Showing papers in "Annals of Mathematics in 1963"

On compact analytic surfaces II

Abstract: THEOREM 11.1. The family 9(,{, G) consists of all analytic fibre spaces Bi, '2C H'(A, f2(B#)). The notion of fibre spaces and their equivalences depends on the sheaf of structure groups.' By a (63-fibre space we mean a fibre space with a sheaf (3 of structure groups, and we say that two fibre spaces are (X-equivalent if they are equivalent as (63-fibre spaces. The fibre space Bi may be considered as an analytic fibre space, as an f2(BO)-fibre space, or as an f2(Bl)-fibre space. The cohomology class C) C H'(Ay, f(BO)) represents the f2(Bl)-equivalence class of Be. Let C& denote the fibre of BI over u c A. It is clear that

Groups of Homotopy Spheres: I

Abstract: DEFINITION. Two closed n-manifolds M, and M2 are h-cobordant1 if the disjoint sum M, + (- M2) is the boundary of some manifold W, where both M1 and (-M2) are deformation retracts of W. It is clear that this is an equivalence relation. The connected sum of two connected n-manifolds is obtained by removing a small n-cell from each, and then pasting together the resulting boundaries. Details will be given in ? 2.

A poisson formula for semi-simple lie groups*

Abstract: for some bounded function h on the boundary of the disc. The function h(z) determines a function h(g) on G by setting h(g) = h(g(O)). If h(z) is harmonic, it may be shown that h(g) is annihilated by a certain class of differential operators on G. The Poisson formula (1) may be used to express h(g), and we find that here it takes on a particularly simple form. Namely, if we denote by m the normalized Lebesgue measure on {j z I = 1}, and by gm, the transform of this measure by the group element g E G, then it can be seen that (1) becomes

Harmonic Integrals on Strongly Pseudo-Convex Manifolds: II

Abstract: In this paper we prove the basic existence and regularity theorems for the a-Neumann problem (see Theorems 6.6 and 6.14). The results presented here were outlined by the author in [8]. In Part I of this work (see [7]) we established some of the fundamental properties of the aNeumann problem and, on the assumption of existence and regularity, we obtained several applications. A variant of the 8-Neumann problem was first formulated in [3]. D. C. Spencer and the author studied the problem by means of singular integral equations in [5]. The starting point for the author's work (see [6] and [7]) is the estimate (1.6), a special case of this estimate was first established by C. B. Morrey in [9]. In his thesis, (see [1]) M. E. Ash has derived estimates relative to moving frames. This method has enabled him to generalize our work (see also [2]). The introduction of moving frames is also very useful in the present work. The method of proving regularity by studying the families of norms depending on the parameters a and zwas suggested to the author by L. Hbrmander; essential use is made of some of the results (stated in Ch. 4) which are developed in his book (see [4]). The T-norms of Ch. 3 have been also introduced for a different purpose by Andreotti and Vesentini. They have obtained, by the argument of [6] for forms with values in a holomorphic vector bundle over a strongly pseudoconvex domain of Cn, an inequality which contains the inequality of Proposition 3.5 as a particular case. In Ch. 7 we show how the solution of the 8-Neumann problem implies the solution of several boundary value problems which were posed in [5]. The methods developed here and in [7] can be used to prove existence and regularity theorems for very general elliptic over determined systems. We shall return to this question in a future publication.

Some integration problems in almost-complex and complex manifolds

Abstract: The theory of almost-complex manifolds leads to a number of problems which properly belong to the field of differential equations. This paper deals with three such problems. The first concerns differential equations in the exterior operator d" (frequently also denoted 8) on differential forms; the other two deal with holomorphic coordinates and holomorphic curves. The latter are first stated in differential-geometric terms, and then re-stated as problems in differential equations. The equivalence of these formulations is established in a separate section. The differential equations problems (Theorems I, II', III') and their proofs can be read independently. Two appendices contain underlying facts that are either not available in the form required, or can be obtained only from widely scattered sources.

