4. Deus ex machina: $l_2$-cohomology

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1 4 Deus ex mchin: $l_$-cohomology Objekttyp: Chpter Zeitschrift: L'Enseignement Mthémtique Bnd (Jhr): 43 (997) Heft 3-4: L'ENSEIGNEMENT MATHÉMATIQUE PDF erstellt m: Nutzungsbedingungen Die ETH-Bibliothek ist Anbieterin der digitlisierten Zeitschriften Sie besitzt keine Urheberrechte n den Inhlten der Zeitschriften Die Rechte liegen in der Regel bei den Herusgebern Die uf der Plttform e-periodic veröffentlichten Dokumente stehen für nicht-kommerzielle Zwecke in Lehre und Forschung sowie für die privte Nutzung frei zur Verfügung Einzelne Dteien oder Ausdrucke us diesem Angebot können zusmmen mit diesen Nutzungsbedingungen und den korrekten Herkunftsbezeichnungen weitergegeben werden Ds Veröffentlichen von Bildern in Print- und Online-Publiktionen ist nur mit vorheriger Genehmigung der Rechteinhber erlubt Die systemtische Speicherung von Teilen des elektronischen Angebots uf nderen Servern bedrf ebenflls des schriftlichen Einverständnisses der Rechteinhber Hftungsusschluss Alle Angben erfolgen ohne Gewähr für Vollständigkeit oder Richtigkeit Es wird keine Hftung übernommen für Schäden durch die Verwendung von Informtionen us diesem Online-Angebot oder durch ds Fehlen von Informtionen Dies gilt uch für Inhlte Dritter, die über dieses Angebot zugänglich sind Ein Dienst der ETH-Bibliothek ETH Zürich, Rämistrsse 0, 809 Zürich, Schweiz, wwwlibrryethzch

2 Let Mbe complex surfce, nd ssume thï its fund mentlgroup G fulfills p(g) > 0 Then the holomorphic Euler chrcteristic PROPOSITION M is > 0 By the Kodir-Enriques clssifiction it follows tht M cnnot be ruled over curve genus > REMARK The formule bove leding to the holomorphic Euler chrc teristicrefer to the orienttion the complex surfce dictted by the complex structure Thus the rgument is vlid only if in tht orienttion (M) < 0 If however (M) > 0 then p(g) > 0 implies tht -/3i(G) + /^ wrong (M) > 0 nd is = (5y(M) Now where Bt orienttion rl wrong refers to the "wronç" /3+(M) > /Ç(M) by ssumption Thus the resuit remins true; the holomor phicchrcteristic is > 0 III) Donldson Theory Finitely presented groups G with p(g) > 0 nd A (G) > 4 do not qulify for the Theorems A,B, nd C Donldson [D] relting to non-simply connected topologicl mnifolds Indeed in thèse or theorems the signture is ssumed to be négtive with /5 + However p(g) >0 mens -A(G) + ft+(m) >0, ie (3+ (M) > /i(g)- + =0, 4 4 Deus ex mchin: $l_$-cohomology We recll in few words the (cellulr) définition nd / -Betti numbers, in the cse -cohomology 4-mnifold M but things pply to ny finite cell-complex Some définitions : For ny countble group G let hg be the Hilbert squre-integrble rel functions on G, with G operting on the left, spce nd NG the lgebr bounded G-equivrint liner opertors on hg A Hilbert- G -module H is Hilbert spce with isometric left G -ction which dmits n isometric G-equivrint imbedding into some lig m (direct sum m copies / G) The projection opertor Gm with imge H is given by mtrix (o*/), (j) ki e NG The "trce" (<fe(l), ) is the von Neumnn dimension dim G H; it is rel number > 0, nd = 0 if nd only if H = 0 Let M be the universl cover M with the cell-decomposition corre spondingtotht chosen in M The squre-integrble rel z'-cochins M constitute Hilbert spce () (M) with isometric G -ction It décomposes into the direct sum i copies / G, i= 0,,4 As before,- dénotes the ll </> C()(M)C

3 number z-cells M ; G is the fundmentl group M cting by permu ttionthe cells M The Cf with the induced coboundry opertors form Hilbert- G -module chin complex The cohomology H tht complex is esily identified with H (M; hg), cohomology with locl coefficients (see, eg [E]) The reduced cohomology group H (ie cocycles modulo the closure coboundries) tht complex cn be imbedded in C[) s G -invrint subspce nd is therefore Hilbert- G -module Its von Neumnn dimension diirig#'(m) is the z-th / -Betti number /3 t (M) It is topologicl, even homotopy, invrint M Since dim G C[ ) =t nd since the von Neumnn dimension behves rnk, the usul Euler-Poincré rgument shows tht the /-Betti numbers compute the Euler chrcteristic exctly s the ordinry Betti numbers do 4 like : Moreover the j3 t closed mnifold fulfill Poincré dulity Thus According to Atiyh's -signture theorem [A], (M) cn lso be ex pressedby pproprite /-Betti numbers: H (M) splits into two comple mentryg -invrint subspces with von Neumnn dimensions f3 (M) nd (3 (M), nd (M) is their différence Thus, s with ordinry Betti numbers, ll one hs / We now ssume G to be infinité Then 0 (G) 0 Indeed 0-cocycle in M is constnt nd if M is n infinité complex cn be l only if it is =0 P0(G)P / THEOREM Pi(G) is If for finitely presented group 0 then the invrints COROLLARY 3 l -Betti number p(g) nd q(g) re non-negtive // /^(G) =G then def(g) G the first ll < COROLLARY 4 If G= it\(complex surfce M) with P {G) holomorphic Euler chrcteristic M is non-negtive X =0 then the 43 There re mny groups for which it is known tht /^(G) =OA good list is given in [B-V] We mention hère three big nd interesting clsses groups with tht property

