Kitahara, Haruo

Yorozu, Shinsuke

Annales de l’institut Fourier (AIF)

ANNALES DE L’INSTITUT FOURIERHARUO KITAHARASHINSUKE YOROZUOn the Cech bicomplex associated withfoliated structuresAnnales de l’institut Fourier, tome 28, no 3 (1978), p. 217-224<http://www.numdam.org/item?id=AIF_1978__28_3_217_0>© Annales de l’institut Fourier, 1978, tous droits réservés.L’accès aux archives de la revue « Annales de l’institut Fourier »(http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé-nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa-tion commerciale ou impression systématique est constitutive d’une in-fraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/Ann. Inst. Fourier, Grenoble28, 3 (1978), 217-224ON THE CECH BICOMPLEX ASSOCIATEDWITH FOLIATED STRUCTURESby H. KITAHARA and S. YOROZU1. We shall be in C°°-category. Let M be a paracompact connected^-dimensional manifold with a foliation ^ of codimension q , andlet ^U = {U^} be a simple covering of M such that each U^ isa flat neighborhood with respect to ^ . Then there exists a decom-posable q-form w = w1 A . . . A wQ on each U^ and, by Frobenius'theorem, there exists a 1-form 17 on eaclwU^ satisfying dw = w A 17,where d denotes the exterior differentiation and A the exteriorproduct. The 1-form 17 is an interesting object; it is well knownthat T? ^(dr])q defines a de Rham class in H^^M.R) ([I] , [2],[3]). Our aim is to show that 77 itself defines a certain cohomologyclass, that is,THEOREM A. - ((-I^-V^OT? defines a D-cohomology classin H^C^^A^M^D) depending only on Si.The above theorem was announced in [4], where it containedmisstatements.THEOREM B. — Supposing M admits filiations ^ , ̂  comple-men tally transversal to each other, 17 defines a D'-cohomology classin H^C^CUiA^M^.D'KCf. [5], [6], [7]).2. Since M has a foliation ^ of codimension q, the tangentbundle TM of M has an integrable subbundle E with fibre di-mension n-q. Let Q = TM/E be a quotient bundle with fibredimension q. Choosing a suitable Riemannian metric on TM,we obtain an isomorphism TM ^ E ® Q (Whitney sum).218 H- KITAHARA AND S. YOROZUThendW = ^ dt[ A w7 4- ^ t\ dw1i ]=2:(^-1:^) A W 7 ,/ v A:and on the other handdw1 = ^ w^ A (^I(-S^)AW/-./ fcThus-S^^-S^+l:/^*: ^ fcwhere ^ are functions on LL H U^Let / s^ 0 \ denote the inverse matrix o f /^ - 0 \ .M) ^^ ^0 ^>'b oThenZ ̂  ̂  ̂  = - S ̂  ̂  + S ̂  ̂  ̂  - S ̂  ̂  ̂/^ / /,fc Afcand we obtainW=- S^^J+S^ - I ^/j.^.» ij / i, j , kFrom (4), dtj = ̂  ^ w^ . . Thus, by (3), we obtain ̂  ̂  = S ̂ 'Ar T TTherefore we obtain ^ = T? on U^HU^ and 77 € C0'1 (CU;A*(M)).Q.E.D.THEOREM B. — Supposing M admits foliations ^,^' complemen-tally transversal to each other, 17 defines a D'-cohomology class inH1 (C^CZl ; A*(M)), D') w/z^ rf is defined by ^.teCH BICOMPLEX ASSOCIATED WITH FOLIATED STRUCTURES 21 9We may suppose that "il = {U^} is a simple covering such thateach U^ is a locally trivial neighborhood of the bundle Q —^ M.Let V0' denote a local connection on Q l r j , and let oj^.a(resp. 12^) denote a connection form (resp. a curvature form) ofV^ on U^. Let A^ be a canonical p-simplex in RP^1 (with coor-dinates (^o, ^ , . . . , t ) ) . We define a connection form 0;̂  ...aon U^...^ x A^ (U^.,^ = U^ 0 U^ H . . . 0 U^) byc^o...^ = ^0^0 + . . .+ ^^p= (l _ ̂  - : . . - t p ) ̂  + r^ + . . . 