Showing papers in "Annals of Mathematics in 1961"

Lie Algebra Cohomology and the Generalized Borel-Weil Theorem

Abstract: The present paper will be referred to as Part I. A subsequent paper entitled, “Lie algebra cohomology and generalized Schubert cells,” will be referred to as Part II.

Recursive Unsolvability of Post's Problem of "Tag" and other Topics in Theory of Turing Machines

Abstract: The equivalence of the notions of effective computability as based (1) on formal systems (e.g., those of Post), and (2) on computing machines (e.g., those of Turing) has been shown in a number of ways. The main results of this paper show that the same notions of computability can be realized within (1) the highly restricted monogenic formal systems called by Post the "Tag" systems, and (2) within a peculiarly restricted variant of Turing machine which has two tapes, but can neither write on nor erase these tapes. From these, or rather from the arithmetization device used in their construction, we obtain also an interesting basis for recursive function theory involving programs of only the simplest arithmetic operations. We show first how Turing machines can be regarded as programmed computers. Then by defining a hierarchy of programs which perform certain arithmetic transformations, we obtain the representation in terms of the restricted two-tape machines. These machines, in turn, can be represented in terms of Post normal canonical systems in such a way that each instruction for the machine corresponds to a set of productions in a system which has the monogenic property (for each string in the Post system just one production can operate). This settles the questions raised

On the Existence of Slices for Actions of Non-Compact Lie Groups

Abstract: If G is a topological group then by a G-space we mean a completely regular space X together with a fixed action of G on X. If one restricts consideration to compact Lie groups then a substantial general theory of G-spaces can be developed. However if G is allowed to be anything more general than a compact Lie group, theorems about G-spaces become extremely scarce, and it is clear that if one hopes to recover any sort of theory, some restriction must be made on the way G is allowed to act. A clue as to the sort of restriction that should be made is to be found in one of the most fundamental facts in the theory of G-spaces when G is a compact Lie group; namely the result, proved in special cases by Gleason 12], Koszul [5], Montgomery and Yang [6] and finally, in full generality, by Mostow [8] that there is a "slice" through each point of a G-space (see 2.1.1 for definition). In fact it is clear from even a passing acquaintance with the methodology of proof in transformation group theory that if G is a Lie group and X a G-space with compact isotropy groups for which there exists a slice at each point, then many of the statements that apply when G is compact are valid in this case also. In ? 1 of this paper we define a G-space X (G any locally compact group) -to be a Cartan G-space if for each point of X there is a neighborhood U such that the set of g in G for which g U n U is not empty has compact closure. In case G acts freely on X (i.e., the isotropy group at each point is the identity) this turns out to be equivalent to H. Cartan's basic axiom PF for principal bundles in the Seminaire H. Cartan of 1948-49, which explains the choice of name. In ? 2 we show that if G is a Lie group then the Cartan G-spaces are precisely those G-spaces with compact isotropy groups for which there is a slice through every point. As remarked above this allows one to extend a substantial portion of the theory of G-space that holds when G is a compact Lie group to Cartan G-spaces (or the slightly more restrictive class of proper G-spaces, also introduced in ? 1) when G is an arbitrary Lie group. Part of this extension is carried out in ? 4, more or less by way of showing what can be done. In particular we prove a generalization of Mostow's equivariant embed-

Generalized Poincare's Conjecture in Dimensions Greater Than Four

Abstract: Poincare has posed the problem as to whether every simply connected closed 3-manifold (triangulated) is homeomorphic to the 3-sphere, see [18] for example. This problem, still open, is usually called Poincare's conjecture. The generalized Poincare conjecture (see [11] or [28] for example) says that every closed n-manifold which has the homotopy type of the nsphere S" is homeomorphic to the n-sphere. One object of this paper is to prove that this is indeed the case if n > 5 (for differentiable manifolds in the following theorem and combinatorial manifolds in Theorem B).

