A Spectral Sequence Connecting Continuous With Locally Continuous Group Cohomology

A SPECTRAL SEQUENCE CONNECTING CONTINUOUS WITH LOCALL Y CONTINUOUS GR OUP COHOMOLOGY MAR TIN FUCHSSTEINER Abstract. W e presen t a spectral sequenc e connecting the con tin uous and ’ lo- cally con tinuo us’ group cohomologies for top ological groups. As an application it is shown that f or contractible top ological gr oups these cohomology concept s coincide. Introduction There exist v arious cohomolog y concepts for top o logical groups G a nd topo logical co efficient groups V which take the top olo gies o f the gro up and that of the co effi- cients into account. One is o btained by restricting o nes elf to t he complex C ∗ c ( G ; V ) contin uous group co chains only whose cohomolo g y is calle d the c ontinuous gr oup c ohomolo gy H c ( G ; V ). F o r abstrac t groups G and G -mo dules V the first cohomo l- ogy group H 1 ( G ; V ) classifies cross ed morphis ms mo dulo principal deriv ations , the second cohomo logy group H 2 ( G ; V ) clas s ifies equiv alence classes of gro up exten- sions V ֒ → ˆ G ։ G a nd the third cohomo logy group H 3 ( G ; V ) class ifies equiv alence classes cro ssed mo dules with k ernel V and cokernel G (cf. [W ei9 4, Theor em 6.4.5, Theorem 6.6.3 and Theorem 6.6 .13]. Analog ous consider ations show that for top o- logical gr oups G a nd G -mo dules V the first co homology group H 1 c ( G ; V ) classifies contin uous crossed morphisms modulo principal der iv ations, the se c ond co ho mol- ogy group H 2 c ( G ; V ) clas s ifies equiv alence classes of top olo gical gr oup extensions V ֒ → ˆ G ։ G which admit a globa l section (i.e. ˆ G ։ G is a tr ivial V -principal bundle) a nd the third cohomo logy gro up H 3 c ( G ; V ) classifies equiv alence classes of topo lo gically split crossed mo dules. The con tin uous group cohomolog y has the drawbac k that for even the compact Hausdorff group G = R / Z the short exact sequence 0 → Z ֒ → R ։ R / Z → 0 of coe fficient s do es not induce a long exact sequence of cohomology groups. (The group H 1 c ( G ; R ) is trivial b eca use the pro jection R ։ R / Z does not admit global sections, H n c ( G ; Z ) = 0 because all contin uous gr oup co chains on G are constant whereas the gro up of H 1 c ( G ; G ) all contin uous endo morphisms of G is non-tr iv ial.) This drawback is r elieved by a s econd more gener al coho mology concept, which is obtained by considering the complex C ∗ cg ( G ; V ) of gro up co chains whic h are contin uous o n so me identit y neighbour ho o d in G . By abuse of la nguage some p eople call the corres po nding coho mology gro ups H cg ( G ; V ) the ‘lo cally contin uo us group cohomolog y. The first cohomo logy group H 1 cg ( G ; V ) classifies contin uous crossed morphisms mo dulo principal deriv ations, the se c ond cohomo logy group H 2 cg ( G ; V ) Date : May 01, 2011. Key wor ds and phr ases. T ransform ation Group, T op ological Group, Contin uous Group Coho- mology , Al exander-Spanier Cohomology , Equiv ariant Cohomology . 1 2 M. F UCHSSTEINER classifies equiv ale nc e classes of top ologica l group extensions V ֒ → ˆ G ։ G which admit loca l sections (i.e. ˆ G ։ G is a lo cally trivial V -principa l bundle) and the third cohomolog y group H 3 cg ( G ; V ) class ifies equiv alence clas ses o f cr ossed mo dules in which all homomor phisms admit lo cal sectio ns . The inc lus ion C ∗ cg ( G ; V ) ֒ → C ∗ c ( G ; V ) o f co c ha in complexes induces a morphism H ∗ cg ( G ; V ) → H ∗ c ( G ; V ) of cohomology groups , which is used to compar e the tw o cohomolog y concepts. As the ab ov e exa mple shows, these coho mology concepts do not even coincide for co nnected compact Hausdorff groups a nd r e al co efficients. In the follo wing w e will show that the contractibilit y o f a top olo gical g roup G for ces the tw o cohomo logies to coincide (cf. Cor ollary 3 .17): Theorem. F or c ontr actible gr oups G the inclusion C ∗ c ( G ; V ) ֒ → C ∗ cg ( X ; V ) induc es an isomorphism in c ohomolo gy. This is prov ed b y constructing a row-exact double complex A cg, eq ( G ; V ) whose rows a nd columns c a n b e augmented by the complexes C ∗ cg ( G ; V ) and C ∗ c ( G ; V ) resp ectively . The contractibilit y of G will be shown to forc e the columns of this double complex to b e exact as well, which then in turn is s hown to imply that the inclusion C ∗ cg ( G ; V ) ֒ → C ∗ c ( G ; V ) induces a n isomo rphism in cohomolog y . In fact we will b e cons idering the more gener al setting of trans formation g roups ( G, X ) and G -equiv aria nt co c ha ins o n X a nd prov e these results in this mor e general s etting. Similar results for k -g roups and smo oth transforma tio n groups will a ls o b e obta ined. 1. Basic Concepts In this section we reca ll the definitions o f v ario us co chain co mplex es and the int erpretation of so me of their co ho mology groups. F or top o lo gical spaces X and ab elian top olog ical groups V one can co nsider v a riations of the exact standar d c omplex A ∗ ( X ; V ) = hom Set ( X ; V ) of abelian gr oups. Definition 1. 1. F o r every top ologica l spa c e X a nd abelia n topolo g ical group V the sub complex A ∗ c ( X ; V ) := C ( X ∗ +1 ; V ) of the standa rd complex is called the c ontinuous standar d c omplex . F o r transfo r mation groups ( G, X ) and G -mo dules V the g r oup G a c ts on the spaces X n +1 via the diag o nal action and the g roups A n ( X ; V ) c an b e endow ed with a G -action via (1.1) G × A n ( X ; V ) → A n ( X ; V ) , [ g .f ]( ~ x ) = g . [ f ( g − 1 .~ x )] . The G -fixed points of this action are the G -equiv aria n t c o chains. Beca us e the dif- ferential of the s ta ndard complex int ertwines the G -a ction, the eq uiv ariant co chains form a sub complex A ∗ ( X ; V ) G of the standard complex a nd the contin uous eq uiv ari- ant co c ha ins form a sub complex A ∗ c ( X ; V ) G of the contin uo us standard complex. These complexes not exact in general. Example 1.2 . F or any gro up G which acts on itself by left trans la tion and G - mo dule V the complex A ∗ ( G ; V ) G is the co mplex of (homogeneous) gr o up co c ha ins; for top olo gical groups G and G -mo dules V the complex A ∗ c ( G ; V ) G is the complex of contin uous (homo geneous) g roup co chains Definition 1.3. The cohomology H eq ( X ; V ) of the complex A ∗ ( X ; V ) G is ca lled the equiv ar iant cohomolo gy of X (with v alues in V ). The co homology H eq,c ( X ; V ) A SPECTRAL SEQUENCE CONNECTING CONTINUOUS WITH LOCALL Y CONTINUOUS GROUP COHOMOLOGY 3 of the sub complex A ∗ c ( X ; V ) G is called the equiv ar iant contin uous cohomolo gy of X (with v alues in V ). Example 1.4 . F or any gro up G which acts on itself by left trans la tion and G - mo dule V the coho mology H eq ( G ; V ) is the gr oup coho mology of G with v alues in V ; for topo lo gical groups G and G -mo dules V th e cohomo lo gy H eq,c ( G ; V ) is the contin uous gro up co homology o f G with v a lues in V . F o r transformatio n gr oups ( G, X ) and G -mo dules V there exis ts a G -inv ariant complex A ∗ cg ( X ; V ) betw een A c ( X ; V ) a nd A ( X ; V ) which w e are g o ing to define now. F o r each op en cov er ing U of X and each n ∈ N one can define an op en neighbourho o d U [ n ] of the dia gonal in X n +1 via U [ n ] := [ U ∈ U U n +1 . These neighbourho ods of the diag onals in X ∗ +1 form a simplicial subspace of X ∗ +1 which allows us to consider the sub co mplex o f A ∗ ( X ; V ) formed by the groups A n cr ( X, U ; V ) :=  f ∈ A n ( X ; V ) | f | U [ n ] ∈ C ( U [ n ]; V )  of co chains whose r estriction to the subspac e s U [ n ] of X n +1 are contin uous. The cohomolog y o f the co chain complex A ∗ cr ( X, U ; V ) is denoted by H cr ( X, U ; V ). If the cov ering U of X is G -in v ariant, then the subspaces U [ ∗ ] is a simplicial G -subspace of the simplicial G -s pace X ∗ +1 . Example 1.5. If G = X is a to p o logical group which acts on itself by left tra ns- lation and U an op en identit y neighbo ur ho o d, then U U := { g .U | g ∈ G } is a G -inv a riant op en cov ering of G and U [ ∗ ] is an o pen simplicial G -subspace of G ∗ +1 . F o r G -inv a riant cov e rings U of X the co homology o f the sub complex A ∗ cr ( X, U ; V ) G of G -equiv aria n t co chains is denoted by H cr,eq ( X, U ; V ). Example 1.6. If G = X is a to p o logical group which acts on itself by left tra ns- lation and U a n open identit y neigh b o urho o d, then the complex A ∗ cr ( X, U U ; V ) G is the co mplex of ho mogeneous g roup co c ha ins whose r estrictions to the s ubspaces U U [ ∗ ] are contin uous. (These a re sometimes called U - contin uous co chains.) F o r directed systems { U i | i ∈ I } of o pen co verings of X one can also consider the colimit co mplex colim i A ∗ cr ( X, U i ; V ). In particula r for the directed system of all o pe n coverings of X one o bserves that the op en diag onal neighbo urho o ds U [ n ] in X n +1 for o p en coverings U of X are cofinal in the directed set of all op en diagona l neighbourho o ds, hence one o btains the complex A ∗ cg ( X ; V ) := colim U is op en cover of X A ∗ cr ( X ; U ; V ) of globa l co chains who s e germs at the diago na l a re co n tinuous. This is a sub complex of the standar d co mplex A ∗ ( X ; V ) which is inv ariant under the G -action (E q. 1.1) and thus a sub complex of G -mo dules. The G -equiv aria nt co chains with contin uous germ form a s ubco mplex A ∗ cg ( X ; V ) G thereof, whose cohomolog y is denoted by H cg, eq ( X ; V ). The latter subco mplex can also b e obtained by taking the colimit ov e r all G -inv ariant op en cov erings o f X o nly: Prop ositio n 1.7. The natur al morphism of c o chain c omplexes A ∗ cg, eq ( X ; V ) := co lim U is G -invariant open cover of X A ∗ cr ( X ; U ; V ) G → A ∗ cg ( X ; V ) G is a natu ra l isomorphism. 