On the Duality between l^1-Homology and Bounded Cohomology

ON THE DUALITY BETWEEN ℓ 1 -HOMOLOGY AND BOUNDED COHOMO LOGY THEO B ¨ UHLER Abstract. W e modify t he definition of ℓ 1 -homology and ar gue why our defi- nition is more adequate than the classical one. While we cannot reconstruct the classical ℓ 1 -homology from the new definition f or v a rious reasons, we can re- construct its Hausdorffification so that no information concerning semi- norm s is lost. W e obtain an axiomatic c haracterization of our ℓ 1 -homology as a uni- v ersal δ - functor and prov e t hat it is pre-dual to our definition of bounded cohomology . W e th us answer a question r aised by L¨ oh in her thesis. More- o v er, we prov e Gr omo v’s theorem and the Matsumoto-Morita conjecture i n our cont ext. Contents 1. Int ro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Cohomolog y in Quasi-Abelian Categories . . . . . . . . . . . . . . . . 4 3. ℓ 1 -Homology and Bounded Cohomolog y . . . . . . . . . . . . . . . . . 6 4. Canonical Res olutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5. Remarks o n our Definition of ℓ 1 -Homology . . . . . . . . . . . . . . . . 11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1. Introduction Gromov intro duced ℓ 1 -homology and b ounded co homology for top olog ical spa ces in the late seven ties [Gro8 2]. The initial purp ose of these exotic (co- )homology the- ories w as to pro vide topolog ical in v ariants which cont rol the minimal v olume of a smo oth manifold which, b y definition, is an inv aria nt of the differentiable structure. One of Gromov’s de e per theore ms asserts that the bo unded cohomo logy of a co unt - able and connected CW- c o mplex is an inv ariant of its fundament al g roup. In order to mak e this statement pr ecise, he needed to introduce ℓ 1 -homology and b ounded cohomolog y for discrete groups, which apparently was developed in unpublished work of T rauber. Matsumoto-Mor ita raised the question whether the a na log of Gr omov’s theorem holds t rue for ℓ 1 -homology [MM8 5, Re ma rk 2.6]. After some flaw ed a ttempts to prov e this tr ue, s e e [Par04] a nd [Bou04], the question was finally answered affir ma- tively by L¨ oh [L¨ oh07] and the prese nt author [B ¨ uh08] indep e nde ntly . The v arian ts of ℓ 1 -homology a nd bounded c ohomolog y for groups were stud- ied by [MM85] and b ounded cohomolo gy w as given a “ functorial appr oach” b y Bro oks [Bro81], Iv ano v [I v a85, Iv a 8 8] and Nosko v [Nos90, Nos9 2], see [Gri95, Gri96] and [L¨ oh07] for fur ther r eferences. The theory was substantially improv ed and ge n- eralized to top ologic al groups by Burger and Mono d, see [BM9 9, Mon01, BM02]. While the Burger -Mono d theory pro ved to b e extremely fruitful in the co nt ext of Date : Nov em ber 3, 2018. 2000 Mathematics Subj e ct Classific ation. 18E30 (18E10, 18G60, 46M18, 20J05, 57T). 1 2 THEO B ¨ UHLER rigidity theor y , the algebr aic underpinning remained r ather undeveloped. In par- ticular, it was unknown whether bounded cohomology could b e in terpreted as a derived functor. The main purpo se o f [B ¨ uh08] is to clo se this gap and to give an int erpretation of ℓ 1 -homology and b ounded co homology in the context o f mo dern homologica l algebr a in order to b enefit from the pow er o f its pr op er languag e , i.e., category theor y . Let us turn to mathematics pro p er. Let Ban b e the additive catego ry of Banach spaces a nd contin uous linear maps. It is well-kno wn that Ban is quasi-ab elia n and that there are enough pro jectives and enough injectiv es. If G is a g roup, we denote the categor y of isometric r e presentations of G on Ba nach spaces and G -equiv a riant contin uous linear maps by G − Ban . It is easy to prov e that G − Ban is quasi-ab elia n and has enough pr o jectives and enough injectives, hence the formalism of derived categorie s allows us to derive functors defined on G − B an . In order to sp eak of homology , the theo ry of t - s tructures and their hearts is virtually fo r ced up on us. F or every quasi-a b elian categor y there a re two canonical t -structures, which we ca ll the left and righ t t -structures, see Definition 2.5. The left t -structure on D ( A op ) ∼ = ( D ( A )) op is dual to the right t -s tructure on D ( A ) in the sense of [BBD82, 1.3.2 (iii)]. In particula r the hear t C ℓ ( A op ) of the left t -str ucture o n D ( A op ) is equiv a lent to the opposite category o f the heart C r ( A ) of the righ t t -structure on D ( A ). W e write H n ℓ : D ( A ) → C ℓ ( A ) and H n r : D ( A ) → C r ( A ) for the associa ted ho mo logical functors. There is t he follo wing