On Gradient Dynamical Systems

Abstract: We consider in this paper a Co vector field X on a Co compact manifold Mn (&M, the boundary of M, may be empty or not) satisfying the following conditions: (1) At each singular point /8 of X, there is a cell neighborhood N and a Co function f on N such that X is the gradient of f on N in some riemannian structure on N. Furthermore /8 is a non-degenerate critical point of f. Let ,81 , m denote these singularities. (2) If x e &M, X at x is transversal (not tangent) to SM. Hence X is not zero on SM. (3) If x e M let p,(x) denote the orbit of X (solution curve) through x satisfying p0(x) = x. Then for each x e M, the limit set of p,(x) as t +-~ oo is contained in the union of the /3i. (4) The stable and unstable manifolds of the /3i have normal intersection with each other. This has the following meaning. The stable manifold Wj* of /3i is the set of all x e M such that limits ...p,(x) = /i. The unstable manifold Wi of 8i is the set of all x e M such that limit,,-,. t(x) = /i. It follows from conditions (1), (2) and a local theorem in [1, p. 330], that if /3i is a critical point of index X, then Wi is the image of a 1-1, Co map pi: U-s M, where Uc Rn A has the property if x e U, tx e U, 0 ? t ? 1 and pi has rank n X everywhere (see [4] for more details). A similar statement holds for Wi* with the U c RA. Now for x e Wi (or Wi*) let Wi2, (or We*) be the tangent space of Wi (or Wi*) at x. Then for each i, j, if x e Wf nWj*, condition (4) means that

Torsion in h-spaces

Abstract: A topological space with a continuous multiplication with unit is called an H-space The topological properties of these spaces have been investigated by many authors, in particular the homology and homotopy groups The case of Lie groups has been investigated intensely and many interesting results have been obtained by special methods for these groups E Cartan [8] proved that the second homotopy group of a Lie group is zero, a result which also follows from Bott's work [4] In this paper we obtain a new proof of Cartan's theorem, using homological methods Unlike the previous proofs which made strong use of the infinitesimal structure of Lie groups, the proof given here depends only on the homological structure and can be applied to H-spaces whose homology is finitely generated If X is a simply connected H-space whose homology is finitely generated, then it follows from Hopf's theorem [12] on Hopf algebras that H2(X; R) = 0 (where R = real numbers) and hence that r2(X) is finite The argument would show that a non-zero element x e H2(X; R) has infinite height (Xn # 0 for all n) which would contradict the hypothesis of finitely generated homology Now if r2(X) # 0, then H2(X; Z) # 0 for some prime p While x e H2(X; Z,) may not in general have infinite height, a slightly weaker statement is proved; ie, that x has a property called cc-implications (see definition in ? 6) which would again contradict the hypothesis that H*(X) is finitely generated The above follows from a general theorem (Theorem 61) which gives a condition ensuring that an element will have cc-implications Many consequences are deduced from this, particularly for H-spaces whose homology is finitely generated If X is an arcwise connected H-space with H*(X) finitely generated, it is proved that the mod p Hurewicz homomorphism h: 7rm(X) 0 Z ) Hm(X; Z) is zero in even dimensions m, for all p, which implies that the first non-vanishing higher homotopy group of X occurs is an odd dimension The known examples of H-spaces whose homology is finitely generated seem to consist of the Lie groups, the seven sphere S7, real projective 7-space P7 and products of these These are all manifolds It is shown here that for an H-space X with H*(X) finitely generated, the highest dimensional non-zero group Hn(X) is isomorphic to the integers Z, and * This paper was written while the author was a National Science Foundation Postdoctoral

On dominance and varieties of commuting matrices

Abstract: [4]), a question first raised by Goto. In the same section it is shown that the algebra generated by a pair of commuting n x n matrices can not have dimension greater than n, and that this algebra can always be embedded in a commutative algebra of dimension exactly n, a result derived by introducing the notion of a specialization of an algebra of matrices. The remainder of Chapter II contains related results of independent interest included mainly for later reference. The present paper was originally intended to be the first part of "On nilalgebras and linear varieties of nilpotent matrices, IV" (in preparation), which study continues the program of determining the structure of linear varieties of nilpotent matrices (" nilvarieties "). It is there shown that if V is a nilvariety of n x n matrices, and if c is the dimension of the algebra of all n x n matrices commuting with a generic element of V, then dim V_? 1/2(n2 c). All nilvarieties V for which dim V l/2(X2-c) are determined and shown to be associative nilpotent algebras of a type which we have elected to call anti-semisimple. These may be described

A Note on Some Contractible 4-Manifolds

Abstract: In a recent note [2], J. H. C. Whitehead exhibits involutions of Sn with fixed point sets non-simply connected, for n > 14. This is based on a certain example of M.H.A. Newman [1] of a contractible 5-manifold U5, with non-simply connected boundary. The purpose of this note is to construct specific 4-manifolds W4 which are contractible and have nonsimply connected boundaries & W, = Mr (which are necessarily homology 3-spheres), and which have the following additional property

