Showing papers in "Annals of Mathematics in 1959"

The Stable Homotopy of the Classical Groups

Abstract: Throughout this paper M shall denote a compact connected Riemann manifold of class Co. Let i2 = (P, Q; h) be the triple consisting of two points P and Q on M together with a homotopy class h of curves joining P to Q. We will refer to such triples as base points on M. Corresponding to v = (P, Q; h) we define Md to be the set of all geodesics of minimal length which join P to Q and are contained in h. There is an obvious map of the suspension of M,' into M: one merely assigns to the pair (s, t), s e M-; t e [0, 1], the point on s which divides s in the ratio t to 1 t. (For fixed small t > 0, this map is 1 to 1 on M" and serves to define a topology on M>.) The induced homomorphism of wk,(MV) into 7krl(M) will be denoted by v*. Let s be an arbitrary geodesic on M from P to Q. The index of s, denoted by X(s), is the properly counted sum of the conjugate points of P in the interior of s. We write I I I for the first positive integer which occurs as the index of some geodesic from P to Q in the class h. In terms of these notions our principal result is the following theorem.

Picard Values of Meromorphic Functions and their Derivatives

Abstract: The theorem of Picard in its simplest form asserts that every nonconstant functionf (z), meromorphic in the plane, assumes there all complex values w with the possible exception of two. A value w which is not assumed by f(z) will be called a Picard value. The problem of our paper concerns the possible relationships between Picard values of f(z) and its derivatives. Let us first consider integral functions. Here the complete result is the following, due to Milloux [2], though special cases go back to Saxer [4].

Some global properties of contact structures

Abstract: transformations in general and to the study of global contact transformations in the special case of euclidean space. In attempting to generalize Lie's results to more general manifolds, it becomes clear that there are intrinsic global differences between the even and odd dimensional cases. In this paper, only the odd dimensional case will be discussed. Intuitively, a manifold carries a contact structure if the coordinate transformations can be chosen to preserve the 1-form dz - y'dx' up to a non-zero, multiplicative factor. We first show that if the manifold is orientable then this property is equivalent to the existence of a globally defined 1form ca of maximal rank, and then we derive some further equivalent conditions. It is well known that the existence of such a 1-form implies that the structure group of the tangent bundle can be reduced to the unitary group. (See, e.g., Chern, [7]). If this can be done, we say that the manifold is an almost-contact manifold. The obstructions to the existence of such a structure are investigated and it is shown that the primary obstruction is the third Stiefel-Whitney class. This solves completely the question of the existence of U(2) structures on five dimensional manifolds. We turn then to a discussion of global contact transformations, i.e., transformations which preserve ca up to a non-zero, multiplicative factor, T. Lie's results are shown to be valid in general by providing intrinsic proofs of his theorems. It should be noted that in this context, in general, analysis occurs only in the definitions, while the proofs consist simply of algebraic manipulations. Sheaves are employed at this point only because they provide a convenient language. Finally, we show that the factors T which can occur in contact transformations are not arbitrary. These results are then applied to the study of deformations (in the

The lefschetz theorem on hyperplane sections

Abstract: is bijective for i < n - 1 and surjective for i = n - 1. Several proofs of this theorem are to be found in the literature (see [5] for an account of the problem). Recently Thom has given a proof (unpublished) which, as far as we know, is the first to use Morse's theory of critical points. We present in ? 3, in a slightly more general setting, an alternate proof inspired by Thom's discovery. Our statement is given in the equivalent language of cohomology. The proof is derived from a theorem on Stein manifolds which is presented in ? 2. Some standard properties of the distance function which we require are assembled in ? 1 for the sake of completeness.

On degrees of unsolvability

Abstract: In [4], Kleene and Post showed that there are degrees between 0 (the degree of recursive sets) and 0' (the highest degree of recursively enumerable sets). Friedberg [1] and Muchnik [5] showed that there are recursively enumerable degrees (i.e., degrees containing a recursively enumerable set) between 0 and 0'. The question then arises: are there degrees between 0 and 0' which are not recursively enumerable? Since the recursively enumerable sets can be enumerated by a function of degree 0', the existence of such degrees follows from the following theorem, which may be roughly stated for d = 0 as: if a > 0', then the degrees < a cannot be enumerated by a function of degree < a. DEFINITION. A sequence {a,} of functions is uniformly of degree < a if an(x), as a function of (n, x), is of degree < a.