explicit descriptio n of t he left heart C ℓ ( A ) on D ( A ): ob jects ar e r epresented b y a monic ( A − 1 → A 0 ) in A while the morphisms are obtained from the morphisms of pairs by dividing out the homotopy equiv alence relation and in v erting quasi-isomorphisms (bica rtesian squares) formally . B y the aforementioned duality , the rig ht heart C r ( A ) has a dual de s cription. There ar e exa ct inclusion functors ι ℓ : A → C ℓ ( A ) and ι r : A → C r ( A ) given on ob jects b y ι ℓ ( A ) = (0 → A ) and ι r ( A ) = ( A → 0). The functor ι ℓ has a left a djoint q ℓ given on o b jects by q ℓ ( d : A − 1 → A 0 ) = Coker A ( A ). Similarly , ι r has a r ight adjoint q r induced by the k ernel functor in A . Let us s pe c ia lize to the category G − Ban . The trivial mo dule functor (augmen- tation) ε ( − ) : Ban → G − Ban has a left a djoint given by the co- inv a r iants ( − ) G and a right adjoint given b y the inv ariants ( − ) G . Underlying o ur definition of ℓ 1 - homology and b o unded coho mology are the deriv ed functors L − ( − ) G : D − ( G − Ban ) → D − ( Ban ) and R + ( − ) G : D + ( G − Ban ) → D + ( Ban ) . By considering G − Ban as the full sub categ ory of complexes concentrated in deg ree zero w e define for eac h M ∈ G − Ban the ℓ 1 -homolo gy o f G with co efficients in M as H ℓ 1 n ( G, M ) := H − n r ( L − ( − ) G ( M )) ∈ C r ( Ban ) and the b ounde d c ohomolo gy of G with co e fficient s in M as H n b ( G, M ) := H n ℓ ( R + ( − ) G ( M )) ∈ C ℓ ( Ban ) . Theorem. (i) The ℓ 1 -homolo gy fu n ctors assemble to a universal homolo gic al δ - functor H ℓ 1 ∗ ( G, − ) : G − Ban → C r ( Ban ) , mor e over, H ℓ 1 0 ( G, M ) = ( M G → 0) . ON THE DUALITY BETWEEN ℓ 1 -HOMOLOGY AND BOUNDED COHOMOLOGY 3 (ii) The b oun de d c ohomolo gy functors assemble to a un iversal c ohomolo gic al δ - functor H ∗ b ( G, − ) : G − Ban → C ℓ ( Ban ) , mor e over, H 0 b ( G, M ) ∼ = (0 → M G ) . Proof. A more precise statement for ℓ 1 -homology is given in Theorem 3.6 and the (en tirely dual) statemen t for b ounded cohomology is given in [B ¨ uh08, p.x iv].  Remark. While it may b e p erfectly plausible th at for dua lity reasons one should choose to use b oth the left heart and the righ t heart for defining ℓ 1 -homology a nd bo unded coho mology , it is natural to wonder whether one could in terchange “ le ft” and “right” in the definition. In brief, the answer is “yes, one could, but only at the cos t of a reasonable dua lity theor y”. W e will discuss this matter in Section 5. The dualit y functor on Ban whic h is ex act by Hahn-Banach, yields an exact duality functor on G − B an and a dualit y functor ( − ) ∗ : C r ( G − Ban ) → C ℓ ( G − Ban ) which is explicitly g iven on ob jects by ( f : A → B ) ∗ = ( f ∗ : B ∗ → A ∗ ). W e will prov e the follo wing result as Prop os ition 3 .7. Proposition. The duality functor ( − ) ∗ : C r ( G − Ban ) → C ℓ ( G − Ban ) is wel l- define d, exact and ther e is a natur al isomorphism of functors on D ( G − B an ) H n ℓ (( − ) ∗ ) ∼ =  H − n r ( − )  ∗ . One principal motiv ation for our definition is that o ne c annot interc ha ng e “left” and “rig ht ” in the previous propositio n, see Remark 3.8. Theor em 3.10 is: Theorem. The duality functor ( − ) ∗ : C r ( Ban ) → C ℓ ( Ban ) yields a natura l isomorphi sm  H ℓ 1 n ( G, M )  ∗ ∼ = H n b ( G, M ∗ ) . T o end this intro ductory sec tion, we pass fr om g roups to spaces. F ollowing Gr o- mov we asso ciate to a to po logical space X its ℓ 1 -singular chain complex C ℓ 1 ∗ ( X ) and its b ounded singular co chain complex C ∗ b ( X ), see [B ¨ uh08, p.xxi] for the pr e cise definition. W e define ℓ 1 -homology of X a s H ℓ 1 n ( X ) := H − n r ( C ℓ 1 ∗ ( X )) ∈ C r ( Ban ) and b ounded c ohomolog y a s H n b ( X ) := H n ℓ ( C ∗ b ( X )) ∈ C ℓ ( Ban ) . If X is a c o untable and connected CW-complex, let G = π 1 ( X ) b e its fundamental group. W e proved that C ℓ 1 ∗ ( e X ) considered as complex in G − Ban is a pro jective resolution o f the ground field k , see [B ¨ uh08, p.xxiii]. Dually , considered as a complex in G − Ban the bounded c o chain complex C ∗ b ( e X ) is an injective res olution o f the ground field. Our proof of these facts r elies on one o f the main r esults of Iv anov’s pro of of Gr omov’s theorem, whence the h yp othes is that X b e co untable. Since C ℓ 1 ∗ ( X ) ∼ = ( C ℓ 1 ∗ ( e X )) G ∼ = L − ( − ) G ( k ) and C ∗ b ( X ) ∼ = ( C ∗ b ( e X )) G ∼ = R + ( − ) G ( k ) we obtain the following v ariant of Gromov’s theorem a nd the Matsumoto- Mo rita conjecture: Theorem. L et X b e a c onne cte d and c ountable CW-c omplex and let G = π 1 ( X ) b e its fun damental gr oup. Th er e ar e c anonic al isomorphisms: H ℓ 1 ∗ ( X ) ∼ = H ℓ 1 ∗ ( G, k ) and H ∗ b ( X ) ∼ = H ∗ b ( G, k ) . 