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Chapter 2 a Homology Theory for Etale Groupoids

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Chapter 2 a Homology Theory for Etale Groupoids Chapter 2 A homology theory for etale group oids 2.1 Intro duction In this chapter weintro duce a homology theory for etale group oids. Etale group oids serve as 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 general- ized spaces. In the literature one nds, roughly sp eaking, two di erent approaches to the study of etale group oids. One approach is based on the construction of the convolution alge- bras asso ciated to an etale group oid, in the spirit of Connes' non-commutative geometry [25, 29 ], and involves the study of cyclic and Ho chschild homology and cohomology of these algebras [20 , 29 ]. 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 [13 , 57 , 72 ]. Our motivation in this chapter is twofold. First, wewant to give a more complete picture of the second approach, by constructing a suitable homology theory which com- plements the existing cohomology theory. Secondly,we use this homology theory as the main to ol to relate the two approaches see Theorems 4.6.3, 4.6.4, 4.6.5 of chapter 4. Let us b e more explicit: In the second approach, one de nes for any etale group oid G natural cohomology groups with co ecients in an arbitrary G -equivariant sheaf. These were intro duced in a direct wayby Hae iger [55]. As explained in [72], they can b e viewed as a sp ecial instance of the Grothendieck theory of cohomology of sites [2], and agree with the cohomology groups of the classifying space of G [74]. Moreover, these cohomology groups are invariant under Morita equivalences of etale group oids. This invariance is of crucial imp ortance, b ecause the construction of the etale group oid mo deling the leaf space of a given foliation involves some choices which determine the group oid only up to Morita equivalence, cf 1.3.3. We complete this picture by con- structing a homology theory for etale group oids, again invariant under Morita equiva- lence, which is dual in the sense of Verdier duality to the existing cohomology theory. Thus, one result of our work is the extension of \the six op erations of Grothendieck"[2] from spaces to leaf spaces of foliations. 42 Chapter 2: A homology theory for etale group oids Our homology theory of the leaf space of a foliation re ects some geometric prop- erties of the 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 [1] and the references cited there. It also shows that the Ruelle-Sullivan currentof a measured foliation see [25] lives in Ha iger's closed cohomology. The results of Chapter 4 imply that our homology is also the natural target for the lo calized Chern character. The homology theory also plays a central role in explaining the relation b etween the sheaf theoretic 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 connec- tion explains several basic prop erties of the cyclic and p erio dic homology groups, and leads to explicit calculations see Chapter 4. The previous work on the Baum-Connes conjecture for discrete groups, or for prop er actions of discrete groups on manifolds, suggest that this homology will play a role in the Baum-Connes conjecture for etale group oids. From an algebraic p oint of view, our homology theory is an extension of the homol- ogy 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 which mo del 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 non-separated non-paracompact manifolds. For example, as a sp ecial case of our results one obtains the Verdier and Poincar e duality for non-separated manifolds. Our notion of compactly supp orted sections is also used in the construction of the convolution algebra of a non-separated etale group oid. We b elieve that this extension to non-separated spaces has a much wider use that the one in this thesis, and wehave tried to give an accessible presentation of it in the app endix. The results in the app endix also play a central role in the calculation concerning the cyclic homology of etale group oids in Chapter 4, and make it p ossible to extend the results of [20] for separated group oids to the non-separated case. We conclude this intro duction with a brief outline. In section 2.2, we present the de nition of our homology theory and mention some of its immediate prop erties. In section 2.3, a covariant op eration ' for any map ' 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 homology. It also plays a crucial role in many calculations in Chapter 4. ! In section 2.4, we prove that the op eration L' has a right adjoint ' at the level ! of derived categories, thus establishing Verdier duality. The Poincar e dualitybetween Hae iger cohomology and our homology of etale group oids is an immediate conse- quence. In an app endix, we showhow to adapt the de nition of the functor X ; A c assigning to a space X and a sheaf A the group of compactly supp orted sections in Section 2.2: Homology/compactly supp orted cohomology 43 suchaway that all the prop erties as expressed in [16], say can b e proved without using Hausdor ness and paracompactness of the space X . 2.2 Homology/compactly supp orted cohomology In this section we will intro duce the homology groups H G ; A for any etale n group oid G and any G -sheaf A. Among the main prop erties to b e proved will b e the invariance of homology under Morita equivalence. For any Hausdor space X , the standard prop erties of the functor which assigns to a sheaf S its group of compactly supp orted sections X ; S are well known and c can b e found in any b o ok on sheaf theory. In the app endix, we showhow to extend this functor to the case where X is not necessarily Hausdor , while retaining all the standard prop erties. We emphasize that throughout this thesis, will denote this c extended functor. n 1 0 and hence the spaces G and G Let us x an etale group oid G . The spaces G for n 0 are assumed to satisfy the general conditions of 1.1.14, but we will not assume 0 0 that G is Hausdor . We write d = cdimG for the cohomological dimension of G . n Thus, for any n 0 and any Hausdor op en set U G , the usual cohomological dimension of U is at most d. 2.2.1 Bar complex. Let A be a G -sheaf, and assume that A is c-soft cf. 2.5.1 0 we will brie y say that A is a \c-soft G -sheaf ". For each n 0, as a sheaf on G n n consider the sheaf A = Aon G constructed by pull-back along : G ! n n n 0 G ; x ::: x =x . It is again a c-soft sheaf b ecause is etale. The groups n 0 n 0 n n G ; A of compactly supp orted sections, intro duced in the App endix, together c n form a simplicial ab elian group: / / / 2 1 0 / / / B G ; A: ::: 2.1 G ; A G ; A G ; A c 2 c 1 c 0 / / / with face maps: n n1 d : G ; A ! G ; A 2.2 i c n c n1 n n1 de ned as follows. First, for the face map d : G !G cf. 1.1.13 there is i ! g 1 g an evident map isomorphism in fact A ! d A , whose stalk at =x n n1 0 i g n ! ::: x is the identity map for i 6= 0 and the action by g :A = A !A = n 0 n x x 0 1 g ! d A if i = 0. The map d in 2.2 is now obtained from this by summation along n1 i 0 g the bres see 2.5.12: d i n1 n / G ; A G ; A c n1 c n O P n1 G ; c d i n n1 / G ; d G ; d d A A c c i ! n1 n1 i i Using our notation 2.5.13, B G ; A is the chain complex asso ciated to the simplicial n ; A with the structure maps: vector space n 7! G c 8 > ag j g ; :::; g if i =0 1 2 n < d a j g ; :::; g = a j g ; :::; g g ; :::; g if 1 i n 1 ; i 1 n 1 i i+1 n > : a j g ; :::; g if i = n 1 n1 44 Chapter 2: A homology theory for etale group oids s a j g ; :::; g = a j :::; g ; 1;g ; ::: ; i 1 n i i+1 The homology groups H G ; A are de ned as the homology groups of the simplicial n ab elian groups 2.1, or, equivalently, as those of the asso ciated chain complex given P i by the alternating sum = 1 d .
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