Arithmetic of Algebraic Tori

Abstract: Introduction ? 0. Notation and conventions ? 1. Arbitrary fields 1.1. Duality 1.2. Splitting fields 1.3. Isogenies 1.4. Raising the field of definition 1.5. A structure theorem 1.6. Differential forms ? 2. Local fields 2.1. Maximal compact groups 2.2. Isogenies 2.3. Reduction modulo p ? 3. Fields of dimension 1 3.1. Adelization 3.2. Haar measures 3.3. Canonical correcting factors 3.4. The number p(X, K/k) 3.5. Definition of z(T) 3.6. Isogenies 3.7. Definition of r(a) 3.8. Explicit formula for C(CJA,Ir, (T)k) (resp. c()A, r)) 3.9. Explicit formula for r(cr) 3.10. Main theorem References

Ultraproducts in the Theory of Models

Abstract: In this paper we shall study an algebraic construction which has become a powerful new tool in the theory of models.1 This construction, called the ultraproduct operation', was first described in Los [20] under the name "champ logique," where its characteristic property of yielding elementary extensions of a given relational system was stated. This paper falls into two main parts, the first part entitled Ultraproducts, and the second Ultralimits. In the first part, the ultraproduct construction is defined and its fundamental model-theoretic property (Theorem 5.1) proved. This theorem is implicit in Los [20]; we give the particularly elegant and general formulation and proof due to D. Scott. After giving those set-theoretic properties of the ultraproduct necessary for model-theoretic applications (? 6), we apply the ultraproduct to obtain many classical results in logic. Chief among these is the completeness of various elementary theories, including the theory of real closed fields. In ? 8 we show that ultraproducts over division rings have a particularly simple algebraic description. In this first part, no attempt has been made to use ultraproducts to characterize logically defined concepts. In ? 9 we combine the ultraproduct operation with the direct limit operation to produce a new construction, the ultralimit. The ultralimit proves adequate in characterizing most concepts in the theory of models, including elementary equivalence, elementary classes, and elementary functions (?? 9-12). As each logical concept is characterized, a number of applications are presented. For instance, an immediate consequence of Theorem 12.1 is Beth's theorem in the theory of definition (Theorem 12.4). In ? 10, we show that there is a connection between ultraproducts and A. Robinson's method of diagrams. We remark here that H. J. Keisler has given an independent generalization of ultraproducts which allows him to obtain a characterization of elementary equivalence closely connected with Theorem 9.3 of this paper.3

Homology of the infinite symmetric group

Abstract: H*(S(co); Z.,) with coefficients in the integers mod p (p :prime). It is proved that the height of any non-zero element is co if p = 2, and is either co or < p if p is odd. If p = 2 this fact and the result in [11, ? 6] enable us to determine the structure of the cohomology algebra H*(S(oo); Z2) by using Borel's theorem on the structure of Hopf algebras. Let S(m) denote the symmetric group of degree m, and S(m, p) a p-Sylow subgroup of S(m). It is also proved that the above result on height is still valid if S(oo) is replaced by S(m) or S(m, p). In fact we derive the result for S(m) and S(oo) from the one for S(m, p). As is well known, S(m, p) is constructed by means of the wreath product. Therefore Part II starts with the consideration of the cohomology of the wreath product. Part III determines the structure of the graded algebra H*(S(cc); Z,) in which the multiplication is given by A* in Part I. It is proved that H*(S(oo); Zp) is isomorphic to the free commutative Z.-algebra generated by a certain set Q(p). Let SPm(Sn) denote the rn-fold symmetric product of an n-sphere, and let u(m) be a generator of the integral cohomology group Hn(SPm(Sn); Z). Assume n is even, and consider the reduced powers u(m)m/a of u(m) with respect to elements a e H,(S(m); G). Then Steenrod states without proof in his lecture notes [14] that if i < n the correspondence a - u(m)m/a yields an isomorphism of H,(S(m); G) onto Hmn-1 (SPm(Sn); G). This fact is basic in the argument of Part III. For the sake of completeness, a proof of the above Steenrod theorem is given in Part IV. The proof is

On imbedding differentiable manifolds in Euclidean space

Abstract: Author(s): Hirsch, MW | Abstract: Assume n,k,m,q are positive integers. Let M^n denote a smooth differentiable n-manifold and R^k Euclidean k-space. (a) If M^n is open it imbeds smoothly in R^k, k=2n-1 (b) If M^n is open and parallelizable it immerses in R^n (c) Assume M^n is closed and (m-1)-connected, 1l 2m-n l n+1. If a neighborhood of the (n-m)-skeleton immerses in R^q, ag2n-2m, then the complement of a point of M^n imbeds smoothly in R^q.