Genus of alternating link types

Abstract: A link projection' is said to be alternating iff it is connected and, as one follows along any component of the link, undercrossings and overcrossings alternate. A projection is trivial iff it is connected and has no crossings; otherwise it is non-trivial. We include trivial projections as alternating. An alternating link type is one which has an alternating projection. The principal result of this paper is Theorem 3.5 which contains the assertion that the degree of the reduced Alexander polynomial of an alternating link type plus one equals twice its genus plus its multiplicity. A second result, obtained as an immediate corollary of the same method which ultimately yields (3.5), is Theorem (2.13): The reduced Alexander polynomial of an alternating link type is an alternating polynomial. This theorem provides the simplest proof of the existence of non-alternating types. The image P of a connected, non-trivial projection has a natural decomposition as a graph. The vertices are the crossings, i.e., the images of the undercrossings, and the edges are the open arcs into which the crossings subdivide P. Since we shall have no reason to distinguish a point at infinity, we regard P as a spherical rather than a planar graph. The results of this paper are obtained by studying the image graph of a non-

Fixed Points and Torsion on Kahler Manifolds

Abstract: When a 1-parameter group acts by isometries on a Riemannian manifold M, the fixed point set F is nicely behaved. It is known that each component Fo0 of F is a totally geodesic submanifold of M whose dimension has the same parity as the dimension of M (see e.g., S. Kobayashi, Fixed points of isometries, Nagoya Math J., 13 (1958), 63-68). When M is compact Kihler theisometries are holomorphic transformations and the FM are compact Kihler submanifolds (which may reduce to points); in particular, as cycles, these components cannot bound in M. This paper is mainly concerned with the structure of the fixed point set in this Kihlerian case; however, the use of the complex structure is mainly for convenience; our results also hold for the special type of symplectic manifold in which the fundamental exterior 2-form is harmonic. As has been pointed out to us by several people, our situation is equivalent to having a toral group acting complex analytically on a compact Kahler M. Bott [2] has given some important results on the homology of certain homogeneous spaces and the loop space to a group. Our main results, the theorem and corollaries of ? 4 can be considered as direct generalizations of the former (see Corollary 3). Our method yields, at the same time, new proofs of his results. Our proofs are simple applications of another phase of Bott's work, namely his extension of the Morse theory of critical points to functions with "non-degenerate critical manifolds" [3]. The following simple example illustrates the method. Let S2 be the 2-sphere and let '1t be the 1-parameter group of rotations of S2 about the z axis. The fixed (or stationary) set F of (D, consists of the north and south poles, i.e., the places where the velocity vector X vanishes. Now

Infinite Abelian Groups and Homological Methods

Abstract: This paper has two sections concerned with the characterization of the algebraic structures of certain types of infinite abelian groups such as compact groups, or tensor products of torsion groups. It also has a section which gives an approach to the problems of mixed abelian groups (those which are neither torsion nor torsion-free) by establishing a duality between torsion groups and those abelian groups which are as " mixed as possible" (that is, they are as far away as possible from being just a direct sum of a torsion group and a torsion-free group). There is a final section containing some miscellaneous results on torsion-free abelian groups. For the reader unfamiliar with homological algebra, we give in Section 1 a brief outline of some results and methods of that theory. Almost all our proofs will depend heavily on homological methods. Section 2 is devoted to the duality of which we have spoken. A reduced group is called co-torsion if it is always a direct summand whenever it appears as a subgroup with a torsion-free quotient group. In the same way that Pontrjagin duality reduces the problems of compact groups to those of discrete groups, the problems of co-torsion groups are reduced to those of torsion groups by a one-to-one duality between the groups of these two classes. A method is then given by which all the mixed groups with a given torsion group can be constructed from the co-torsion group dual to that torsion group. The main result of Section 3 is that an abelian group can be a compact topological group if and only if it is isomorphic to a direct product (unrestricted direct sum) of copies of finite cyclic groups, p-adic integers, the reals, and the groups Z(p-) where, for each prime p, the number of copies of Z(p-) does not exceed the number of copies of the reals. It is also proved that if G is an abelian group with the multiples n! G considered as neighborhoods of the identity, then essentially G is a direct summand of a direct product of finite cyclic groups if and only if it is complete in the metric defined by these neighborhoods, while G is a direct sum of cyclic groups if and only if it is as far from complete as possible in this metric. At the very end of Section 3 it is shown that a torsion-free group is a direct summand of a direct product of finite 366

