Shibata, Katsuyuki

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AstérisqueKATSUYUKI SHIBATASullivan-Quillen mixed type model for fibrations and theHaefliger model for the Gelfand-Fuks cohomologyAstérisque, tome 113-114 (1984), p. 292-297<http://www.numdam.org/item?id=AST_1984__113-114__292_0>© Société mathématique de France, 1984, tous droits réservés.L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/Publications/Asterisque/) implique l’accord avec les conditions générales d’uti-lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ouimpression systématique est constitutive d’une infraction pénale. Toute copieou impression de ce fichier doit contenir la présente mention de copyright.Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/K. SHIBATA SULLIVAN-QUILLEN MIXED TYPE MODEL FOR FIBRATIONS AND THE HAEFLIGER MODEL FOR THE GELFAND-FUKS COHOMOLOGY by Katsuyuki SHIBATA (Saitama University) 1. Introduction (The Bott "Conjecture"). oo Let M be a paracompact Hausdorff C -manifold of dimension n > 1 and L be the topological Lie algebra of C -vector fields on M. Gelfand-Fuks [1J considered the differential graded algebra (DGA for brevity) C (L ) of continuous j£ C M cochains of L̂ , and its cohomolgy H (Ĉ CL̂ )) is called the Gelfand-Fuks cohomo- logy of M. On the other hand, let EU ( 2 n ) n B U( 2 n > n be the universal U -bundle restric-n ted over the (homotopical) 2n-skelton of the base and (1.1.) Yn: EU ( 2 n ) n EU ( 2 n ) n x EU •> BU U n n be the associated fiber bundle over BU with fiber EU^n^ . And let be the n n C complexification of the tangent bundle of M classified by a map f̂  : M •> B U n-Consider the cross-section space T((f5) * (Y )) °f tne induced bundle C (f̂ ) * (yn) equipped with the compact open topology. Then the Bott "Conjecture" asserts ; (1.2.) H*(C*(LM)) s H*(r((f£) * (yn));R). A. Haefliger [3] , [_4] affirmatively solved this conjecture by constructing a Sullivan-Quillen mixed type model for the fibration (fM) * (y ) . Here, by a. Sullivan-Quillen mixed type model for a fibration, we mean a DG Lie algebra 292 SULLIVAN-QUILLEN MIXED TYPE MODEL L = A 0 L over a DGA A with a differential d, whose restriction (A ,d A = d̂ ) is a model for the base space in the sense of Sullivan and whose quotient (L = R ® L, 1 8 d) is a model for the fiber in the sense of Quillen. A A The superiority of the mixed type model lies in the following fact. The cochain complex Ĉ (L) over A of L is a Sullivan model for the total space of the fibration while the cochain complex C*(L) over R of L is a Sullivan model for the cross-section space of the fibration. But if we want to construct a mixed type model on the universal level, i.e. C * >s a model for y itself instead of the induced one (fw) * (Y ) » we have no 'n M n longer a differential on L but a pair (D,x) of a derivation D on L and the Euler element x ^ n L_2 * X being the obstruction for D to be a diffe­rential and, at the same time, being a representative for the obstruction class to the existence of a cross-section of the fibration. In this note we give a sketch of the following two subjects, the details of which will appear elsewhere. First we present a general view of the Sullivan-Quillen mixed type model in section 2, generalizing the Haefliger-Silveira theory of mixed type model for fibrations with a given cross-section [7]. In section 3, we exhibit a very explicit description of the mixed type model for the fibration (1.1.), and thus give a complete