On discontinuous groups operating on the product of the upper half planes

Abstract: Let H" be the direct product of the n upper half planes, and let G be the connected component of the identity of the group of all analytic automorphisms of Hn. G is the direct product of n subgroups G1, G2, . . .* Gq each of which is isomorphic to the group of all analytic automorphisms of the upper half plane H. Consider a discrete subgroup F of G such that the factor space 17\G is of finite measure. (The notations Hn, G or F will keep, unless otherwise stated, these meanings throughout this paper.) For the study of the groups of this type, it is important to investigate the case where F is an irreducible subgroup of G in the following sense. A discrete subgroup F of G is said to be irreducible if F is not commensurable' with any direct product F' x F", where F' and F" are respectively discrete subgroups of the partial products G' and G" of G =G, x G2 x ... x Ga with G = G' x G", G' I {1}, G" # {1}. The main purpose of this paper was originally to calculate the dimension of the space of cusp foruis for an irreducible group by means of Selberg's trace formula; however, for the sake of completeness, and in view of the fact that no proof has been published as yet for the results stated by Pyatetzki-Shapiro in [6], it has been found desirable to prove these results here, following the ideas indicated by Pyatetzki-Shapiro himself. Therefore, no new results will be found in our ?? 1-3 except for some supplementary results such as Theorem 1, and for Theorems 6 and 7, which are proved under an additional condition on the fundamental domain of F (Assumption (F) in No. 11, ?3). Theorems 2, 3, 4, 5 are the restatements of the results of Pyatetzki-Shapiro; but, for the sake of simplicity, we state here the latter three theorems only for irreducible case. For these results, Pyatetzki-Shapiro has indicated in [6] only a sketch of proof for Theorem 3 as well as the implications between them. However, our proofs will be probably more or less the same as the ones he has. In ??4-5, we shall calculate the dimension of the space of cusp forms for an irreducible group F under the assumption (F). The main result is

Fonctions composées différentiables

Abstract: © Séminaire Lelong. Analyse (Secrétariat mathématique, Paris), 1962-1963, tous droits réservés. L’accès aux archives de la collection « Séminaire Lelong. Analyse » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

On the tamagawa number of algebraic tori

Abstract: Introduction 1. Extensions and cross sections 1.1. Extensions 1.2. Cross sections 2. Nakayama's duality 2.1. Preliminaries 2.2. Nakayama's duality 2.3. Some sequences 3. The numbers T(T), h(T) and i(T) 3.1. Functorial properties 3.2. The number r(T) 3.3. The number h(T) 3.4. The number i(T) 4. The numbers T(E), h(E) and i(E) 4.1. A function f (E) 4.2. The number T(E) 4.3. Proof of (Al), (A2) 4.4. Relation among r(E), h(E) and i(E) 4.5. Cyclic splitting field 5. Main theorem 6. Examples 6.1. The sequence (N) 6.2. Construction of T with T(T) E Z 6.3. An example of semi-simple group G with z(G) E Z References

On topologically unknotted spheres

Abstract: 1 B Mazur [10] and M Brown [2] have contributed to a general theorem on the unknotting of (n - 1)-spheres in the n-sphere In its most recent form [4], it states that if X is a subset of the n-sphere Sn, where X is homeomorphic to Sn-1, and if X satisfies a topological condition of local smoothness, then there is a homeomorphism of Sn onto itself which takes X onto an equatorial (n - 1)-sphere of Sn One wonders whether an analogous statement is true about subsets Y of Sn, where Y is homeomorphic to Sk, for values of k other than k = n - 1 It is a classical fact that knotted smooth spheres occur when k = n - 2 But otherwise there are no known examples of truly knotted spheres Sk in Sn, except when there is some sort of local pathology Here we shall show that a locally smooth k-sphere embedded in So is indeed topologically unknotted, provided k 5 The case of a locally smooth 1-sphere in S4 is unsolved A condition will be given which insures that an (n - 2)-sphere X contained smoothly in Sn is unknotted It is that n > 5 and Sn - X have the homotopy type of S1 The case of a 2-sphere in S4 is unsolved As for 1-spheres in S3, the problem has been settled by a special result in 3-dimensional topology [1] and by Dehn's lemma [11] When the k-sphere X contained in Sn is smooth except possibly at one point, and k 5, then it is shown that X is unknotted The case of knots which may fail to be smooth at two points cannot be handled by this method This odd state of affairs appears related to the difficulties in the isotopy conjecture, that a homeomorphism of degree one of Sn on itself should be isotopic to the identity map The proof of these results is rather complicated It is necessary to study in detail homotopy properties of the complement of a smooth knot A delicate application of the engulfing theorem [12] is made The proof is completed by applying a result about the union of open cones [13] Comparion with the piecewise-linear unknotting theorem of E C Zeeman [15] shows this Zeeman's theorem, and ours, have an analogous appearance and a region of overlap Furthermore, both are proved by somewhat similar piecewise-linear methods Zeeman's result is much stronger than ours when applied to piecewise-linear embeddings of