4 THEO B ¨ UHLER Remark. Notice that we deduced the theor em from the fact that the c omplexes computing ℓ 1 -homology and b ounded cohomolo gy ar e in v aria nts of the fundamen tal group in the derived catego ry D ( Ban ). Remark. F or co nnected (countable) CW-complexes, L¨ oh intro duce d ℓ 1 -homology and b ounded cohomology with twiste d c o efficients , see [L¨ oh0 7, p.27]. Let M be a Banach G -mo dule, equip the pro jectiv e tensor pro duct co mplex C ℓ 1 ∗ ( e X ) b ⊗ M with the diag o nal G -action and a pply the co-in v a riants. I n other words, she considers C ℓ 1 ∗ ( e X ) b ⊗ G M ∼ = k b ⊗ L − G M , where the r ight hand s ide sho w s that this complex is an in v a riant of the fundamen tal group in D ( Ban ). Similar ly , for b ounded cohomology , she considers the complex Hom G ( C ℓ 1 ∗ ( e X ) , M ) ∼ = R + Hom G ( k , M ) . Using the facts that k b ⊗ G − ∼ = ( − ) G and Hom G ( k , − ) ∼ = ( − ) G as well a s the bala nce of the derived tenso r pr o duct and der ived Hom, we immediately conclude that these complexes co mpute H ℓ 1 ∗ ( G, M ) and H ∗ b ( G, M ). Remark. The previous remar k and our duality theor em constitute a rather trivial universal c o efficient the or em fo r ℓ 1 -homology and b o unded coho mo logy with twisted co efficients of coun ta ble and connected CW-complexe s —provided that o ne is willing to accept our definition of ℓ 1 -homology as the correct one. 2. Cohomol ogy in Quasi-Abelian Ca tegories Let A b e a n ab elian ca tegory and co nsider a complex A • = ( A ′ f − → A g − → A ′′ ) in A , that is, g f = 0. Since the comp os itions Im f ֌ A → A ′′ and A ′ → A ։ Im g are b oth zero we obtain a commutativ e diagra m Im f / / ϕ / / # # # # G G G G G G G G Ker g } } } } z z z z z z z A ′ ; ; ; ; w w w w w w w w w f / / A { { { { w w w w w w w w w ! ! ! ! D D D D D D D D g / / A ′′ Coker f ψ / / / / Im g < < < < y y y y y y y and the (co )homology of A • is defined to be an y one of the isomorphic ob jects H ( A • ) ∼ = Coker ϕ ∼ = Ker ψ ∼ = Im u, where u is the morphism K er g → Coker f , see e.g. [KS0 6, p.178 ]. Recall the no tion of a quas i- ab elian c a tegory in the s ense of Y oneda [Y on6 0] (see also Pro smans [Pro00] and Schneiders [Sch99 ]): an additive category A is called quasi-ab elian if (i) every mo rphism has a kernel and a cokernel, (ii) the class o f all kernel-cokernel pair s in A is an e x act structure in the sense of Quillen [Qui73]: every kernel is the k ernel of its cokernel, the class of kernels is clo sed under comp ositio n and push-o uts alo ng arbitrar y morphisms and, dually , every cokernel is the cokernel of its kernel, the clas s o f cokernels is closed under composition a nd pull-backs along a rbitrary morphisms. ON THE DUALITY BETWEEN ℓ 1 -HOMOLOGY AND BOUNDED COHOMOLOGY 5 If A is q ua si-ab elian but not ab elian, the situation is no longer as s traightforw ard as befo re. Assume for simplicit y that A has enough pr o jectives and enough injectiv es. W e obtain the diagram Coim f / / ϕ / / # # # # Ker g } } } } z z z z z z z A ′ ; ; ; ; w w w w w w w w w f / / A { { { { w w w w w w w w w ! ! ! ! g / / A ′′ Coker f ψ / / / / Im g < < < < y y y y y y y in which the do tted arr ows are categ orical monics or epics (here we use that there are enough pr o jectives a nd enoug h injectiv es ) that may or ma y not be kernels or cokernels. Remark 2.1 (Hub er) . The morphism u : Ker g → Coker f is strict in the sense that it factors as Ker g ։ X ֌ Coker f , so tha t X ∼ = Coim u ∼ = Im u . Since f factor s over Im f and g f = 0, the morphism Im f ֌ A factors ov er Ker g ֌ A . The mo rphism v : Im f → Ker g is an admissible monic by Q uillen’s “obscure axiom”, see [Kel90, A.1, c) op ]. Let X = Coker v and for m the following push-out diagr am Ker g     / / / / X     A / / Y which b y [Kel90, A.1, 1st s tep] is bicartesian. It is ea sy to see that A → Y is the cokernel of Im f ֌ A s o that Y ∼ = Coker f (it is a genera l fact that in an exact category the push-o ut of an a dmissible epic a long an admiss ible monic yields a n admissible epic). F rom this diagram one readily reads off that Ker u ∼ = Im f and Coker u ∼ = Coim g , so X ∼ = Coim u ∼ = Im u as c la imed. Remark 2.2 . The ob ject X constructed in the prev io us r emark is a t the same time the cokernel of Im f ֌ Ker g a nd the kernel of Coker f ։ Coim g . If the quasi- ab elian ca tegory A is such that for each morphism h the morphism Coim h → Im h is ca tegorica lly monic a nd epic then it follows that Coker ϕ ∼ = X ∼ = Ker ψ . This is the case if A