answer to the algebraic computational problem posed by Haefliger [3]. We remark that partial results to this problem permitted us to deduce the following result. THEOREM (1.3)([6]) : A closed connected orientable manifold M of dimension ̂  1 has finitely generated Gelfand-Fuks cohomology (as an R-algebra) if and only if M - S1. I am greatly indebted to S. Hurder's suggestion for accomplishing my compu­tations of the differential in Haefliger*s model. I also owe a great deal to A. Haefliger for suggesting me to generalize the mixed type model theory to fibra­tions without cross-section. Finally the discussions with H. Sliga clarified me the role of the Euler element in the mixed type model. 2. Sullivan-Quillen mixed type model for fibrations. Let A (= A ) be a positively graded DGA with a differential d and L id be a graded Lie algebra over A with the grading deg (a.y) = deg(y) - deg(a) for a e A and y e L. 293 K. SHIBATA DEFINITION 2.1. : A graded Lie algebra L over A is an algebraic fibration * * of mixed type over A if (R 8 L ) = 0 for p ̂  0 and if it is equipped A P with a pair (x>D) , where x is a n element of L « called an Euler element and D : L̂  •> L̂  is an A -Lie derivation of degree -1 , i.e. (2.2) D< a -[ypy2]> = d A ( a )-&py 2] + (_ 1 )deg(a) a > { [D(yi),y2] + (-l)deg(yl)[y1,D(y2)]}, satisfying the following trace formulas ; (2.3) D(x) = 0 , and (2.4) (D)2(y) - |_X,Y] for every y e L̂  . The quotient DG Lie algebra (R 8 L ,1 8 D) is called the fiber of this A A algebraic fibration. DEFINITION 2.5. : Let (L̂  , X»D) be an algebraic fibration of mixed type over A . A Y * A Its chain complex C ,A(L^) over A is the DG coalgebra (SAoL),d = dT+D+d ), 1 1 1 ^ • *F ?n ^ _L, ^ where aL is the suspension of L̂  (the shift of degree by +1), S (aL ) denotes the symmetric coalgebra of oL taken over A , d is the usual diffe-A rential on s^aL^) arising from the Lie bracket of L̂ , D is the coderivation on Ŝ (aL̂ ) induced by the derivation on L̂  denoted by the same symbol, and d is the differential which is nothing but the multiplication by the suspension ax A Y of x> i-e. d (x) = aX*x« W e c a l 1 d the Euler differential in C 'A(L^). X X 2 A The trace formulas (2.3) and (2.4) are equivalent to ; d = 0 in SQ(aL̂ ). The cochain complex C* (Lv) over A* of L̂  is the A*-dual of C **(L^), • A,X — *r ^ + namely (2.6) Horn ̂(Ŝ (aL̂ ),A*) = A* 8 S*(R 0 aL̂ ) A A Horn *(d,l). A This is an algebraic fibration over A in the sense of Sullivan. Conversely, starting from an algebraic fibration A E in the sense of Sullivan, we can construct a mixed type fibration (L̂  , D, x) over A with x being a representative of the obstruction class to the existence for a cross-section in the minimal model for the fibration above. 3. The Haefliger model. Now we return to the fibration Z of (1.1). The minimal model for the base 'n 294 SULLIVAN-QUILLEN MIXED TYPE MODEL space BU^ is given by (3.1) I n = R Cl,c2,...,cnJ deg ĉ  « 2i, dCĉ ) = 0. A model for the fiber EU(2n> n is given by (3.2) W n = E(hj,h2,...,hn) 0 (R[c1,c2,...,cn]/(deg > 2n)) with deg h. = 2i-l, deg c. = 2i, d(hi) - c^ d(ci) « 0. A model (in the sense of Sullivan) for the total space is given by (3.3) I n 0 W ; n d(h.) = c. - c. , d(c.) = d(c.) = 0. l l l l l The fiber EU^11^ has the rational homotopy type of a bouquet of spheres and its minimal model (in the sense of Quillen) is (3.4) L^H*^)') d = 0 • A convenient basis { [he ] ; partitions I and J satisfy certain inequalities} for