Acyclicity of certain homeomorphism groups.

*(English)*Zbl 0711.57022A 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) |