has enough pro jective and eno ug h injectiv e ob jects, how ever, the author do es not know whether this is true in ge ner al. Example 2.3 . Let A = Ban be the category of Bana ch spaces and co nsider the complex ℓ 1 [ i 0 ] − − → c 0 ⊕ ℓ 1 [ 0 i ] − − − → c 0 where i : ℓ 1 → c 0 is the o bvious inclusion. W e have Coim [ i 0 ] = ℓ 1 , Ker [ 0 i ] = c 0 , Coker [ i 0 ] = ℓ 1 , Im [ 0 i ] = c 0 , which shows that the dotted mor phis ms a re indee d not kernels or cokernels in general. By the theor y of t -structure s , b o th ϕ and ψ yield legitimate notions of cohomol- ogy: ϕ r epresents H 0 ℓ ( A • ) in the left hea rt C ℓ ( A ) and ψ represents H 0 r ( A • ) in the right heart C r ( A ) of the deriv ed c ategory D ( A ) if A • is co nsidered as a co mplex concentrated in degrees − 1 , 0 , 1. T o b e more spe cific, we need tw o definitions. 6 THEO B ¨ UHLER Definition 2.4 . Let A • = ( · · · − → A − 2 d − 2 − − → A − 1 d − 1 − − → A 0 d 0 − → A 1 − → · · · ) b e a complex in the quasi-ab elian categ o ry A . The left trunc ation functors are defined by τ ≤ 0 ℓ A • = ( · · · − → A − 2 d − 2 − − → A − 1 − → Ker d 0 − → 0 − → 0 − → · · · ) and τ ≥ 0 ℓ A • = ( · · · − → 0 − → Coim d − 1 − → A 0 d 0 − → A 1 − → · · · ) while the righ t t runc ation functors are given by τ ≤ 0 r A • = ( · · · − → A − 1 d − 1 − − → A 0 − → Im d 0 − → 0 − → · · · ) and τ ≥ 0 r A • = ( · · · − → 0 − → Coker d − 1 − → A 1 d 1 − → A 2 − → · · · ) . The truncatio n functors yield endofunctors of the derived ca tegory D ( A ). As us ua l, we put for n ∈ Z τ ≤ n ℓ = Σ − n ◦ τ ≤ 0 ℓ ◦ Σ n , etc. Definition 2.5 . Denote by D ≤ 0 ℓ ( A ) the essential image o f τ ≤ 0 ℓ , et c. It is not diffi- cult to pr ov e that ( D ≤ 0 ℓ ( A ) , D ≥ 0 ℓ ( A )) is a t - structur e , see [BBD82, 1 .3.1, 1.3.2 2 ], which we call the le ft t -s tr ucture. By dua lit y ( D ≤ 0 r ( A ) , D ≥ 0 r ( A )) is a t -structure as w ell and we ca ll it the right t -structure. The corr esp onding (left and right) he arts are C ℓ ( A ) = D ≤ 0 ℓ ( A ) ∩ D ≥ 0 ℓ ( A ) and C r ( A ) = D ≤ 0 r ( A ) ∩ D ≥ 0 r ( A ) , they ar e admissible ab elian sub categor ies of D ( A ). The as so ciated homo logical functors ar e H 0 ℓ = τ ≤ 0 ℓ τ ≥ 0 ℓ : D ( A ) → C ℓ ( A ) and H 0 r = τ ≤ 0 r τ ≥ 0 r . There is the following explicit description of C ℓ ( A ): ob jects are represented by a mo nic ( A − 1 ֒ → A 0 ) in A while the mor phisms are obtained fro m the morphisms of pair s by dividing out the homo topy equiv alence rela tion a nd inv er ting q uasi- isomorphisms (bicar tes ian s quares) formally , s ee [BBD82, 1.3.22], [Lau83, 1.5.7] o r [B ¨ uh08, Constr uction 2.2.1 , p.35 ]. Proposition 2.6 . The inclusion functor ι ℓ : A → C ℓ ( A ) given on obje cts by A 7→ (0 ֒ → A ) pr eserves monics, is ful ly faithful, exact and r efle cts exactness. Its image is close d under extensions in C ℓ ( A ) . It has a left adjoint q ℓ given on obje cts by Coker ( A − 1 ֒ → A 0 ) . Every exact and monic-pr eserving functor A → B to an ab elian c ate gory factors uniquely over an exact functor C ℓ ( A ) → B . Proof. This is all well-kno wn, see e.g. [B ¨ uh0 8, Chapter II I.2].  3. ℓ 1 -Homology and Boun ded Cohomology Let G be a gro up and let G − Ban b e the category of isometric representations of G on Banach spaces and G -equiv ariant b ounded linear ma ps . It is a simple consequence o f the op en mapping theorem that G − Ban is quasi-ab elian. Not a tion 3.1 . Let ℓ 1 ( G ) b e the Bana ch group algebr a and let E b e a Ba nach space. The induc e d Banach G -mo dule is ↑ E = ℓ 1 ( G ) b ⊗ E ∼ = ℓ 1 ( G, E ) with the left G -a c tio n on the facto r ℓ 1 ( G ). The c oinduc e d Banach G -mo dule is ⇑ E := Hom Ban ( ℓ 1 ( G ) , E ) ∼ = ℓ ∞ ( G, E ) with the action co ming from the righ t action of G on ℓ 1 ( G ). ON THE DUALITY BETWEEN ℓ 1 -HOMOLOGY AND BOUNDED COHOMOLOGY 7 Not a tion 3.2 . Let M ∈ G − Ban b e a Banach G -mo dule. The mo dule of c oinvari- ants of M is the Ba nach space M G = M / span { m − g m : m ∈ M , g ∈ G } and the mo dule of invariants is the Banach space M G = { m ∈ M : g m = m for all g ∈ G } . A t the heart of the homolog ic al algebr a of ℓ 1 -homology and b ounded cohomolo gy is the following simple result which is pro ved by dir ect insp ectio n: Theorem 3.3 (F undamental Adjunctions [B¨ uh0 8, p.xviii]) . Le t ↓ : G − Ban → Ban b e the for getful fun ctor and let ε ( − ) : Ban → G − Ban b e the trivial mo dule functor. Ther e ar e two adjoint triples of funct ors G − Ban ↓   Ban ⇑ b b ↑ < < and G − Ban ( − ) G # # ( − ) G { { Ban ε ( − ) O O that is to say ↑ is left adjoint t o ↓ and ↓ is left adjoint to ⇑ , etc. The for getful functor, induction, c oinduction ar e al l exact as wel l as the t rivial mo dule functor.  