H (Wn) was found by J. Vey [2J . Now I n 0 L(a 1H*(Wn)f) has the natural graded Lie algebra structure over In. We define the Euler element x *-n by (3.6) X = CO c 0 ö U) h c c . . .1 . wl 2̂ œ3 J where the summation runs over all the partitions a) = (o)j ,o) 2,ai 3,...) such that 1 £ (A)J £ 0)2 £ — , + CJÜ̂  + .. . £ n, and that 03j + o)2 + U)̂  + ... > n. And we define In-Lie derivation D as a sum of two differentials dj and d2 ; (c.f. [6j, p. 398 for the notations) (3.7) D = d. + d2 : ln 0 L(a ̂ (W )») n ln 0 L(a 1H*(wn)'), (3.8) d (1 8 y(I,J)) = (1) sign I<v (a)1-î )ĉ  8 y(a)1+I;a)-a)1+J) (2) sign I<v (Jt"i )ca) 0 y(a)1+I-i1+Jt;(jo-co1+i1+J-Jt) , where y(I;J) = a 1 [hj-Cj] 1 > and 295 K. SHIBATA (3.9) d„(l 0 y(l;J)) - I (1) (-1)|I(1) sign Ky,v (i^-i^hc^ 0 [y(I(l);J(l)), y(aJ1+I(2)-i[2) ;a)-a)1 + i^2)+J(2))] (2) (-1)|I(1)sign Ky,v (iu(1)-iv(2))cw(1)cw(2) y(w1(1)+I(1) - i;i);a)(l)-a);i>+i;i)+J(l)),y(a)<2)+I(2)-i(i2);aJ(2)^2)+i;2)+J(2))]. One checks by direct computations that x a nd D defined above satisfy the trace formulas (2.3) and (2.4). Thus (in 0 L(a_1H*(W ) f ) , x,D) is an algebraic fibration of mixed type. Its cochain complex C*n (ln 0 L(a ̂ H*(W )f)) I »X is proved to be the minimal model for the fibration (3.3). Now let M be an n-dimensional manifold as stated in the introduction, and fi*(M) be its de Rham algebra. A choice of Pontrjagin forms p̂  e ft^(M) makes fi*(M) an in-algebra via the homomorphism defined by ĉ ^ ^i,C2i-l "̂ B^ the scalor extension, we obtain a DG Lie algebra over ft*(M) (3.10) (ft*(M) 0 (in 0 L(a 1H*(w )') = n*(M) 0 L(a lE*(Q )');1 0 D) n̂ n n .̂n ^ ¥ — 1 ¥ — whose cochain complex C (ft (M) 0 L(a H (W )') over R is a model for the cross-K n C ~ section space T((f ) * (y )). This is the Haefliger model for the Gelfand-Fuks M n *•* cochain complex Cc(LM). Notice that (f̂ ) * (ŷ ) admists a unique homotopy class of cross-sections since the fiber EU^2n^ is 2n-connected. n REMARK 3.11. : The minimal model for the algebraic fibration (3.3) is isomorphic to that of DGA 1 ^ = in/(deg > 2n) . So the minimal model above can also be - 3 regarded as the minimal model M . In fact, the modulo (M ) -reduction of I(n) I(n) the formulas (3.6)-(3.9) gives rise to formulas (2.15)-(2.19) of Hurder-Kamber [5J . Since R[PJ ,P 2 > • • • >Pn/2-l ~ ̂ n/(c2i-P» we °bta.in tne minimal model (= the Postnikov decomposition) for the algebraic fibration P̂  -> P̂  0 Wr by putting c2i_i *-n t*ie mo(iel above. This is a complete answer to the computational problem posed in [3] . 296 SULLIVAN-QUILLEN MIXED TYPE MODEL REFERENCES [1] I. GELFAND and D. FUKS, The cohomology of the Lie algebra of tangent vector fields... (I), (II), Funct. Anal., 3 (1969), 32-52, and 4 (1970), 23-32. [2] C. G0DBILL0N, Cohomologie d'algèbre de Lie..., Lect. Note Math. 383. [3] A. HAEFLIGER, Sur la cohomologie de Gelfand-Fuks, Lect. Note Math. 484. [4] , Sur la cohomologie..., Ann. Sci. E.N.S., 9 (1976), 503-532. [5] S. HURDER and F. KAMBER, Homotopy invariants..., Lect. Note Math. 788. [6] K. SHIBATA, On Haefliger's model..., Japan. J. Math., 7 (1981), 379-415. [7] F. SILVEIRA, Homotopie rationnelle..., Thèse, Univ. Genève, 1979. 297 

Sullivan-Quillen mixed type model for fibrations and the Haefliger model for the Gelfand-Fuks cohomology

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Sullivan-Quillen mixed type model for fibrations and the Haefliger model for the Gelfand-Fuks cohomology

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