The symplectic geometry of cotangent bundles from a categorical viewpoint

THE SYMPLECTIC GEOMETR Y OF COT ANGENT BUNDLES FR OM A C A TEGORIC AL VIEWPOINT KENJI FUKA Y A, P AUL SEIDEL, IV AN SMITH Abstra ct. W e describe v arious a pproaches to understanding F uka y a categories of cotangen t bu ndles. All of the approaches rely on in tro- ducing a suitable class of noncompact Lagrangi an submanifolds. W e review the work of Nadler-Zaslo w [30 , 29] and the aut h ors [15], b efore discussing a new approach using family Flo er cohomology [11] and the “wrapped F uk ay a ca tegory”. The latter, i nspired by Viterb o’s symplec- tic homology , emphasises t h e connection t o loop spaces, h ence seems particularly suitable when trying to extend the existing th eory b eyond the simply-connected case. A classical p roblem in symplectic topology is to d escrib e exact Lagrangian submanifolds inside a cotangen t bundle. The main conjecture, u sually at- tributed to Arn ol’d [4], asserts that any (compact) submanifold of this kind should b e Hamiltonian isotopic to the zero-sectio n. In this sharp form, the result is kno wn only for S 2 and RP 2 , and the pr o of uses metho ds whic h are sp ecifically four -dimensional (b oth cases are d ue to Hind [17]; concerning the state of the art f or surfaces of genus > 0, see [18]). In higher dimen- sions, w ork has concen trated on trying to establish top ologi cal r estrictions on exact Lagrangian sub m anifolds. T h ere are many resu lts deali ng with as- sorted partial asp ects of this question ([26, 37 , 38, 7 , 35] and others), using a v ariet y of tec h niques. Qu ite recen tly , more cate gorical metho d s h a v e b een added to the toolkit, and these ha v e led to a resu lt co vering a fairly general situation. The basic statemen t (wh ic h we will later generalise somewhat) is as follo ws: Theorem 0.1 (Nadler, F uk a y a-Seidel-Smith) . L et Z b e a close d, simply c onne cte d manifold which is spin, and M = T ∗ Z its c otangent b und le. Sup- p ose that L ⊂ M is an exact close d L agr angian submanifold, which is also spin, and additional ly has vanishing Maslov class. Then: (i) the pr oje ction L ֒ → M → Z has de gr e e ± 1 ; (ii) pul lb ack by the pr oje ction is an isomor- phism H ∗ ( Z ; K ) ∼ = H ∗ ( L ; K ) for any c o efficient field K ; and (iii) given any two L agr angian submanifolds L 0 , L 1 ⊂ M with these pr op erties, which me et tr ansversal ly, one has | L 0 ∩ L 1 | ≥ dim H ∗ ( Z ; K ) . 1 2 KENJI FUKA Y A, P AUL SEIDEL, IV AN SMITH Remark ably , three wa ys of arriving at this goal ha ve emerged, wh ic h are essen tially ind ep endent of eac h other, but share a b asic philosophical out- lo ok. On e pro of is due to Nadler [29], building on earlier work of Nadler and Zaslo w [30] (the result in [29] is form ulated f or K = C , b ut it seems that the pro of go es th rough for an y K ). Another one is giv en in [15], and in v olv es, among other things, to ols from [36] (for tec hn ical reasons, this actually w orks only for char( K ) 6 = 2). The thir d one, whic h is collabora- tiv e work of the three authors of this pap er, is not complete at the time of writing, mostly b ecause it relies on ongoing dev elopmen ts in general Flo er homology theory . In spite of th is, we included a description of it, to round off the o v erall picture. The b est starting p oint ma y actually b e the end of th e pro of, which can b e tak en to b e roughly the same in all three cases. Let L b e as in Theorem 0.1. The Flo er cohomology groups H F ∗ ( T ∗ x , L ), where T ∗ x ⊂ M is the cotangen t fibre at some p oin t x ∈ Z , form a fl at bundle of Z -graded v ector spaces o v er Z , whic h we denote b y E L . There is a sp ectral sequence con v erging to H F ∗ ( L, L ) ∼ = H ∗ ( L ; K ), w hose E 2 page is (0.1) E r s 2 = H r ( Z ; E nd s ( E L )) , E nd ∗ ( E L ) = H om ∗ ( E L , E L ) b eing the graded en d omorphism ve ctor bun- dle. Beca use of the assump tion of simp le connectivit y of Z , E L is actually trivial, so the E 2 page is a “b o x” H ∗ ( Z ) ⊗ E nd ( H F ∗ ( T ∗ x , L )). Th e E 2 lev el differen tial go es from ( r , s ) to ( r + 2 , s − 1): ❣ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❣ and similarly for the higher pages. Hence, the b ottom left and top righ t cor- ners of the b ox n ecessarily sur viv e to E ∞ . Just by lo oking at their d egrees, it follo ws that H F ∗ ( T ∗ x , L ) ∼ = K m ust b e one-dimensional (and, w e ma y assume after c hanging the gradin g of L , concent rated in degree 0). Giv en that, the sp ectral sequence degenerat es, yielding H ∗ ( L ; K ) ∼ = H ∗ ( Z ; K ). On the other hand, we ha v e also s h o wn that the pro jection L → Z h as degree ± 1 = ± χ ( H F ∗ ( T ∗ x , L )). This means that the in d uced m ap on cohomology is in jectiv e, hence necessarily an isomorphism. Finally , there is a similar sp ectral sequence for a pair of Lagrangian submanifolds ( L 0 , L 1 ), whic h can b e used to deriv e th e last part of Th eorem 0.1. COT ANGENT BUNDLES 3 A t this p oin t, we already need to insert a cautionary note. Namely , the ap- proac h in [15] leads to a sp ectral s equ en ce whic h only appro ximates the one in (0.1) (the E 1 term is an analogue of the expression ab ov e, replacing the cohomolog y of Z b y its Morse co chain complex, and the differen tial is only partially kno wn ). In spite of this hand icap, a slight ly mo dified versio n of the previous argumen t can b e carried out successfully . T he other tw o strategies ([29] and the unpublished approac h) do not su ffer from this deficiency , since they directly pro du ce (0.1 ) in the form stated ab o v e. F rom the description we h a v e just giv en, one can already infer one basic philosophical p oint , namely the in terpretation of Lagrangia n submanifolds in M as (some kind of ) shea v es on the base Z . T his can b e view ed as a limit of standard ideas ab out Lagrangian torus fib rations in m irror symmetry [12, 25 ], where the v olume of the tori b ecomes in fi nite (there is no algebro- geometric mirror of M in the usual sense, so w e b orrow only h alf of the mirror symmetry argument). The main problem is to pro v e that the sheaf-theoret ic ob jects accuratel y reflect the Flo er cohomology groups of Lagrangian sub- manifolds, hence in particular repro duce H F ∗ ( L, L ) ∼ = H ∗ ( L ; K ). In formally sp eaking, this is ensu red b y pro viding a suitable “resolution of the diago nal” in the F uk a y a categ ory of M , whic h r educes the questio n to one ab out cotan- gen t fib res L = T ∗ x . In saying that, we ha v e implicitly already int ro d u ced an enlargemen t of the ordinary F uk a y a categ ory , namely one which allo ws noncompact Lagrangian submanifolds. There are sev eral p ossible w a ys of treating su c h su bmanifolds, leading to categories with sub s tantia lly differen t prop erties. This is where the three approac hes dive rge: Characteristic cycles. [30] considers a class of Lagrangian su bmanifolds whic h, at in finit y , are in v arian t und er rescaling of the cotangen t fibres (or more generally , asymptotically inv ariant ). Intersecti ons at infinit y are dealt with b y small p erturbations (in a distinguish ed direction giv en by the nor- malized geod esic flo w; this requires the choi ce of a real analytic structure on Z ). An imp ortant source of inspiration is Kashiw ara’s construction [22] of charac teristic cycles for constru ctible shea v es on Z ; and indeed, Nadler pro v es that, once deriv ed, the resu lting v ersion of the F uk a y a categ ory is equiv alen t to the constructible deriv ed category . (A similar p oin t of view w as tak en in earlier pap ers of Kasturirangan and Oh [23, 24, 3 1]). Generally sp eaking, to get a finite resolution of the d iagonal, this category has to b e mo dified further, b y restricting the b ehavi our at infin it y; ho w ev er, if one is only in terested in applications to closed Lagrangian submanifolds, this step can b e greatly simplified. Lefsc hetz thim bles. The idea in [15] is to em b ed the F uk a y a cat egory of M in to th e F uk ay a category of a Lefsc hetz fibration π : X → C . The latter class of categorie s is kno w n to admit fu ll exceptional collectio ns, giv en by an y ba- sis of Lefsc hetz thim bles. Results from homologica l algebra (more precisely , 4 KENJI FUKA Y A, P AUL SEIDEL, IV AN SMITH the theory of m u tations, see for instance [16]) then ensure the existence of a resolution of the diagonal, in terms of K oszul dual bases. T o apply this ma- c hinery one has to construct a Lefsc hetz fib ration, with an an tiholomorphic in v olution, whose real part π R is a Morse f unction on X R = Z . T his can b e done easily , although not in a canonical w a y , b y usin g tec hniques f rom real alge br aic geo metry . Roughly sp eaking, the resulting F uk ay a category lo oks similar to the category of shea v es constru ctible with resp ect to the stratificatio n giv en by the