4 M. F UCHSSTEINER Pr o of. W e sho w that this mo rphism is s ur jective and injective. Every eq uiv alence class in A n cg ( X ; V ) G can be represented by a co chain f ∈ A n cr ( X, U ; V ) G , where U is an o pen cov er of X . The cochain f is co ntin uous on U [ n ] b y definition. Its equiv aria nce implies, that it als o is cont inuous o n G. U [ n ] = ( G. U )[ n ], hence an element o f A n eq ( X, ( G. U )[ n ]; V ). The equiv a le nce cla s s [ f ] ∈ A n cg, eq ( X ; V ) is mapp ed onto [ f ] A n cg ( X ; V ) G . This proves surjectivity . Every equiv alence class in A n cg, eq ( X ; V ) G can b e r epresented by an eq uiv ariant n -co chain f in A n cr ( X, U ; V ) G , where U is a G -inv ariant op en cov er of X . If the image of the cla s s [ f ] ∈ A ∗ cg ( X ; V ) G is trivial, then the co chain f itself is tr ivial and so is its clas s [ f ] ∈ A n cg, eq ( X ; V ) G . This prov es injectivity .  Corollary 1.8. The c ohomolo gy H cg, eq ( X ; V ) is the c ohomolo gy of the c omplex of e quivariant c o chains which ar e c ontinuous on some G -invariant neighb ourho o d of the dia gonal. Example 1. 9. If G = X is a top ologica l gr o up which acts on itself by left transla - tion, then the complex A ∗ cg ( G ; V ) G is the complex of homog e neous gro up co chains whose germs at the dia gonal are contin uous. (By abus e of languag e these a re some- times c a lled ’lo cally co nt inuous’ g roup co chains.) 2. The Spectral Sequence Let ( G, X ) b e a transfo rmation g r oup, V be G -mo dule and U b e an op en cov ering of X . W e will show (in Section 3) that the inclusion A ∗ cr ( X, U ; V ) ֒ → A ∗ c ( X ; V ) induces an isomorphis m in cohomolo gy pro vided the space X is contractible. F o r this pur po se we cons ider the a b elian gr o ups (2.1) A p,q cr ( X, U ; V ) :=  f : X p +1 × X q +1 → V | f | X p +1 × U [ q ] is contin uous  . The ab elian g r oups A p,q cr ( X, U ; V ) fo r m a firs t quadra n t double complex whose ver- tical and hor iz o ntal differentials are given by d p,q h : A p,q cr → A p +1 ,q cr , d p,q h ( f p,q )( ~ x, ~ x ) = p +1 X i =0 ( − 1) i f p,q ( x 0 , ..., b x i , ..., x p +1 , ~ x ′ ) d p,q v : A p,q cr → A p,q +1 cr , d p,q v ( f p,q )( ~ x, ~ x ′ ) = ( − 1) p q +1 X i =0 ( − 1) i f p,q ( ~ x, x ′ 0 , ..., b x i ′ , ..., x ′ q +1 ) . The double c o mplex A ∗ , ∗ cr ( X, U ; V ) can b e filtrated column-wis e to obtain a sp ectra l sequence E ∗ , ∗ cr, ∗ ( X, U ; V ) (cf. [McC0 1, Theor em 2 .15]). Since the double complex is a firs t quadrant double complex, the sp ectr al seq uenc e E ∗ , ∗ cr, ∗ ( X, U ; V ) conv erges to the cohomolo gy of the total complex o f A ∗ , ∗ cr ( X, U ; V ). The rows of the double complex A ∗ , ∗ cr ( X, U ; V ) can b e augmented by the complex A ∗ cr ( X, U ; V ) for the cov ering U and the columns can be augmented b y the exact A SPECTRAL SEQUENCE CONNECTING CONTINUOUS WITH LOCALL Y CONTINUOUS GROUP COHOMOLOGY 5 complex A ∗ c ( X ; V ) of contin uous co chains: . . . . . . . . . . . . A 2 cr ( X, U ; V ) / / d v O O A 0 , 2 cr ( X, U ; V ) d h / / d v O O A 1 , 2 cr ( X, U ; V ) d h / / d v O O A 2 , 2 cr ( X, U ; V ) d h / / d v O O · · · A 1 cr ( X, U ; V ) / / d v O O A 0 , 1 cr ( X, U ; V ) d h / / d v O O A 1 , 1 cr ( X, U ; V ) d h / / d v O O A 2 , 1 cr ( X, U ; V ) d h / / d v O O · · · A 0 cr ( X, U ; V ) / / d v O O A 0 , 0 cr ( X, U ; V ) d h / / d v O O A 1 , 0 cr ( X, U ; V ) d h / / d v O O A 2 , 0 cr ( X, U ; V ) d h / / d v O O · · · A 0 c ( X ; V ) d h / / O O A 1 c ( X ; V ) d h / / O O A 2 c ( X ; V ) d h / / O O · · · W e denote the to tal complex of the do uble co mplex A ∗ , ∗ cr ( X, U ; V ) by T ot A ∗ , ∗ cr ( X, U ; V ). The augment ations of the rows and columns of this double complex induce mor- phisms i ∗ : A ∗ cr ( X, U ; V ) → T o t A ∗ , ∗ cr ( X, U ; V ) and j ∗ : A ∗ c ( X ; V ) → T ot A ∗ , ∗ cr ( X, U ; V ) of co chain complexes re sp e ctively . Lemma 2. 1. The morphism i ∗ : A ∗ cr ( X, U ; V ) → T ot A ∗ , ∗ cr ( X, U ; V ) induc es an isomorphi sm in c ohomolo gy. Pr o of. O n each a ugmented row A q cr ( X, U ; V ) ֒ → A ∗ ,q cr ( X, U ; V ) one can de fine a contraction h ∗ ,q via (2.2) h p,q : A p,q cr ( X, U ; V ) → A p − 1 ,q cr ( X, U ; V ) , h p,q ( f )( ~ x, ~ x ′ ) = f ( x 0 , . . . , x p − 1 , x ′ 0 , ~ x ′ ) . Therefore the augmented rows are exact and the aug men tation i ∗ induces an iso- morphism in coho mology .  R emark 2.2 . Note that for non- trivial U this constr uction do es not work for the col- umn complexes, b ecause the so constructed co chains would not fulfil the c ontin uity condition in Def. 2.1. F o r G -inv ariant op en cov erings U of X one can consider the s ub do uble complex A ∗ , ∗ cr ( X, U ; V ) G of A ∗ , ∗ cr ( X, U ; V ) whose rows a re augmented by the c o c ha in complex A ∗ cr ( X, U ; V ) G for the cov ering U a nd the columns can b e augmented by the complex A ∗ c ( X ; V ) G of contin uous equiv a riant co chains (,which is not exa ct in general). Lemma 2.3. F or G -invariant c overings U of X the morphism i ∗ eq := i ∗ G induc es an isomorphism in c ohomolo gy. Pr o of. The co ntraction h ∗ ,q of the augmented rows A q cr ( X, U ; V ) ֒ → T ot A ∗ ,q cr ( X, U ; V ) defined in Eq . 2.2 is G -eq uiv ariant and thus restricts to a row co ntraction of the augmented sub-row A q cr ( X, U ; V ) G ֒ → T ot A ∗ ,q cr ( X, U ; V ) G .  So the morphism H ( i eq ) : H cr,eq ( X, U ; V ) → H (T ot A ∗ , ∗ cr ( X, U ; V ) G ) is inv er tible. F o r the co mpo sition H ( i eq ) − 1 H ( j eq ) : H c,eq ( X ; V ) → H cr,eq ( X, U ; V ) we obser ve: 6 M. F UCHSSTEINER Prop ositio n 2. 4. The image j n ( f ) of a c ont inu ous e quivariant n -c o cycle f on X in T ot A ∗ , ∗ cr ( X, U ; , V ) G is c ohomolo gous to t he image i n eq ( f ) of t he e quivariant n -c o cycle f ∈ A n cr ( X, U ; V ) G in T ot A ∗ , ∗ cr ( X, U ; V ) G . Pr o of. The pro of is a v ariation of the pro o f of [F uc10, Pro p o sition 14.3 .8 ]: Let f : X n +1 → V b e a contin uous equiv a riant n - co cycle o n X and (for p + q = n − 1) define equiv a riant co chains ψ p,q : X p +1 × X q +1 ∼ = X n +1 → V in A p,q cr ( X, U ; V ) G via ψ p,q ( ~ x, ~ x ′ ) = ( − 1) p f ( ~ x, ~ x ′ ). The vertical cob oundar y of the co chain ψ p,q is given by [ d v ψ p,q ]( ~ x, x ′ 0 , . . . , x ′ q +1 ) = ( − 1) p X ( − 1) i f ( ~ x, x ′ 0 , . . . , ˆ x ′ i , . . . , x ′ q ) = − X ( − 1) p +1+ i f ( x 0 , . . . , ˆ x i , . . . , x p , ~ x ′ ) = [ d h ψ p − 1 ,q +1 ]( x 0 , ..., x p , ~ x ′ ) . The anti-comm utativity of the ho rizontal and the vertical differential ensures that the cobo undary of the co chain P p + q = n − 1 ( − 1) p ψ p,q in the total complex is the co chain j n ( f ) − i n ( f ). Thus the co cycles j n ( f ) and i n eq ( f ) are cohomolo gous in T o t A ∗ , ∗ cr ( X, U ; V ) G .  Corollary 2.5. The c omp osition H ( i eq ) − 1 H ( j eq ) : H c,eq ( X ; V ) → H cr,eq ( X, U ; V ) is induc e d by the inclusion A ∗ c ( X, U ; V ) G ֒ → A ∗ cr ( X, U ; V ) G . Corollary 2. 6. If the morphism j ∗ eq := j ∗ G : A ∗ c ( X ; V ) G → T ot A ∗ , ∗ cr ( X, U A ) G induc es a m onomorphism, epimo rphism or isomorphism in c ohomolo gy, then the inclusion A ∗ c ( X ; V ) G ֒ → A ∗ cr ( X, U ; V ) G induc es a monomorphism, epimorp hism or isomorphi sm in c ohomolo gy r esp e ct ively. F o r any directed system { U i | i ∈ I } of op en cov er ings of X o ne can also con- sider the cor resp onding augmented colimit double complex es. In par ticular for the directed system of all op en cov erings of X one obta ins the double complex complex A ∗ , ∗ cg ( X ; V ) := co lim U is op en cov er of X A ∗ , ∗ cr ( X ; U ; V ) whose r ows a nd c o lumns are a ugmented by the colimit complex A ∗ cg ( X ; V ) and by the complex A ∗ c ( X ; V ) respec tiv ely . Lemma 2.7. F or any dir e cte d syst em { U i | i ∈ I } of op en c overings of X t he morphism colim i i ∗ : colim i A ∗ cr ( X, U i ; V ) → T ot colim i A ∗ , ∗ cr ( X, U i ; V ) induc es an isomorphi sm in c ohomolo gy. Pr o of. The pa ssage to the colimit pr eserves the exactness of the augmented row complexes (Lemma 2.1).  As a consequence the colimit morphism i ∗ cg : A ∗ cg ( X ; V ) → T ot A ∗ , ∗ cg ( X ; V ) in- duces an iso morphism in cohomology . The c o limit do uble co mplex A ∗ , ∗ cg ( X ; V ) is a double complex o f G -mo dules and the G -equiv ariant co chains in form a sub dou- ble complex A ∗ , ∗ cg ( X ; V ) G , whose rows and columns are augmen ted by the co limit complex A ∗ cg, eq ( X ; V ) and b y the complex A ∗ c ( X ; V ) G resp ectively . Lemma 2. 8. F or any dir e cte d system { U i | i ∈ I } of G -invariant op en c overings of X the morphism colim i i ∗ eq : colim i A ∗ cr ( X, U i ; V ) G → T ot colim i A ∗ , ∗ cr ( X, U i ; V ) G induc es an isomorphism in c ohomolo gy. A SPECTRAL SEQUENCE CONNECTING CONTINUOUS WITH LOCALL Y CONTINUOUS GROUP COHOMOLOGY 7 Pr o of. The pa ssage to the colimit pr eserves the exactness of the augmented row complexes (Lemma 2.3).  