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A Homology Theory for Etale Groupoids*

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A Homology Theory for Etale Groupoids* CORE Metadata, citation and similar papers at core.ac.uk Provided by Utrecht University Repository A Homology Theory for Etale Group oids by Marius Crainic and Ieke Mo erdijk UtrechtUniversity Department of Mathematics Netherlands Abstract Etale group oids arise naturally as mo dels for leaf spaces of foliations for orbifolds and for orbit spaces of discrete group actions In this pap er weintro duce a sheaf homology theory for etale group oids Weprove its invariance under Morita equivalence as well as Verdier duality between Haeiger cohomology and this homologyWe also discuss the relation to the cyclic and Ho chschild homologies of Connes convolution algebra of the group oid and derive some sp ectral sequences whichserve as a to ol for the computation of these homologies Keywords etale group oids homology duality sp ectral sequences cyclic homology foliations In this pap er weintro duce a homology theory for etale group oids Etale group oids serveas mo del for structures like leaf spaces of foliations orbifolds and orbit spaces of actions by discrete groups In this sense etale group oids should b e viewed as generalized spaces In the literature one nds roughly sp eaking two dierent approaches to the study of etale group oids One approach is based on the construction of the convolution algebras asso ciated to an etale group oid in the spirit of Connes noncommutative geometry and involves the study of cyclic and Ho chschild homology and cohomology of these algebras The other approach uses metho ds of algebraic top ology such as the construction of the classifying space of an etale group oid and its sheaf cohomology groups Our motivation in this pap er is twofold First wewanttogive a more complete picture of the second approach by constructing a suitable homology theory which complements the existing cohomology theory Secondlywe use this homology theory as the main to ol to relate the two ap proaches Let us b e more explicit In the second approach one denes for anyetale group oid G natural cohomology groups with co ecients in an arbitrary G equivariant sheaf These were intro duced in a direct wayby Haeiger As explained in they can b e viewed as a sp ecial instance of the Grothendieck theory of cohomology of sites and agree with the cohomology groups of the classifying space of G Moreover these cohomology groups are invariant under Morita equiva lences of etale group oids This invariance is of crucial imp ortance b ecause the construction of the etale group oid mo delling the leaf space of a given foliation involves some choices which determine the group oid only up to Morita equivalence We complete this picture by constructing a homology theory for etale group oids again invariant under Morita equivalence which is dual in the sense of Verdier duality to the existing cohomology theoryThus one result of our work is the extension of the six op erations of Grothendieck from spaces to leaf spaces of foliations Our homology theory of the leaf space of a foliation reects some geometric prop erties of the Research supp orted byNWO foliation For example byintegration along the b ers leaves it is related to the leafwise cohomology theory studied by Alvarez Lop ez Hector and others see and the references cited there It also shows that the RuelleSullivan current of a measured foliation see lives in Haigers closed cohomology The results in see also Prop osition imply that our homology is also the natural target for the lo calized Chern character We plan to describ e some of these connections more explicitly in a future pap er The homology theory also plays a central role in explaining the relation b etween the sheaf the oretic and the convolution algebra approaches to etale group oids already referred to ab ove Indeed the various cyclic homologies of etale group oids can b e shown to b e isomorphic to the homology of certain asso ciated etale group oids it extends the previous results of Burghelea Connes Feigin Karoubi Nistor Tsygan This connection explains several basic prop erties of the cyclic and p erio dic homology groups and leads to explicit calculations The previous work on the BaumConnes conjecture for discrete groups or for prop er actions of discrete groups on manifolds suggest that this homology will play a role in the BaumConnes conjecture for etale group oids From an algebraic p oint of view our homology theory is an extension of the homology of groups while from a top ological p oint of view it extends compactly supp orted cohomology of spaces In this context we should emphasize that even in the simplest examples the etale group oids whichmodel leaf spaces of foliations involve manifolds which are neither separated nor paracompact Thus an imp ortant technical ingredient of our work is a suitable extension of the notions related to compactly supp orted section of sheaves to nonseparated nonparacompact manifolds For example as a sp e cial case of our results one obtains the Verdier and Poincare duality for nonseparated manifolds Our notion of compactly supp orted sections is also used in the construction of the convolution alge bra of a nonseparated etale group oid We b elieve that this extension to nonseparated spaces has amuch wider use that the one in this pap er and wehave tried to give an accessible presentation of it in the app endix The results in the app endix also playacentral role in the calculation concerning the cyclic homology of etale group oids in and make it p ossible to extend the results of for separated group oids to the nonseparated case We conclude this intro duction with a brief outline of the pap er In the rst section we review the basic denitions and examples related to etale group oids and in the second section we summarize the sheaf cohomology of etale group oids These two sections serve as background and do not contain any new results Readers familiar with this background should immediately go to section and consult the earlier sections for notational conventions In section we present the denition of our homology theory and mention some of its imme diate prop erties In section a covariant op eration for anymap between etale group oids is intro duced which can intuitively b e thought of as a kind of integration along the b er at the level of derived categories We then prove a Leray sp ectral sequence for this op eration This sp ectral sequence is extremely useful For example we will use it to prove the Morita invariance of homologyItalso plays a crucial role in many calculations in In section we prove that the op eration L has a right adjoint at the level of derived categories thus establishing Verdier duality The Poincare dualitybetween Haeiger cohomology and our homology of etale group oids is an immediate consequence In section we summarize the main asp ects of the relation to cyclic homology This section is based on to whichwe refer the reader for detailed pro ofs and further calculations In an app endix weshowhow to adapt the denition of the functor X A assigning to a c space X and a sheaf A the group of compactly supp orted sections in suchaway that all the prop erties as expressed in say can b e proved without using Hausdorness and paracompactness of the space X This app endix can b e read indep endently from the rest of the pap er Contents Etale group oids Sheaves and cohomology Homology Leray sp ectral sequence Morita invariance Verdier duality Relation to cyclic homology App endix Compact supp orts in nonHausdor spaces Etale group oids In this section we review the denition of top ological group oids x the notations and mention some of the main examples Recall rst that a groupoid G is a small category in whichevery arrowisinvertible We will for the set of ob jects and the set of arrows in G resp ectively and denote the and G write G structure maps by s m i u G G G G G G G t Here s and t are the source an target m denotes comp osition m g h g h i is the inverse g i g g and for any x G u x is the unit at xWe write g x y or x y to x indicate that g G is an arrow with s g x and t g y A topological groupoid G is similarly given by top ological spaces G and G and bycontinuous structure maps as in For a smo oth group oid G and G are smo oth manifolds and these structure maps are smo oth moreover one requires s and t to b e submersions so that the b ered G in is also a manifold pro duct G G Denition A top ological smo oth group oid G as ab oveiscalledetale if the source map s G G is a lo cal homeomorphism lo cal dieomorphism This implies that all other structure maps in are also lo cal homeomorphisms lo cal dieomorphisms Germs Anyarrow g x y in an etale group oid induces a germg U xV yfrom a neighborhood U of x in G to a neighborhood V of y Indeed we can deneg t where x U G is so small that s G G has a section U G with x g If U is so small that t j is also a homeomorphism resp dieomorphism theng U V is U also a homeomorphism resp dieomorphism We will also writeg for the germ at x of this map g g U V Note that is the identity germ and that hg hg if g x y and h y z x Examples of etale group oids Note that in examples and the space G is in general not Hausdor Any top ological space manifold X can b e viewed as an etale group oid X with identity arrows only X X X etc We will often simply denote this group oid by X again If a discrete group acts from the right on a space
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