## Cyclic Cohomology of Étale Groupoids; The General Case (1999)

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Venue: | K-theory |

Citations: | 21 - 1 self |

### BibTeX

@TECHREPORT{Crainic99cycliccohomology,

author = {Marius Crainic},

title = {Cyclic Cohomology of Étale Groupoids; The General Case},

institution = {K-theory},

year = {1999}

}

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### Abstract

We give a general method for computing the cyclic cohomology of crossed products by 'etale groupoids, extending the Feigin-Tsygan-Nistor spectral sequences. In particular we extend the computations performed by Brylinski, Burghelea, Connes, Feigin, Karoubi, Nistor and Tsygan for the convolution algebra C 1 c (G) of an 'etale groupoid, removing the Hausdorffness condition and including the computation of hyperbolic components. Examples like group actions on manifolds and foliations are considered. Keywords: cyclic cohomology, groupoids, crossed products, duality, foliations. Contents 1 Introduction 3 2 Homology and Cohomology of Sheaves on ' Etale Groupoids 4 2.1 ' Etale Groupoids : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 \Gamma c in the non-Hausdorff case : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.3 Homology and Cohomology of ' Etale Groupoids : : : : : : : : : : : : : : : : : : : : : 8 3 Cyclic Homologies of Sheaves ...

### Citations

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Citation Context ...( 1 ) = X \Theta G, s ( x; g ) = xg; t ( x; g ) = x; u ( x ) = ( x; 1 ) ; m( ( x; g ) ; ( y; h ) ) = ( x; gh ) ; i ( x; g ) = ( xg; g \Gamma1 ) . It is a good replacement for the orbit space X=G (see =-=[12]-=-). 4. Many examples of groupoids arise in foliation theory: Haefliger's groupoid \Gamma q , or the holonomy groupoid Hol(M;F) of a foliated manifold (M; F). The latter is 'etale if one reduces the spa... |

450 |
Théorie des topos et cohomologie étale des schémas, Séminaire de Géometrie Algébrique du Bois-Marie 1963–1964
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Citation Context ...with is an orbit space of an 'etale groupoid; in particular, any 'etale groupoid can be viewed as such a non-commutative space. This fits in with Grothendieck's idea of what a "generalized space&=-=quot; is ([1, 31]-=-), and includes examples like leaf spaces of foliations, orbit spaces of group actions on manifolds, orbifolds. To say what the groups HPs(C 1 c (G)) look like is an important step in solving index pr... |

444 |
L.Tu: Differential Forms in Algebraic Topology
- Bott
- 1982
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Citation Context ...s \Gamma! s is given by: s(x) = germ x (s) for x 2 U ; and 0 otherwise: The basic property which enables us to extend the usual results from the Hausdorff case is the Mayer-Vietoris sequence (like in =-=[5], pp.-=-139,186): for any open covering U of X, there is a long exact sequence ([14], Prop.7:4): : : : \Gamma! M U;V \Gamma c (U " V ; A) \Gamma! M U \Gamma c (U ; A) \Gamma! \Gamma c (X; A) \Gamma! 0 : ... |

405 |
An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge
- Weibel
- 1994
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Citation Context ...V s denotes the category of complex vector spaces) by: Ext G (A; B) = R HomSh ( G )(A; \Gamma)(B); with the particular case Ext G (C ; \Gamma) = H (G; \Gamma) (R stands for the right derived functors =-=[23, 42]-=-). Homological algebra provides us an alternative description of the vector spaces Ext p G (A; B) by means of Yoneda extensions (see [30], [42]). For ps1, the elements in Ext p G (A; B) are represente... |

330 | Higher algebraic K-theory I - Quillen - 1973 |

297 |
Cohomology of groups, Graduate texts
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- 1982
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Citation Context ...ich is a quasi-isomorphism (of complexes of sheaves on G ( 0 ) ), induces an isomorphism Hs(G; S ffl ) \Gamma! Hs(G; S 0 ffl ). 2.22 Example: If G = G is a group, we get the usual homology of groups (=-=[8]-=-). If G = X is a space we get H k (G; \Gamma) = H \Gammak c (X; \Gamma). In general, Hs(G; \Gamma) lives in degreess\Gammacohdim(G ( 0 ) ). 2.23 The long exact sequence([14]): For any short exact sequ... |