un stable manifolds of π R , compare [21]. Ho w- ev er, b ecause th e constructio n of X is not precisely con trolled, one d o es not exp ect these tw o categorie s to agree. Whilst this is n ot a p roblem for the pro of of Theorem 0.1, it m ay b e æ s thetical ly un satisfacto ry . On e p ossibilit y for impro ving the situation would b e to fi nd a wa y of d irectly pro ducing a Lefschetz fibration on the cota ngent bun dle; steps in that direction are tak en in [20]. W rapping at infinit y . The third approac h remains w ithin M , and again uses Lagrangian su bmanifolds whic h are scaling-i nv ariant at infinity . Ho w- ev er, in tersections at infinity are dealt with b y flo wing along the (not nor- malized) geo desic fl o w, whic h is a large p erturbation. F or instance , after this p erturb ation, th e in tersections of an y t wo fibres will b e giv en b y all geo desics in Z connecting the relev an t t w o p oin ts. In con trast to the previous con- structions, this one is in trinsic to the differen tiable m an ifold Z , and do es not r equire a real-analytic or real-alg ebraic structure (there are of course tec h nical c h oices to b e made, suc h as the Riemannian metric and other p er- turbations b elonging to standard pseud o-holomorphic curve theory; but the outcome is indep enden t of those up to quasi-isomorphism). Conjecturally , the resulting “wrapp ed F uk a ya category” is equiv alen t to a full sub category of the cat egory of m o dules o ver C −∗ (Ω Z ), the dg (differen- tial graded) algebra of chai ns on the based lo op space (actuall y , the Mo ore lo op space, with the Pon tryagin pro duct). Note th at the classical bar con- struction establishes a r elatio n b et w een this and the dg algebra of cochains on Z ; for K = R , one can tak e th is to b e the alge br a of differen tial forms; for K = Q , it could b e Su lliv an’s m o del; and for general K one can use singu- lar or Cec h cohomology . If Z is simp ly conn ected, this relation leads to an equiv alence of suitably defined mod ule categories, and one can reco v er (0.1) in this w a y . In fact, w e pr op ose a more geometric version of this argument, whic h in v olv es an explicit functor from the wrapp ed F uk a y a catego ry to th e catego ry of mo d ules o v er a dg algebra of Cec h co chains. Remark 0.2 . It is inter esting to c omp ar e (0.1) with the r e sult of a naive ge ometric ar gument. Supp ose L is close d and exact; under fibr ewise sc aling L 7→ cL , as c → ∞ , cL c onver ges (in c omp act subsets) to a disjoint union of c otangent fibr es S x ∈ L ∩ i ( Z ) T ∗ x , wher e i : Z ֒ → M denotes the zer o-se ction. In p articular, for lar ge c ther e is a c anonic al bije ction b etwe en p oints of L ∩ cL COT ANGENT BUNDLES 5 and p oints of { L ∩ T ∗ x | x ∈ L ∩ i ( Z ) } . Starting fr om this identific ation, and filtering by ene r gy, one exp e c ts to obtain a sp e ctr al se quenc e (0.2) M x ∈ L ∩ i ( Z ) H F ( L, T ∗ x ) ⇒ H ∗ ( L ) using exactness to identify H F ( L, cL ) ∼ = H ∗ ( L ) . This would (r e )pr ove that L ∩ i ( Z ) 6 = ∅ and that H F ( L, T ∗ x ) 6 = 0 ; and, in an informal fashion, this pr ovides motivation for b elieving that L is “gener ate d by the fibr e s”. It se ems har d, however, to c ontr ol the homolo gy class of L starting f r om this, b e c ause it se ems har d to gain sufficie nt c ontr ol over L ∩ i ( Z ) . The follo w ing three sections of the pap er are eac h devo ted to explaining one of these ap p roac hes. Then, in a concluding sectio n, w e tak e a lo ok at the non-simply-connected case. First of all, there is a u seful tric k inv olving the sp ectral sequence (0.1) and finite co ve rs of the base Z . In principle, this tric k can b e applied to an y of th e three app roac h es outlined ab ov e, but at the present state of the literature, the necessary prerequisites hav e b een fully established on ly for the theory fr om [30]. Applying that, one arrive s at the follo win g consequence, wh ic h app ears to b e n ew: Corollar y 0.3 . The assumption of simple-c onne ctivity of Z c an b e r e- move d fr om The or em 0.1. This me ans that for al l close d spin manifolds Z , and al l exact L agr angian submanifolds L ⊂ T ∗ Z which ar e spin and have zer o M aslov class, the c onclusions (i)–(iii) hold. F rom a more fund amen tal p ersp ectiv e, the app roac h via w rapp ed F uk ay a catego ries seems particularly suitable for inv estigating cotange nt bun dles of non-simply-connected manifolds, since it retains in formation that is lost when passing from c hains on Ω Z to co c hains on Z . W e end by d escribing what this wo uld mean (mo dulo one of our conjectures, 3.6) in the sp ecial case when Z is a K (Γ , 1). In the sp ecial case of the torus Z = T n , Conj ecture 3.6 can b e sidestepp ed b y direct geometric argument s, at least when c h ar( K ) 6 = 2. Imp osing that condition, one fin ds that an arbitrary exact , orient ed and spin Lagrangian subm anifold L ⊂ T ∗ T n = ( C ∗ ) n satisfies the conclusions of Theorem 0.1, with no assump tion on the Maslo v class. Finally , it is w orth p oin ting out that for any orien ted and spin Lagrangian submanifold L ⊂ T ∗ Z , and any closed spin manifold Z , the theory pro duces a Z / 2 Z -graded sp ectral sequence (0.3) H ( Z ; E n d( E L )) ⇒ H ( L ) whic h h as applications in its own righ t, for instance to the classificati on of “local Lagrangian knots” in Eu clidean sp ace. Eliash b erg and Polte ro vic h [9] pro v ed that an y exact L agrangian L ⊂ C 2 whic h co-incides with the 6 KENJI FUKA Y A, P AUL SEIDEL, IV AN SMITH standard R 2 outside a compact subset is in fact Lagrangian isotopic to R 2 . Again, their pro of relies on exclusiv ely f our-dimensional machinery . Corollar y 0.4 . L et L ⊂ C n b e an exact L agr angian sub manifold which c o- incides with R n outside a c omp act set. Supp ose that L is oriente d and spin. Then (i) L is acyclic and (ii) π 1 ( L ) has no non-trivial finite- dimensional c omplex r epr esentations. Using a result of Viterb o [38] and standard facts from 3-manifold top ology , when n = 3 this implies that an orient ed exact L ⊂ C 3 whic h co-i ncides with R 3 outside a compact set is actually diffeomorphic to R 3 . T o p ro v e Corollary 0.4, one em b eds the giv en L (view ed in a Darb oux ball) in to th e zero-sect ion of T ∗ S n , obtaining an exact L agrangian submanifold L ′ of the latter whic h co-incides with the zero-sect ion on an op en s et. This last f act immediately implies that E L ′ has rank one, so w e see End( E L ′ ) is the trivial K -line bundle, and the sequence (0.3) imp lies rk K H ∗ ( S n ) ≥ rk K H ∗ ( L ′ ). On the other hand, pro jection L ′ → S n is (obvio usly) degree ± 1, whic h giv es the rev erse inequalit y . Going bac k to L ⊂ L ′ , we deduce that L is acyclic with K -co efficien ts. Since K is arbitrary , one deduces the first part of the Corollary . A t this p oin t, one kn o ws a p oste ori that the Maslo v class of L v anishes; hence so d o es that of L ′ , and the final statemen t of the Corollary then follo w s from an older result of Seidel [35]. When n = 3, it follo ws that L ′ is simply connected or a K ( π , 1), at whic h p oint one can app eal to Viterb o’s work. (It is straigh tforw ard to d educe the acycli cit y ov er fields of c haracteristic other than tw o in the other framew orks, for instance that of [15], b ut then the conclusion on th e Maslo v class do es n ot follo w.) A cknow le dgements. I.S. is grateful to K evin Costello for encouragemen t and man y helpful con v ersations. Some of this material w as present ed at the “Dusafest” (Ston y Bro ok, Oct 2006). P .S. was partially supp orted by NSF gran t DMS-040 5516. I.S. ac knowledge s EPSRC gran t EP/C5359 95/1. 1. Constructible shea ves This section should b e considered as an introdu ction to the t wo pap ers [30, 29]. Our aim is to present ideas from those pap ers in a w a y whic h is familiar to sy m plectic geometers. With that in mind, we ha ve tak en some lib erties in th e present ation, in p articular omitting the ( nontrivia l) tec hnical w ork in v olv ed in smo othing out c haracteristic cycles. 