On Maximal Subgroups of Real Lie Groups

Abstract: 1.1. Let V denote a finite dimensional vector space over a field F. If x is an endomorphism of V and a e F, we denote by Va(x) the subset of V annihilated by some power of x a. A semi-simple endomorphism x (i.e., x has simple elementary divisors) is called F-diagonal if its eigenvalues are in F. A set S of endomorphisms is called F-diagonal if there is a base B in V such that the matrix of each element of S with respect to B is diagonal. If G is a real Lie algebra and h e G, we denote by Ga(h), G+(h), G_(h) and G*(h), respectively, the subspaces Ga(ad h), Ia>O Ga(h), Ia

Holomorphically convex sets in several complex variables

Abstract: In this paper we study holomorphic functions from the point of view of function algebras. By a function algebra we mean an algebra A of continuous (complex-valued) functions on a compact Hausdorff space X which is closed in the sup norm, 11 * Ix. We assume also that A contains the constants and separates the points of X. In this case, it is well-known, the space of maximal ideals S(A) of A is a compact Hausdorff space in the weak topology determined by A; A can be represented as a closed algebra of continuous functions on S(A), and X can be embedded as a closed subset of S(A) [11]. Further, for any f e A, I I f I x = I I f I 1s(,). Any subset B of S(A) such that for all f eA, Iif lB = lIfIIS(A) is called a boundary for A. The intersection of all closed boundaries is a boundary, and is called the Silov boundary, denoted by 1(A) [12]. The Silov boundary is not necessarily the smallest boundary for A; the latter need not even exist. But if A is separable (as a Banach space), then S(A) is metric, and this minimal boundary does exist. In this case 1(A) is just the closure of the minimal boundary [4]. Let M be a complex analytic manifold, and let K be a compact subset of M. Let H(K) be the algebra of all functions holomorphic in a neighborhood of K, and let A(K) represent the closure of H(K) in the norm II IlK. How can we determine S(A(K)), 1(A(K)), and, as it makes sense in this case, since A(K) is separable, the minimal boundary? These questions have been discussed for certain types of compact sets in Cn (complex n-dimensional vector space) by K. de Leeuw [10], and K. Hoffman [14]. The papers of S. Bergman on distinguished boundary domains [3] are the first to indicate the significance of the Silov boundary in several complex variables (see also D. Lowdenslager [17] and H. Bremermann [8]). The Silov boundary is the smallest subset of K on which we can hope to represent holomorphic functions by an integral formula. If M is of dimension one (i.e., a Riemann surface), then K is the space of maximal ideals, and OK is the Silov boundary of A(K) [1, 4, 22]. In higher dimensions, neither of these is in any sense generally true. For example, if 470

On the homotopy groups of the classical groups

Abstract: which was conjectured by Serre [2, p. 428]. The theory of [8] showed that the groups in (3) had isomorphic p-primary components for p > 2n, and so we show that this actually holds for p > 2 (it does not hold for p = 2). We prove (1) and (2) by showing that, although the above fibrations do not have a cross-section, there is a map which behaves like a cross-section in homology with coefficients any field of characteristic = 2. If G = su(2n), K = sp(n) or if G = su(2n + 1), K = so(2n + 1) and if a is the automorphism of G leaving K fixed (complex conjugation in the second case, complex conjugation followed by an inner automorphism in the first) then the map in question is the map q: G/K , G given by q(gK) = g(g)-1. Although this map q has been noted previously in the theory of symmetric spaces, its action on cohomology has been determined only recently (in our note 15], which considers general symmetric and other homogeneous spaces). In a later paper we will show that this map is the generalization to symmetric spaces of the characteristic maps for fibre bundles over spheres considered in [9, ?? 23, 24]. We also show that the direct sum decompositions (1) and (2) are simply the decompositions into the +1 and -1 eigenspaces of the map induced