155 |
Topologie algébrique et théorie des faisceaux
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- 1958
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Citation Context ...softness, compactly supported cohomology, the compactly supported sections functor \Gamma c or, more generally, the functor f ! : Sh(X) \Gamma! Sh(Y ) induced by a continuous map f : X \Gamma! Y (see =-=[17, 24]-=-) will have the usual meaning only on Hausdorff spaces. For locally Hausdorff spaces (i.e. spaces which have a Hausdorff open covering) a good extension of these notions is developed in [14]. In this ... |

135 |
Produits tensoriels topologiques et espaces nucléaires
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Citation Context ...ga 2 b fflffl A\Omega 2 b 0 1\Gamma A\Omega 2 b N A\Omega 2 b 0 1\Gamma : : : N A A 1\Gamma oo A N oo A 1\Gamma oo : : : N oo When A is a locally convex algebra, we use the projective tensor product (=-=[18]). 3.4 Exa-=-mple: If A = C 1 M is the sheaf of smooth functions on a manifold M , we define A " as in 3.3 by taking into account the topology, i.e. A " ( n ) := \Delta n+1 (A \Theta ( n+1 ) ) where A \T... |

113 |
Cyclic homology, Grundlehren der mathematischen Wissenschaften
- Loday
- 1992
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Citation Context ... = n X i=0 (\Gamma1) i h i ; h i (ajh 1 ; :::; hn ) = (ajh 1 ; :::; h i ; F (s ( h i ) ); l ( h i+1 ) ; :::; l ( hn ) ):s3 Cyclic Homologies of Sheaves on ' Etale Groupoids 3.1 Cyclic Objects Recall (=-=[15, 27, 28]-=-) some basic definitions concerning cyclic objects in an abelian category M. 3.1 Mixed Complexes [27]: A mixed complex in M is a family fXn : ns0g of objects in M, equipped with maps of degree \Gamma1... |

110 |
A course in homological algebra
- Hilton, Stammbach
- 1971
(Show Context)
Citation Context ...V s denotes the category of complex vector spaces) by: Ext G (A; B) = R HomSh ( G )(A; \Gamma)(B); with the particular case Ext G (C ; \Gamma) = H (G; \Gamma) (R stands for the right derived functors =-=[23, 42]-=-). Homological algebra provides us an alternative description of the vector spaces Ext p G (A; B) by means of Yoneda extensions (see [30], [42]). For ps1, the elements in Ext p G (A; B) are represente... |

71 | Cyclic cohomology and the transverse fundamental class of a foliation, in Geometric Methods in Operator Algebras - CONNES - 1983 |

70 |
Homologie cyclique et K-théorie, Astérisque
- Karoubi
- 1987
(Show Context)
Citation Context ...; gn\Gamma1 ) : 3.8 Definition: If A ffl is a cyclic G-sheaf, define its Hochschild and cyclic hyperhomology by HHs(G; A ffl ) = Hs(G; (A ffl ; b)); HCs(G; A ffl ) = Hs(G; (A ffl ; B; b)) (compare to =-=[25, 26]-=-). If An is c-soft for all n, define HPs(G; A ffl ) = Hs(G; lim r (A ffl ; B; b)[\Gamma2r]) (in the general case one can define HPsusing resolutions; see [43]). This is an extension of the definition ... |

55 | Cyclic homology and the Lie algebra homology of matrices - Loday, Quillen - 1984 |

54 | A survey of foliations and operator algebras - Connes - 1980 |

48 | CONNES – Geometric K-theory of Lie groups and foliations (Preprint - BAUM, A - 1982 |

37 |
V.: Cyclic cohomology of étale groupoids. K-Theory 8
- Brylinski, Nistor
- 1994
(Show Context)
Citation Context ...more difficult to compute and involve in a deeper way the combinatorics of the groupoid. In the general setting of smooth 'etale groupoids the results were partially extended by Brylinski and Nistor (=-=[6]-=-): for a Hausdorff 'etale groupoid G, the localized homologies HPs(C 1 c (G)) O are defined for any invariant closed-open set O of loops; the elliptic components are computed in terms of double comple... |