1.1. F uk ay a categories of W einstein manifolds. Let M b e a W ein- stein manifold whic h is of finite t yp e and complete. Recall that a symplectic COT ANGENT BUNDLES 7 manifold ( M , ω ) is W einstein if it comes with a distinguished Liouville (sym- plectical ly expanding) ve ctor field Y , and a prop er b ound ed b elo w f u nction h : M → R , suc h that dh ( Y ) is p ositiv e on a sequence of leve l sets h − 1 ( c k ), with lim k c k = ∞ . The stronger finite type assumption is th at dh ( Y ) > 0 outside a compact su bset of M . Finally , completeness m eans that the flow of Y is defined for all times (for negativ e times, this is automati cally tru e, but for p ositiv e times it is an additional constrain t). Note that the Liouville v ector field defines a one-form θ = i Y ω with dθ = ω . At infi nit y , ( M , θ ) has the form ([0; ∞ ) × N , e r α ), where N is a con tact m an ifold with con tact one-form α , and r is the radial co ordinate. In other w ords, the end of M is mo delled on th e p ositiv e h alf of th e s y m plectiza tion of ( N , α ). The obvious examples are cotangen t bundles of closed manifolds, M = T ∗ Z , wh ere Y is the radial rescaling vec tor field, and N the un it cotangen t bun dle. W e will consider exact Lagrangian submanifolds L ⊂ M whic h are Legen- drian at infinity . By defin ition, this means that θ | L is the deriv ativ e of some compactly supp orted function on L . Outside a compact sub s et, an y su c h L will b e of the form [0; ∞ ) × K , where K ⊂ N is a Legendrian sub mani- fold. Now let ( L 0 , L 1 ) b e t wo such submanifolds, whose structure at infinity is mo delled on ( K 0 , K 1 ). T o define their Floer cohomology , one needs a w a y of resolving the intersec tions at infinit y b y a suitable small p erturba- tion. The details ma y v ary , dep ending on w hat kind of L egendrian subman- ifolds one w an ts to consider. Here, w e make the assumption that ( N , α ) is real-analyti c, and allo w only those K whic h are r eal-analytic submanifolds. Then, Lemma 1.1 . L et ( φ t R ) b e the R e eb flow on N . F or any p air ( K 0 , K 1 ) , ther e is an ǫ > 0 such that φ t R ( K 0 ) ∩ K 1 = ∅ for al l t ∈ (0 , ǫ ) . This is a consequence of the Curv e Selection Lemma [28, Lemma 3.1], com- pare [30, Lemma 5.2.5]. Recall that, when defining th e Floer cohomology of tw o Lagrangian submanifolds, one often adds a Hamilto nian p erturba- tion H ∈ C ∞ ( M , R ) (for tec h nical reasons, this Hamiltonian is usu ally also tak en to b e time-dep enden t, but w e sup press that here). T he associated Floer co c hain complex is generated by the fl o w lines x : [0; 1] → M of H going fr om L 0 to L 1 ; equiv alen tly , these are the intersect ion p oints of φ 1 X ( L 0 ) ∩ L 1 , where X is the Hamiltonian vecto r field of H . W e denote this co chain complex b y (1.1) C F ∗ ( L 0 , L 1 ; H ) = C F ∗ ( φ 1 X ( L 0 ) , L 1 ) . In our case, we tak e an H whic h at infin it y is of the f orm H ( r , y ) = h ( e r ), where h is a f unction with h ′ ∈ (0; ǫ ). Then, X is h ′ ( e r ) times the Reeb v ector field R , hen ce φ 1 X ( L 0 ) ∩ L 1 is compact b y Lemma 1.1. Sta nd ard argumen ts sh o w that the resulting Floer cohomolog y group H F ∗ ( L 0 , L 1 ) = H F ∗ ( L 0 , L 1 ; H ) is ind ep endent of H . I t is also inv ariant under compactly 8 KENJI FUKA Y A, P AUL SEIDEL, IV AN SMITH supp orted (exact L agrangian) isotopies of either L 0 or L 1 . Note that in the case w here K 0 ∩ K 1 = ∅ , one can actually set h = 0, wh ic h yields Flo er cohomolog y in the ordin ary (un p erturb ed) sense. Finally , for L 0 = L 1 = L one has the u sual redu ction to Morse theory , so that H F ∗ ( L, L ) ∼ = H ∗ ( L ), ev en for noncompact L . A t this p oin t, w e need to m ak e a few more tec hnical remarks. F or simplic- it y , all our Flo er cohomology group s are with co efficient s in some field K . If char ( K ) 6 = 2, one needs (relativ e) S pin structur es on all Lagrangian sub - manifolds in v olv ed, in order to address the u sual orientat ion pr oblems for mo duli spaces [14]. Next, Flo er cohomology groups are generally only Z / 2- graded. One can upgrade this to a Z -grading by requir in g th at c 1 ( M ) = 0, and c ho osing gradin gs of eac h Lagrangia n subm anifold. F or the momen t, w e d o not need this, but it b ecomes imp ortan t wh enev er one wa nts to mak e the connection with ob jects of classical homologic al algebra, as in Theorem 1.3 b elo w , or in (0.1). Example 1.2 . Consider the c ase of c otangent b und les M = T ∗ Z (to satisfy the gener al r e quir ements ab ove, we should imp ose r e al-analyt icity c onditions, but that is not actual ly ne c essary for the sp e cific c omputatio ns we ar e ab out to do). A typic al example of a L agr angian submanifold L ⊂ M satisfying the c onditions set out ab ove is the c onorma l bund le L = ν ∗ W of a close d submanifold W ⊂ Z . If ( L 0 , L 1 ) ar e c onormal bund les of tr ansversal ly in- terse cting submanifolds ( W 0 , W 1 ) , then (1.2) H F ∗ ( L 0 , L 1 ) ∼ = H ∗− codim ( W 0 ) ( W 0 ∩ W 1 ) . This is e asy to se e (exc ept p erhaps for the gr ading), sinc e the only interse c- tion p oints of the L k lie in the zer o-se ction. Al l of them have the sam e value of the action functional, and standar d Morse-Bott te chniques appl y. As a p ar al lel b ut slightly differ ent example, let W ⊂ Z b e an op en subset with smo oth b oundary. T ake a function f : W → R which is strictly p ositive in the interior, zer o on the b oundary, and has ne gative normal derivative at al l b oundary p oints. We c an then c onsider the gr aph of d (1 /f ) , which is a L agr angian sub manifold of M , asymptotic to the p ositive p art of the c onormal bu nd le of ∂ W . By a suitable isotop y, one c an deform the gr aph so that it agr e es at infinity with that c onorma l bund le. Denote the r esult by L . Given two such subsets W k whose b oundaries interse ct tr ansversal ly, one then has [23 , 24, 31] (1.3) H F ∗ ( L 0 , L 1 ) ∼ = H ∗ ( W 1 ∩ W 0 , W 1 ∩ ∂ W 0 ) . Note that in b oth these c ases, the L agr angian submanifolds under c onsider- ation do admit natur al gr adings, so the isomorphisms ar e ones of Z -gr ade d gr oups. COT ANGENT BUNDLES 9 W e will need m ultiplicativ e structures on H F ∗ , realized on the c hain lev el b y an A ∞ -categ ory stru cture. The tec hnical obstacle, in th e first nontrivia l case, is that the natural triangle pr o duct (1.4) C F ∗ ( L 1 , L 2 ; H 12 ) ⊗ C F ∗ ( L 0 , L 1 ; H 01 ) = C F ∗ ( φ 1 X 12 ( L 1 ) , L 2 ) ⊗ C F ∗ ( φ 1 X 12 φ 1 X 01 ( L 0 ) , φ 1 X 12 ( L 1 )) − → C F ∗ ( φ 1 X 12 φ 1 X 01 ( L 0 ) , L 2 ) do es not qu ite land in C F ∗ ( L 0 , L 2 ; H 02 ) = C F ∗ ( φ 1 X 02 ( L 0 ) , L 2 ). F or instance, if one tak es the same H for all pairs of Lagrangian submanifolds, the output of the pro du ct has φ 2 X = φ 1 2 X instead of the desired φ 1 X . The s olution adopted in [30] is (roughly sp eaking) to c ho ose all functions h in v olv ed to b e v ery small, in which case the deformation from X to 2 X indu ces an actual isomorphism of Flo er coc hain groups. The do wnsid e is that this can only b e done for a fin ite num b er of Lagrangian su bmanifolds, and more imp ortan tly , for a finite num b er of A ∞ -pro ducts at a time. Hence, what one gets is a partially defined A d -structure (for d arb itrarily large), from which one then has to p ro duce a prop er A ∞ -structure; for some relev an t algebraic results, see [14, Lemma 30.163]. As an alternativ e, one can tak e all fun ctions h to satisfy h ( t ) = log t , whic h means H ( r , y ) = r . Then, φ 2 X is conjugate to a compactly sup p orted p erturbation of φ 1 X (the conjugating diffeomorphism is the Liouville flo w φ t Y for time t = − log(2)). By making the other choi ces in a careful wa y , one can th en arrange that (1.4) tak es v alues in a Floer co chain group whic h is isomorph ic to C F ∗ ( L 0 , L 2 ; H 02 ). In either wa y , one ev en tually ends up with an A ∞ -categ ory , whic h w e denote by F ( M ). 1.2. Characteristic cyc les. Giv en a real-analyti c m anifold Z , one can con- sider shea v es of K -ve ctor spaces wh ic h are constru ctible (with r esp ect to some real analytic stratification, which ma y dep end on the sheaf ). Denote b y D c ( Z ) the fu ll sub category of the b ounded derive d categ ory of s h ea ves of K -v ector spaces compr ising complexes with constructible cohomolo gy . Kashiw ara’s charac teristic cycle construction [22] asso ciates to an y ob ject G in this catego ry a L agrangian cycle C C ( G ) insid e M = T ∗ Z , whic h is a cone (inv ariant u nder rescaling of cotangen t fib res). If G is th e stru cture sheaf of a closed s u bmanifold, this cycle is just the conormal bundle, bu t otherwise it tends to b e singular. Nadler and Zaslo w consider the stru cture shea v es of submanifolds W ⊂ Z whic h are (real-analyt ic but) not necessar- ily closed. F or eac h suc h “standard ob ject” , they construct a smo othing of C C ( G ), wh ic h is a Lagrangian sub manifold of M . In the sp ecial case where W is an op en sub s et w ith smo oth b oundary , this is essenti ally equiv alen t to the construction indicated in Examp le 1.2. The singular b oundary case is considerably more complicated, and leads to Lagrangian submanifolds whic h are generally only asymptotically inv arian t under the Liouville flo w (their limits at infin it y are singular Legendrian cycles). Still, one can use 10 KENJI FUKA Y A, P AUL SEIDEL, IV AN SMITH them as ob jects of a F uk a ya-t yp e categ ory , whic h is a v arian t of the pr evi- ously describ ed construction. W e denote it by A ( M ), where A stands for “asymptotic” . Th e main result of [30] is Theorem 1.3 . The smo othe d char acteristic cycle c onstruction gives rise to a ful l emb e dding of derive d c ate gories, D c ( Z ) − → D A ( M ) . The pro of relies on t wo ideas. One of them, namely , that the standard ob jects generate D c ( Z ), is m ore element ary (it can b e view ed as a fact ab out decomp ositions of r eal subanalytic sets). F o r pu rely algebraic reasons, this means that it is enough to d efine the em b edd ing only on standard ob jects. The other, more geome tric, tec hn ique is the reduction of Flo er cohomolog y to Morse theory , pro vided b y the work of F uk a y a-Oh [13]. 