4- t^and let ?2^ a denote a corresponding curvature form on^o-p^- ' pFor the set A^(U^...a x A^) of all fc-forms on VaQ...a x A ^ ,f : A^CU^ a xA^)—^A^-^CU^ ^ ) denotes the integra-^PJ 0 ' " p 0 ' " ption along the fibre. Then we obtain Stokes' theoremto d= (-IT d o f + fo /*A^ ^PJ ^PJwhere / : U^ ^ x bA? ——> U^ ...a x Ap denotes the inclusion.We consider Cech bicomplex C^^U ;A*(M)) : LetCP'^ == n A^(U^ ^ ), and let D ' l C ^ — ^ ^+1^ denoteOiQ...Oip 0 Pthe ordinary simplicial differential and D" = (- \)P d: C^^ —^ C"^^the de Rham differential. A multiplication-; 6^ ® C^''^'—> C P ^ ' ' ^ 'is defined by.(^ • ̂ k-..̂  = (-D-'^o...^ <..,̂  - S...̂  <...̂  •For ^CU ;A*(M)) = ^ 6''̂  and D = D' + D": C^ -^C^1),p+<?=fcwe obtain a graded algebra (C^CU ; A*(M)), D, •).T.et I*(g(,) denote a graded algebra of invariant polynomialson a Lie algebra gl^. A caracteristic homomorphism7: I*(flI,)—>C<*)(<U;A*(M))is defined by^=^^Y'^k~'> ^ei^)p220 H- KITAHARA AND S. YOROZUwhere(̂ C ;̂ = ,J ^o-"p' • • • ' ^o-v> •Then we obtain,LEMMA 1 (Cf. [8]). - For ^ G 1 )̂,(_1)[P(P-1)]/2 ^-^! . )P,2^-pv 7 fc ! ^lv)ao•••apf dt^ A . . . A dtp A ̂ (̂  - G^o . ̂  - ^ao - • - <^ - ^OQ . ̂ ^.p^)A p <A;0 ' p > kwhere I\...^ = ^0...^ - S ̂  A (a;a, - ^ao) •Remark. — j induces the Chem-Weil homomorphism7*: I*(g^)—>H*(6^,D)^^H*(M).The following lemma is easily proved.LEMMA 2. - Let w 1 , . . . , W'7 be 1-forms on U^ such thatw1 A . . . A w^ =^ 0 on Va. P^r w = w1 A . . . A w^. TTz^z (i) fl/^rf(ii) are equivalent:(i) 7%^ exists a (q,q)-mqtrix (<^1) of 1-forms on Uo: ^cArtar dw1 = S w7 A (^f .(ii) r/zere exists a 1-form r] on \Ja such that dw = w A T? .Remark. - The existance of the matrix (<^f) doesn't dependon choice of q 1-forms w1,. . . , wQ on Ua .Remark. — In the proof of this lemma, we obtain^(-i)9-1 S^ - a)iLet r(-) denote the space of all sections of bundle. The Bottconnection ^ : F(E) x F(Q) —> F(Q) is defined byv^z = 7r^([x, z]) x E F(E) , z e F(Q)where Z G F(TM) such that 7i^(Z) = Z and TT : TM —> Q. LetCECH BICOMPLEX ASSOCIATED WITH FOLIATED STRUCTURES 221^/. ^} (1 < i < < ? , ^ + 1 < a < n) be a local basis dual to {w1, w°}on U^ satisfying ^ ^ r ( Q j ^ ) and ^ ^ r ( E | ^ ) with respect tothe isomorphism TM ^ E ® Q . Hereafter, we suppose that the indicesrun the following ranges : 1 < ;, /, k , . . . < q, q + 1 < a, b, . . . < n.We define a connection ^a on U^ byV^Z = V^Z + S XQ(ZO^ + S Z'^(XQ)^ (2)» i,kwhere X=Xg + Xp G F(E |^) ® r(Q[u^) and Z= ^Z^,er(Q|u )./We put V^ <?, = ^ G;'̂ . (X)e, , that is, ,̂ denotes the connectioniform of V" on U^.LEMMA 3. - < .̂ = î : o« U^.Proo/ -Weput VXE e/ = S ^(XE) ̂ , thenf^/X) == o5;:(XE) + ^'(XQ ) . Now we obtaindw' (e,, €,) = -^ {e,(w'(e,)) - e, (w'(ej) -w' ([e,,<?,])}=-^-S;(^).On the other hand,dw'(e,,e,)= (Sw^^)^,^))^=-^^).Thus we obtain ^(Xg) = ^(XE ) . Therefore, for anyX.G r(TMj^), c^,(X) = ^(XE) + ^(XQ) = ^(X). Q.E.D.LEMMA 4. — // we consider the connection V0'0 defined by( 2 ) on U^ and a Riemannian connection V0 1 OM U^ , then, for^i e I1 <91,), (7^i )^^ = ((-)'7- l /2 f f )»? . 1222 H- KITAHARA AND S. YOROZUProof. — By Lemma 1,(Wi)^! = / dtt A ^1(0;^ -co^)A1""^(^i-^o)-Since ^ £ 1̂  (g(g) is defined bydet(XL, -(l/2ir)X) = ̂  <^(X) X^ , X € a1, and trace (a;^.) = 0 ,we obtain (7^1 )<^Li ^ (l/27r) trace (c^oy) • By (!) and Lemma 3,(Wi)^ = ((-D^'^ir) T? . Q.E.D.From this lemma, we obtainTHEOREM A. - ((—l) ' t ~ l /27r )T? defines a D-cohomology class inH2 (6" (11 ; A* (M)), D) depending only on S>.Proof. - If 7^1 € 6<2) (11 ; A*(M)) is D-closed, then particularobject (7^1 )^^ is D-closed and, by Lemma 4,((-ly-'^TOT?