30 |
Homotopy and integrability
- Haefliger
- 1971
(Show Context)
Citation Context ...ariant (i.e. p ( xg ) = p ( x ) ). It is called principal if p is an open surjection and E \Theta G ( 0 ) G \Gamma! E \Theta B E; ( e; g ) 7! ( e; eg ) is a homeomorphism. 2.5 Morphisms of Groupoids (=-=[19]-=-,[31]): Let G and H be two groupoids. A morphism P : G \Gamma! H from G to H (or Hilsum-Skandalis map cf. [36]) consists of a space P , continuous maps (source and target): s P : P \Gamma! G ( 0 ) ; t... |

27 |
Cyclic homology. Comodules and mixed complexes
- Kassel
- 1987
(Show Context)
Citation Context ... = n X i=0 (\Gamma1) i h i ; h i (ajh 1 ; :::; hn ) = (ajh 1 ; :::; h i ; F (s ( h i ) ); l ( h i+1 ) ; :::; l ( hn ) ):s3 Cyclic Homologies of Sheaves on ' Etale Groupoids 3.1 Cyclic Objects Recall (=-=[15, 27, 28]-=-) some basic definitions concerning cyclic objects in an abelian category M. 3.1 Mixed Complexes [27]: A mixed complex in M is a family fXn : ns0g of objects in M, equipped with maps of degree \Gamma1... |

27 |
Group cohomology and the cyclic cohomology of crossed products
- Nistor
- 1990
(Show Context)
Citation Context ...ted by Connes for the case where G = M is a manifold ([11]), Burghelea and Karoubi for the case where G = G is a group ([9, 25]) and by Feigin, Tsygan and Nistor for crossed products by groups ([15], =-=[37]-=-). The general strategy is to decompose these homology groups as direct sums of localized homologies; there are two different kinds of components which behave differently. Following the terminology in... |

26 |
The cyclic homology of the group rings
- Burghelea
- 1985
(Show Context)
Citation Context ...ar the Baum-Connes assembly map [2]). The computation of HPs(C 1 c (G)) was started by Connes for the case where G = M is a manifold ([11]), Burghelea and Karoubi for the case where G = G is a group (=-=[9, 25]-=-) and by Feigin, Tsygan and Nistor for crossed products by groups ([15], [37]). The general strategy is to decompose these homology groups as direct sums of localized homologies; there are two differe... |

26 |
Classifying spaces and spectral sequences, Pub
- Segal
(Show Context)
Citation Context ...n [31] or [32]). Because of this we need another technical tool when we deal with 'etale categories (see also [14] Prop.3:8); it is a variant of a well known principle due to Segal (see Prop.(2:I) in =-=[40]-=-): 2.30 Lemma and definition: Let G and H be 'etale categories. A continuous functor ' : G \Gamma! H is called a strong deformation retract of H if there is a continuous functor / : H \Gamma! G (calle... |

25 |
On the Chern-Weil homomorphism and the continuous cohomology of Liegroups
- Bott
- 1973
(Show Context)
Citation Context ...gives us more freedom in constructing cocycles. For instance, any G-vector bundlesgives its Chern classes cs() 2 H (G) in explicit cocycles (we can repeat the construction of Chern classes as done in =-=[6]-=-). 4.3 Group Actions on Manifolds We use the previous results to describe the homologies of the cross-product (locally convex) algebra C 1 c (M) ? / G. Here G is a discrete group acting smoothly on th... |

23 |
Stability and Invariants of Hilsum–Skandalis Maps
- Mrčun
- 1996
(Show Context)
Citation Context ...amma! E \Theta B E; ( e; g ) 7! ( e; eg ) is a homeomorphism. 2.5 Morphisms of Groupoids ([19],[31]): Let G and H be two groupoids. A morphism P : G \Gamma! H from G to H (or Hilsum-Skandalis map cf. =-=[36]-=-) consists of a space P , continuous maps (source and target): s P : P \Gamma! G ( 0 ) ; t P : P \Gamma! H ( 0 ) , a left action of G on P with the moment map s P , a right action of H on P with the m... |