1.3. Decomposing the diagonal. In D c ( Z ), the structure shea ve s of any t w o distinct p oin ts are algebraica lly disjoin t (there are no morphisms b e- t w een them). O f course, the same holds for cotangen t fibres in the F uk a y a catego ry of M , which are the images of such structure shea ve s un der the em b eddin g from Th eorem 1.3. As a consequence, in F ( M ) (or A ( M ), the difference b eing irrelev ant at this lev el) one cann ot exp ect to hav e a finite resolution of a closed exact L ⊂ M in terms of cotangen t fib r es. Ho w ev er, using the W ehrheim-W o o dwa rd formalism of Lagrangian corresp ond ences [40], Nadler pr o v es a m o dified v ersion of this stat ement , wh ere the fi b res are replaced b y stand ard ob jects asso ciated to certain con tractible sub sets of Z . Concretely , fi x a real analytic triangulation of Z . Denote by x i the v ertices of the triangulation, by U i their stars, and b y U I = T i ∈ I U i the in tersections of suc h stars, indexed by finite sets I = { i 0 , . . . , i d } . There is a standard C ec h resolution of the constan t s h eaf K Z in terms of the K U I (w e will encounter a similar construction again later on, in Section 3.1). Rather th an applying this to Z itself, w e tak e the diagonal inclusion δ : Z → Z × Z , and consider the induced r esolution of δ ∗ ( K Z ) b y the ob jects δ ∗ ( K U I ). Consider the em- b edding from T h eorem 1.3 applied to Z × Z . T h e image of δ ∗ ( K Z ) is the conormal b undle of the diagonal ∆ = δ ( Z ), and eac h δ ∗ ( K U I ) maps to the standard ob ject asso ciated to δ ( U I ) ⊂ Z × Z . Sin ce eac h U I is con tractible, one can deform δ ( U I ) to U I × { x i d } (as lo cally close d su b manifolds of Z × Z ), and this induces an isotop y of the asso ciated sm o othed c haracteristic cycles. A prio ri , this isotop y is not compact ly supp orted, hence not we ll-b ehav ed in our category (it do es n ot preserv e the isomorphism type of ob jects). Ho w- ev er, this is n ot a problem if one is only interest ed in morphisms from or to a giv en closed Lagrangian su bmanifold. T o formalize this, tak e A cpt ( M ) to b e the su b category of A ( M ) consisting of closed Lagrangia n submanifolds (this is in fact the F u k a y a category in the COT ANGENT BUNDLES 11 classical sens e). Dually , let mod ( A ( M )), mod ( A cpt ( M )) b e the asso ciated catego ries of A ∞ -mo dules. There is a c hain of A ∞ -functors (1.5) A cpt ( M ) − → A ( M ) − → mod ( A ( M )) − → mod ( A cpt ( M )) . The fi rst and second one, whic h are inclusion and the Y oneda em b edding, are full and faithful. T he last one, restriction of A ∞ -mo dules, will not generally ha v e that pr op ert y . Ho wev er, the comp osition of all three is just the Y oneda em b eddin g for A cpt , which is again full and faithful. In view of [40], and its c hain-lev el analogue [27], eac h ob ject C in A ( M × M ) (and more generally , t wisted complex bu ilt out of such ob jects) induces a con v olution fun ctor (1.6) Φ C : A cpt ( M ) − → mod ( A cpt ( M )) (usually , one reve rses the sign of the sym plectic form on one of the t wo fac- tors in M × M , but for cotangen t bun dles, this can b e comp ensated by a fibrewise reflection σ : M → M ). First, tak e C to b e the conormal bundle of the diagonal, whic h is the same as the graph of σ . Then, conv olution with C is isomorphic to the embed d ing (1.5). On the other hand, if C is the smo othed c h aracteristic cycle of some pro duct U × { x } , then Φ C maps eac h ob ject to a dir ect sum of copies of T ∗ x (the image of that fibr e under the functor A ( M ) → mod ( A cpt ( M )), to b e precise). Finally , if C is just a Lagrangian submanifold, Φ C is in v arian t und er Lagrangian isotopies which are not necessarily compac tly sup p orted. By com b ining those facts, one ob- tains the desired resolution of a closed exact L ⊂ M . Nadler actually pu shes these ideas somewh at further, using a refined version of this argument, to sho w that: Theorem 1.4 . The emb e dding D c ( Z ) → D A ( M ) fr om The or em 1.3 is an e quivalenc e. Remark 1.5 . If one is only inter e ste d i n the sp e ctr al se quenc e (0.1) , ther e may b e a p otential simplific ation, which would byp ass some of the c ate gor- ic al c onstructions ab ove. First of al l, r ewrite H F ∗ ( L, L ) ∼ = H ∗ ( L ; K ) as H F ∗ ( L × L, ν ∗ ∆) . Then, u sing the r esolution of ν ∗ ∆ in A ( M × M ) de- scrib e d ab ove, one gets a sp e c tr al se quenc e c onver ging towar ds that gr oup, whose E 1 p age c omprises the Flo er c ohomol o gy gr oups b etwe en L × L and the smo othe d char acteristic cycles of δ ( U I ) . Sinc e L × L is c omp act, one c an deform δ ( U I ) to U I × { x i d } , and then further isotop its smo othe d char acteris- tic cycle to T ∗ x i d ⊗ T ∗ x i d . In the terminolo gy use d in (0.1) , this brings the terms in the E 1 p age into the form E nd ( E L ) x i d . T o get the desir e d E 2 term, one would further have to che ck that the differ entials r epr o duc e the ones in the Ce ch c omplex with twiste d c o efficients in E nd ( E L ) . This of c ourse fol lows fr om The or em 1.4, but ther e ought to b e a mor e dir e ct ge ometric ar gu ment, just by lo oking at the r elevant sp ac es of holom orphic triangles; this se ems a worth while ende avour, bu t we have not attempte d to study it i n detail. 12 KENJI FUKA Y A, P AUL SEIDEL, IV AN SMITH 2. Lefsche tz thimbles This section giv es an o ve rview of the pap er [15], and an accoun t – emp h a- sising geometric rather than algebraic asp ects – of some of the underlying theory from the b o ok [36]. 2.1. F uk ay a categories of Lefsc hetz fibrations. In p rinciple, the notion of Lefschet z fibr ation can b e defined in a purely symp lectic w a y . How eve r, w e will limit ourselv es to the more traditional algebro-ge ometric con text. Let X b e a smo oth affine v ariety , and (2.1) π : X − → C a p olynomial, whic h h as only nondegenerate critical p oin ts. F or con ve nience, w e assume that no t w o s u c h p oin ts lie in the same fibr e. Additionally , w e imp ose a cond ition whic h excludes singularities at infinit y , namely: let ¯ X b e a pro jectiv e completion of X , such that D = ¯ X \ X is a divisor with normal crossing. W e then require that (for an approriate c hoice of ¯ X ) the closure of π − 1 (0) should b e s m o oth in a neigh b ourho o d of D , and in tersect eac h comp onen t of D transversall y . Finall y , for Flo er-theoretic reasons, we require X to b e Calabi-Y au (ha v e trivial canonical bun dle). T ak e an y K¨ ahler f orm on ¯ X whic h comes fr om a metric on O ( D ). Its restric- tion to X mak es that v ariet y in to a W einstein m anifold (of finite t yp e, but not complete; the latter deficiency can, h o w ev er, b e cured easily , by att ac h- ing the missin g part of the conical end). Moreo v er, parallel transp ort for π is well -defined a wa y f rom the singular fib res, in spite of its non-prop erness. A v anishing path γ : [0 , ∞ ) → C is an em b edding starting at a critical v alue γ (0) of π , and suc h that for t ≫ 0, γ ( t ) = c onst. − it is a half-line going to − i ∞ . T o eac h suc h path one can associate a L e f schetz thimble ∆ γ ⊂ X , whic h is a Lagrangian su b manifold diffeomorphic to R n , pr o jecting prop erly to γ ([0 , ∞ )) ⊂ C . More precisely , γ − 1 ◦ π | ∆ γ is the standard prop er Morse function on R n with a sin gle min imum (placed at the unique critic al p oin t of π in the fi bre o ver γ (0)). When defining the Flo er cohomology b et ween t wo Lefsc hetz thimbles, the con ve ntio n is to rotate the semi-infinite part of the first path in antic lo c kwise direction for some small angle. Om itting certain tec h nical p oin ts, th is can b e interpreted as adding a Hamiltonia n term as in Section 1.1. In particular, one again has (2.2) H F ∗ ( L γ , L γ ) ∼ = H ∗ ( L γ ; K ) = K . No w supp ose that ( γ 0 , . . . , γ m ) is a basis (sometime s also called a distin- guished basis) of v anishing paths. W e w ill not recall the d efinition here; for a sket c h, see Figure 1 . In that situation, if one tak es ( γ j , γ k ) with j > k and applies the rotatio n describ ed ab o ve to γ j , the result remains disjoin t from COT ANGENT BUNDLES 13 γ k . Hence, (2.3) H F ∗ ( L γ j , L γ k ) = 0 for all j > k . γ 1 r r r γ 0 γ 2 γ 3 r Figure 1. As usual, rather than wo rking on the lev el of Flo er cohomolo gy , we