Moreov er, since the op en diagonal neighbourho o ds U [ n ] in X n +1 for op en cov er- ings U o f X are cofinal in the dir ected set of all op en diago nal neighbourho o ds, we observe: Lemma 2.9. The natur al morphism of double c omplexes A ∗ , ∗ cg, eq ( X ; V ) := co lim U is G -invariant op en cover of X A ∗ , ∗ cr ( X ; U ; V ) G → A ∗ cg ( X ; V ) G is a natu ra l isomorphism. Pr o of. The pro of is a nalogous to that of Prop ositio n 1.7.  As a co nsequence the colimit morphism i ∗ cg, eq : A ∗ cg, eq ( X ; V ) → T ot A ∗ , ∗ cg ( X ; V ) G induces an iso mo rphism in cohomo logy , and the morphism H ( i cg, eq ) is inv ertible. F o r the co mp os ition H ( i cg, eq ) − 1 H ( j eq ) : H c,eq ( X ; V ) → H cg, eq ( X, U ; V ) we observe: Prop ositio n 2.10. The image j n ( f ) of a c ontinuous e quivariant n -c o cycle f on X in T ot A ∗ , ∗ cg ( X ; , V ) G is c ohomolo gous to t he image i n cg, eq ( f ) of t he e quivariant n -c o cycle f ∈ A n cg, eq ( X ; V ) in T ot A ∗ , ∗ cg ( X ; V ) G . Pr o of. The pro of is a nalogous to that of Prop ositio n 2.4.  Corollary 2. 11. The c omp osition H ( i cg, eq ) − 1 H ( j eq ) : H c,eq ( X ; V ) → H cg, eq ( X ; V ) is induc e d by the inclusion A ∗ c ( X ; V ) G ֒ → A ∗ cg ( X ; V ) G . Corollary 2. 12. If the morphism j ∗ eq := j ∗ G : A ∗ c ( X ; V ) G → T o t A ∗ , ∗ cg ( X ; V ) G induc es a m onomorphism, epimo rphism or isomorphism in c ohomolo gy, then the inclusion A ∗ c ( X ; V ) G ֒ → A ∗ cg, eq ( X ; V ) induc es a m onomorphism, epimorphism or isomorphi sm in c ohomolo gy r esp e ct ively. 3. Continuo us and U -Continuo us Cochains In this s ection w e cons ider trans formation g roups ( G, X ) and G -modules V for which we show that the inclusion A ∗ c ( X, U ; V ) G ֒ → A ∗ cr ( X, U ; V ) G of the complex of cont inuous equiv a riant co chains into the complex of equiv a riant U -contin uous co chains induces an isomorphism H ∗ c ( X, U ; V ) ∼ = H ∗ cr ( X, U ; V ) provided the top o- logical space X is contractible. The pro of relies on the row exa c tness of the double complexes A ∗ , ∗ c ( X, U ; V ) G and A ∗ , ∗ cr ( X, U ; V ) G . A t first we r educe the pr oblem to the non-equiv a riant c ase: Prop ositio n 3.1. If the augmente d c olumn c omplexes A p c ( X ; V ) ֒ → A p, ∗ cr ( X, U ; V ) ar e exact, then t he augmente d sub c olumn c omplexes A p c ( X ; V ) G ֒ → A p, ∗ cr ( X, U ; V ) G of e quivariant c o chains ar e exact as wel l. Pr o of. Assume that the augmented column complexes A p c ( X, U ; V ) ֒ → A p, ∗ cr ( X, U ; V ) are exact. Then ea ch equiv aria nt vertical co cycle f p,q eq ∈ A p,q r c ( X, U ; V ) G is the v er- tical cob oundary d v f p,q − 1 of a co cycle f p,q − 1 ∈ A p,q − 1 cr ( X, U ; V ) (which is not nec- essary equiv ar iant). Define an equiv a riant co chain f p,q − 1 eq of bidegree ( p, q − 1) via f p,q − 1 eq ( ~ x, ~ x ′ ) := x 0 .f p,q − 1 ( x − 1 0 .~ x, x − 1 0 .~ x ′ ) . This equiv ariant c o c hain is contin uous on X p +1 × U [ q − 1 ] b ecause f p,q − 1 is contin- uous on X p +1 × U [ q − 1 ]. W e asse r t that the vertical cob oundary d v f p,q − 1 eq of f eq is 8 M. F UCHSSTEINER the equiv aria n t vertical co cycle f p,q eq . Indeed, since the differential d v is equiv ariant, the vertical cob oundar y of f p,q − 1 eq computes to d v f p,q − 1 eq ( ~ x, ~ x ′ ) = x 0 .  d v f p,q − 1 ( x − 1 0 .~ x, x − 1 0 .~ x ′ )  = f p,q eq ( ~ x, ~ x ′ ) . Thu s every equiv ar iant vertical co cycle f p,q eq in A ∗ , ∗ cr ( X, U ; V ) G is the vertical cob ound- ary of an equiv a riant co c ha in f p,q − 1 eq of bidegree ( p, q − 1).  Corollary 3.2. If the augmente d c olumn c omplexes A p c ( X ; V ) ֒ → A p, ∗ cr ( X, U ; V ) ar e exact, then the inclusion j ∗ eq : A ∗ c ( X ; V ) G ֒ → T ot A ∗ , ∗ cr ( X, U , V ) G induc es an isomorphi sm in c ohomolo gy. Corollary 3. 3 . If the augmente d c olumn c omplexes A p c ( X ; V ) ֒ → A p, ∗ cr ( X, U ; V ) ar e exact, then the inclusion A ∗ c ( X ; V ) G ֒ → A ∗ cr ( X, U ; V ) G induc es an isomorphism in c ohomolo gy. T o a chiev e the announced result it rema ins to show that for contractible spaces X the colimit augmented columns A p c ( X ; V ) ֒ → A p, ∗ cg ( X ; V ) are exact. F or this purp ose we first consider the co chain complex asso ciated to the cosimplicial abelia n g r oup A p, ∗ ( X ; V ) :=  f : X p +1 × X ∗ +1 → V | ∀ ~ x ′ ∈ X ∗ +1 : f ( − , ~ x ′ ) ∈ C ( X p +1 , V )  of global co chains, its subcomplex A p, ∗ cr ( X, U ; V ) and the co chain c o mplexes a sso ci- ated to the co simplicial ab elia n gro ups A p, ∗ ( U ; V ) := { f : X p +1 × U [ ∗ ] →| ∀ ~ x ′ ∈ U [ ∗ ] : f ( − , ~ x ′ ) ∈ C ( X p +1 , V ) } and A p, ∗ c ( X, U ; V ) := C ( X p +1 × U [ ∗ ] , V ) . Restriction of glo ba l to loca l co c hains induces morphisms of co chain complexes Res p, ∗ : A p, ∗ ( X ; V ) ։ A p, ∗ ( X, U ; V ) and Res p, ∗ cr : A ∗ cr ( X, U ; V ) ։ A p, ∗ c ( X, U ; V ) int ertwining the inclusions of the subco mplexes A p, ∗ cr ( X, U ; V ) ֒ → A p, ∗ ( X ; V ) and A p, ∗ c ( X, U ; V ) ֒ → A p, ∗ ( X, U ; V ), so one obtains the follo wing co mmutative diagram (3.1) 0 − → ker(Res p, ∗ cr ) − → A p, ∗ cr ( X, U ; V ) − → A p, ∗ c ( X, U ; V ) − → 0 ↓ ↓ ↓ 0 − → ker(Res p, ∗ ) − → A p, ∗ ( X ; V ) − → A p, ∗ ( X, U ; V ) − → 0 of co chain complexes who se r ows are exact. The kernel ker(Res p,q ) is the subspace of those ( p, q )-co chains which ar e trivial o n X p +1 × U [ q ]. Since thes e ( p, q )-cochains are co nt inuous on X p +1 × U [ q ] we find that b oth kernels coincide. W e a bbreviate the complex ker(Res p, ∗ ) = ker (Res p, ∗ r c ) by K p, ∗ and denote the coho mology gro ups of the complex A p, ∗ cr ( X, U ; V ) by H p, ∗ cr ( X, U ; V ), the cohomolog y g roups o f the complex A p, ∗ c ( X, U ; V ) of contin uo us co chains by H p, ∗ c ( X, U ; V ) and the cohomolo g y groups of the complex A p, ∗ ( X, U ; V ) by H p, ∗ ( X, U ; V ). Lemma 3.4. The c o chain c omplexes A p, ∗ ( X ; V ) ar e exact. Pr o of. F or a ny p oint ∗ ∈ X the homo morphisms h p,q : A p,q ( X ; V ) → A p,q − 1 ( X ; V ) given by h p,q ( f )( ~ x, ~ x ′ ) := f ( ~ x, ∗ , ~ x ′ ) for m a contraction o f the complex A p, ∗ ( X ; V ).  The morphism of shor t ex act sequences of co chain co mplex es in Diagra m 3.1 gives rise to a morphism of lo ng exact coho mology sequences, in which the cohomo logy A SPECTRAL SEQUENCE CONNECTING CONTINUOUS WITH LOCALL Y CONTINUOUS GROUP COHOMOLOGY 9 of the complex A p, ∗ ( X ; V ) is trivia l: (3.2) / / H q ( K p, ∗ )) / / H p,q cr ( X, U ; V ) / /   H p,q c ( X, U ; V ) / /   H q +1 ( K p, ∗ ) / / ∼ = / / H q ( K p, ∗ ) / / 0 / / H p,q ( X, U ; V ) ∼ = / / H q +1 ( K p, ∗ )) / / Lemma 3.5. The augmente d c omplex A p c ( X ; V ) ֒ → A p, ∗ cr ( X, U ; V ) is exact if and only if the inclusion A p, ∗ c ( X, U ; V ) ֒ → A p, ∗ ( X, U ; V ) induc es an isomorphism in c ohomolo gy. Pr o of. This is an immediate consequence of Diagram 3.2  Prop ositio n 3. 6 . If the inclusion A p, ∗ c ( X, U ; V ) ֒ → A p, ∗ ( X, U ; V ) induc es an iso- morphism in c ohomolo gy, then t he inclus ions j ∗ eq : A ∗ c ( X ; V ) G ֒ → T ot A ∗ , ∗ cr ( X, U , V ) G and A ∗ c ( X, U ; V ) G ֒ → A ∗ cr ( X, U ; V ) G also induc es an isomorphism in c ohomolo gy. Pr o of. This follows from the preceding Lemma and Corollaries 3 .2 and 3.3.  The passage to the colimit ov er all op en cov erings of X yields the co rresp onding results for the co mplexes of co chains with co n tinuous g erms: Prop ositio n 3. 7. If the au gmen te d c olumn c omplexes A p c ( X ; V ) ֒ → A p, ∗ cg ( X ; V ) ar e ex act, then the augmente d sub c olumn c omplexes A p c ( X ; V ) G ֒ → A p, ∗ cg ( X ; V ) G of e quivariant c o chains ar e exact as wel l. Pr o of. The pro of is simila r to that of P rop osition 3.1.  Corollary 3.8. If the augmente d c olumn c omplexes A p c ( X ; V ) ֒ → A p, ∗ cg ( X ; V ) ar e exact, then the inclusion j ∗ eq : A ∗ c ( X ; V ) G ֒ → T ot A ∗ , ∗ cg ( X ; V ) G induc es an isomor- phism in c ohomolo gy. Corollary 3.9. If the augmente d c olumn c omplexes A p c ( X ; V ) ֒ → A p, ∗ cg ( X ; V ) ar e exact, then the inclusion A ∗ c ( X ; V ) G ֒ → A ∗ cg ( X ; V ) G induc es an isomorphism in c ohomolo gy. R emark 3.10 . Alternatively to tak ing the co limit ov e r a ll op en c ov er ings U of X one may consider G -in v ariant op en co verings only to o bta ins the sa me res ults. (This was s hown in Pro p o sition 1 .7 and Lemmata 2.9.) Example 3 .11. If G = X is a top olo gical gr oup which acts on itself by left tra nsla- tion and the augmen ted columns A p c ( X ; V ) ֒ → A p, ∗ cg ( X ; V ) := colim A p, ∗ ( X, U U ; V ) (where U runs o ver all open identit y neighbo urho o ds in G ) ar e exact, then the inclusion A ∗ c ( X ; V ) G ֒ → A ∗ cg ( X ; V ) G induces an isomo rphism in c ohomolog y . The complex A p, ∗ ( X, U ; V ) is isomor phic to the complex A ∗ ( U ; C ( X p +1 , V )). The colimit A ∗ AS ( X ; C ( X p +1 , V )) := co lim A ∗ ( U ; C ( X p +1 , V )), where U runs o ver all op en coverings of X is the complex of Alexander-Spa nier c o chains on X . There- fore the co limit complex colim A p ( X ; A ∗ ( U ; V )) is isomorphic to the co chain com- plex A ∗ AS ( X ; C ( X p +1 , V )). A similar observ ation can b e made for the co chain complex A p, ∗ c ( X, U ; V ) if the exp onential law C ( X p +1 × U [ q ] , V ) ∼ = C ( X, C ( U [ q ] , V )) holds for a cofina l set of o pen coverings U of X . Passing to the colimit in Diagra m 3.1 y ields the mor phism 10 M. F UCHSSTEINER (3.3) 0 − → ker(Res p, ∗ cg ) − → A p, ∗ cg ( X ; V ) − → colim A p, ∗ c ( X, U ; V ) − → 0 ↓ ↓ ↓ 0 − → ker(Res p, ∗ ) − → A p, ∗ ( X ; V ) − → A ∗ AS ( X ; C p +1 ( X, V )) − → 0 of short exact sequences of co c ha in complexes. The kernel k er(Res p,q ) is the sub- space o f those ( p, q )-cochains which are tr iv ial on X p +1 × U [ q ] for some op en covering U of X . Since these ( p, q )-co chains ar e contin uous on X p +1 × U [ q ] we find that b oth kernels coincide. W e a bbreviate the complex ker(Res p, ∗ ) = ker (Res p, ∗ cg ) by K p, ∗ cg and denote the coho mology g roups of the complex A p, ∗ cg ( X ; V ) by H p, ∗ cg ( X ; V ). The morphism of short e x act sequences of co chain complexe s in Diagram 3.3 g ives rise to a morphism of lo ng exact coho mology se q uences: (3.4) / / H q ( K p, ∗ cg )) / / H p,q cg ( X, U ; V ) / /   H q (colim A p, ∗ c ( X, U ; V ) / /   H q +1 ( K p, ∗ cg ) / / ∼ = / / H q ( K p, ∗ ) / / 0 / / H q AS ( X ; C p +1 ( X, V )) ∼ = / / H q +1 ( K p, ∗ )) / / Lemma 3.12. Th e augmente d c omplex A p c ( X ; V ) ֒ → A p, ∗ cg ( X ; V ) is ex act if and only if the inclusion colim A p, ∗ c ( X, U ; V ) ֒ → A ∗ AS ( X ; C p +1 ( X, V )) of c o chain c om- plexes induc es an isomorphism in c ohomolo gy. Pr o of. This is an immediate consequence of Diagram 3.4  Prop ositio n 3. 