23 |
A bivariant Chern–Connes character
- Nistor
- 1993
(Show Context)
Citation Context ...e projections1 \Gamma!sand the S-boundary as an Euler class e( 1 ; T ) 2 H 2 (; C ). Our description of S in terms of extensions (as described in the previous proof) is very close to the one given in =-=[38]-=-, pp. 565. 3.20 Definition: Let (G; `) be a cyclic category. A `-cyclic G-sheaf is a 1-cyclic object A ffl in Sh(G) (i.e. a contravariant functors1 \Gamma! Sh(G) cf 3.2) such that, for any c 2 G ( 0 )... |

17 |
Higher index theorems and the boundary map in cyclic cohomology
- Nistor
- 1997
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Citation Context ...the two cohomologies described in 4.10. Remark that we chose the maximal definition such that we keep the pairing with the cyclic homology and such that it is a receptacle for "Chern character&qu=-=ot; maps ([12, 39]). And, as-=- we shall see, it is "computable". Theorem 4.12 : For any smooth 'etale groupoid G, and any elliptic open O ae B ( 0 ) which is a topological manifold (not necessarily Hausdorff) of dimensio... |

15 |
The graph of a foliation, Ann
- Winkelnkemper
- 1983
(Show Context)
Citation Context ...(M;F) of a foliated manifold (M; F). The latter is 'etale if one reduces the space of objects to a complete transversal, and is a good replacement for the leaf space of the foliation. See e.g. [21] , =-=[44]-=- , [10] . 5. Orbifolds can be modelled by 'etale groupoids ; they correspond to the 'etale groupoids G with the property that (s; t) : G \Gamma! G ( 0 ) \Theta G ( 0 ) is a proper map (see [34]) . 2.3... |

12 | Groupoides d’holonomie et espaces classifiants, Astérisque 116 - Haefliger - 1984 |

12 |
An Asset Theory
- Iversen, Soskice
- 2001
(Show Context)
Citation Context ...softness, compactly supported cohomology, the compactly supported sections functor \Gamma c or, more generally, the functor f ! : Sh(X) \Gamma! Sh(Y ) induced by a continuous map f : X \Gamma! Y (see =-=[17, 24]-=-) will have the usual meaning only on Hausdorff spaces. For locally Hausdorff spaces (i.e. spaces which have a Hausdorff open covering) a good extension of these notions is developed in [14]. In this ... |

10 |
Classifying spaces and classifying topoi
- Moerdijk
- 1995
(Show Context)
Citation Context ...Connes (M ); which is not injective in general (of course, if M is Hausdorff, the two definitions coincide and have the usual meaning). 2.3 Homology and Cohomology of ' Etale Groupoids 2.15 G-sheaves(=-=[1, 32]-=-): Let G be an 'etale groupoid. A G-sheaf is a sheaf A on the space G ( 0 ) , equipped with a continuous right action of G. This means that for any arrow g : c \Gamma! d in G, there is a morphism betw... |

10 | The Hodge filtration and cyclic homology. K-theory 12 - Weibel - 1997 |

8 |
Calssifying topos and foliations
- Moerdijk
- 1991
(Show Context)
Citation Context ...with is an orbit space of an 'etale groupoid; in particular, any 'etale groupoid can be viewed as such a non-commutative space. This fits in with Grothendieck's idea of what a "generalized space&=-=quot; is ([1, 31]-=-), and includes examples like leaf spaces of foliations, orbit spaces of group actions on manifolds, orbifolds. To say what the groups HPs(C 1 c (G)) look like is an important step in solving index pr... |

8 | Differentiable cohomology, “Differential Topologi (CIME, Varenna - Haefliger - 1976 |

7 |
Homologie cyclique des groupes et des algebres
- Karoubi
- 1983
(Show Context)
Citation Context ...; gn\Gamma1 ) : 3.8 Definition: If A ffl is a cyclic G-sheaf, define its Hochschild and cyclic hyperhomology by HHs(G; A ffl ) = Hs(G; (A ffl ; b)); HCs(G; A ffl ) = Hs(G; (A ffl ; B; b)) (compare to =-=[25, 26]-=-). If An is c-soft for all n, define HPs(G; A ffl ) = Hs(G; lim r (A ffl ; B; b)[\Gamma2r]) (in the general case one can define HPsusing resolutions; see [43]). This is an extension of the definition ... |