w ant to ha v e und erlying A ∞ -structures. There is a conv enien t shortcut, w h ic h eliminates noncompact Lagrangian submanifolds from the foundations of the theory (but w hic h unfortunately requires char ( K ) 6 = 2). Namely , let ˜ X → X b e the double co v er b ranc hed ov er some fibre π − 1 ( − iC ), C ≫ 0. Roughly s p eaking, one tak es th e ordinary F uk a y a catego ry F ( ˜ X ), wh ic h con tains only compact Lagrangian submanifolds, and defines F ( π ) to b e its Z / 2-in v arian t part (only in v arian t ob jects and morphisms; obvi ously , get- ting this to w ork on the co chain lev el r equires a little care) . T h is time, let ’s allo w only v anishing paths whic h satisfy γ ( t ) = − it for t ≥ C (whic h is no p roblem, since eac h path can b e br ough t into this form b y an isotop y). One then truncates the Lefsc hetz thim ble asso ciated to s uc h a path, so that it b ecomes a Lagrangian disc with b oundary in π − 1 ( − iC ), and tak es its preimage in ˜ X , whic h is a closed Z / 2-in v arian t Lagrangian sphere ˜ S γ ⊂ ˜ X . On the cohomological lev el, the Z / 2-in v arian t parts of the Floer cohomolo- gies of these sp heres still satisfy the same prop erties as b efore, in particular repro duce (2.2) and (2.3). Theorem 2.1 . If ( γ 0 , . . . , γ m ) is a b asis of v anishing p aths, the asso c iate d ˜ S γ j form a fu l l exc eptional c ol le ction in the derive d c ate gory D F ( π ) . The fact that we get an exceptional collect ion is elemen tary; it ju st re- flects the tw o equations (2.2) and (2.3) (or rather, their count erparts for the mo dified definition of Flo er cohomolo gy in vo lving doub le cov ers). F ullness, whic h is the prop ert y that this collect ion generates th e deriv ed categ ory , is rather more in teresting. The pro of giv en in [36 ] relies on the fact that the pro duct of Dehn t wists along the ˜ L γ k is isotopic to th e co v ering inv olu- tion in ˜ X . Hence, if L ⊂ X is a closed L agrangian sub manifold w hic h lies in π − 1 ( { im( z ) > − C } ), this p ro duct of Dehn t w ists will exc hange the tw o 14 KENJI FUKA Y A, P AUL SEIDEL, IV AN SMITH comp onen ts of the preimage ˜ L ⊂ ˜ X . The rest of the argument essenti ally consists in applying the long exact sequence from [34]. 2.2. P ostnik o v decomp ositions. W e will use some purely algebraic prop- erties of exceptional collections, s ee for instance [16] (the sub ject has a long history in algebraic geometry; readers in terested in this migh t fi nd the col- lectio n [32] to b e a go o d starting p oint). Namely , let C b e a triangulated catego ry , linea r o v er a field K , and let ( Y 0 , . . . , Y m ) b e a full exceptional collec tion of ob jects in C . Then, for any ob ject X , there is a collection of exact triangles (2.4) Z k ⊗ Y k → X k → X k − 1 [1] − → Z k ⊗ Y k where X m = X , X − 1 = 0, an d Z k = H om ∗ C ( Y k , X k ) (morphisms of all degrees; b y assu mption, this is finite-dimensional, s o Z k ⊗ Y k is the direct sum of finitely m any sh ifted copies of Y k ). The map Z k ⊗ Y k → X k is the canonical ev aluation map, and X k − 1 is defined (by d escending induction on k ) to b e its mapping cone. T o get another description of Z k , one can use the unique (right) Koszul d ual exceptional collecti on ( Y ! m , . . . , Y ! 0 ), w hic h satisfies (2.5) H om ∗ C ( Y j , Y ! k ) = ( K (concen trated in degree zero) j = k , 0 otherwise . It then follo ws by rep eatedly applying (2.4) that Z ∨ k ∼ = H om ∗ C ( X, Y ! k ). No w, giv en an y cohomologic al functor R on C , w e get an ind uced sp ec- tral sequence con ve rging to R ( X ), wh ose starting page h as columns E r s 1 = ( Z ∨ m − r ⊗ R ( Y m − r )) r + s . In p articular, taking R = H om ∗ C ( − , X ) and usin g the exp r ession for Z k explained ab o v e, we get a sp ectral sequence con verging to H om ∗ C ( X, X ), whic h starts with (2.6) E r s 1 =  H om ∗ C ( X, Y ! m − r ) ⊗ H om ∗ C ( Y m − r , X )  r + s . W e no w return to the concrete setting where C = D F ( π ). In this case, the Koszul dual of an exceptional collection giv en by a basis of Lefsc hetz thim bles is another suc h basis { γ ! 0 , . . . , γ ! m } . This is a consequence of the more general relation b et w een mutat ions (algebra) and Hurwitz mov es on v anishing paths (geometry). Applying (2.5), and going bac k to the original definition of Floer cohomology , w e therefore get the follo wing result: for ev ery (exact , graded, spin) closed L agrangian su bmanifold L ⊂ M , there is a sp ectral sequ en ce con v erging to H F ∗ ( L, L ) ∼ = H ∗ ( L ; K ), whic h starts with (2.7) E r s 1 = ( H F ( L, ∆ γ ! m − r ) ⊗ H F (∆ γ m − r , L )) r + s . COT ANGENT BUNDLES 15 2.3. Real algebraic approx imation. The existence of (2.7) is a general statemen t ab out Lefschetz fib rations. T o mak e the connection with cotan- gen t bund les, we use a form of the Nash-T ognoli theorem, see for instance [19], n amely: Lemma 2.2 . If Z is a close d manifold and p : Z → R is a Morse function, ther e is a L e f schetz fibr ation π : X → C with a c omp atible r e al structur e, and a diffe omorph ism f : Z → X R , such that π ◦ f is C 2 -close to p . The diffeomorphism f can b e extended to a symplectic emb edding φ of a neighbourh o o d of the zero-sectio n of Z ⊂ M = T ∗ Z in to X . Hence (p erhaps after a preliminary radial rescaling) we can transp ort closed exact Lagrangian submanifolds L ⊂ M o ver to X . Th e critical p oin ts of π fall in to t w o classes, namely real and purely complex ones, and the ones in the first class corresp ond canonically to critical p oin ts of p . By a suitable c hoice of v anishing paths, one can ensur e that (2.8) H F ∗ (∆ γ k , φ ( L )) ∼ = ( H F ∗ ( T ∗ x k , L ) if γ (0) ∈ X R corresp onds to x k ∈ C r it ( p ) , 0 otherwise . Here, the Floer cohomolo gy on the righ t hand side is tak en inside T ∗ Z . The same statemen t holds for the other groups in (2.7), up to a shift in the grad- ing whic h dep ends on the Morse in dex of x k . As a consequence, the starting page of that sp ectral sequence can b e though t of as C ∗ M orse ( Z ; E nd ∗ ( E L )), where the Morse complex is take n with resp ect to the function p , and using the (graded) lo cal coefficien t system E L . This is one page earlier than our usual starting term (0.1), bu t is already go o d enough to deriv e Theorem 0.1 by app ealing to some cla ssical manifold top ology (after taking the pro d- uct with a sphere if necessary , one can assu me that dim ( Z ) > 5, in wh ich case simple connectivit y of Z implies that one can choose a Morse fun ction without critical p oint s of in dex or co-index 1). Remark 2.3 . The differ ential on the E 1 -p age of (2.7) is give n in [36, Corol- lary 18.2 7] , in terms of holomor phic triangle pr o ducts b etwe en adjac ent L ef- schetz thimbles in the e xc eptional c ol le ction. In the sp e cial situation of (2.8), identifying the E 1 p age with C ∗ M orse ( Z ; E nd ( E L )) , ther e is also the Morse differ ential δ c oming fr om p ar al lel tr ansp ort in the lo c al system E L → Z (c omp ar e R emark 1.5). F or L efschetz fibr ations arising f r om r e al algebr aic appr oximation, r ather than some mor e c anonic al c onstruction, ther e se ems to b e no r e ason for these to agr e e in gene r al; but one do es exp e ct the p arts of the differ ential le ading out of the first c olumn, and into the last c olumn, to agr e e. F or instanc e, by deforming the final vanishing p ath to lie along the r e al axis, one c an ensur e that the entir e thimble le ading out of the maximum 16 KENJI FUKA Y A, P AUL SEIDEL, IV AN SMITH x max of the Morse function is c ontaine d in the r e al lo cu s, after which the interse ction p oints b etwe en this thimble and one c oming fr om a c ritic al p oint x of index one less c orr e sp ond bije ctively to the gr adient lines of the M orse function b etwe en x max and x (the situation at the minimum x min is anal- o gous). If it was known that this p art of the E 1 -differ ential did r epr o duc e the c orr esp onding pie c e of δ , one c ould hop e to study non-simply-c onne cte d c otangent bund les in this appr o ach. 3. F amil y Floer cohomology This sectio n co v ers the third p oin t of view on Theorem 0.1. This time, the presen tation is less linear, and o ccasionally sev eral wa ys of reac hing a par- ticular goal are sk etc hed. The reader should keep in mind the preliminary nature of this discussion. In some parts, this means that there are com- plete but u npublished constructions. F or others, only outlines or strategi es of pro of exist, in wh ic h case we will b e careful to formulate the relev an t statemen ts as conjectures. 