1 3. If the inclusion colim A p, ∗ c ( X, U ; V ) ֒ → A ∗ AS ( X ; C ( X p +1 , V )) induc es an isomorp hism in c ohomolo gy, then j ∗ eq : A ∗ c ( X ; V ) G ֒ → T ot A ∗ , ∗ cg ( X ; V ) G and A ∗ c ( X ; V ) G ֒ → A ∗ cg ( X ; V ) G also induc e an isomorphism in c ohomolo gy. Pr o of. This follows from the preceding Lemma and Corollaries 3 .8 and 3.9.  As observed be fore (cf. Remark 3 .10) one may r estrict oneself to the directed system of G -inv ariant open c overings only to achiev e the sa me result. Th us we observe: Corollary 3.14. If G = X is a lo c al ly c ontr actible top olo gic al gr oup which acts on itself by left t r anslation and t he inclusion colim A p, ∗ c ( X, U ; V ) ֒ → A ∗ AS ( X ; C ( X p +1 , V )) (wher e U ru ns over al l op en identity neighb ourho o ds in G ) induc es an isomorphism in c ohomolo gy, t hen t he inclusion A ∗ c ( X ; V ) G ֒ → A ∗ cg ( X ; V ) G induc es an isomor- phism in c ohomolo gy as wel l. Pr o of. It has b een shown in [vE6 2] that the cohomolo gy of the colimit co chain complex colim A ∗ ( U ; C ( X p +1 , V )) is the Alexa nder-Spanier cohomolo gy o f X .  Lemma 3.15. If the top olo gic al sp ac e X is c ontr actible, t hen t he c ohomolo gy of the c omplex colim A p, ∗ c ( X, U ; V ) is t r ivial. Pr o of. The reas oning is analo gous to tha t for the Alexander -Spanier pres heaf. The pro of [F uc10, Theo rem 2.5.2] carrie s ov er a lmost in verbatim.  Theorem 3.16 . F or c ontr actible X the inclusion A ∗ c ( X ; V ) G ֒ → A ∗ cg ( X ; V ) G in- duc es an isomorphism in c ohomolo gy. A SPECTRAL SEQUENCE CONNECTING CONTINUOUS WITH LOCALL Y CONTINUOUS GROUP COHOMOLOGY 11 Pr o of. If the top olog ical space X is co nt ractible, then the Alexander-Spa nier coho- mology H AS ( X ; C p +1 ( X, V )) is trivia l a nd the cohomo lo gy of the c o c ha in complex colim A p, ∗ c ( X, U ; V ) is trivial by Lemma 3.15. By Pr op osition 3.13 the inclusion A ∗ c ( X ; V ) G ֒ → A ∗ cg ( X ; V ) G then induces an iso morphism in cohomology .  Corollary 3. 17. F or c ontr actible top olo gic al gr oups G the c ontinuous gr oup c oho- molo gy is isomorphic to the c ohomo lo gy of homo gene ous gr oup c o chains with c on- tinuous germ at the diag onal. 4. working in the ca tegor y of k -sp ace s In this section w e consider transformation gro ups ( G, X ) in the category kT op of k -space s and G -mo dules V in kT op . W orking o nly in the ca tegory kT op w e construct a s pec tr al sequence analogo us ly to tha t in Section 2 and der ive results analogo us to those obta ined ther e. Definition 4.1. F or ev ery k -spac e X and ab elian k -gr oup V t he subco mplex A ∗ kc ( X ; V ) := C (k X ∗ +1 ; V ) of the standa rd complex is called the c ontinuous stan- dar d c omplex in kT op . F o r op en cov erings U of a k - space X we a lso cons ider the sub complex of A ∗ ( X ; V ) formed by the gro ups A n kcr ( X, U ; V ) :=  f ∈ A n ( X ; V ) | f | k U [ n ] ∈ C (k U [ n ]; V )  of co chains who se restrictio n to the op en subspaces k U [ n ] of k X n +1 are contin uo us. The cohomolo gy of the co chain complex A ∗ kcr ( X, U ; V ) is denoted by H kcr ( X, U ; V ). If the cov e ring U o f X is G -in v ariant, then the subspa ces k U [ ∗ ] is a simplicial G - subspace o f the simplicial G -spa ce k X ∗ +1 . Example 4.2. If G = X is a k -g roup which acts on itself by left tra nslation and U an op e n identit y neighbourho o d, then U U := { g .U | g ∈ G } is a G -inv ariant op en cov ering of G a nd k U [ ∗ ] is a simplicial G -s ubspace of k G ∗ +1 . F o r G -inv a riant cov e rings U of X the co homology o f the sub complex A ∗ kcr ( X, U ; V ) G of G -equiv aria n t co chains is denoted by H kcr,eq ( X, U ; V ). Example 4.3 . If G = X is a k -g roup whic h acts on itse lf by left translatio n and U an open identit y neigh bo urho o d, then the complex A ∗ kcr ( X, U U ; V ) G is the complex of homogeneo us group co chains whose res trictions to the subspaces k U U [ ∗ ] are contin uous. (These are sometimes ca lle d U -contin uous co chains.) F o r directed systems { U i | i ∈ I } of o pen co verings of X one can also consider the colimit co mplex colim i A ∗ kcr ( X, U i ; V ). In particular , if the op en diag o nal neigh- bo urho o ds k U [ n ] in k X n +1 for o pen coverings U o f X are co final in the dir ected set of all op en diago nal neighbo urho o ds, one obtains the c omplex A ∗ kcg ( X ; V ) := colim U is op en cover of X A ∗ kcr ( X ; U ; V ) of global co chains whose g e rms at the diag onal are co nt inuous. This ha pp ens for all k -spa ces X for which the finite pr o ducts X n +1 in T op are already compa ctly Hausdorff generated, e.g. metrisa ble spaces, loc ally compact spaces or Hausdo rff k ω -spaces. The complex A ∗ kcg ( X ; V ) is then a s ubco mplex of the standard complex A ∗ ( X ; V ) which is inv ariant under the G -action (Eq. 1.1) and thus a sub co mplex of G -mo dules. The G -equiv ar iant co chains with cont inuous germ form a sub complex 12 M. F UCHSSTEINER A ∗ kcg ( X ; V ) G thereof, whose c o homology is de no ted by H kcg ,eq ( X ; V ). The latter sub c omplex ca n also b e obtained by taking the colimit o ver all G -in v ariant op e n cov erings of X only: Prop ositio n 4.4. If t he op en diago nal neighb ourho o ds k U [ n ] in k X n +1 for op en c overings U of X ar e c ofinal in the dir e cte d set of al l op en diagonal neighb ourho o ds then t he natur al m orphism of c o chain c omplexes A ∗ kcg ,eq ( X ; V ) := colim U is G -invariant open cover of X A ∗ kcr ( X ; U ; V ) G → A ∗ kcg ( X ; V ) G is a natu ra l isomorphism. Pr o of. The pro of is a nalogous to p that of Pr op osition 1 .7.  Corollary 4.5. If the op en diagonal neighb ourho o ds k U [ n ] in k X n +1 for op en c ov- erings U of X ar e c ofinal in the dir e cte d set of al l op en diago nal neighb ourho o ds then the c ohomolo gy H kcg ,eq ( X ; V ) is the c ohomolo gy of the c omplex of e quivaria nt c o chains which ar e c ontinuous on some G -invariant n eighb ourho o d of the diagonal. Example 4.6. If G = X is a metrisable or lo cally compact top ologica l group or a real or c omplex Kac-Mo ody gr oup which acts o n itself by left tra nslation, then the complex A ∗ kcg ( G ; V ) G is the complex o f ho mogeneous group co chains whos e g erms at the diago nal are co nt inuous. (By abuse of language these ar e sometimes called ’lo cally contin uous’ gr o up c o c ha ins.) Analogously to the pro cedure in Section 2 we can construct a sp ectral seq uenc e relating A ∗ kcr ( X, U ; V ) and A ∗ kc ( X ; V ). F or this purp ose we consider the a belia n groups (4.1) A p,q kcr ( X, U ; V ) :=  f : X p +1 × X q +1 → V | f | k X p +1 × k k U [ q ] is contin uous  . The ab elia n gr oups A p,q kcr ( X, U ; V ) form a fir s t quadr ant double complex whose vertical and horizontal differ e ntials are given by the sa me formulas as for the double complex A p,q cr ( X, U ; V ) int ro duced in Section 2. Analogously to the latter double complex the rows of the double complex A ∗ , ∗ kcr ( X, U ; V ) can b e augmented b y the complex A ∗ kcr ( X, U ; V ) for the covering U and the columns can be augmented b y the e x act co mplex A ∗ kc ( X ; V ) o f co nt inuous co chains. W e denote the total complex of the double complex A ∗ , ∗ kcr ( X, U ; V ) by T ot A ∗ , ∗ kcr ( X, U ; V ). The aug ment ations of the r ows and columns induce mo r phisms i ∗ k : A ∗ kcr ( X, U ; V ) → T ot A ∗ , ∗ kcr ( X, U ; V ) and j ∗ k : A ∗ kc ( X ; V ) → T ot A ∗ , ∗ kcr ( X, U ; V ) of co chain complexes resp ectively . Lemma 4. 7. The morphism i ∗ k : A ∗ kcr ( X, U ; V ) → T ot A ∗ , ∗ kcr ( X, U ; V ) induc es an isomorphi sm in c ohomolo gy. Pr o of. The pro of of Lemma 2 .1 also works in the ca tegory kT op o f k - spaces.  F o r G -inv a riant open cov e r ings U of X one can consider the sub do uble com- plex A ∗ , ∗ kcr ( X, U ; V ) G of A ∗ , ∗ kcr ( X, U ; V ) whose rows are augmented by the co chain complex A ∗ kcr ( X, U ; V ) G for the c overing U a nd the columns can b e augmented by the c o mplex A ∗ kc ( X ; V ) G of contin uous equiv a riant co chains (,which is not exa ct in general). Lemma 4.8 . F or G -invariant c overings U of X the m orphism i ∗ k,eq := i ∗ k G induc es an isomorphism in c ohomolo gy. Pr o of. The pro of is a nalogous to that of Lemma 2.3.  