6 | Proof of a conjecture of A. Haefliger, Topology 37 - Moerdijk - 1998 |

5 |
Algebras associated with group actions and their homology, Brown University preprint
- Brylinski
- 1987
(Show Context)
Citation Context ... \Theta G : xg = xg and is usually denoted by c M ([3]). Any g 2 G defines an invariant open O g = f(x; h) 2 c M : hsgg and c M = ` g2!G? O g . In particular we have the well-known decomposition (see =-=[7]-=-, [15], [37]): HHs(C 1 c (M ) ? / G) = M g2!G? HHs(C 1 c (M ) ? /G)(g ) ; and the analogues for HCs; HPs. For any g we have obvious Morita equivalences ZOg ' M g ? / Z g , NOg ' M g ? / N g . In the e... |

4 |
Differentiable cohomology, "Differential Topologi (CIME, Varenna
- Haefliger
- 1976
(Show Context)
Citation Context ... fact, the cohomology of sheaves on G can be defined and used via elementary homological algebra tools; using standard resolutions, it can be computed by some kind of standard bar-complexes; see e.g. =-=[14, 20, 32, 1]-=- (a brief review will be given in 4.8). More generally, one can define the bi-functors Ext G (\Gamma; \Gamma) : Sh(G) \Theta Sh(G) \Gamma! V s (V s denotes the category of complex vector spaces) by: E... |

3 |
Stabilité des algebres de feuiletages, Ann. Inst. Fourier Grenoble 33
- Hilsum, Skandalis
- 1983
(Show Context)
Citation Context ...]) that G and H are Morita equivalent if and only if there is a groupoid K and essential equivalences G /\Gamma K \Gamma! H. 2.If (M; F) is a foliated manifold, T \Gamma! M is a complete transversal (=-=[10, 13, 22]-=-), there is an obvious functor HolT (M; F) \Gamma! Hol(M;F) from the holonomy groupoid restricted to T to the holonomy groupoid. It is a standard fact that this is a Morita equivalence. 3. If E \Gamma... |

3 |
Foliations, groupoids and Grothendieck etendues, Preprint 17
- Moerdijk
(Show Context)
Citation Context ...t all the sheaves on Hol(M ; F) (see [32]). In particular, the right side is the cohomology of the orientation sheaf of M inside the category of sheaves of complex vector spaces on Hol(M;F). See also =-=[33]-=- for the connection with De Rham (or basic) cohomology of the leaf-space ([37]). 4.19 Examples: 1. Consider the foliation of the open Moebius band and the complete transversal T as in the picture. The... |

2 |
Cohomologie cyclique et functeur Ext n
- Connes
- 1983
(Show Context)
Citation Context ... = n X i=0 (\Gamma1) i h i ; h i (ajh 1 ; :::; hn ) = (ajh 1 ; :::; h i ; F (s ( h i ) ); l ( h i+1 ) ; :::; l ( hn ) ):s3 Cyclic Homologies of Sheaves on ' Etale Groupoids 3.1 Cyclic Objects Recall (=-=[12, 17, 29]-=-) some basic definitions concerning cyclic objects in an abelian category M. 3.1 Mixed Complexes: A mixed complex in M is a family fXn : ns0g of objects in M, equipped with maps of degree \Gamma1, b :... |

1 |
character for discrete groups, Féte de topology
- Baum, Connes
- 1988
(Show Context)
Citation Context ... that the pairing between HPsand HP is a Poincar'e-duality pairing, so it is highly non-trivial. See 4.12, 4.13 . 7) For group-actions on manifolds we get the old results for the elliptic components (=-=[3]-=-, [6]), and a new description of the hyperbolic ones (see Corollary 4.15) ; 8) For foliations we prove that the cyclic homology is a well defined invariant of the leaf space of the foliation, in the s... |