Lie Algebra Cohomology and Generalized Schubert Cells

Abstract: This paper is referred to as Part II. Part I is [4], The numerical I used as a reference will refer to that paper. A third and final part, Clifford algebras and the intersection of Schubert cycles is also planned.

On the c-functions of a division algebra

Abstract: In this paper, we will develop a theory of C-functions with characters in a division algebra. The ordinary C-function of a division algebra was introduced by K. Hey [4], and generalized by M. Eichler [1] to L-functions with abelian characters. The first attempt to generalize these theories to C-functions with non-abelian characters is due to H. Maass [5]. Later, R. Godement [2] gave a method to get the most general formulations on these matters. In this note, we will define a type of C-functions of a division algebra over an algebraic number field which are included in Godement's work as a special case, and for which one can develop the theory of Euler products. The latter theory has its own meaning as an application of the theory of spherical functions on p-adic algebraic groups. Here we have a generalization of Hecke's theory of so-called Heckeoperators. (Theorem 1-6). One can expect that there exist similar theories for other simple algebraic groups defined over p-adic number fields, and that there will be applications of these theories to non-commutative number theory. The author wishes to express his thanks to Professor Shimura for some valuable suggestions about the first part of this paper.

On isomorphic groups and homeomorphic spaces

Abstract: Suppose X and Y are topological spaces, and H(X) is the group of all homeomorphisms of X onto itself. We are concerned, in this paper, with the following question: (a) if p is a group isomorphism between H(X) and H(Y), does there exist a homeomorphism w of X onto Y such that (p(h) = h-1 for all h e H(X)? A more restricted question, which is also of interest, would be: (b) is every group automorphism of H(X) inner? The answer to (a) is, in general, no, for we may let X be the closed interval [0, 1], Y= (0, 1), and, for each h e H(X), 9(h) be the restriction of h to Y. To show that (b) also has a negative answer in general, let X = [O, 1] U (2, 3), w be a homeomorphic mapping of (0, 1) onto (2, 3) and (2, 3) onto (0, 1), and 9(h) be the extension of wha-v to X. Then 9 is an outer automorphism of H(X), for w can not be extended to an element of H(X). If X and Y are discrete spaces, then the answer to both questions is known to be yes, except in (b) when X has six elements (cf. [2] for a short historical survey). Question (a) has been studied in a more general setting by Gerstenhaber [2]. Fine and Schweigert [1] proved that (b) has an affirmative answer when X is a 1-cell, either open or closed, and Gerstenhaber [4] proved the same when X is a closed 2-cell. In the present paper, we will show that (a) has an affirmative answer when X and Y are compact, locally euclidean manifolds, with or without boundary. By setting X= Y, we also obtain an affirmative answer to (b) for the same class of spaces. The results obtained actually show that an isomorphism between any two sufficiently large subgroups of H(X) and H( Y) implies that X and Y are homeomorphic. In the course of this work, we develop a representation for the minimal normal subgroups of any sufficiently large subgroup of H(X).

On Products and Algebraic Families of Jacobian Varieties

Abstract: 1. The first theorem of this paper (? 2) is an extension to products of jacobian varieties of Matsusaka's characterization of a jacobian variety [6, Th. 3]. As a corollary (? 3) we obtain a result which was proved in the author's thesis (Chicago, 1958) and which is cited by Baily in the proof of the following: Let E denote the set of points corresponding to period matrices for Riemann surfaces of genus n in the quotient space VE = H,, / F,,. Then E is an irreducible locally closed subset of Vn in the Zariski topology determined on V. by the modular forms for L'n on Hn. [1, Cor. to Th., p. 313.] Our second theorem (? 5) is an extension of this result of Baily to algebraic families of polarized abelian varities (Weil's generalizations of the spaces Vs)'. The author wishes to express his appreciation to Professor Matsusaka for his advice and encouragement.