3.1. Cec h complex es. A t the start of the p ap er, we men tioned that to an L ⊂ M = T ∗ Z one can asso ciate the bu ndle E L of Flo er cohomolo gies E x = H F ∗ ( T ∗ x , L ). One naturally wan ts to replace E L b y an underlying co chain lev el ob ject E L , whic h should b e a “sheaf of complexes” in a suitable sense. In our int erpretation, this w ill b e a d g mo dule ov er a dg algebra of Cec h co c hains (there are sev eral other p ossibilities, with v arying degrees of tec h nical difficult y; see [11, Section 5] and [29, Section 4] for sketc hes of t wo of these). Fix a smo oth triangulation of Z , with vertic es x i , and let U I b e the in- tersectio ns of sets in th e associated op en cov er, just as in Section 1.3 (but omitting th e real analytic it y cond ition). This time, we wan t to write do wn the asso ciated Cec h complex explicitly , h ence fix an ordering of the i ’s. Let Γ( U I ) b e the space of lo cally constan t K -v alued fu nctions on U I , whic h in our case is K if U I 6 = ∅ , and 0 otherwise. The Cech complex is (3.1) C = M I Γ( U I )[ − d ] where d = | I | − 1. This carries the u sual differenti al, and also a natural asso ciativ e pro duct making it in to a dg algebra. Namely , for every p ossible splitting of I ′′ = { i 0 < · · · < i d } into I ′ = { i 0 < · · · < i k } and I = { i k < · · · < i d } , one tak es (3.2) Γ( U I ) ⊗ Γ( U I ′ ) restriction − − − − − − → Γ( U I ′′ ) ⊗ Γ( U I ′′ ) multiplicat ion − − − − − − − − → Γ( U I ′′ ) . W e wan t to consider (unital righ t) dg mo dules o v er C . Denote the dg cat- egory of such mo dules by M = mod ( C ). This is not a dg m o del for the COT ANGENT BUNDLES 17 deriv ed catego ry: there are acycli c mo dules whic h are nontrivia l in H ( M ), and as a consequence, quasi-isomorphism do es not imply isomorphism in that category . All ob jects w e will consider actually b elong to a more restricted class, dis- tinguished b y a suitable “lo calit y” pr op ert y; w e call these dg mo dules of pr eshe af typ e . The defin ition is that suc h a dg mo dule E n eeds to admit a splitting (3.3) E = M I E I [ − d ] , where the su m is ov er all I = { i 0 < · · · < i d } s u c h that U I 6 = ∅ . Th is splitting is required to b e compatible w ith the differenti al and mo du le stru ctur e. This means that the differentia l maps E I to the direct su m of E I ′ o v er all I ′ ⊃ I ; that 1 ∈ Γ( U i ′ ) acts as th e identit y on E I [ − d ] for all I = { i 0 < · · · < i d } with i d = i ′ , and as zero otherwise; and that th e comp onen t E I ⊗ Γ( U I ′ ) − → E I ′′ of the mo d ule structure can only b e nonzero if I ′′ ⊃ I ∪ I ′ . In p articular, the “stalks” E I themselv es are sub quotien ts of E , and inherit a dg mo dule structure from that. The stalks asso ciated to the smallest subsets of Z (i.e. to m aximal ind ex sets I ) are actually c hain complexes, with all c hain homomorphisms b eing mo d ule endomorphisms, f rom wh ic h one can sho w: Lemma 3.1 . L et E b e a dg mo dule of pr eshe af typ e. If e ach E I is acyclic, E itself is i somorp hic to the zer o obje ct in H ( M ) . Here is a first example. Let P → Z b e a flat K -v ector b undle (or lo cal co efficient system). F or eac h I = { i 0 < · · · < i d } suc h that U I 6 = ∅ , define ( E P ) I = P x i d . F orm the d irect s u m E P as in (3.3). Equ iv alen tly , one can think of th is as the sum of P x i d ⊗ Γ( U I ) o v er all I , including empt y ones. The differentia l consists of terms ( E P ) I → ( E P ) I ′ for I = { i 0 < · · · < i d } , I ′ = I ∪ { i ′ } . If i ′ < i d , these are giv en (up to sign) by restriction maps P x i d ⊗ Γ( U I ) → P x i d ⊗ Γ( U I ′ ). In the remaining nontrivia l case where i ′ > i d and U I ′ 6 = ∅ , there is a unique edge of our triangulation going from x i d to x i ′ , and one com bines restriction with parallel transp ort along that edge. The mo dule stru ctur e on E P is defined in the ob vious w a y , f ollo wing the mo del (3.2); compatibilit y of the differen tial and the righ t C -mo dule structure is ensured b y our choic e ( E P ) I = P x i d , taking the fi b re of P o ve r the ve rtex corresp onding to the last index i d of I . Clearly , H ∗ ( E P ) ∼ = H ∗ ( Z ; P ) is ordinary cohomolog y with P -coefficien ts. Mo reo ve r, using the fact that ev ery E P is free as a mo du le (ignoring the different ial), one sees that (3.4) H ∗ ( hom M ( E P 0 , E P 1 )) ∼ = H ∗ ( Z ; H om ( P 0 , P 1 )) . In particular, referring b ack to the remark made ab ov e, this leads to quite concrete examples of acyclic dg mo dules w h ic h are n ev ertheless non trivial ob jects. 18 KENJI FUKA Y A, P AUL SEIDEL, IV A N SMITH Still within classical top ology , but getting somewhat closer to the in tended construction, supp ose n o w that Q → Z is a differentia ble fibre b undle, equipp ed with a connection. F or I = i 0 < · · · < i d with U I 6 = ∅ , d efine (3.5) ( E Q ) I = C −∗ ( Q x i d ) , where C −∗ stands for cubical co c hains, with the grading rev ersed (to go with our general cohomological con v en tion). As b efore, w e w an t to tu r n the (shifted) direct su m of these into a dg mo du le o v er C . The mod ule structure is straigh tforwa rd , b ut the differen tial is a little more in teresting, b eing the sum of thr ee terms. The first of these is the ordinary b ound ary op erator on eac h ( E Q ) I . The second on e is th e Cec h differen tial ( E Q ) I → ( E Q ) I ′ , where I = { i 0 < · · · < i d } and I ′ = I ∪ { i ′ } with i ′ < i d . The final term consists of maps (3.6) C −∗ ( Q x i d ) − → C −∗ ( Q x i ′ e )[1 − e ] where I = { i 0 < · · · < i d } , I ′ = I ∪ { i ′ 1 < · · · < i ′ e } with i d < i ′ 1 . T ak e the standard e -dimensional s implex ∆ e . It is a classical observ ation [3] that there is a natural family of piecewise smo oth paths in ∆ e , parametrized by an ( e − 1)-dimensional cub e, which join the first to the last ve rtex. I n our situation, the triangulation of Z con tains a u n ique simplex with vertice s { i d , i ′ 1 , . . . , i ′ e } , and w e therefore get a f amily of paths joining x i d to x i ′ e . T he resulting parametrized parallel transp ort map (3.7) [0; 1] e − 1 × Q x i d − → Q x i ′ e induces (3.6 ) (up to sign; the app earance of a cub e here motiv ated our u se of cubical c hains). I t is not difficult to sho w that the resulting tota l differen tial on E Q indeed squares to zero, and is compatible with the mo du le structure. Finally , let’s turn to th e sym p lectic analogue of this, in whic h one starts with a closed exact Lagrangian sub manifold L ⊂ M = T ∗ Z (sub ject to the usual conditions: if one wa nts Z -graded m o dules, L should b e graded; and if one wan ts to use coefficien t fi elds char ( K ) 6 = 2, it should b e relativ ely spin ). The appropr iate v ersion of (3.5) is (3.8) ( E L ) I = C F ∗ ( T ∗ x i d , L ) and again, E L is the sum of these. In the different ial, we no w use the Floer differentia l on eac h C F ∗ summand (replaci ng the d ifferen tial on cu b i- cal c hains), and con tin uation maps or their parametrized analogues (rather than parallel transp ort maps), whic h go v ern moving one cota ngent fi bre into the other. (Related con tin uation maps app eared in [24].) Of course, the de- tails are somewhat differen t from the previous case. T o get a c h ain map C F ∗ ( T ∗ x i d , L ) → C F ∗ ( T ∗ x i ′ 1 , L ), one needs a path (3.9) γ : R − → Z, γ ( s ) = x i d for s ≪ 0, γ ( s ) = x i ′ 1 for s ≫ 0. COT ANGENT BUNDLES 19 More generally , families of suc h paths parametrized b y [0; 1) e − 1 app ear. I n the limit as one (or more) of the parameters go to 1, the path b reaks up in to t wo (or more) pieces s ep arated b y increasingly long constan t stretc hes. This ensures that the us u al comp osition la ws for conti nuatio n maps apply , compare [33]. Still, with these tec hnical mo difi cations tak en into accoun t, the argument remains essen tially the same as b efore. 3.2. W rapp ed Floe r cohomolo gy. One n aturally w an ts to extend the previous construction to allo w L to b e noncompact (for instance, a cotan- gen t fib re). Th is w ould b e imp ossible using the v ersion of Floer cohomolo gy from Section 1.1, since that do es not hav e sufficien tly s tr ong isotopy in v ari- ance prop erties: H F ∗ ( T ∗ x , L ) generally dep ends strongly on x . Instead, we use a mo dified ve rsion called “wrapp ed Flo er cohomology”. This is not fu n- damen tally new: it app ears in [1 ] for the case of cotangen t bun dles, and is actually the op en string analogue of Viterb o’s symplectic cohomolog y [38]. T ak e a W einstein manifold M (complete and of fi n ite t yp e), and consider exact Lagrangian submanifolds inside it which are Legendrian at infinity; this time, n o r eal analyticit y conditions will b e necessary . W e again u se Hamiltonian functions of the f orm H ( r , y ) = h ( e r ) at infi nit y , b ut no w r e- quire that lim r →∞ h ′ ( e r ) = + ∞ . Th is means that as one go es to infin it y , the asso ciated Hamiltonian fl o w is an unb oundedly accelerat ing version of the Reeb flo w. Denote by C W ∗ ( L 0 , L 1 ) = C F ∗ ( L 0 , L 1 ; H ) = C F ∗ ( φ 1 X ( L 0 ) , L 1 ) the resulting Flo er complex, and by (3.10) H W ∗ ( L 0 , L 1 ) = H ( C W ∗ ( L 0 , L 1 )) its cohomology , wh ic h w e call wr app e d Flo er c ohomolo gy . This is indep en- den t of the c hoice of H . Moreo ve r, it remains in v arian t under isotopies of either L 0 or L 1 inside the relev an t class of sub m anifolds. Suc h isoto pies need no longer b e compactly sup p orted; the Legendrian submanifolds at infinit y ma y m o v e (whic h is exactly the prop ert y we w an ted to ha v e). By exploiting this, it is easy to define a triangle pro du ct on wrapp ed Flo er cohomolog y . In the case where L 0 = L 1 = L , w e ha ve a natural map from the ordinary cohomolog y H ∗ ( L ) to H W ∗ ( L, L ), whic h ho we ve r is generally n either in jec- tiv e nor surj ectiv e. F or in stance, for L = R n inside M = C n , H W ∗ ( L, L ) v anishes. Another, far less trivial, example is the follo wing one: Theorem 3.2 (Abb ondand olo-Sc hw arz) . L et M = T ∗ Z b e the c otangent bund le of a close d oriente d manifold, and take two c otangent fibr es L 0 = T ∗ x 0 Z , L 1 = T ∗ x 1 Z . Then H W ∗ ( L 0 , L 1 ) ∼ = H −∗ ( P x 0 ,x 1 ) is the (ne gatively gr ade d) homolo gy of the sp ac e of p aths in Z going fr om x 0 to x 1 . 