1 |
Baci, Différentielle de la suite spectrale de Feigin-Tsygan-Nistor
- Bella
- 1992
(Show Context)
Citation Context ...se of crossed products by groups, this is 2.6 in [37]. We also know (from 2.25) the form of the boundaries d 2 p;q . This generalizes a similar result for crossed products by groups (see Prop. 3.2 in =-=[4]-=-). 3.41 Hyperbolic case: We call O ae B ( 0 ) hyperbolic if it is G-invariant and ord(fl) = 1, for all fl 2 O. Denote by e O 2 H 2 (NO ; C ) the Euler class of the (hyperbolic) cyclic groupoid ZO . Fr... |

1 |
Noncommutative differential geometry Ch. II: de Rham homology and non commutative algebra, Publ
- Connes
(Show Context)
Citation Context ...nd the analysis of "leaf spaces" (here we have in mind in particular the Baum-Connes assembly map [2]). The computation of HPs(C 1 c (G)) was started by Connes for the case where G = M is a =-=manifold ([11]-=-), Burghelea and Karoubi for the case where G = G is a group ([9, 25]) and by Feigin, Tsygan and Nistor for crossed products by groups ([15], [37]). The general strategy is to decompose these homology... |

1 |
A homology theory for 'etale groupoids, Utrecht Univ
- Crainic, Moerdijk
- 1998
(Show Context)
Citation Context ... a more conceptual meaning of the results. In this paper we answer all the questions above, among some others. The main tool we use is the homology theory for 'etale groupoids which was introduced in =-=[14]-=-; in particular the results are stated in terms of this homology. This leads to various models (DeRham, Alexander-Spanier, Cech, etc) for representing cyclic cocycles. As immediate consequences we der... |

1 | The cyclic cohomology of crossed product algebras, J.Reine.Angew.Math - Getzler, Jones - 1993 |

1 | Lectures at "Non-commutative geometry - Moscovici |

1 |
Homology of étale categories
- Crainic, Moerdijk
(Show Context)
Citation Context ...tations made in [7]) was to answer the last question; this led to the definition of a homology theory for 'etale groupoids (which could be called cohomology with compact support as well) developed in =-=[16]-=-. This homology turned out to be extremely helpful in answering all the questions above, among others. As it is very flexible, we expect it to be useful also in understanding the Baum-Connes assembly ... |

1 |
Cohomology theory for étale groupoids
- Haefliger
- 1992
(Show Context)
Citation Context ...ap t P , is principal. 5 A nice intuitive motivation of this definition is that P can be viewed as a continuous map between the orbit spaces of G and H, described by its graph (see II.8.fl in [11] or =-=[24]-=-). A nice theoretical motivation is that these morphisms are exactly the topos-theoretic morphisms between the orbit spaces of G and H viewed as toposes (i.e. between the classifying toposes of G and ... |

1 |
Homologie cyclique des groupes et des algébres
- Loday
- 1983
(Show Context)
Citation Context ... A ffl ) = Hs(G; lim r (A ffl ; B; b)[\Gamma2r]) (in the general case we can define HPsby the same methods as in [47]). This is an extension of the definition given by Loday for groups (section II in =-=[28]-=-). 3.9 SBI-sequences: From the general considerations in 3.1, there is a long exact sequence in Sh(G): : : : B \Gamma! g HH n (A ffl ) I \Gamma! g HC n (A ffl ) S \Gamma! g HC n\Gamma2 (A ffl ) B \Gam... |

1 | Proof of a conjecture of A. Haefliger - Moerdijk |

1 |
la geométrie transverse des feuilletages
- Molino, Sur
- 1975
(Show Context)
Citation Context ...he cohomology of the orientation sheaf of M inside the category of sheaves of complex vector spaces on Hol(M;F). See also [33] for the connection with De Rham (or basic) cohomology of the leaf-space (=-=[37]-=-). 4.19 Examples: 1. Consider the foliation of the open Moebius band and the complete transversal T as in the picture. Then G = Hol T (M; F) has G ( 0 ) = (\Gamma1; 1); G ( 1 ) = (\Gamma1; 1) [D (\Gam... |