Einstein Spaces Admitting a One-Parameter Group of Conformal Transformations

Abstract: Publisher Summary This chapter considers M to be connected complete Einstein space of dimension n > 2 and of class C ∞ and suppose that a vector field on M generates globally a one parameter group of non-homothetic conformal transformations. Then M is isometric to a simply connected space of positive constant curvature. In particular M is homeomorphic to the sphere S n . If M is an irreducible and complete Riemannian manifold of class C ∞ , then A(M) is equal to I(M), except the case M is the one-dimensional euclidean space, where A(M) (I (M)) is the group of a fine (isometric) transformations of M. If a simply connected complete Riemannian manifold of class C ∞ admits a one-parameter group of non-isometric homothetic transformations, then it is a euclidean space.

Some remarks on chern classes

Abstract: In this paper, we wish to study the problem of classifying all complex n-plane bundles over CW-complexes M, or equivalently, of classifying (topologically) all principal fibre bundles over M with structural group the unitary group U(n). It is well-known [6] that this set of equivalence classes is in one-to-one correspondence with the set of homotopy classes of maps from M into the classifying space for U(n), the complex Grassmannian n-manifold Gn. (Gn is the set of all complex n-planes through the origin in complex cc-space.) Thus we have reduced our geometric problem to the problem of computing r(M, G), the set of homotopy classes of maps of M into Gn In order to study r(M, G), we shall partially construct the Postnikov system of Gn. With this information, we shall show that if dim M ? 2n and if the only torsion in HIJ(M) is relatively prime to (j 1)!, then a complex n-plane bundle e is trivial if and only if cl(e) = 0, . ., Cn(s) = 0, where ci() e H2`(M) is the ith Chern class of I. We shall also study the problem of determining which cohomology classes of M can be Chern classes of an n-plane bundle.'

An Interpretation of G. Whitehead's Generalization of H. Hopf's Invariant

Abstract: In the present paper', the generalized H. Hopf's invariant H: 7rdwn++(Sn+l) w7rd +n+i(Sqn+ ), due to G. Whitehead [10], is given a new definition which has some similarity with the original H. Hopf's definition [5]. The invariant H(f ) appears as depending in particular on the position in Sd+n+l of the inverse images M, Md byf: Sdln+l -_ Sn+1 of two regular values q, q' in Sn+1. Arnold Shapiro has defined the linking coefficient of two spheres Sp, Sq imbedded (without common point) in Em+i for p + q > m.2 In ? 5, the notion of linking coefficient is extended to (p + q m)-connected 7rmanifolds Mp, Mq in Em+i. It is an element of the stable homotopy group 7rW+N(SN), where r p + q m (7r-manifold = manifold which can be imbedded in some euclidean space with a trivial normal bundle). In the definition of H given in ? 3, M, and M' are 7r-manifolds but need not be (d n)-connected. As a consequence, H will in general also depend on the fields of normal vectors over M and M'. Therefore, it cannot be considered strictly as a linking coefficient which should be uniquely determined by the position in space of the two manifolds. It is an open question whether the method can be used to define the linking coefficient of (non-necessarily (p + q m)-connected) w-manifolds Mp, Mq in Em+i by going over to the quotient of 7Wp+qm,+N(SN) by some suitable subgroup. As an application of the new definition of H, it is proved that any regular imbedding (without self-intersection) of the d-sphere into euclidean (d + n)-space induces over Sd the trivial normal bundle provided that 2n > d + 1. A partial result in this direction was announced in [6].

