Trivial Compact Lie Group. There Exists an Orthogonal Representation Of

VOL. 47, 1961 MATHEMATICS: D. C. SPENCER 1205

trivial compact Lie group. There exists an orthogonal representation of dimension n > 10 which, considered as action on En via a homomorphism into O(n), has only the origin as stationary point. By lemmas 6 and 7, we obtain differentiable actions -y of G on Dn+5 and on Sn+4 for i = 1, 2, 3, . . .; the stationary set of yi on Sn+4 = Mi X Dn U Y2 X Sn-, is M, X 0. By lemmas 1 and 3, the actions -yi are pairwise topologically inequivalent, and the theorem is proved, with k = n + 4. It is clear that we can require all transformations to be orientation-preserving by using SO(n) instead of O(n). * The second author was partially supported by NSF grant G-14113. 1 Cairns, S. S., "Homeomorphisms between topological manifolds and analytic manifolds," Ann. Math., 41, 796-808 (1940). 2 Cairns, S. S., "The manifold smoothing problem," Bull. Amer. Math. Soc., 67, 237-238 (1961). 3Hirsch, M., "On combinatorial submanifolds of differentiable manifolds," to appear in Comm. Math. Helv. 4Kurosh, A. S., The Theory of Groups, 2nd English ed. (New York: Chelsea Publishing Com- pany, 1956). 5 Milnor, J., "Differentiable manifolds which are homotopy spheres, ' Princeton University, 1959, mimeographed notes. 6 Newman, M. H. A., "Boundaries of ULC sets in Euclidean n-space," these PROCEEDINGS, 34, 193-196 (1948). 7Palais, Bull. Amer. Math. Soc. (to appear). 8 Smale, S., "Generalized Poincare conjecture in dimensions greater than 4," Ann. of Math. (to appear). 9Steenrod, N. E., The Topology of Fibre Bundles (Princeton University Press, 1951). 10 Whitehead, J. H. C., "Simplicial spaces, nuclei and m-groups," Proc. London Math. Soc., 45, 243-327 (1939). 11 Whitehead, J. H. C., "On involutions of spheres," Ann. of Math., 66, 27-29 (1957). 12 Whitehead, J. H. C., "Manifolds with transverse fields in Euclidean space," Ann.. Math., 73, 154-212 (1960). 13 Whitney, H., "Differentiable manifolds." Ann. Math., 37, 645-680 (1936).

DEFORMATION OF PSEUDOGROUP STRUCTURES BY D. C. SPENCER DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY Communicated July 3, 1961 In their paper, K. Kodaira and the author have described a general pattern into which the deformation of structures on manifolds, defined by pseudogroups of bidifferentiable transformations, seems to fit; at least, the special types of structure so far investigated (see, for example, refs. 2, 3). The purpose of this note is to describe briefly a method of deformation which fits all the requirements of the general pattern and generalizes the method of the paper ;3 and which applies to structures defined by arbitrary transitive, continuous pseudogroups of local bidifferentiable or biholomorphic transformations of real or complex n-space, respectively. However, as in the case of the structures considered heretofore, the principal results have been obtained for complex pseudogroups only (i.e., pseudogroups of biholomorphic transformations). Downloaded by guest on September 24, 2021 1206 MA THEMAT'ICS: D. C. SPENCER PROC. N. A. S.

