta$$ of B into G and an inner automorphism $$\gamma$$ of G such that $$\sigma =\delta \gamma$$, and $$b\delta =b(b\sigma)$$ for all $$b\in B$$, and $$[b',b\sigma]=1$$ for all $$b,b'\in B$$. A group is said to be acyclic if all its homology groups over the integers with trivial group action vanish in all dimensions $$\geq 1$$. It was proved by the second author [J. Pure Appl. Algebra 37, 205-213 (1985; Zbl 0569.20039)] that pseudo-mitotic groups are acyclic and that the group $$G_ n$$ of homeomorphisms of $${\mathbb{R}}^ n$$ with compact support is pseudo-mitotic. In the present paper techniques are developed to prove pseudo-mitoticity of certain other homeomorphism groups. It is proved among others that the group of homeomorphisms of the rationals $${\mathbb{Q}}$$ (resp. the irrationals) with bounded support is pseudo-mitotic, in particular acyclic. This result is inspired by a result of D. M. Kan and W. P. Thurston [Topology 15, 253-258 (1976; Zbl 0355.55004)]. It is also proved that the group of homeomorphisms of the Cantor set which are the identity in a neighbourhood of 0 and 1 is pseudo-mitotic and hence acyclic.
Reviewer: V.L.Hansen

##### MSC:
 57S05 Topological properties of groups of homeomorphisms or diffeomorphisms 20J99 Connections of group theory with homological algebra and category theory 54H15 Transformation groups and semigroups (topological aspects)
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