4 generted menble groups [C-G] We recll tht this clss includes the virtully solvble groups, thus in prticulr the finitely generted Abelin groups (whence exmple ) in ) [Actully for n menble finite ) G with K(G, type, ie there is K(G, ) with finite group ) Ail finitely Z\ m-skelet, il / -Betti numbers re o] THEOREM 5 If G is finitely presenîed menble group then p(g) nd q(g) re non-negtive ) [Ll] Ail finitely presented groups G contining n infinité finitely generted norml subgroup TV such tht there is in G/N n élément infinité order For thèse "Liick groups" one hs the sme conclusions s in the menble cse In [Ll] the subgroup N is ssumed to be finitely presented Lùck hs shown lter [L] tht the weker ssumption bove is sufficient 3) The sttement Theorem 5 lso holds more generlly for finitely presented group G which contins finitely generted norml subgroup N such tht G/N is infinité nd menble [E] The pro is somewht différent : It mkes use not the universl cover but the cover belonging to N The menble group G/N opérâtes on tht cover nd one cn use the /-Betti numbers reltive to G/N A simple exmple is given by group with finitely generted commuttor subgroup nd infinité Abelinistion 44 Remrks for finitely presented infinité menble groups, nd lso for groups s in 43, 3) bove, the deficiency is < This cn lso be proved without 4-mnifolds : It suffices to consider K(G, ) with -skeleton corresponding to présenttion G ) We note tht ) since It is well-known tht group with deficiency > cnnot be menble it contins free subgroups rnk > ; see [B-P], where stronger resuit is proved 3) There is clss groups for which f3 is positive: The groups l I G with infinitely mny ends (ie with (G\ZG) infinité rnk; hère HI(G\ZG)H one tkes ordinry cohomology with locl coefficients) A nice pro for this cn be found in [B-V] Another pproch is to use Stllings' structure theorem from which it follows tht thèse groups contin free subgroups rnk > nd thus re non-menble For non-menble the Guichrdet groups menbility criterion [G] tells tht h\g\ G) = Hl(G;H l (G;l G) The coefficient

5 mp HI(G;ZGH I (G;ZG) >H l (G\l G) induced by the imbedding seen to be injective Since we hâve ssumed The vnishing 5 H (G; ZG) ZG > G 0 the resuit is esily follows q(g) 5 Hère we mention in few words wht hppens when for finitely presented group G the invrint q(g) is 0 For the détils nd more comments we refer to the pper [E] We thus consider 4-mnifold M with m (M) = G nd x(m) = 0 Since we restrict ttention to groups with /^(G) = 0 the vnishing (M) = 0 As shown in [E] by spectrl séquence rgument it follows tht H (M\ ZG) is isomorphic to H (G;ZG), ordinry cohomology with locl coefficients ZG By Poincré dulity H (M; ZG) = H (M\ ZG) which cn be identified with H (M\ Z) Since M is simply connected, H (M; Z) is isomorphic to the second homotopy group X(M) implies = 0, / (M) whence HH 7T (M) = 7T (M) Wht bout #3 (M;Z) It cn be identified with H3 (M;ZG) which, by Poincré dulity, is H\M\ZG) = H\G\ZG) This group, the "endpoint group" G, is known to be either 0 or Z or infinité rnk As mentioned in 44, remrk 3) the ltter cse is excluded by our ssumption /^(G) = 0 The cse H (G; ZG) = Z is exceptionl : it mens tht G is virtully infinité cyclic, nd we exclude this Then H-S(M\Z) 0 We now dd the ssumption tht // (G;ZG) =0 This is property shred by mny groups (eg dulity groups) Then the homology groups,, 3,4 (/ =4 becuse M is n open mnifold) Hi(M;Z) re =0 for Thus il homotopy groups M re = 0, M is contrctible, M is K(G, ), 5 /= nd the group G fulfllls Poincré dulity Let G be n infinité, finitely presented group, not virtully cyclic, infinité fulfilling J3 (G) = 0 nd H (G\ZG) = 0, nd let M be mnifold with fundmentl group G If the Euler chrcteristic x(m) = 0, then M is n Eilenberg-McLne spce for G nd G is Poincré dulity group dimension 4 THEOREM 6 X We recll tht for knot groups nd -knot groups q(g) =0, see exmples 3) nd 4) in Theorem 6 cn only be pplied to -knot groups which re not clssicl knot groups since the ltter hâve cohomologicl dimension