The most important co nsequence for the pr esent work is: Corollar y 3.4 ([B ¨ uh08, p.xviii]) . Ther e ar e enough pr oje ctives a nd enough inje c- tives in G − Ban .  This allows us to co nsider the deriv ed fun ctors L − ( − ) G : D − ( G − Ban ) → D − ( Ban ) and R + ( − ) G : D + ( G − Ban ) → D + ( Ban ) which underlie ℓ 1 -homology and bounded c o homology . Definition 3.5 . Let M ∈ G − Ban . W e define ℓ 1 -homolo gy as H ℓ 1 n ( G, M ) := H − n r ( L − ( − ) G ( M )) and b ounde d c ohomolo gy as H n b ( G, M ) := H n ℓ ( R + ( − ) G ( M )) . Theorem 3 .6 . Up to unique isomorphism of δ -functors ther e is a unique family of functors H ℓ 1 n ( G, − ) : G − Ban → C r ( Ban ) , n ∈ Z , having the f ol lowing pr op erties: (i) (Normalization) H ℓ 1 0 ( G, M ) = ( M G → 0) for al l M ∈ G − Ban . (ii) (V anishing) H ℓ 1 n ( G, P ) = 0 for al l pr oje ctive obje cts P ∈ G − Ban and al l n > 0 . (iii) (L ong exact se quen c e) Asso ciate d to e ach short exact se quenc e M ′ ֌ M ։ M ′′ in G − Ban ther e ar e morphisms δ n +1 : H ℓ 1 n +1 ( G, M ′′ ) → H ℓ 1 n ( G, M ′ ) de- p en ding natur al ly on the se quenc e and fitting into a long exact se quenc e · · · δ n +1 − − − → H ℓ 1 n ( G, M ′ ) − → H ℓ 1 n ( G, M ) − → H ℓ 1 n ( G, M ′′ ) δ n − → H ℓ 1 n − 1 ( G, M ′ ) − → · · · in C r ( Ban ) . 8 THEO B ¨ UHLER On the P roof. This follows from dua lizing the pro of o f the theo rem on pa ge xiv of [B¨ uh08]. Notice that ( − ) G and ι r : Ban → C r ( Ban ) , E 7→ ( E → 0) b oth hav e a right adjoint. An ex is tence pro of is also given in Sectio n 4.  Now consider the duality functor ( − ) ∗ : G − Ban → G − Ban and recall that it is exact, hence it extends to the der ived catego ry D ( G − Ban ). It induces a (contra v ar ia nt) duality functor ( − ) ∗ : C r ( G − Ban ) → C ℓ ( G − Ban ) which is explicitly given on ob jects by ( e : A → B ) ∗ = ( e ∗ : B ∗ → A ∗ ). Proposition 3.7 . The du ality fun ctor ( − ) ∗ : C r ( G − Ban ) → C ℓ ( G − Ban ) is wel l-define d, exact and ther e is a natura l isomorphism of functors on D ( G − Ban ) H n ℓ (( − ) ∗ ) ∼ =  H − n r ( − )  ∗ . Proof. First, the dua lity functor C r ( G − Ban ) → C ℓ ( G − Ban ) is well-defined since the dua lity functor on G − Ban (i) maps epics (morphisms with dense range) to monics (injective morphisms) by [Rud91, 4.12 , Co rollar ies (b), p.99], (ii) prese r ves the homotopy equiv alence r elation since it is additiv e, (iii) prese r ves bica rtesian s quares b ecause it is exact. Let us prove that the duality functor C r ( G − Ban ) → C ℓ ( G − Ban ) is exact. Points (i) and (iii) yield that the duality functor ( − ) ∗ : G − Ban op → G − Ban is exact a nd preserves monics. The s ame holds true for ι ℓ : G − Ban → C ℓ ( G − Ban ), he nc e also for the comp os ition F = ι ℓ ◦ ( − ) ∗ . By [B ¨ uh08, 2.2.3, p.37] the universal prop er t y of the inclusio n functor ι ℓ : G − Ban op → C ℓ ( G − Ban op ) yields a uniq ue exa ct prolonga tion e F : C ℓ ( G − Ban op ) → C ℓ ( G − Ban ). The construction of e F given in [B ¨ uh08, p.40] together with [B¨ uh08, 2 .2.8, p.39 ] yield that e F ( f : A → B ) = ( f ∗ : B ∗ → A ∗ ) . so that e F coincides with the ab ov e description o f the duality functor under the equiv a lence C r ( G − Ban ) op ∼ = C ℓ ( G − Ban op ). In order to see that there is a na tural isomorphism H n ℓ (( − ) ∗ ) ∼ = ( H − n r ( − )) ∗ , it suffices to no tice that for a morphism f of G − Ban ther e are natur al isomorphisms (Coker f ) ∗ ∼ = Ker ( f ∗ ) and (Im f ) ∗ ∼ = Coim ( f ∗ ) , which is a stra ightforw ar d conseq uenc e of [Rud91, 4.12, Theor e m, p.99].  Remark 3.8 . The dual of a monic in G − Ban is not in gener al an epic, the rang e is weak ∗ -dense by [Rud9 1, 4.1 2, Corollar ies, (c), p.99] but not necessar ily norm-dense: consider for instance the inclusion ℓ 1 ֒ → c 0 whose dual is the inclusio n ℓ 1 ֒ → ℓ ∞ the range of whic h is clearly not norm-dense. It follo ws in particular that there is no duality functor C ℓ ( G − Ban ) → C r ( G − Ban ) a s c o nstructed abov e. In a similar vein, (Coim f ) ∗ do es not in ge ner al coincide with Im ( f ∗ ) but ra ther with its weak ∗ -closure and (K er f ) ∗ is isomorphic to the c o domain mo dulo the weak ∗ -closure of Im ( f ∗ ), hence it ma y be distinct from Coker ( f ∗ ). Recall the main prop er ties of the dua lity functor on G − Ban : Proposition 3.9 ([B ¨ uh08, p.65]) . The duality functor ( − ) ∗ : G − Ban → G − Ban is exact, r efle cts exactness and sends pr oje ct ive obje cts t o inje ctive obje ct s . Mor e- over, ther e is a natur al isomorphism ( − ) ∗ ◦ ( − ) G ∼ = ( − ) G ◦ ( − ) ∗ .  