Details will be given in a paper to appear in the Annals of Mathematics. 1. Fundamental Sheaves.-By "differentiable" we shall always mean "differentiable of class C'." Let r be a transitive, continuous pseudogroup of local differentiable transformations of n-space (see ref. 4), and let M be a compact r-manifold, i.e., the r- structure of M is represented by a (locally finite) covering of M by open sets, each of which is covered by a -coordinate, and the transformation from one r- coordinate into another belongs to r. Then r induces over M a pseudogroup of local (differentiable) transformations which, for economy of language, will also be called r. For each non-negative integer 1A, we have the differentiable fibre bundle L-- M X M, where LO may be identified with M X M and L", u > 1, consists of all jets of order g (in the sense of C. Ehresmann) of transformations of M belonging to r. Let O: L'-y M be the projection of Lgonto M sending each jet into its source, and denote by To the bundle over LI of tangent vectors along the fibres of d: LI' M. Let I- be the submanifold of L", over the diagonal of M X M, composed of the jets of order s of the identity map of M. If the diagonal of M X M is identified with M, the restriction of TA to IS may be identified with a vector bundle RA over M. The structure group of RI is a (finite dimensional) Lie group GA+', and its fibre is R' @ A", where gA is the Lie algebra of GM. In particular, RI may be identified, in the case u = 0, with the tangent bundle of M. Denote by PA+' the principal bundle of R", where P+' has fibre and group G$+'; then RI' may be identified with the bundle T(PM)/G, obtained (from the tangent bundle T(PA) of Pa) by identifying points which are transforms of each other under the operations of GI on T(PA) induced by the right operations on Pa. Let SC(J) be the sheaf over M of germs of local differentiable deformations (fibre- preserving transformations) of PI which depend on a parameter t varying over a neighborhood of the origin 0 of m-space and reduce to the identity at t = 0; and let W(m) be the sheaf over M of germs of local transformations of M belonging to r which depend on t and reduce to the identity at t = 0. We have the projection of 3C(A) onto JC(m) where C(°) is the sheaf of germs of all local bidifferentiable transformations of M which depend on t and reduce to the identity at t = 0; and we have the injection P:5 (m) 3C(1) of 5(m) into 3C(A) as sheaf of left operators, where tA is the prolongation of the transformations of 1(fm) into their jets of order ,. Let MAO, /o > 1, be the order of P; then, for u . 1o- 1, we have the following exact sequence of sheaves (with distinguished or unit subsheaves which we denote by 1): ,A+1 1 y W(m) 3CIL+ + 1 1, where OD is a differential operator mapping 3C(A)+ 1 into a sheaf J(Z1` of germs of "jet forms" of degree 1 depending on t and vanishing at t = 0 and where the image aD5C(, 1 of SC(y) 1 in J(A'1 is isomorphic to the quotient SC(#) 1/LP+1Gf)) Then the corresponding exact cohomology sequence is

1 HO(M, >(m) HO(M,3C(#J+ 1) > HMW HI(MJ(m)) 1, (1) where, by a theorem of A. Haefliger', H'(M, 5(m)) may be identified with the set of germs of deformations of the restructure of M depending on the parameter t; and Downloaded by guest on September 24, 2021 VOL. 47, 1961 MATHEMATICS: D. C. SPENCER 1207

the dependence of the germs of 5(m) and 3C(,.) 'on t is differentiable if F is a pseudogroup of (local) non-analytic differentiable transformations, and holomorphic if r is complex, i.e., a pseudogroup of (local) holomorphic transformations. In the first case, 5(m) and 3C(ji+ 1 are sheaves of groups, H1(M, i(m)) and H1(M, 3C(r)A 1) are both defined, and H'(M, 3C()A+1) = 1; hence, in this case, (1) is an exact sequence (in the sense of sets with distinguished elements). In the second case, where r is a complex pseudogroup of holomorphic transformations and the germs of 5(m), (3()+ 1 depend holomorphically on the point t of complex m-space, 5(m) is a sheaf of groups, Hl(M,f(m)) is therefore defined, +l5(m) operates on 3C(-)+ 1 on the left, and O3C(#)+ 1is composed of germs depending holomorphically on t, hence DJC()j is a subsheaf of a sheaf J( `1 of germs of "jet forms" depending holomorphically on t. However, the sheaf ae(IC 1 of germs of local differentiable deformations, depending holomorphically on t, of the structure of PI+' of complex analytic fibre space, is obviously not a sheaf of groups; and H1(M,3C()+ 1) is therefore not even defined. Never- theless, it is easily verified that the coboundary map H°(M, w34 + ii H(Ml 5 ) is surjective. Similarly, this map is surjective in the case of a transitive, continuous pseudogroup of real analytic transformations where the germs ;(m), 3C(,)+ 1 depend real analytically on the point t of real m-space. Hence, in either the real (analytic or non-analytic) case or the complex (analytic) case, H'(M, 5(m)) is composed of germs of deformations which are trivial in the differentiable sense. Next, let 2; = ,2;Ar be the graded sheaf over M of germs of differential forms on M (of class C') with values in R". Then 2 ,O is the sheaf of infinitesimal transformations of 3C(A) i.e., we have the surjective map 3C() .>- 2AO of C(") onto 2PO sending the germ h(t) of 3C(" into dh(t)/1t~to. Let ,+1,r+ ,JIr be the map of 2;+ 1,r onto the sheaf J"'r of "jet forms" of degree r, where the germ +J of 2P + 17 mapped into the pair u = (a, t), o = o" is the projection of a" + 1, in Z"p, and t = ba"+', where 6 is an operator of formal exterior differentiation. Namely, let -"+', expressed in terms of a local r-coordinate (xl, . . ., xi, . .. , xn) of M, have the components craj1 ... j,, O < v < ,u + 1; then 6a"y' has the components n ity=k...j= dx k.ki, iy 0 < v < ,u.