20 KENJI FUKA Y A, P AUL SEIDEL, IV A N SMITH This is p ro v ed in [1]; the follo wup pap er [2] sh ows that this isomorphism sends the triangle pro duct on H W ∗ to the P on trya gin pro du ct (induced b y comp osition of paths) on path space h omology . W e w an t our wrapp ed Flo er co c hain groups to carry an A ∞ -structure, r e- fining the cohomology lev el pro duct. When defin ing this, one enco unters the same difficult y as in (1.4). Again, there is a solution b ased on a rescal- ing tric k: one tak es h ( t ) = 1 2 t 2 , and uses the fact that φ 2 X differs from φ − log (2) Y φ 1 X φ log(2) Y ( φ Y b eing the Liouville flo w) only by a compactly su p - p orted isotop y . This is p articularly int uitive for cotangen t bundles, where the radial co ord inate at infin it y is e r = | p | ; then, H = h ( e r ) = 1 2 | p | 2 giv es rise to the standard geod esic flo w ( φ t X ), and Y = p∂ p is the rescaling v ector field, meaning that φ log(2) Y doubles the lengt h of cotangen t v ectors. An alte r- nativ e (not iden tical, but ultimately qu asi-isomorphic) approac h to the same problem is to define wrapp ed Flo er cohomolo gy as a direct limit o v er func- tions H k with more mo derate gro wth H k ( r , y ) = k e r at infin it y . Ho w ev er, the d etails of this are quite in tricate, and w e will not describ e them here. In either wa y , one gets an A ∞ -categ ory W ( M ), which w e call the wr app e d F ukaya c ate g ory . F or cotange nt bundles M = T ∗ Z , a plausible cochai n lev el refinemen t of Theorem 3.2 is the follo win g Conjecture 3.3 . L et L = T ∗ x b e a c otangent fibr e. Then, the A ∞ -structur e on C W ∗ ( L, L ) should b e quasi-isomorphic to the dg algebr a structur e on C −∗ (Ω x ) , wher e Ω x is the (Mo or e) b ase d lo op sp ac e of ( Z, x ) . Returning to the general case, w e recall that a f undament al prop ert y of sym- plectic cohomology , established in [38], is V i terb o functoriality with resp ect to em b eddings of W einstein manifolds. On e naturally exp ects a corresp ond - ing pr op ert y to hold f or wrapp ed F uk a y a categories. Namely , tak e a b oun ded op en su bset U ⊂ M with smo oth b oundary , such that Y p oin ts out w ards along ∂ U . One can then attac h an infinite cone to ∂ U , to form another W einstein manifold M ′ = U ∪ ∂ U ([0; ∞ ) × ∂ U ). Supp ose that ( L 1 , . . . , L d ) is a fin ite family of exact Lagrangian submanifolds in M , whic h are Legen- drian at infinit y , and with the f ollo wing additional prop ert y: θ | L k = dR k for some compactly supp orted function R k , whic h in addition v anishes in a neigh b ourho o d of L k ∩ ∂ U . T his implies that L k ∩ ∂ U is a Legendrian submanifold of ∂ U . Again, by attac hing infinite cones to L k ∩ U , one gets exact Lagrangian su bmanifolds L ′ k ⊂ M ′ , whic h are Legendrian at infinit y . Let A ⊂ W ( M ) b e the full A ∞ -sub category with ob jects L k , and similarly A ′ ⊂ W ( M ′ ) for L ′ k . Then, Conjecture 3.4 (Ab ouzaid-Seidel) . Ther e is a natur al (up to isomor- phism) A ∞ -functor R : A → A ′ . COT ANGENT BUNDLES 21 Note that, ev en th ou gh we ha v e n ot menti oned th is explicitl y so far, all A ∞ - catego ries un der consideration ha v e units (on the cohomologi cal leve l), and R is supp osed to b e a (cohomol ogically) unital functor. Hence, the conjec- ture implies that v arious r elatio ns b et w een ob jects, suc h as isomorphism or exact triangles, p ass from A to A ′ , which is a nontrivial state ment in itself. 3.3. F amily Flo er cohomology revisited. F or M = T ∗ Z , there is a straigh tforw ard v ariation of the construction from S ectio n 3.1, using wr app ed Floer cohomology ins tead of ordinary Flo er cohomology . This associates to an y exact Lagrangian subm anifold L ⊂ M , whic h is Legendrian at infi nit y , a dg mo du le E L o v er C . In fact, it giv es rise to an A ∞ -functor (3.11) G : W ( M ) − → M = mod ( C ) . While little has b een rigorously p ro v ed so far, there are plausible exp ec- tations for ho w th is f unctor should b ehav e, w h ic h w e w ill no w formula te precisely . T ak e Q → Z to b e the p ath fibration (wh ose total sp ace is con- tractible). Ev en though this is not s tr ictly a fibre bund le, the construction from Section 3.1 app lies, and yields a dg mo d u le E Q , wh ic h has H ∗ ( E Q ) ∼ = K (this can b e view ed as a resolution of the simple C -mo du le). Conjecture 3.5 . L et L = T ∗ x b e a c otangent fibr e. Then, E L is isomorphic to E Q in H ( M ) . Mor e over, if Z is simply-c onne cte d, G g ives rise to a quasi- isomorp hism C −∗ (Ω x ) ∼ = C W ∗ ( L, L ) → hom M ( E L , E L ) . The first stat emen t can b e seen as a parametrized extension of Theorem 3.2. A p ossible pro of would b e to consist of c hec king that the chai n lev el maps constructed in [1] can b e made compatible with p arallel transp ort (resp ec- tiv ely con tinuat ion) maps, u p to a suitable hierarc h y of c hain homotopies. This w ould yield a map of d g mo dules; to pr o v e that it is an isomorphism , one would then app ly L emm a 3.1 to its mapping cone. The second part of the conjecture is less in tuitiv e. Th e assu mption of simple connectivit y is necessary , since otherwise the endomorph isms of E Q ma y n ot rep r o duce the homology of the based lo op space (see Section 3.4 for further discussion of this); ho wev er, it is not enti rely clear ho w that w ould enter int o a pro of. Conjecture 3.6 . Any one fibr e L = T ∗ x gener ates the derive d c ate gory (taken, as usual, to b e the homoto py c ate gory of twiste d c omplexes) of W ( M ) . There are tw o app aren tly quite viable approac hes to this, arising from the con texts of Sections 1 and 2, resp ectiv ely . T o explain the first one, w e go bac k to the general situation wh ere M is a fin ite t yp e complete W einstein manifold, whose end is mo delled on the con tact manifold N , and where r eal analyticit y conditions are imp osed on N and its Legendrian submanifolds. 22 KENJI FUKA Y A, P AUL SEIDEL, IV A N SMITH Then, if ( L 0 , L 1 ) are exact Lagrangian submanifolds w hic h are Legendr ian at infi nit y , w e hav e a natural homomorphism (3.12) H F ∗ ( L 0 , L 1 ) − → H W ∗ ( L 0 , L 1 ) , whic h generalizes the map H ∗ ( L ) → H W ∗ ( L, L ) mentio ned in Section 3.2. These maps are compatible with triangle pr o ducts, and ev en though th e details ha v e not b een c hec k ed, it s eems plausible that they can b e lifted to an A ∞ -functor F ( M ) → W ( M ). Actually , what one would lik e to u se is a v arian t of this, where F ( M ) is replaced b y the Nadler-Zaslo w category A ( M ), or at least a sufficien tly large f ull sub catego ry of it. Assuming that this ca n b e done, one can tak e the generators of A ( M ) pro vided b y the pro of of T heorem 1.4, and then map them to W ( M ), w h ere isotopy inv ariance ough t to ensur e that they all b ecome isomorphic to cotangen t fib res (n ote that in the wrapp ed F u k a ya category , any t w o cotangen t fibres are m u tually isomorphic). The second strategy f or attac king Conj ecture 3.6 is fundamen tally similar, but based on Lefsc hetz fibrations. Recall that with our d efinition, the total space of a Lefsc hetz fibration π : X → C is itself a finite t yp e W einstein manifold. One then exp ects to h a v e a canonical fun ctor F ( π ) → W ( X ). Giv en a Lefsc hetz fibration constructed as in Section 2.3 b y complexifying a Morse fu nction on Z , one wo uld then co mbine F ( π ) → W ( X ) with the func- tor from Conjecture 3.4, applied to a small neigh b ourh o o d of Z em b edd ed in to X . T he outcome would b e that the restrictio ns of Lefsc hetz th imbles generate the wrapp ed F u k a ya catego ry of th at neigh b ourho o d. One can easily c hec k th at all suc h r estrictions are isomorphic to cotangen