ON THE DUALITY BETWEEN ℓ 1 -HOMOLOGY AND BOUNDED COHOMOLOGY 9 Theorem 3 .10 . The duality functor ( − ) ∗ : C r ( Ban ) → C ℓ ( Ban ) yields a natu r al isomorphi sm  H ℓ 1 n ( G, M )  ∗ ∼ = H n b ( G, M ∗ ) . Proof. T o compute H ℓ 1 n ( G, M ) c ho o se a pr o jective r esolution P • ։ M , apply the coinv a r iants ( − ) G to P • and then the rig ht coho mology functor H − n r to the resulting complex . Now the t wo pr e vious pro po sitions give na tural isomorphisms ( H − n r (( P • ) G )) ∗ ∼ = H n ℓ ((( P • ) G ) ∗ ) ∼ = H n ℓ ((( P • ) ∗ ) G ) and it remains to notice that M ∗ ֌ ( P • ) ∗ is a n injectiv e resolution of M ∗ , so that the rig ht hand side computes b ounded cohomology in degr ee n .  4. Canonical Resolutions Using the cano nical resolution as s o ciated to the induction comonad we give a relatively ele mentary pro o f o f the existence o f the ℓ 1 -homology functors as describ ed in Theorem 3.6. In the next section w e will mak e us e of this construction in order to relate o ur theory to the classical one. Recall the fundamental adjunction o f induction ↑ = ℓ 1 ( G ) b ⊗ − : B an → G − Ban to the forgetful functor ↓ : G − Ban → Ban , see Theo r em 3 .3. The latter functor is obviously exact while the former is exact since ℓ 1 ( G ) is pro jective and hence flat as a Banach space. Every adjo int pair of functors gives rise to a c omonad and a monad, see [W ei94 , 8.6, 8 .7], as f ollows: Let L : A ↔ B : R be an adjoint pa ir a nd let ε : LR ⇒ id B and η : id A ⇒ R L be the adjunction mor phisms. W rite ⊥ = LR and ⊤ = R L , as well as δ B = L ( η RB ) and µ A = R ( ε LA ), it is then a simple fac t that ( ⊥ , ε, µ ) is a co monad and ( ⊤ , η , δ ) is a monad, see [W ei94 , 8.6 .2]. The simplicia l ob ject a sso ciated to the comona d ⊥ is describ ed in [W ei94, 8 .6.4], it giv es r ise to a simplicial r e solution ⊥ ∗ B → B , where ⊥ n B := ( ⊥ ) n +1 B . Suppo se A and B are a dditive. By taking the alternating sum o f the face maps one o btains a complex which we still denote b y ⊥ ∗ B , and it yields the c anonic al r esolution ⊥ ∗ B → B . This parla nce is ju stified since it is well-known and easy to chec k [W ei94, 8.6 .8, 8.6.10] that R ( ⊥ ∗ B ) → R ( B ) as well a s ⊥ ∗ L ( A ) → L ( A ) are chain homotopy eq uiv a le nc e s for a ll B ∈ B and all A ∈ A . W e apply this to the induction c omonad ⊥ = ↑↓ and obtain in particular for each M ∈ G − Ban the c anonic al r esolution ⊥ ∗ M → M , which has the prop er ty that for all M ∈ G − Ban and all E ∈ Ban th e complexes ↓ ( · · · → ⊥ 1 M → ⊥ 0 M → M ) and · · · → ⊥ 1 ↑ E → ⊥ 0 ↑ E → ↑ E are split ex act in B an and G − Ban , resp ectively . Since ⊥ is ex a ct, we obtain for ea ch shor t exact sequence M ′ ֌ M ։ M ′′ a sho r t exact sequence of complexes ⊥ ∗ M ′ ֌ ⊥ ∗ M ։ ⊥ ∗ M ′′ in Ch ≤ 0 ( G − Ban ). W riting temporarily ⊥ − 1 = id G − Ban we hav e for all n ≥ 0 ( ⊥ n M ) G ∼ = ℓ 1 ( G ) b ⊗ G ε ( ↓⊥ n − 1 M ) ∼ = ↓⊥ n − 1 M , so we g et a short exact sequence of complexes in Ch ≤ 0 ( Ban ) ( ⊥ ∗ M ′ ) G ֌ ( ⊥ ∗ M ) G ։ ( ⊥ ∗ M ′′ ) G . 10 THEO B ¨ UHLER Since the inclusion functor ι r : Ban → C r ( Ban ) is exa ct, the snake lemma provides us with a long exact sequenc e · · · → H n r ( ι r ( ⊥ ∗ M ′ ) G ) → H n r ( ι r ( ⊥ ∗ M ) G ) → H n r ( ι r ( ⊥ ∗ M ′′ ) G ) → H n +1 r ( · · · ) → · · · which is obviously natura l in the short exact seq uence M ′ ֌ M ։ M ′′ so that w e hav e constructed a δ -functor. Because the c o mplexes inv o lved are co nc e n trated in non-p ositive degr ees and bec ause ι r and ( − ) G are left adjoints a nd hence comm ute with taking cokernels, we hav e that H 0 r ( ι r (( ⊥ ∗ M ) G )) = Co ker ( ι r (( ⊥ 1 M → ⊥ 0 M ) G )) ∼ = ι r ◦ ( − ) G ◦ Co ker ( ⊥ 1 M → ⊥ 0 M ) ∼ = ( M G → 0) . F or eac h Banac h space E the s e quence · · · → ⊥ 1 ↑ E → ⊥ 0 ↑ E → ↑ E is split e x act, s o the map ( ⊥ ∗ ↑ E ) G → ( ↑ E ) G ∼ = E is a quasi-isomorphism and hence the cohomology of the complex ι r ( ⊥ ∗ ↑ E ) G v a n- ishes outside deg ree zero. Finally , the morphism ↓ ε M : ↓⊥ M → ↓ M is a split epic for each M ∈ G − Ban , hence ⊥ M → M is an admissible epic and it fo llows that every pro jective P ∈ G − Ban is a direct summand of ⊥ P = ↑↓ P . Co ns equently , our δ -functor v anishes on pro jectiv es outside deg ree zer o a nd we conclude from Theorem 3.6 that: Theorem 4 .1 . Ther e is a c anonic al isomorphism H ℓ 1 ∗ ( G, − ) ∼ = H −∗ r ( ι r ( ⊥ ∗ ( − )) G ) . Remark 4.2 . The co mplex ⊥ ∗ M is of cour se nothing but the bar reso lution as given e.g. in [L¨ oh0 7, (2.13), p.20 ]. Call a Ba nach G -mo dule induc e d if it is of the form ↑ E for some E ∈ Ban . By [W ei94, 8.6.7, E xercise 8.6 .3] the direct summands of induced modules are precisely the ⊥ -pro jective ob jects, or, equiv alently , the pro jective ob jects with resp ect to the exact structure E G rel on G − Ban co ns isting of short sequences σ such that ↓ σ is split e xact. This notion is clo sely rela ted to r elative pr oje ctivity as defined in [L¨ o h07, (A.1), p.10 4] but it is somewhat les s restric tive. In particular w e have shown: Corollar y 4.3 . Every ⊥ -pr oje ctive obje ct is H ℓ 1 ∗ ( G, − ) -acyclic.  