A SPECTRAL SEQUENCE CONNECTING CONTINUOUS WITH LOCALL Y CONTINUOUS GROUP COHOMOLOGY 13 So the mor phism H ( i k,eq ) : H kcr,eq ( X, U ; V ) → H (T ot A ∗ , ∗ kcr ( X, U ; V ) G ) is inv ert- ible. F or the co mpo s ition H ( i k,eq ) − 1 H ( j k,eq ) : H kc,eq ( X ; V ) → H kcr,eq ( X, U ; V ) we observe: Prop ositio n 4. 9. The image j n k ( f ) of a c ont inu ous e quivariant n -c o cycle f on X in T ot A ∗ , ∗ kcr ( X, U ; , V ) G is c ohomolo gous to the image i n k,eq ( f ) of the e qu ivariant n -c o cycle f ∈ A n kcr ( X, U ; V ) G in T ot A ∗ , ∗ kcr ( X, U ; V ) G . Pr o of. The pro of is a nalogous to that of Prop ositio n 2.4.  Corollary 4.10. The map H ( i k,eq ) − 1 H ( j eq ) : H kc,eq ( X ; V ) → H kcr,eq ( X, U ; V ) is induc e d by the inclusion A ∗ kc ( X, U ; V ) G ֒ → A ∗ kcr ( X, U ; V ) G . Corollary 4.11. If the morphi sm j ∗ k,eq := j ∗ G : A ∗ kc ( X ; V ) G → T o t A ∗ , ∗ kcr ( X, U A ) G induc es a m onomorphism, epimo rphism or isomorphism in c ohomolo gy, then the inclusion A ∗ kc ( X ; V ) G ֒ → A ∗ kcr ( X, U ; V ) G induc es a monomorphism, epimorphism or isomorp hism in c ohomolo gy r esp e ctively. Lemma 4.12. F or any dir e cte d system { U i | i ∈ I } of op en c overings of X the morphism colim i i ∗ : co lim i A ∗ kcr ( X, U i ; V ) → T ot colim i A ∗ , ∗ kcr ( X, U i ; V ) induc es an isomorphi sm in c ohomolo gy. Pr o of. The pa ssage to the colimit pr eserves the exactness of the augmented row complexes (Lemma 4.7).  Lemma 4.13. F or any dir e cte d system { U i | i ∈ I } of G -invariant op en c overings of X the morphism co lim i i ∗ k,eq : colim i A ∗ kcr ( X, U i ; V ) G → T o t colim i A ∗ , ∗ cr ( X, U i ; V ) G induc es an isomorphism in c ohomolo gy. Pr o of. The pa ssage to the colimit pr eserves the exactness of the augmented row complexes (Lemma 4.8).  If the op en dia gonal neighbourho o ds k U [ n ] in k X n +1 for op en cov erings U of X are cofinal in the directed se t of a ll op en dia g onal neighbo ur ho o ds then o ne obtains the double complex complex A ∗ , ∗ kcg ( X ; V ) := co lim U is op en cov er of X A ∗ , ∗ kcr ( X ; U ; V ) whose r ows and columns are augmented by the complexes A ∗ kcg ( X ; V ) and A ∗ kc ( X ; V ) resp ectively . In this case the colimit morphism i ∗ kcg : A ∗ kcg ( X ; V ) → T ot A ∗ , ∗ kcg ( X ; V ) induces a n isomorphism in cohomolog y . F urthermore the colimit double complex A ∗ , ∗ kcg ( X ; V ) then is a double complex of G -mo dules and the G -equiv ar iant co c ha ins in for m a sub double complex A ∗ , ∗ kcg ( X ; V ) G , whose r ows a nd co lumns are augmented by the co limit complex A ∗ kcg ,eq ( X ; V ) a nd by the complex A ∗ kc ( X ; V ) G resp ectively . In a ddition we obser ve: Lemma 4.14. If the op en diagonal neighb ourho o ds k U [ n ] in k X n +1 for op en c ov- erings U of X ar e c ofinal in the dir e cte d set of al l op en diago nal neighb ourho o ds then t he natur al m orphism of double c omplexes A ∗ , ∗ kcg ,eq ( X ; V ) := colim U is G -invariant open cover of X A ∗ , ∗ kcr ( X ; U ; V ) G → A ∗ kcg ( X ; V ) G is a natu ra l isomorphism. Pr o of. The pro of is a nalogous to that of Prop ositio n 4.4.  14 M. F UCHSSTEINER As a conseq uenc e the colimit mor phism i ∗ kcg ,eq : A ∗ kcg ,eq ( X ; V ) → T ot A ∗ , ∗ kcg ( X ; V ) G then induces an iso morphism in cohomolog y , and the morphism H ( i kcg ,eq ) is inv e r t- ible. F or the compos ition H ( i kcg ,eq ) − 1 H ( j k,eq ) : H kc,eq ( X ; V ) → H kcg ,eq ( X, U ; V ) we observe: Prop ositio n 4.15. If the op en dia gonal neighb ourho o ds k U [ n ] in k X n +1 for op en c overings U of X ar e c ofinal in the dir e cte d set of al l op en diagonal neighb ourho o ds then the image j n ( f ) of a c ontinuous e quivariant n -c o cycle f on X in T ot A ∗ , ∗ kcg ( X ; , V ) G is c ohomolo gous to the image i n kcg ,eq ( f ) of the e qu ivariant c o cycle f ∈ A n kcg ,eq ( X ; V ) in T ot A ∗ , ∗ kcg ( X ; V ) G . Pr o of. The pro of is a nalogous to that of Prop ositio n 2.4.  Corollary 4.16. If the op en diagonal neighb ourho o ds k U [ n ] in k X n +1 for op en c ov- erings U of X ar e c ofinal in the dir e cte d set of al l op en diagonal neighb ourho o ds then the c omp osition H ( i kcg ,eq ) − 1 H ( j k,eq ) : H kc,eq ( X ; V ) → H kcg ,eq ( X ; V ) is induc e d by the inclusion A ∗ kc ( X ; V ) G ֒ → A ∗ kcg ( X ; V ) G . Corollary 4.17. If the op en diagonal neighb ourho o ds k U [ n ] in k X n +1 for op en c overings U of X ar e c ofin al in t he dir e cte d set of al l op en diagonal n eighb our- ho o ds and t he morphism j ∗ k,eq := j ∗ k G : A ∗ kc ( X ; V ) G → T ot A ∗ , ∗ kcg ( X ; V ) G induc es a monomorp hism, epimorphism or isomorphi sm in c ohomolo gy, then the inclusion A ∗ kc ( X ; V ) G ֒ → A ∗ kcg ,eq ( X ; V ) induc es a monomorphism, epimorphi sm or isomor- phism in c ohomolo gy re sp e ctively. 5. Continuo us and U -Continuo us Cochains on k -sp a ces In this section w e cons ide r transformatio n k -gro ups ( G, X ) and G -modules V in kT op for which we show that the inclusion A ∗ kc ( X, U ; V ) G ֒ → A ∗ kcr ( X, U ; V ) G of the complex of contin uous equiv a riant co chains into the complex o f equiv ariant U -contin uous co chains induces a n isomor phism H ∗ kc ( X, U ; V ) ∼ = H ∗ kcr ( X, U ; V ) pr o - vided the k -space X is contractible. The pro ceeding is s imilar to tha t in Section 5. A t first we reduce the problem to the no n-equiv aria nt case : Prop ositio n 5.1. If the augment e d c olumn c omplexes A p kc ( X ; V ) ֒ → A p, ∗ kcr ( X, U ; V ) ar e exact, then the augmente d su b c olumn c omplexes A p kc ( X ; V ) G ֒ → A p, ∗ kcr ( X, U ; V ) G of e quivariant c o chains ar e exact as wel l. Pr o of. The pro of is a nalogous to that of Prop ositio n 3.1.  Corollary 5.2. If the augmente d c olumn c omplexes A p kc ( X ; V ) ֒ → A p, ∗ kcr ( X, U ; V ) ar e exact, t hen t he inclusion j ∗ k,eq : A ∗ kc ( X ; V ) G ֒ → T ot A ∗ , ∗ kcr ( X, U , V ) G induc es an isomorphi sm in c ohomolo gy. Corollary 5.3. If the augmente d c olumn c omplexes A p kc ( X ; V ) ֒ → A p, ∗ kcr ( X, U ; V ) ar e exact, then the inclusion A ∗ kc ( X ; V ) G ֒ → A ∗ kcr ( X, U ; V ) G induc es an isomor- phism in c ohomolo gy. T o achiev e the a nno unced result it remains to show that for contractible k -spa ces X the co limit augmented co lumns A p kc ( X ; V ) ֒ → A p, ∗ kcg ( X ; V ) a re exact. F or this purp ose we fir st co ns ider the co chain complex asso ciated to the cos implicia l ab elia n group A p, ∗ k ( X ; V ) :=  f : X p +1 × X ∗ +1 → V | ∀ ~ x ′ ∈ X ∗ +1 : f ( − , ~ x ′ ) ∈ C (k X p +1 , V )  A SPECTRAL SEQUENCE CONNECTING CONTINUOUS WITH LOCALL Y CONTINUOUS GROUP COHOMOLOGY 15 of global cochains, its sub complex A p, ∗ kcr ( X, U ; V ) and the co chain co mplex es a sso- ciated to the co simplicial a b elia n gro ups A p, ∗ k ( U ; V ) := { f : X p +1 × U [ ∗ ] →| ∀ ~ x ′ ∈ U [ ∗ ] : f ( − , ~ x ′ ) ∈ C (k X p +1 , V ) } and A p, ∗ kc ( X, U ; V ) := C (k X p +1 × k k U [ ∗ ] , V ) . Restriction of glo ba l to loca l co c hains induces morphisms of co chain complexes Res p, ∗ k : A p, ∗ k ( X ; V ) ։ A p, ∗ k ( X, U ; V ) and Res p, ∗ kcr : A ∗ kcr ( X, U ; V ) ։ A p, ∗ kc ( X, U ; V ) int ertwining the inclusio ns of the s ub co mplexes A p, ∗ kcr ( X, U ; V ) ֒ → A p, ∗ k ( X ; V ) and A p, ∗ kc ( X, U ; V ) ֒ → A p, ∗ k ( X, U ; V ), so one obtains the follo wing co mmutative diagram (5.1) 0 − → ker(Res p, ∗ kcr ) − → A p, ∗ kcr ( X, U ; V ) − → A p, ∗ kc ( X, U ; V ) − → 0 ↓ ↓ ↓ 0 − → ker( Res p, ∗ k ) − → A p, ∗ k ( X ; V ) − → A p, ∗ k ( X, U ; V ) − → 0 of co chain complexes who se r ows are exact. The kernel ker(Res p,q k ) is the subspace of those ( p, q )-co chains which are trivial on k X p +1 × k k U [ q ]. Since these ( p, q )- co chains are contin uous on k X p +1 × k k U [ q ] we find that b oth kernels coincide. W e abbrev ia te the complex ker(Res p, ∗ k ) = ker(Res p, ∗ kr c ) by K p, ∗ k and deno te the cohomolog y gro ups of the complex A p, ∗ kcr ( X, U ; V ) by H p, ∗ kcr ( X, U ; V ), the coho mology groups of the complex A p, ∗ kc ( X, U ; V ) of contin uous co chains by H p, ∗ kc ( X, U ; V ) and the cohomolo gy gro ups of the complex A p, ∗ k ( X, U ; V ) by H p, ∗ k ( X, U ; V ). Lemma 5.4. The c o chain c omplexes A p, ∗ k ( X ; V ) ar e exact. Pr o of. F or a ny p oint ∗ ∈ X the homo morphisms h p,q : A p,q k ( X ; V ) → A p,q − 1 k ( X ; V ) given by h p,q ( f )( ~ x, ~ x ′ ) := f ( ~ x, ∗ , ~ x ′ ) for m a contraction o f the complex A p, ∗ k ( X ; V ).  The morphism of shor t ex act sequences of co chain co mplex es in Diagra m 5.1 gives rise to a morphism of lo ng exact coho mology sequences, in which the cohomo logy of the complex A p, ∗ k ( X ; V ) is trivia l: (5.2) / / H q ( K p, ∗ k )) / / H p,q kcr ( X, U ; V ) / /   H p,q kc ( X, U ; V ) / /   H q +1 ( K p, ∗ k ) / / ∼ = / / H q ( K p, ∗ ) / / 0 / / H p,q k ( X, U ; V ) ∼ = / / H q +1 ( K p, ∗ )) / / Lemma 5.5. The augment e d c omplex A p kc ( X ; V ) ֒ → A p, ∗ cr ( X, U ; V ) is exact if and only if the inclusion A p, ∗ kc ( X, U ; V ) ֒ → A p, ∗ k ( X, U ; V ) induc es an isomorphism in c ohomolo gy. Pr o of. This is an immediate consequence of Diagram 5.2  Prop ositio n 5. 6 . If the inclusion A p, ∗ kc ( X, U ; V ) ֒ → A p, ∗ k ( X, U ; V ) induc es an iso- morphism in c ohomolo gy, then the inclusions A ∗ c ( X ; V ) G ֒ → T o t A ∗ , ∗ kcr ( X, U , V ) G and A ∗ kc ( X, U ; V ) G ֒ → A ∗ kcr ( X, U ; V ) G also induc es an isomorphism in c ohomolo gy. Pr o of. This follows from the preceding Lemma and Corollaries 5 .2 and 5.3.  