A note on homology in categories

Abstract: exact category, although this was not the object (nor the result) of that paper.' Yoneda showed (using the existence of projectives) that Extn(A, C) is in 1-1 correspondence with the set of "n-fold extensions" of A by C:

On Some Arithmetic Properties of Linear Algebraic Groups

Abstract: Let Q be the field of rational numbers and k an extension of Q of finite degree. The multiplicative group k* of k, considered as a group of linear transformations of the vector space k over Q, forms an algebraic group, i.e., a Q-torus in the sense of A. Borel. As is well known in the algebraic number theory, the properties of k* and of its related structures, in particular that of the group Jk of id6les of k, play important roles. On the other hand, let f be a quadratic form on a vector space V over Q of finite dimension. The orthogonal group O( V, f ) composed of all linear transformations of V leaving invariant the form f forms an algebraic group. The properties of the group O( V, f) have essential relations to the arithmetic of the quadratic form f, and the study of these relations has been one of the principal themes in M. Eichler's book " Quadratische Formen und Orthogonale Gruppen ". Recently the theory of algebraic groups of linear transformations has been systematized by C. Chevalley on the basis of fundamental concepts of algebraic geometry, and the classical mechanism of the Lie theory (correspondence between groups and Lie algebras) has been generalized to the case where the basic field K is an arbitrary field of characteristic 0 (cf., Chevalley [2], [3]). By specializing K to Q, we may apply his methods and results to the study of arithmetic properties of algebraic groups. Thus it could be said that the above two theories, i. e., the arithmetic of k* and that of O( V, f) are two profiles of a kind of unified theory which we might call the arithmetic of algebraic groups. In the present paper, we shall formulate some fundamental concepts for algebraic groups from this point of view and prove some results which might possibly give us some suggestions for further developments in this direction. Thus, in Section 1, we shall introduce the notion of rational characters of algebraic groups and determine the structure of the group of rational characters for a special algebraic group, i. e., a Q-torus (Theorem 1). In Section 2, we shall introduce the notion of G-id6les for an algebraic group G which generalizes the usual notion of id6les of an algebraic number field k and define a subgroup J1(G) of the group J(G) of G-id6les in con266

On the theory of automorphic functions

Abstract: We know that the elliptic modular function j(z) gives the birational invariant of the elliptic curve with the analytic modulus z, and that the elliptic modular functions belonging to the congruence-subgroups are obtained from the values of elliptic functions at the points of finite order on elliptic curves. This is not only the origin of those functions, but one of the most essential points to which we may ascribe the significance of elliptic modular functions in number-theory. It is important to generalize these facts, namely, to investigate in a more general case the relation between automorphic functions with one or more variables and systems of algebraic varieties, especially, systems of abelian varieties. The object of this paper is to give some results on this subject. We shall deal with certain systems of polarized abelian varieties parametrized by holomorphic functions and show that there exist meromorphic functions whose values are considered as "moduli" of the members of the systems (Theorem 1). The theory developed here will be chiefly concerned with systems of abelian varieties with non-trivial endomorphisms, since I have already given elsewhere a theory for Siegel's modular functions [11]. I am particularly interested, in the present paper, in the determination of the fields of definition for fields of automorphic functions. This is the first problem which confronts us when we proceed beyond a formal treatment in the arithmetic theory of automorphic functions. We obtain a criterion (Theorem 2) applicable even in the case of compact fundamental domain where one can not employ Fourier expansions. The last part of the paper is devoted to the theory of a certain type of automorphic functions of one variable known in the literature as functions belonging to indefinite ternary quadratic forms [8], [5]; they occur as moduli of abelian varieties of dimension 2 whose endomorphismrings are isomorphic to an order of an indefinite quaternion algebra. We get in this case the field of rational numbers as a field of definition for the function-field. We can also prove the congruence formulae for the modular correspondences, which give a generalization of what is obtained in [3], [10]. We shall treat these formulae in a subsequent paper together with the theory of the functions belonging to congruence-subgroups. Our method can be applied to other types of automorphic functions, for example, to Hilbert's modular functions, by considering a system of

Periodic Solutions of a Perturbed Autonomous System

Abstract: for small s. We shall obtain results pertaining to existence, uniqueness, and stability of solutions of (1.1) in the form (1.3). Some of the results are illustrated with van der Pol's equation. Levinson [4] has discussed the case of (1.1) when x0(t) is strongly stable (cf., Section 4). He also makes no assumption about the ratio of T to Lo. His results show that for small 6 all solutions of (1.1), regarded as trajectories in (n + 1)-dimensional space, tend to a hyper-surface which lies near the hyper-cylinder having x = xo(t) as base, provided that at some finite time they are near enough to the hyper-surface. This result shows that even though some of the periodic solutions we find in the present paper are unstable, their instability is of a very weak type when x0(t) is strongly stable. The author [5] has made a study similar to the present paper of the second-order scalar equation