Remark 4.4 . The acyclicity of ⊥ -pro jective ob jects implies b y dimension-shifting that one may compute ℓ 1 -cohomolo gy with coefficie nt s in M using any resolution P • ։ M with ⊥ -pro jective comp onents. Requiring that ։ is more than just a quasi- isomorphism (e.g., a s t r ong r esolution ) is o nly neces sary if one is concer ned with ensuring that the resolution can b e used to compute the canonical semi-no rms. Remark 4.5 . The constructio n given here s hows in particular that ℓ 1 -homology is the derive d functor of t he induct ion c omonad with c o efficient fun ctor ι r in the sense of Barr a nd Beck, s e e e.g. [W ei94 , 8.7.1]. Remark 4.6 . Putting ⊤ = ⇑↓ we obtain the c oinduction monad which we will not discuss further because the a rguments given in this section ar e straightf orward to dualize. ON THE DUALITY BETWEEN ℓ 1 -HOMOLOGY AND BOUNDED COHOMOLOGY 11 5. Remarks on our Definition of ℓ 1 -Homology Our fir st and main mo tiv atio n fo r our definition of ℓ 1 -homology is pure ly utilitar- ian in nature: we want to hav e a smo o th duality betw een ℓ 1 -homology and b ounded cohomolog y in order to sa ve a lot of work. Second, we wan t to show that no information conce r ning semi-norms is lo st: F or this we need to desc r ib e the classical ℓ 1 -homology as defined e.g. in [L¨ oh07]. An o b ject o f C ℓ ( Ban ) can be co ns idered as a mor phism of the catego r y Csn of c omplete seminorme d sp ac es and contin uous linear maps. T aking the cokernel in Csn gives a rea lization functor real : C ℓ ( Ban ) → Csn which is exa ct in the sense that it transfor ms exact sequence s to sequences in Csn whose under lying sequenc e of vector spac es is exa ct, see [B ¨ uh08, p.xv, Lemma]. It is thus easy to see that ℓ 1 -homology as defined e.g. in [L¨ oh07] coincides with H ℓ 1 n ( G, M ) = real H − n ℓ ( ⊥ ∗ M ) Notice that w e use the left homology functor H ∗ ℓ instead of the r ight one. W e hav e H n b ( G, M ) ∼ = real H n ℓ ( ⊤ ∗ M ) ∼ = real H n b ( G, M ) . The c omplications in volv ed in the developmen t o f a r easonable dualit y betw een the t wo class ical theories is discussed a t length in [L¨ oh0 7, Chapter 3 ]. Recall that the inclusion functor ι ℓ : Ban → C ℓ ( Ban ) has a left adjoint q ℓ defined on ob jects b y taking the co kernel in Ban , see Pr op osition 2.6. Dually , the inclusion functor ι r has a r ight adjoint g iven by taking the kernel in Ban . Remark 2.2 implies that there is a natural isomorphism q ℓ H n ℓ ∼ = q r H n r on the derived ca teg ory D ( Ban ). F rom a ll this we deduce easily: Theorem 5.1 . The funct or q r ◦ H ℓ 1 ∗ ( G, − ) c oincides with Hausdorffific ation of classic al ℓ 1 -homolo gy H ℓ 1 ∗ ( G, − ) . Similarly, q ℓ ◦ H ∗ b ( G, − ) c oincides with the Haus- dorffific ation of classic al b ounde d c ohomolo gy H ∗ b ( G, − ) .  Remark 5.2 . The main interest of the theorem is of cours e that it shows tha t as far as semi-norms a r e concerned o ne may as w e ll w o rk with our version of the ℓ 1 -homology functors since Hausdor ffification only consists of quotienting out the space of vectors of semi-norm zero in H ℓ 1 n ( G, M ). Remark 5.3 . It is impo rtant to no tice that Hausdorffification as well as q r bo th fail to be “exact”, so that the Hausdorffified long exact sequence of ℓ 1 -homology and b ounded c ohomolog y is no long er exa c t in g eneral. Remark 5.4 . The dua l of the kernel o f f : M → N in Ban is not the cokernel o f the dua l ma p f ∗ : N ∗ → M ∗ in g eneral but the quotient of M ∗ by the weak ∗ -closure of the r ange of f ∗ . So th ere is only a natural quotien t map q ℓ ◦ H ∗ b ( G, M ∗ ) ։ ( q r ◦ H ℓ 1 ∗ ( G, M )) ∗ as is well-known in the cla s sical co nt e xt, see e.g. [MM85, p.540 ]. This map is of course not an iso morphism in general. References [BBD82] A . A. Be ˘ ılinson, J. Bernstein, and P . Deligne, F aisc e aux p e rvers , Analysis and topology on singular spaces, I (Lumin y , 1981) , A st ´ erisque, v ol. 100, So c. Math. F r ance, Paris, 1982, pp. 5–171. MR751966 (86g:32015) [BM99] M. Burger and N. Mono d, Bounde d c ohomolo g y of lattic e s in higher r ank