F o r k -space s X for which the op en diagonal neighbour ho o ds k U [ n ] in k X n +1 for op en cov erings U o f X ar e cofinal in the directed set o f all op en diago na l neigh- bo urho o ds the passag e to the colimit over all op en coverings of X yields the cor re- sp onding results for the c o mplexes of co chains with co ntin uous ger ms: 16 M. F UCHSSTEINER Prop ositio n 5.7. If t he op en diago nal neighb ourho o ds k U [ n ] in k X n +1 for op en c overings U of X ar e c ofinal in the dir e cte d set of al l op en diagonal neighb ourho o ds and t he augmente d c olumn c omplexes A p kc ( X ; V ) ֒ → A p, ∗ kcg ( X ; V ) ar e exact, t hen the augmente d sub c olumn c omplexes A p kc ( X ; V ) G ֒ → A p, ∗ kcg ( X ; V ) G of e quivariant c o chains ar e exact as wel l. Pr o of. The pro of is simila r to that of P rop osition 3.1.  Corollary 5.8. If the op en diagonal neighb ourho o ds k U [ n ] in k X n +1 for op en c ov- erings U of X ar e c ofinal in the dir e cte d set of al l op en diago nal neighb ourho o ds and t he augmente d c olumn c omplexes A p kc ( X ; V ) ֒ → A p, ∗ kcg ( X ; V ) ar e exact, t hen the inclusion j ∗ k,eq : A ∗ kc ( X ; V ) G ֒ → T ot A ∗ , ∗ kcg ( X ; V ) G induc es an isomorph ism in c ohomolo gy. Corollary 5.9. If the op en diagonal neighb ourho o ds k U [ n ] in k X n +1 for op en c ov- erings U of X ar e c ofinal in the dir e cte d set of al l op en diago nal neighb ourho o ds and the augmente d c olumn c omplexes A p kc ( X ; V ) ֒ → A p, ∗ kcg ( X ; V ) ar e exact, then the inclusion A ∗ kc ( X ; V ) G ֒ → A ∗ kcg ( X ; V ) G induc es an isomorph ism in c ohomolo gy. R emark 5.10 . Alternatively to tak ing the co limit ov e r a ll op en c ov er ings U of X one may consider G -in v ariant op en co verings only to o bta ins the sa me res ults. (This was s hown in Pro p o sition 4 .4 and Lemmata 4.1 4.) Example 5.11. If G = X is a metrisable, lo ca lly c ompact or Hausdo rff k ω top o- logical g roup which acts on itse lf b y left translation and the augmented co lumns A p kc ( X ; V ) ֒ → A p, ∗ kcg ( X ; V ) := co lim A p, ∗ k ( X, U U ; V ) (where U runs ov er all op en ident ity neig hbourho o ds in G ) are exact, then A ∗ kc ( X ; V ) G ֒ → A ∗ kcg ( X ; V ) G induces an isomorphism in coho mo logy . The complex A p, ∗ ( X, U ; V ) is iso morphic to the complex A ∗ ( U ; C (k X p +1 , V )). If the o p e n diagonal neighbour ho o ds U [ n ] in X n +1 for o p e n cov erings U of X are cofina l in the directed set of all op e n diagonal neighbo ur ho o ds then the col- imit A ∗ AS ( X ; C ( X p +1 , V )) := colim A ∗ ( U ; C ( X p +1 , V )), where U r uns over all op en cov erings of X is the complex of Alex ander-Spanier co chains o n X (with v a lues in C ( X p +1 , V )). In this case the colimit complex colim A p ( X ; A ∗ ( U ; V )) is iso- morphic to the co chain complex A ∗ AS ( X ; C (k X p +1 , V )). A similar observ a tion can b e made for the cochain complex A p, ∗ kc ( X, U ; V ) becaus e the exp onential law C (k X p +1 × k k U [ q ] , V ) ∼ = C ( X, k C ( U [ q ] , V )) holds in kT op . Passing to the colimit in Diag ram 5.1 yields the morphism (5.3) 0 − → ker(Res p, ∗ kcg ) − → A p, ∗ kcg ( X ; V ) − → colim A p, ∗ kc ( X, U ; V ) − → 0 ↓ ↓ ↓ 0 − → ker(Res p, ∗ k ) − → A p, ∗ k ( X ; V ) − → A ∗ AS ( X ; C p +1 ( X, V )) − → 0 of short exact sequences of co c ha in complexes. The kernel k er(Res p,q k ) is the sub- space of those ( p, q )-co chains which ar e triv ial on k X p +1 × k k U [ q ] for s o me op en cov- ering U of X . Since these ( p, q )-co chains are co ntin uous o n k X p +1 × k k U [ q ] we find that b oth kernels coincide. W e abbrevia te the complex ker(Res p, ∗ k ) = ker(Res p, ∗ kcg ) by K p, ∗ kcg and denote the coho mology groups of the co mplex A p, ∗ kcg ( X ; V ) b y H p, ∗ kcg ( X ; V ). The morphism o f sho r t exact sequences of co chain complexes in Diag ram 5 .3 g ives A SPECTRAL SEQUENCE CONNECTING CONTINUOUS WITH LOCALL Y CONTINUOUS GROUP COHOMOLOGY 17 rise to a morphism of long exa ct cohomolog y sequences: (5.4) / / H q ( K p, ∗ kcg )) / / H p,q kcg ( X, U ; V ) / /   H q (colim A p, ∗ kc ( X, U ; V ) / /   H q +1 ( K p, ∗ kcg ) / / ∼ = / / H q ( K p, ∗ kcg ) / / 0 / / H q AS ( X ; C p +1 ( X, V )) ∼ = / / H q +1 ( K p, ∗ kcg )) / / Lemma 5.12. The augmente d c omplex A p kc ( X ; V ) ֒ → A p, ∗ kcg ( X ; V ) is exact if and only if the inclusion colim A p, ∗ kc ( X, U ; V ) ֒ → A ∗ AS ( X ; C p +1 ( X, V )) of c o chain c om- plexes induc es an isomorphism in c ohomolo gy. Pr o of. This is an immediate consequence of Diagram 5.4  Prop ositio n 5.13. If the op en dia gonal neighb ourho o ds k U [ n ] in k X n +1 for op en c overings U of X ar e c ofin al in t he dir e cte d set of al l op en diagonal n eighb our- ho o ds and the inclusion colim A p, ∗ kc ( X, U ; V ) ֒ → A ∗ AS ( X ; C ( X p +1 , V )) induc es an isomorphi sm in c ohomolo gy, the n j ∗ k,eq : A ∗ kc ( X ; V ) G ֒ → T o t A ∗ , ∗ kcg ( X ; V ) G and A ∗ kc ( X ; V ) G ֒ → A ∗ kcg ( X ; V ) G also induc e an isomorphism in c ohomolo gy. Pr o of. This follows from the preceding Lemma and Corollaries 5 .8 and 5.9.  As observed be fore (cf. Remark 5 .10) one may r estrict oneself to the directed system of G -inv ariant open c overings only to achiev e the sa me result. Th us we observe: Corollary 5.14. If G = X is a lo c al ly c ontra ctible metrisable, lo c al ly c ont r actible lo c al ly c omp act or lo c al ly c ontr actible H au s dorff k ω top olo gic al gr oup which acts on itself by left t r anslation and t he inclusion colim A p, ∗ kc ( X, U ; V ) ֒ → A ∗ AS ( X ; C ( X p +1 , V )) (wher e U ru ns over al l op en identity neighb ourho o ds in G ) induc es an isomorphism in c ohomolo gy, then the inclusion A ∗ kc ( X ; V ) G ֒ → A ∗ kcg ( X ; V ) G induc es an isomor- phism in c ohomolo gy as wel l. Pr o of. It has b een shown in [vE6 2] that the cohomolo gy of the colimit co chain complex colim A ∗ ( U ; C (k X p +1 , V )) is the Alexander- Spanier coho mology of X with co efficients C (k X p +1 , V ).  Lemma 5.15. If the top olo gic al sp ac e X is c ontr actible, t hen t he c ohomolo gy of the c omplex colim A p, ∗ kc ( X, U ; V ) is t r ivial. Pr o of. The reas oning is analo gous to tha t for the Alexander -Spanier pres heaf. The pro of [F uc10, Theo rem 2.5.2] carrie s ov er a lmost in verbatim.  Theorem 5.16 . F or c ontr actible X the inclusion A ∗ kc ( X ; V ) G ֒ → A ∗ kcg ( X ; V ) G induc es an isomorphism in c ohomolo gy. Pr o of. If the k -spa ce X is contractible, then the Alexander- Spanier cohomo logy of X is trivial and the coho mology of the co chain complex colim A p, ∗ kc ( X, U ; V ) is trivial by Lemma 5 .15. By P rop osition 5.13 the inclusio n A ∗ kc ( X ; V ) G ֒ → A ∗ kcg ( X ; V ) G then induces an iso morphism in cohomology .  Corollary 5. 17. F or metrisable, lo c al ly c omp act or Hau s dorff k ω top olo gic al gr oups G whic h ar e c ontr actible the c ontinuous gr oup c ohomolo gy H kc,eq ( G ; V ) is isomor- phic to the c ohomolo gy H kcg ,eq ( G ; V ) of homo gene ous gr oup c o chains with c ontinu - ous germ at the diago nal. 18 M. F UCHSSTEINER 6. Complexes of S mooth Cochains In this Section we int ro duce the sub (double)complexes for smooth tra ns forma- tion g roups ( G, M ) and smo oth G -mo dules V , where V is an ab elian Lie group. (W e use the general infinite dimensional c a lculus in tro duced in [BGN04].) Let ( G, M ) be a smoo th tr ansformation group, V be a smo oth G -mo dule and U b e an op en cov ering of M . Definition 6. 1. F or ev ery manifold M and ab elian Lie gro up V the sub complex A ∗ s ( M ; V ) := C ∞ ( M ∗ +1 ; V ) of the s tandard complex is called the smo oth standar d c omplex . The cohomo logy H eq,s ( M ; V ) o f the sub complex A ∗ s ( M ; V ) G is called the equiv aria nt smo oth coho mo logy of M (with v alues in V ). Example 6.2. F o r any Lie group G which a cts on itself by left transla tion and smo oth G -mo dule V the complex A ∗ s ( G ; V ) G is the complex of s mo oth (homog e- neous) group co chains; its cohomolo gy H eq,s ( G ; V ) is the smo o th gr oup co homology of G with v alues in V . F o r Lie gr oups G and G -mo dules V the first cohomolo gy g roup H 1 eq,s ( G ; V ) classifies s mo o th crossed morphis ms mo dulo principal deriv ations, the second co- homology group H 2 eq,s ( G ; V ) classifies equiv alence classes of Lie group ex tens ions V ֒ → ˆ G ։ G which admit a smooth glo bal section (i.e. ˆ G ։ G is a trivia l smo oth V -pr incipal bundle) and the thir d cohomolo gy g roup H 3 eq,c ( G ; V ) classifies e q uiv a- lence classes of smo othly split crossed mo dules. F o r each op e n covering U of M one can consider the sub complex of A ∗ ( M ; V ) formed by the gro ups A n sr ( M , U ; V ) :=  f ∈ A n ( M ; V ) | f | U [ n ] ∈ C ∞ ( U [ n ]; V )  of co chains whose restrictio n to the subspace s U [ n ] of M n +1 are smo o th. The cohomolog y of the co chain complex A ∗ sr ( M , U ; V ) is denoted by H sr ( M , U ; V ). If the cov ering U of M is G -inv ariant, then the subs paces U [ ∗ ] is a simplicia l G - subspace of the simplicia l G -space M ∗ +1 . F or G -inv a riant coverings U of M the cohomolog y of the sub complex A ∗ sr ( M , U ; V ) G of G -equiv ar iant co chains is denoted by H cr,eq ( M , U ; V ). Example 6.3. If G = M is a Lie gro up which acts on itself by left translatio n and U an o pe n identit y neighbourho o d, then the complex A ∗ sr ( M , U U ; V ) G is the complex o f homogeneo us group co chains whose restric tions to the subspaces U U [ ∗ ] are smo oth. (These are so metimes c alled U -smo oth co chains.) F o r dir ected systems { U i | i ∈ I } of op en cov er ings of M one can also consider the colimit complex colim i A ∗ sr ( M , U i ; V ). In particular for the dir ected system of all op en coverings of M o ne observes that the op en dia gonal neighbourho ods U [ n ] in M n +1 for o pen coverings U o f M are co final in the directed set of all op en diag onal neighbourho o ds, hence one o btains the complex A ∗ sg ( M ; V ) := colim U is op en cover of M A ∗ sr ( M ; U ; V ) of globa l co chains who s e germs at the diago na l a re co n tinuous. This is a sub complex of the s tandard complex A ∗ ( M ; V ) which is inv ariant under the G -action (Eq. 1.1) and thus a sub complex of G -mo dules. The G -equiv aria nt co chains with contin uous germ form a sub c omplex A ∗ sg ( M ; V ) G thereof, whose cohomology is deno ted by H cg, eq ( M ; V ). The latter subco mplex can als o be obtained b y tak ing the colimit ov e r all G -inv ariant op en cov erings o f M only: A SPECTRAL SEQUENCE CONNECTING CONTINUOUS WITH LOCALL Y CONTINUOUS GROUP COHOMOLOGY 19 Prop ositio n 6.4. The natur al morphism of c o chain c omplexes A ∗ cg, eq ( M ; V ) := co lim U is G -invariant open cover of M A ∗ sr ( M ; U ; V ) G → A ∗ sg ( M ; V ) G is a natu ra l isomorphism. Pr o of. The pro of is a nalogous to that of Prop ositio n 1.7.  