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**A homology theory for étale groupoids.**
*(English)*
Zbl 0954.22002

Étale groupoids arise in a variety of important areas. Given a foliation, the space of leaves is usually very awkward to study using the tools of classical algebraic topology, but the associated étale groupoid is much more amenable to analysis and can serve as a model for the leaf space. Orbifolds and other orbit spaces of discrete group actions also yield étale groupoids. They provide an important step in the construction of the convolution algebra of a foliation, etc. in Connes’ theory of non-commutative geometry. In great generality, étale groupoids model quite general Grothendieck toposes.

A cohomology theory for such objects has existed for some time in various flavours. This can be seen as a particular case of Grothendieck’s cohomology theory for sites and toposes, but also was directly defined by Haefliger in his work on foliations and the groupoids associated with them.

In this paper, the authors introduce a well-behaved sheaf homology theory for étale groupoids and show its relationship with the above cohomology theories. Their homology theory is importantly Morita invariant. They apply their theory to leaf spaces of foliations and to Hochschild and cyclic homology. They also derive several potentially useful spectral sequences.

A cohomology theory for such objects has existed for some time in various flavours. This can be seen as a particular case of Grothendieck’s cohomology theory for sites and toposes, but also was directly defined by Haefliger in his work on foliations and the groupoids associated with them.

In this paper, the authors introduce a well-behaved sheaf homology theory for étale groupoids and show its relationship with the above cohomology theories. Their homology theory is importantly Morita invariant. They apply their theory to leaf spaces of foliations and to Hochschild and cyclic homology. They also derive several potentially useful spectral sequences.

Reviewer: T.Porter (Bangor)

### MSC:

22A22 | Topological groupoids (including differentiable and Lie groupoids) |

57T99 | Homology and homotopy of topological groups and related structures |

18F99 | Categories in geometry and topology |

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\textit{M. Crainic} and \textit{I. Moerdijk}, J. Reine Angew. Math. 521, 25--46 (2000; Zbl 0954.22002)

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