^ C^CUiA^M)) define a D-cohomology class inH2 ((^ ('U ; A* (M)), D). Now we prove that 7^1 is D-closed.(D(7<Pi))^\ = (D'(w.) + D"(7^))^-(D^,))^ +(D"(7^))Lo2./= «Wi )2,2 - ̂ i )So2 }+(-!) (^(7^i ))^, .From Stokes' theorem,(-Dd. /^(n^,,)- f°d^(n^^)- /^.(ft,^)A 1 A1 3A1and the left side of this is equal to (—l)^^^!),^ , the first termof the right side vanishes and the second term of the right side is equalto ^(^i)-^"^- Jhusd'^^^=^(^al)-{P^ao)•From this and (7^1 )^2 = ̂  (Sl ̂ ), we obtain(D(Wi))^i = {^(^)-^ (^)}+(-l){<Pi(n^)-^(S2^)}= 0 .From (7^1 ) '̂2 = ^i(^oo) and do^(n^) = o , we obtain(D(7^))^3 =(D"(7^.))^3=(-l)o(rf(7^))o^= 0 .CECH BICOMPLEX ASSOCIATED WITH FOLIATED STRUCTURES 223Now, from lemma 1, (7^) e 6°.2 4- ^u . Thus we obtain(D(7^))^^^3 -Oand(D(7^))^., =(D'(7^))^^= ^1(^02) ~"^l(°^i) -^(^^^(^ao)-^(^i)-^! (<^o)= 0 .Therefore we obtain D(7<^ ) = 0. Q.E.D.3. For q 1-forms w 1 , . . . , w^ on U^ (w1 A ... A w^ =^ 0) wemay choose n—q 1-forms w^',..., w" on V^ such thatW 1 A . . . A W ^ AW^'^1 A . . . A W " ^ 0 On UThus we obtain expressions^ == S ̂  ̂  + I ^a ̂  -A: a^ = S ̂  ̂  + S ̂  ̂  •A: aUsing same letters ^ , T? to simplify, we put^ =11^^ ° and T? = S ̂  ^a (3)w"a aon U- .Hereafter, we suppose that the^manifold M admits foliations ^, ̂complementally transversal to each other and that ^ (resp. ^ ' ) is ofcodimension q (resp. n - q). Then we may consider that w1 ,..., wqare defined by ^ and that w^ 1 , . . . , w" are defined by ^ .Let 1-forms w', w0 , ( '̂, T? on U^^ correspond to 1-formsw1, wa , ̂ ?, T? on U^ respectively. Then we obtainLEMMA 5. - On U^ n U^ (^ ^) , rf = 77 .Proo/ - On U^ H U ^ ^ , we may putw1 = S rj w7 , w0 = ^ ̂  w\ (4)/ b224 H- KITAHARA AND S. YOROZUProof. - We take 7?^ (resp. 7^) for 77 on U^ (resp. 77 on U^).Then we obtain^ - ^o = ° on u^ n ua! -and^^-^-^o'0-Thus T? is D'-closed. Q.E.D.BIBLIOGRAPHY[1] R. BOTT, Lectures on characteristic classes and foliations, LectureNotes in Math.,Springer, 279 (1972), 1-94.[2] C. GODBILLON and J. VEY, Un invariant des feuilletages de codi-mension 1, C.R. Acad. Sci., Paris, 273 (1971), A92-95.[3] F.W. KAMBER and P. TONDEUR, Foliated bundles and characteristicclasses, Lecture Notes in Math., Springer,493 (1975).[4] H. KITAHARA and S. YOROZU, Sur Fhomomorphisme de Chern-Weillocal et ses applications au feuilletage, C.R. Acad. Sci., Paris,281 (1975), A703-706.[5] G. REEB , Sur certaines proprietes topologiques des varietes feuil-letees, Hermann (1952).[6] Y. SHIKATA, On the spectral sequences associated to foliated struc-tures, Nagoya Math. /., 38 (1970), 53-61.[7] Y. SHIKATA, On the cohomology of bigraded forms associated withfoliated structures,^//. Soc. Math. Grece, 15(1974), 68-76.[8] A. So, J.C. THOMAS and C. WATKISS, Sur la multiplicativite deThomomorphisme de Chem-Weil local, C.R. Acad. Sci., Paris,280 (1975), A369-371.Manuscrit re^u Ie 12 avril 1977Propose par G. Reeb.H. KITAHARA and S. YOROZU,Department of MathematicsCollege of Liberal ArtsKanazawa UniversityKanazawa 920, Japan.

On the Cech bicomplex associated with foliated structures

Numérisation de Documents Anciens Mathématiques

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On the Cech bicomplex associated with foliated structures

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Annales de l’institut Fourier (AIF)

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