t fibr es. Supp ose no w that Z is simp ly-connected. In that case, if one acce pts Con- jectures 3.5 and 3.6, it f ollo ws b y pur ely algebraic means that (3.11) is full and faithful. T ak e the dg mo dule E L = L I ( E L ) I [ − d ] obtained from a closed exact L ⊂ M , and equip it with the decreasing filtration by v alues of d = | I | − 1. The associated graded space is precisely the dg modu le E P con- structed fr om the local co efficien t system P x = H W ∗ ( T ∗ x , L ) = H F ∗ ( T ∗ x , L ). Hence, one gets a sp ectral sequence whic h starts with H ∗ ( Z ; E nd ( P )) ac- cording to (3.4), and conv erges to the group H ( hom M ( E L , E L )), whic h w ould b e equal to H F ∗ ( L, L ) ∼ = H ∗ ( L ; K ) as a consequence of the previously made conjecture. This explains ho w (0.1) arises from this particular approac h. 3.4. The (co)bar construction. There is a m ore algebraic p ersp ectiv e, whic h provides a shortcut through part of the argument ab o v e. T o explain this, it is helpful to recall the classical bar construction. Let C b e a dg algebra o ver our co efficient field K , with an augmen tation ε : C → K , whose k ernel w e denote by I . One can then equip the free tensor coalgebra T ( I [1]) with a differenti al, and then du alize it to a dg algebra B = T ( I [1]) ∨ . Con- sider the standard resolution R = T ( I [1]) ⊗ C of the simple C -mo du le C / I . COT ANGENT BUNDLES 23 One can pro v e that the endomorp hism dga of R as an ob ject of mod ( C ) is qu asi-isomorphic to B . F or standard algebraic reasons, this indu ces a quasi-equiv alence b etw een the s ub category of mod ( C ) generated by R , and the triangulated su b category of m od ( B ) generated b y the free mo dule B . Denote that categ ory b y mod f ( B ). The relev ance of this dualit y to our discussion is a basic connection to loop spaces, which go es bac k to Adams [3]. He observ ed that if Z is simply- connected, and C is the dg algebra of Cec h co c hains, then B is quasi- isomorphic to C −∗ (Ω Z ). Hence, rev ersing the functor ab o ve , w e get a full and faithful em b edding (3.13) H ( mod f ( C −∗ (Ω Z ))) − → H ( M ) , where M = mod ( C ) as in (3.11). If one moreo ver assumes that Conjec- tures 3.3 and 3.6 h old, it follo ws that W ( M ) itself is d erived equiv alen t to H ( mod f ( C −∗ (Ω Z ))). Hence, one wo uld get a full em b edding of the wrapp ed F u k a ya categ ory into H ( M ) for algebraic reasons, a v oiding the use of Cec h complexes altogether. 4. The non-simpl y-connected case In this final section, we d iscuss exac t Lagrangian sub manifolds in non- simply-connected cotangen t bundles. Sp ecifically , we prov e Corollary 0.3, and then mak e a few more observ ations ab out the wrapp ed F uk a y a cate - gory . 4.1. A finite co v ering t ric k. W e start by recalling the setup f rom Section 1. Fix a real analytic structure on Z , and th e asso ciated categ ory A ( M ). F or an y closed exact Lagrangian subm anifold L ⊂ M = T ∗ Z w hic h is spin and has zero Maslo v class, w e ha v e the sp ectral s equence E pq 2 = H p ( Z ; E nd q ( E L )) ⇒ H ∗ ( L ; K ) arising from the r esolution of L by “standard ob jects” in A ( M ). F rom no w on, we assum e that char ( K ) = p > 0. I n that case, one can certainly find a fin ite co v ering b : ˜ Z → Z such that b ∗ E L is trivial (as a graded bundle of K -v ector spaces). In fact, E L giv es rise to a represent ation ρ : π 1 ( Z ) → GL r ( K ), and one tak es ˜ Z to b e the Galois co v ering asso ciated to ker ( ρ ) ⊂ π 1 ( Z ). Set ˜ M = T ∗ ˜ Z , denote b y β : ˜ M → M the induced co vering, and by ˜ L = β − 1 ( L ) the preimage of our Lagrangian submanifold. This inh erits all the pr op erties of L , hence we ha v e an analogous sp ectral sequence for ˜ L . Note also that for ob vious reasons, the asso ciated b undle of Floer cohomologie s satisfies (4.1) E ˜ L ∼ = b ∗ E L , 24 KENJI FUKA Y A, P AUL SEIDEL, IV A N SMITH hence is trivial by definition. No w, the discussion after (0.1) carries o v er with no problems to the non-simply-connected case if one assu mes that the Lagrangiaan su bmanifold is connected, and its bun dle of Floer cohomologies is trivial. In particular, one gets that ˜ L is in fact connected: if it weren’t, the analogue of Theorem 0.1(iii) would apply to its connected comp onent s (since their Flo er cohomology bun dles are also trivial), implying that an y t w o w ould ha v e to inte rsect eac h other, whic h is a cont radiction. With connectedness at hand, it then follo w s by the same argumen t that (4.1) has one-dimensional fibres. Hence, E nd ( E L ) is trivial, whic h means that the sp ectral sequence for the original L degenerates, yielding H ∗ ( L ; K ) ∼ = H ∗ ( Z ; K ). In fact, b y b orro wing argumen ts from [29] or from [15], one sees that in the F uk ay a category of M , L is isomorphic to the zero-section equipp ed w ith a suitable spin structure (the difference b et w een that and the a priori c hosen sp in structure on Z is p recisely d escrib ed b y the bun dle E L , whic h, note, has stru cture group ± 1). Using that, one also gets the analogue of part (iii) of T heorem 0.1. Finally , the cohomologi cal restricti ons (i)-(iii) of Theorem 0.1 for arbitrary fields of p ositiv e c haracteristic imply them for K with char ( K ) = 0, whic h complete s the pro of of Corollary 0.3. 4.2. The Eilenberg-MacLane case. F or general algebraic reasons, there is an A ∞ -functor (4.2) W ( M ) − → mod ( C W ∗ ( T ∗ x , T ∗ x )) . Here, the righ t hand side is the dg category of A ∞ -mo dules ov er the en- domorphism A ∞ -algebra of the ob ject T ∗ x . No w s u pp ose that Z = K (Γ , 1) is an Eilenberg-MacLane space, so that the cohomology of C W ∗ ( T ∗ x , T ∗ x ) is concent rated in degree zero. By the Homologica l P erturbation Lemma, this imp lies that the A ∞ -structure is formal, hence b y Theorem 3.2 quasi- isomorphic to the group algebra K [Γ] (this argumen t allo w s us to a v oid Con- jecture 3.3). Ho w ev er, in order to mak e (4.2) useful, w e do w an t to app eal to (the current ly un pro ve n) C onj ecture 3.6. Ass u ming from now on th at that is true, one finds that (4.2) is a full em b edding, and actually lands (up to functor isomorphism) in the sub category mod f ( K [Γ]) generated by the one- dimensional free mo dule. As far as th at s u b category is concerned, one could actually replace A ∞ -mo dules b y ordinary chai n complexe s of K [Γ]-mo d ules. This giv es a picture of W ( M ) in classical algebraic terms. T o see ho w this might b e usefu l, let’s dr op the assumption that the Maslo v class is zero, and consider general closed, exact, and spin Lagrangian sub- manifolds L ⊂ M . Neither of the t wo arguments in fa v our of Conjecture 3.6 sketc hed in Section 3.3 actually u ses th e Maslo v index. It is therefore plausible to assume that the d escription of W ( M ) explained ab o v e still ap- plies. Denote b y N L the Z / 2-graded A ∞ -mo dule o v er K [Γ] corresp onding to L , and b y N L its cohomology mo dule. I n fact, N L is nothing other than COT ANGENT BUNDLES 25 our previous family Flo er cohomology bundle E L , now considered as a mo d - ule o v er Γ = π 1 ( Z ). T h ere is a purely algebraic obstruction theory , which determines when an A ∞ -mo dule is formal (isomorphic to its cohomolog y). The obstructions lie in (4.3) E xt r K [Γ] ( N L , N L [1 − r ]) , r ≥ 2 where the [1 − r ] no w has to b e in terpreted mo d 2. In particular, if all those groups v anish, it w ould follo w directly that N L is formal. Ho we ve r, there is no particular a priori reason wh y that should happ en. Returning to the trick from Section 4.1, assu m e no w that char ( K ) = p > 0. Then, after passing to a finite co v er, one can assume that N L is a direct sum of trivial represen tations, hence (4.3) is a direct sum of copies of H r (Γ; K ). One can try to kill the relev ant obstructions by passing to further fin ite co vers. Generally , this is u nlik ely to b e successful (there are examples of mo d p cohomolo gy classes wh ic h sur v ive pullbac k to an y finite co ver, s ee for instance [10, T heorem 6.1]). Ho w ev er, in the sp ecial case w here H ∗ (Γ; K ) is generated by degree 1 classes, suc h as for surfaces or tori, it is ob viously p ossible. In those cases, one could then find a finite co ve r b : ˜ Z → Z , inducing β : ˜ M → M , su c h that the A ∞ -mo dule N ˜ L asso ciated to ˜ L = β − 1 ( L ) is isomorphic to a direct sum of ordinary trivial mo dules. Just by lo oking at th e endomorphism ring of this ob ject, it b ecomes cle ar that there can actually b e only one su mmand, so ˜ L is isomorphic to the zero-sect ion. One can then r eturn to M b y the s ame argument as b efore, and obtain the same consequences as in the original Theorem 0.1. This would b e a p oten tial application of the mac hinery from Section 3 whic h has no ob vious coun terpart in the other app roac hes. 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