Corollary 6.5. The c ohomol o gy H cg, eq ( M ; V ) is the c ohomolo gy of t he c omplex of e quivariant c o chains which ar e c ontinuous on some G -invariant neighb ourho o d of the dia gonal. Example 6.6. If G = M is a Lie group which acts on itself by left translation, then the complex A ∗ sg ( G ; V ) G is the complex of ho mogeneous g roup co chains whose germs a t the dia gonal a re smo oth. (By abuse of lang ua ge these ar e so metimes ca lled ’lo cally smo oth’ gro up co chains.) W e w ill show (in Sec tio n 7) that the inclusion A ∗ sr ( M , U ; V ) ֒ → A ∗ s ( M ; V ) induces an isomorphism in coho mo logy provided the ma nifold M is smo o thly contractible. F o r this purp ose w e co nsider the abelian groups (6.1) A p,q sr ( M , U ; V ) :=  f : M p +1 × M q +1 → V | f | M p +1 × U [ q ] is contin uous  . The ab elian gr oups A p,q sr ( M , U ; V ) form a first qua drant sub double complex of the double complex A p,q cr ( M , U ; V ). The rows of the do uble co mplex A ∗ , ∗ sr ( M , U ; V ) can be augmented by the complex A ∗ sr ( M , U ; V ) for the covering U a nd the columns can be augmented b y the exact complex A ∗ s ( M ; V ) of contin uous co chains: . . . . . . . . . . . . A 2 sr ( M , U ; V ) / / d v O O A 0 , 2 sr ( M , U ; V ) d h / / d v O O A 1 , 2 sr ( M , U ; V ) d h / / d v O O A 2 , 2 sr ( M , U ; V ) d h / / d v O O · · · A 1 sr ( M , U ; V ) / / d v O O A 0 , 1 sr ( M , U ; V ) d h / / d v O O A 1 , 1 sr ( M , U ; V ) d h / / d v O O A 2 , 1 sr ( M , U ; V ) d h / / d v O O · · · A 0 sr ( M , U ; V ) / / d v O O A 0 , 0 sr ( M , U ; V ) d h / / d v O O A 1 , 0 sr ( M , U ; V ) d h / / d v O O A 2 , 0 sr ( M , U ; V ) d h / / d v O O · · · A 0 s ( M ; V ) d h / / O O A 1 s ( M ; V ) d h / / O O A 2 s ( M ; V ) d h / / O O · · · W e denote the total complex of the do uble co mplex A ∗ , ∗ sr ( M , U ; V ) by T ot A ∗ , ∗ sr ( M , U ; V ). The augment ations of the rows and columns of this double complex induce mor- phisms i ∗ : A ∗ sr ( M , U ; V ) → T ot A ∗ , ∗ sr ( M , U ; V ) and j ∗ : A ∗ s ( M ; V ) → T ot A ∗ , ∗ sr ( M , U ; V ) of co chain complexes re sp e ctively . Lemma 6. 7. The morphism i ∗ : A ∗ sr ( M , U ; V ) → T ot A ∗ , ∗ sr ( M , U ; V ) induc es an isomorphi sm in c ohomolo gy. Pr o of. The row co nt raction given in the pro of of Lemma 2.1 restricts to one of the sub r ow complex A ∗ s ( M , U ; V ) ֒ → A ∗ , ∗ sr ( M , U ; V ).  R emark 6.8 . Note that this co nstruction do es not w ork for the co lumn co mplex es. 20 M. F UCHSSTEINER F o r G -inv ariant o p en coverings U of M o ne can consider the s ub do uble complex A ∗ , ∗ sr ( M , U ; V ) G of A ∗ , ∗ sr ( M , U ; V ) whose r ows ar e a ugmented by the co chain complex A ∗ sr ( M , U ; V ) G for the covering U and the co lumns ca n b e augmented by the complex A ∗ s ( M ; V ) G of smo oth equiv a riant co chains (,which is not exa c t in general). Lemma 6. 9. F or G -invaria nt c overings U of M the morphism i ∗ eq := i ∗ G induc es an isomorphism in c ohomolo gy. Pr o of. The co ntraction h ∗ ,q of the augmented rows A q cr ( M , U ; V ) ֒ → T ot A ∗ ,q sr ( M , U ; V ) defined in Eq . 2.2 is G -eq uiv ariant and thus restricts to a row co ntraction of the augmented sub-row A q sr ( M , U ; V ) G ֒ → T ot A ∗ ,q sr ( M , U ; V ) G .  So the mor phism H ( i eq ) : H cr,eq ( M , U ; V ) → H (T ot A ∗ , ∗ sr ( M , U ; V ) G ) is inv ert- ible. F o r the comp osition H ( i eq ) − 1 H ( j eq ) : H c,eq ( M ; V ) → H cr,eq ( M , U ; V ) we observe: Prop ositio n 6.10. The image j n ( f ) of a smo oth e quivariant n -c o cycle f on M in T ot A ∗ , ∗ sr ( M , U ; , V ) G is c ohomolo gous to t he image i n eq ( f ) of the e quivariant n - c o cycle f ∈ A n sr ( M , U ; V ) G in T ot A ∗ , ∗ sr ( M , U ; V ) G . Pr o of. The pro of is a nalogous to that of Prop ositio n 2.4.  Corollary 6. 11. The c omp osition H ( i eq ) − 1 H ( j eq ) : H s,eq ( M ; V ) → H sr ,eq ( M , U ; V ) is induc e d by the inclusion A ∗ s ( M , U ; V ) G ֒ → A ∗ sr ( M , U ; V ) G . Corollary 6.12. If the morphism j ∗ eq := j ∗ G : A ∗ s ( M ; V ) G → T ot A ∗ , ∗ sr ( M , U A ) G induc es a m onomorphism, epimo rphism or isomorphism in c ohomolo gy, then the inclusion A ∗ s ( M ; V ) G ֒ → A ∗ sr ( M , U ; V ) G induc es a monomorph ism, epimorphism or isomorp hism in c ohomolo gy r esp e ctively. F o r any directed sy stem { U i | i ∈ I } of o p e n coverings of M one can also consider the corresp onding augmented c olimit do uble complexes. In particular for the directed system of all open cov erings of M one obtains the double complex complex A ∗ , ∗ sg ( M ; V ) := co lim U is op en cov er of M A ∗ , ∗ sr ( M ; U ; V ) whose rows and columns are a ugmented by the colimit co mplex A ∗ sg ( M ; V ) a nd by the complex A ∗ s ( M ; V ) re s pe c tively . Lemma 6. 13. F or any dir e cte d system { U i | i ∈ I } of op en c overings of M t he morphism colim i i ∗ : colim i A ∗ sr ( M , U i ; V ) → T o t co lim i A ∗ , ∗ sr ( M , U i ; V ) induc es an isomorphi sm in c ohomolo gy. Pr o of. The pa ssage to the colimit pr eserves the exactness of the augmented row complexes (Lemma 6.7).  As a conseq ue nc e the colimit mo rphism i ∗ sg : A ∗ sg ( M ; V ) → T ot A ∗ , ∗ sg ( M ; V ) in- duces an iso morphism in cohomo logy . The colimit do uble c o mplex A ∗ , ∗ sg ( M ; V ) is a double complex o f G -mo dules and the G -equiv ariant co chains in form a sub dou- ble co mplex A ∗ , ∗ sg ( M ; V ) G , whose r ows a nd columns ar e augmented by the colimit complex A ∗ cg, eq ( M ; V ) and by the complex A ∗ s ( M ; V ) G resp ectively . Lemma 6.14. F or any dir e cte d system { U i | i ∈ I } of G -invariant op en c overings of M the morphism co lim i i ∗ eq : colim i A ∗ sr ( M , U i ; V ) G → T o t colim i A ∗ , ∗ sr ( M , U i ; V ) G induc es an isomorphism in c ohomolo gy. A SPECTRAL SEQUENCE CONNECTING CONTINUOUS WITH LOCALL Y CONTINUOUS GROUP COHOMOLOGY 21 Pr o of. The pa ssage to the colimit pr eserves the exactness of the augmented row complexes (Lemma 6.9).  Moreov er, since the op en diagonal neighbourho o ds U [ n ] in X n +1 for op en cov er- ings U o f X are cofinal in the dir ected set of all op en diago nal neighbourho o ds, we observe: Lemma 6.15. The natur al morphism of double c omplexes A ∗ , ∗ cg, eq ( X ; V ) := colim U is G -invariant op en cover of X A ∗ , ∗ sr ( X ; U ; V ) G → A ∗ sg ( X ; V ) G is a natu ra l isomorphism. Pr o of. The pro of is a nalogous to that of Prop ositio n 1.7.  As a co nsequence the colimit mor phism i ∗ cg, eq : A ∗ cg, eq ( M ; V ) → T ot A ∗ , ∗ sg ( M ; V ) G induces an iso mo rphism in cohomo logy , and the morphism H ( i cg, eq ) is inv ertible. F o r the c omp osition H ( i cg, eq ) − 1 H ( j eq ) : H c,eq ( M ; V ) → H cg, eq ( M , U ; V ) we ob- serve: Prop ositio n 6.16. The image j n ( f ) of a c ontinuous e quivariant n -c o cycle f on M in T o t A ∗ , ∗ sg ( M ; , V ) G is c ohomolo gous to the image i n cg, eq ( f ) of the e quivariant n -c o cycle f ∈ A n cg, eq ( M ; V ) in T ot A ∗ , ∗ sg ( M ; V ) G . Pr o of. The pro of is a nalogous to that of Prop ositio n 2.4.  Corollary 6. 17. The c omp osition H ( i cg, eq ) − 1 H ( j eq ) : H c,eq ( M ; V ) → H cg, eq ( M ; V ) is induc e d by the inclusion A ∗ s ( M ; V ) G ֒ → A ∗ sg ( M ; V ) G . Corollary 6. 18. If the morphism j ∗ eq := j ∗ G : A ∗ s ( M ; V ) G → T ot A ∗ , ∗ sg ( M ; V ) G induc es a m onomorphism, epimo rphism or isomorphism in c ohomolo gy, then the inclusion A ∗ s ( M ; V ) G ֒ → A ∗ cg, eq ( M ; V ) induc es a monomorphism, epimorp hism or isomorphi sm in c ohomolo gy r esp e ct ively. 7. S mooth and U -Smo oth Cochains In this Section we derive r esults for smooth transfor mation groups ( G, M ) a nd smo oth G -mo dules V , which ar e analogo us to those concerning contin uous co chains. Let ( G, M ) b e a smo oth tra nsformation group, V b e a s mo o th G -mo dule and U b e an op en cov ering of M . Prop ositio n 7.1. If the augmente d c olumn c omplexes A p s ( M ; V ) ֒ → A p, ∗ sr ( M , U ; V ) ar e exact, then t he augmente d sub c olumn c omplexes A p s ( M ; V ) G ֒ → A p, ∗ sr ( M , U ; V ) G of e quivariant c o chains ar e exact as wel l. Pr o of. The pro of is a nalogous to that of Prop ositio n 3.1.  