Lie gr oups , J. Eur. M ath. Soc. (JEMS) 1 (1999), no. 2, 199–235. MR1694584 (2000g:5705 8a) [BM02] , Continuous b ounde d c ohomolo gy and applic ations t o rigidity the ory , Geom. F unct. Anal. 12 (2002), no. 2, 219–280. MR1911660 (2003d:53065a) 12 THEO B ¨ UHLER [Bou04] Abdesselam Bouarich , Th´ eor ` emes de Zilb er-Eilemb er g [si c!] et de Br own en hom olo gie ℓ 1 , Proy ecciones 23 (2004), no. 2, 151–186. MR2142264 (2006h:5500 8) [Bro81] Rob ert Brooks, Some r emarks on b ounde d co homolo gy , Ri emann surfaces and related topics: Pr oceedings of the 1978 Ston y Br ook Confer ence (State Univ. New Y ork, Ston y Bro ok, N.Y ., 1978) (Princeton, N.J.), A nn. of M ath. Stud., vol. 97, Princeton Univ. Press, 1981, pp. 53–63. MR624804 (83a:57038) [B ¨ uh08] Theo B ¨ uhler, O n the algebr aic foundation of b ounde d c ohomolo gy , Ph.D. thesis, ETH Z¨ urich, 2008. [Gri95] R. I. Grigorc huk, Some r esults on b ounde d c ohomolo gy , Com bi natorial and geometric group theory (Edin burgh, 1993), London Math. So c. Lecture Note Ser., vol. 204, Cam- bridge Univ. Press, Cambridge, 1995, pp. 111–163 . MR1320279 (96j:20073) [Gri96] , Bound ed c ohomolo g y of gr oup c onstructions , Mat. Zamet ki 59 (19 96), no. 4 , 546–550. MR1445197 (98f:20037) [Gro82] Michae l Gromov, V olume and b ounde d c ohomolo gy , Inst. Hautes ´ Etudes Sci. Publ. Math. (1982), no. 56, 5–99 (1983). MR686042 (84h:53053) [Iv a85] N. V. Iv anov, F oundations of the the ory of b ounde d c ohomolo g y , Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklo v. (LOMI) 143 (1985) , 69–109, 177–178 . M R806562 (87b:53070 ) [Iv a88] , The sec ond b ounde d c ohomolo gy gr oup , Zap . Nauchn . Sem. Leningrad. Otde l . Mat. Inst. Steklo v. (LOMI) 167 (1988), no. Issled. T opol. 6, 117–120, 191. M R964260 (90a:55015 ) [Kel90] Bernhard Keller, Chain c omplexes and stable cate gories , Manuscripta Math. 67 (1990), no. 4, 379–417. MR 1052551 (91h:18006) [KS06] Masak i Kashiwara and Pierr e Sc hapira, Cate gories and she aves , Grundlehren der Math- ematisc hen Wis sensc haften [ F undamen tal Principles of M athematical Sciences], vol. 332, Springer-V erlag, Berl in, 2006. MR2182076 (2006k:18001) [Lau83] G. Laumon, Sur la c at´ egorie d´ eriv´ ee des D -mo dules filtr´ es , A lgebraic geometry (T oky o/Kyoto, 1982), Lecture Notes in Math., v ol . 1016, Springer, Ber lin, 1983, pp. 151– 237. MR726427 (85d:32022) [L¨ oh07] Clara L¨ oh, ℓ 1 -Homolo gy and Simp lic ial Volume , Ph.D. thesis, W estf¨ alis c he Wilhelms- Unive rsit¨ at M ¨ unster, 2007. [MM85] Shigenori Matsumoto and Shigeyuki Morita, Bounde d c ohomolo g y of c ertain gr oups of home omorphisms , Pro c. Amer. Math. Soc. 94 (19 85), no. 3, 539–544. MR787909 (87e:55006 ) [Mon01] Nicolas Mono d, Continuous b ounde d c ohomolo gy of lo c al ly co mp act gr oups , Lec- ture Notes in Mathematics, v ol. 1758, Springer-V erlag, Berlin, 2001. MR1840942 (2002h:461 21) [Nos90] G. A. Nosko v, Bounde d co homolo gy of discr ete gr oups with c o efficient s , Algebra i Analiz 2 (1990), no. 5, 146–164. M R1086449 (92b:57005) [Nos92] , The Ho chschild-Serr e sp e ctr al se quenc e for b ounde d c ohomolo gy , Pro ceedings of the In ternational Conference on Algebra, Pa rt 1 (No vosibirsk, 1989) (Pro vidence, RI), Contemp. Math., vol. 131, Amer. Math. So c., 1992, pp. 613–629. MR1175809 (93g:20096 ) [Par04] HeeSook Pa rk, F oundations of the t he ory of l 1 homolo gy , J. Kor ean M ath. So c. 4 1 (2004), no. 4, 591–615. MR2068142 (2005c:55011) [Pro00] F abienne Prosmans, D erive d c ate gories for functional analysis , Publ. Res. Inst. M ath. Sci. 36 (2000), no. 1, 19–83. MR 1749013 (2001g:46156) [Qui73] Daniel Quillen, Higher algebr aic K - the ory. I , Algebraic K -theory , I: Higher K -theories (Proc. Conf., Battelle Memori al Inst., Seattle, W ash., 1972), Spri nger, Berlin, 1973, pp. 85–147. Lecture Notes in Math., V ol. 341. MR0338129 (49 #2895) [Rud91] W alter Rudin, F unctional analysis , Internat ional Series in Pure and Applied Mathemat- ics, McGraw-Hill Inc., New Y ork, 1991. MR1157815 (92k:46001) [Sc h99] Jean-Pierre Schne i ders, Quasi-ab elian c ate gories and she aves , M´ em. Soc. Math. F r. (N.S.) (1999), no. 76, vi+134. MR 1779315 (2001i:18023 ) [W ei94] Charles A . W eib el, An intr o duction to homolo gic al algebr a , Cambridge Studies in Adv anced Mathematics, v ol. 38, Camb ridge University Press, Cambridge, 1994. MR1269324 (95f:18001) [Y on60] Nobuo Y oneda, On Ext and exact se quences , J. F ac. Sci. Univ. T okyo Sect. I 8 (1960), 507–576 (1960). M R0225854 (37 #1445) Dep ar tment of Ma thematic s, ETH Z ¨ urich, Switzerland E-mail addr ess : theo@math.e thz.ch