We have the map D: Jur __, jur+l which sends u = (o,) into Du = D(o-, ) (do-,- d), where do has the components

0 (do-)S . dx l 0 < v < J k=1 xk J.. and the components of dt are defined in a similar manner. The sheaf J" = ~r j#'r has a structure of graded Lie algebra over the real numbers R, defined by a map j;&r ® JIAS _. Ju r+8 sending u 0 v into a bracket [u, v] satisfying the following rela- tion D[u, v] = [Du, v] + (-1)' [u, Dv]. (2) Finally, let 0 be the sheaf over M of germs of r-vector fields (or infinitesimal transformations of r), and let i = i" :e -> J`' be the injection mapping the germ Downloaded by guest on September 24, 2021 1208 MATHEMATICS: D. C. SPENCER PROC. N. A. S.

1' O of 0 into i(O) = (l"O, &P+1O), where L1:0 2 is induced by Ld: (,) 'C(1)5. If u,2 0- 1, the image of 0 under i is the kernel of the map D:JO°J.1. We remark that the cohomology H*(M, 0) = @7rD(M, 0) has a structure of gYraded Lie algebra over R which is induced by the structure of Lie algebra of 0 defined by the Poisson bracket of vector fields. 2. Complex Pseudogroup Structures.-Let F be an arbitrary complex, transitive, continuous pseudogroup and M a compact F-manifold. We suppose that the germs of the sheaves 5(m), KC(), and Jo' depend holomorphically on the parameter t of complex m-space Cm, where J(",' is the sheaf of germs of J('1 depending (holomorphically) on t and vanishing for t = 0. Our main result is the following THEOREM. There exists a positive integer Ml = p,1(r), where Iu > ,uO (order of r), such that,forM 2 pM: (i) The sequence i D D DD 0 --I o >#b Jz2 >o is exact and hence, by (2), we have the isomorphism (actually "anti-isomorphism" in the sense of reference 3 of the graded Lie algebra H*(M, 0) with the graded Lie algebra of the D-cohomology of sections over M of the sheaf J." = Ar. (ii) The set HO(M, D3("+1) is the subset of HO(M, J"1) composed of the elements v(@) satisfying the non-linear differential equation (integrability condition) Dv(t) - 1/2[v(t), v(t)] = 0. (3) (iii) If H2(M, 0) = 0, there exists an effective, complete, complex analytic family of deformations Mt, teCmX, Nt < E, of the restructure of M = Mo, where m = dimc H1(M, 0), i.e., the number of moduli of M is equal to the dimension of H'(M, 0). Let v(t) be an element of HO(M, J"1) satisfying (3). In order to prove (ii), it is sufficient to show that each point of M has a neighborhood on which the equation SDh(t) = v(t) (4) has a solution h(t) which is a section of aE+l over the neighborhood. The author is indebted to Professor L. Nirenberg for suggesting that the equation (4) can be solved in two steps by writing h(t) as the composition h1(t) h2(t) of two local deformations of PU+1, where h2(t) is a deformation of the structure of complex analytic fibre space and h1(t) is holomorphic, i.e., a deformation in which the complex analytic structure is held fixed. I Haefliger, A., "Structures feuilletees et cohomologie a valeur dans un faisceau de groupoldes," Comm. Math. Helv., 32, 248-329 (1958). 2 Kodaira, K., "On deformations of some complex pseudo-group structures," Ann. Math., 71, 224-302 (1960). 3 Kodaira, K., and D. C. Spencer, "Multifoliate structures," Ann. Math., 74, 52-100 (1961). 4 Spencer, D. C., "Some remarks on homological analysis and structures," Proceedings of Sym- posia in Pure Mathematics (Differential Geometry), 3, 56-86 (1961). Downloaded by guest on September 24, 2021