Corollary 7. 2. If the augmente d c olumn c omplexes A p s ( M ; V ) ֒ → A p, ∗ sr ( M , U ; V ) ar e exact, then the inclusion j ∗ eq : A ∗ s ( M ; V ) G ֒ → T ot A ∗ , ∗ sr ( M , U , V ) G induc es an isomorphi sm in c ohomolo gy. Corollary 7. 3. If the augmente d c olumn c omplexes A p s ( M ; V ) ֒ → A p, ∗ sr ( M , U ; V ) ar e exact, t hen t he inclusion A ∗ s ( M ; V ) G ֒ → A ∗ sr ( M , U ; V ) G induc es an isomorphism in c ohomolo gy. 22 M. F UCHSSTEINER It r emains to show that for smo othly contractible manifolds M the colimit augmented columns A p s ( M ; V ) ֒ → A p, ∗ sg ( M ; V ) are exact. F or this purpo se we first co nsider the co chain complex ass o ciated to the cos implicial a belia n group A p, ∗ ( M ; V ) :=  f : M p +1 × M ∗ +1 → V | ∀ ~ x ′ ∈ M ∗ +1 : f ( − , ~ m ′ ) ∈ C ∞ ( M p +1 , V )  of global co chains, its subcomplex A p, ∗ sr ( M , U ; V ) and the cochain co mplexes asso- ciated to the co simplicial a b elia n gro ups A p, ∗ ( U ; V ) := { f : M p +1 × U [ ∗ ] →| ∀ ~ m ′ ∈ U [ ∗ ] : f ( − , ~ m ′ ) ∈ C ∞ ( M p +1 , V ) } and A p, ∗ s ( M , U ; V ) := C ∞ ( M p +1 × U [ ∗ ] , V ) . Restriction of glo ba l to loca l co c hains induces morphisms of co chain complexes Res p, ∗ : A p, ∗ ( M ; V ) ։ A p, ∗ ( M , U ; V ) and Res p, ∗ sr : A ∗ sr ( M , U ; V ) ։ A p, ∗ s ( M , U ; V ) int ertwining the inclus io ns of the subco mplex e s A p, ∗ sr ( M , U ; V ) ֒ → A p, ∗ ( M ; V ) and A p, ∗ s ( M , U ; V ) ֒ → A p, ∗ ( M , U ; V ), s o one obtains the following commutativ e dia gram (7.1) 0 − → ker(Res p, ∗ sr ) − → A p, ∗ sr ( M , U ; V ) − → A p, ∗ s ( M , U ; V ) − → 0 ↓ ↓ ↓ 0 − → ker(Res p, ∗ ) − → A p, ∗ ( M ; V ) − → A p, ∗ ( M , U ; V ) − → 0 of co chain complexes whose rows ar e exa ct. The k ernel ker( Res p,q ) is the sub- space o f those ( p, q )-co chains which are tr ivial on M p +1 × U [ q ]. Since these ( p, q )- co chains are smo oth on M p +1 × U [ q ] we find tha t b oth kernels coincide. W e a bbre- viate the co mplex k er(Res p, ∗ ) = ker(Res p, ∗ r c ) b y K p, ∗ and denote the cohomology groups of the complex A p, ∗ sr ( M , U ; V ) by H p, ∗ sr ( M , U ; V ), the cohomolo g y groups of the complex A p, ∗ s ( M , U ; V ) of co ntin uous co chains by H p, ∗ s ( M , U ; V ) and the coho- mology groups of the c omplex A p, ∗ ( M , U ; V ) by H p, ∗ ( M , U ; V ). Lemma 7.4. The c o chain c omplexes A p, ∗ ( M ; V ) ar e exact. Pr o of. F or a n y p oint ∗ ∈ M the homomo rphisms h p,q : A p,q ( M ; V ) → A p,q − 1 ( M ; V ) given b y h p,q ( f )( ~ x, ~ x ′ ) := f ( ~ x, ∗ , ~ x ′ ) for m a co nt raction o f the complex A p, ∗ ( M ; V ).  The mo r phism o f sho rt exact sequences of co chain c o mplexes in diagr am 3.1 gives rise to a morphism of lo ng exact coho mology sequences, in which the cohomo logy of the complex A p, ∗ ( M ; V ) is trivia l: (7.2) / / H q ( K p, ∗ )) / / H p,q sr ( M , U ; V ) / /   H p,q s ( M , U ; V ) / /   H q +1 ( K p, ∗ ) / / ∼ = / / H q ( K p, ∗ ) / / 0 / / H p,q ( M , U ; V ) ∼ = / / H q +1 ( K p, ∗ )) / / Lemma 7.5. If the inclusion A p, ∗ s ( M , U ; V ) ֒ → A p, ∗ ( M , U ; V ) induc es an isomor- phism in c ohomolo gy, then the augmente d c omplex A p s ( M ; V ) ֒ → A p, ∗ sr ( M , U ; V ) is exact. Pr o of. This is an immediate consequence of Diagram 7.2  Prop ositio n 7. 6. If t he inclusion A p, ∗ s ( M , U ; V ) ֒ → A p, ∗ ( M , U ; V ) induc es an iso- morphism in c ohomolo gy, then t he inclus ions j ∗ eq : A ∗ s ( M ; V ) G ֒ → T ot A ∗ , ∗ sr ( M , U , V ) G and A ∗ s ( M , U ; V ) G ֒ → A ∗ sr ( M , U ; V ) G also induc es an isomo rphism in c ohomolo gy. Pr o of. This follows from the preceding Lemma and Corollaries 7 .2 and 7.3.  A SPECTRAL SEQUENCE CONNECTING CONTINUOUS WITH LOCALL Y CONTINUOUS GROUP COHOMOLOGY 23 The pass age to the colimit over all op en cov erings of M yields the co rresp onding results for the co mplexes of co chains with co n tinuous g erms: Prop ositio n 7.7. If the augmente d c olumn c omplexes A p s ( M ; V ) ֒ → A p, ∗ sg ( M ; V ) ar e ex act, then the augmente d sub c olumn c omplexes A p s ( M ; V ) G ֒ → A p, ∗ sg ( M ; V ) G of e quivariant c o chains ar e exact as wel l. Pr o of. The pro of is simila r to that of P rop osition 3.1.  Corollary 7.8. If the augmente d c olumn c omplexes A p s ( M ; V ) ֒ → A p, ∗ sg ( M ; V ) ar e exact, then the inclusion j ∗ eq : A ∗ s ( M ; V ) G ֒ → T ot A ∗ , ∗ sg ( M ; V ) G induc es an isomor- phism in c ohomolo gy. Corollary 7.9. If the augmente d c olumn c omplexes A p s ( M ; V ) ֒ → A p, ∗ sg ( M ; V ) ar e exact, then the inclusion A ∗ s ( M ; V ) G ֒ → A ∗ sg ( M ; V ) G induc es an isomorphism in c ohomolo gy. R emark 7.10 . Alternatively to taking the colimit ov er all op en coverings U of M one may consider G -in v ariant op en co verings only to o bta ins the sa me res ults. (This was s hown in Pro p o sition 6 .4 and Lemmata 6.1 5.) Example 7 .11. If G = M is a Lie gro up which a cts on itself by left tr anslation a nd the augment ed c o lumns A p s ( M ; V ) ֒ → A p, ∗ sg ( M ; V ) := co lim A p, ∗ ( M , U U ; V ) (where U runs ov er all open iden tit y neig hbourho o ds in G ) are ex act, then the inclusio n A ∗ s ( M ; V ) G ֒ → A ∗ sg ( M ; V ) G induces an isomo rphism in c ohomolog y . The complex A p, ∗ ( M , U ; V ) is isomorphic to the complex A ∗ ( U ; C ( M p +1 , V )). The colimit A ∗ AS ( M ; C ( M p +1 , V )) := colim A ∗ ( U ; C ( M p +1 , V )), where U r uns over all op en coverings of M is the complex of Alex a nder-Spanier co chains on M . There- fore the colimit co mplex colim A p ( M ; A ∗ ( U ; V )) is isomorphic to the co chain com- plex A ∗ AS ( M ; C ( M p +1 , V )). A similar observ a tion can b e made for the co chain com- plex A p, ∗ s ( M , U ; V ) if the exponential law C ( M p +1 × U [ q ] , V ) ∼ = C ( M , C ( U [ q ] , V )) holds for a cofinal set of op en cov erings U of M . Passing to the c o limit in Diag ram 7.1 y ields the mor phism (7.3) 0 − → ker(R es p, ∗ sg ) − → A p, ∗ sg ( M ; V ) − → colim A p, ∗ s ( M , U ; V ) − → 0 ↓ ↓ ↓ 0 − → ker(R es p, ∗ ) − → A p, ∗ ( M ; V ) − → A ∗ AS ( M ; C p +1 ( M , V )) − → 0 of short exact sequences of co c ha in complexes. The kernel k er(Res p,q ) is the sub- space of those ( p, q )-co chains which ar e trivial o n M p +1 × U [ q ] for some op en c ov- ering U of M . Since these ( p, q )-co chains ar e con tinuous on M p +1 × U [ q ] we find that b oth kernels coincide. W e a bbr eviate the co mplex ker (Res p, ∗ ) = ker(Res p, ∗ sg ) by K p, ∗ sg and denote the cohomolo gy gr oups of the co mplex A p, ∗ sg ( M ; V ) by H p, ∗ sg ( M ; V ). The morphism o f sho r t exact sequences of co chain complexes in Diag ram 7 .3 g ives rise to a morphism of long exa ct cohomolog y sequences: (7.4) / / H q ( K p, ∗ sg )) / / H p,q sg ( M , U ; V ) / /   H q (colim A p, ∗ s ( M , U ; V ) / /   H q +1 ( K p, ∗ sg ) / / ∼ = / / H q ( K p, ∗ ) / / 0 / / H q AS ( M ; C p +1 ( M , V )) ∼ = / / H q +1 ( K p, ∗ )) / / 24 M. F UCHSSTEINER Lemma 7.12. If t he inclusion colim A p, ∗ s ( M , U ; V ) ֒ → A ∗ AS ( M ; C p +1 ( M , V )) of c o chain c omplexes induc es an isomorphism in c ohomolo gy, then the augmente d c om- plex A p s ( M ; V ) ֒ → A p, ∗ sg ( M ; V ) is exact. Pr o of. This is an immediate consequence of Diagram 7.4  Prop ositio n 7. 1 3. If the inclusion co lim A p, ∗ s ( M , U ; V ) ֒ → A ∗ AS ( M ; C ( M p +1 , V )) induc es an isomorphism in c ohomolo gy, then j ∗ eq : A ∗ s ( M ; V ) G ֒ → T o t A ∗ , ∗ sg ( M ; V ) G and A ∗ s ( M ; V ) G ֒ → A ∗ sg ( M ; V ) G also induc e an isomo rphism in c ohomolo gy. Pr o of. This follows from the preceding Lemma and Corollaries 7 .8 and 7.9.  As observed be fore (cf. Remark 7 .10) one may r estrict oneself to the directed system of G -inv ariant open c overings only to achiev e the sa me result. Th us we observe: Corollary 7.14. If G = M is a Lie gr oup which acts on its elf by left tr anslation and the inclusion colim A p, ∗ s ( M , U ; V ) ֒ → A ∗ AS ( M ; C ( M p +1 , V )) (wher e U runs over al l op en identity n eighb ourho o ds in G ) induc es an isomorphism in c ohomolo gy, then the inclusion A ∗ s ( M ; V ) G ֒ → A ∗ sg ( M ; V ) G induc es an isomorphism in c ohomolo gy as wel l. Pr o of. It has b een shown in [vE6 2] that the cohomolo gy of the colimit co chain complex co lim A ∗ ( U ; C ( M p +1 , V )) is the Alexander-Spa nier cohomolo gy o f M .  Lemma 7.15. If the manifold M is c ontr actible, t hen the c ohomolo gy of t he c om- plex colim A p, ∗ s ( M , U ; V ) is tr ivial. Pr o of. The reas oning is analo gous to tha t for the Alexander -Spanier pres heaf. The pro of [F uc10, Theo rem 2.5.2] carrie s ov er a lmost in verbatim.  Pr o of. If the manifold M is contractible, then the Alexander-Spa nier cohomol- ogy H AS ( M ; C p +1 ( M , V )) is trivia l and the coho mology o f the co chain complex colim A p, ∗ s ( M , U ; V ) is trivial by Lemma 7.15. By Prop osition 7.13 the inclusio n A ∗ s ( M ; V ) G ֒ → A ∗ sg ( M ; V ) G then induces an iso morphism in cohomology .  Corollary 7. 16. F or smo othly c ontra ctible Lie gr oups G the c ontinuous gro up c o- homolo gy is isomorphic to the c ohomolo gy of homo gene ous gr oup c o chains with c on- tinuous germ at the diag onal. References [BGN04] W. Bertram, H. Gl¨ oc kner, and K.-H. Neeb, D i ffer ent ial ca lculus over ge ner al b ase fields and rings , Exp o. Math. 22 (2004), no. 3, 213–282. 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