Showing papers in "Annals of Mathematics in 1957"

Complex Analytic Coordinates in Almost Complex Manifolds

Abstract: A manifold is called a complex manifold if it can be covered by coordinate patches with complex coordinates in which the coordinates in overlapping patches are related by complex analytic transformations. On such a manifold scalar multiplication by i in the tangent space has an invariant meaning. An even dimensional 2n real manifold is called almost complex if there is a linear transformation J defined on the tangent space at every point (and varying differentiably with respect to local coordinates) whose square is minus the identity, i.e. if there is a real tensor field h' satisfying

Elementary Structure of Real Algebraic Varieties

Abstract: A real (or complex) algebraic variety V is a point set in real n-space R n (or complex n-space C n ) which is the set of common zeros of a set of polynomials. The general properties of a real V as a point set have not been the subject of much study recently (but see for instance [2], [3] and [4]); attention has turned more to the complex case, the complex projective case, and especially the abstract algebraic theory. Facts about the real case are sometimes needed in the applications; proofs are commonly very difficult to locate.

On Dehn's Lemma and the Asphericity of Knots

Abstract: The present paper contains a 'roof of Dehn's lemma and an analogous result that we call the sphere theorem, from which other theorems follow.' DEHN'S LEMMA. Let M be a 3-manifold, compact or not, with boundary which may be empty, and in M let D be a 2-cell with self-intersections (singularities), having as boundary the simple closed polygonal curve C and such that there exists a closed neighborhood of C in D which is an annulus (i.e. no point of C is singular). Then there exists a 2-cell Do with boundary C, semi-linearly imbedded in M. SPHERE THEOREM. Let M be an orientable 3-manifold, compact or not, with boundary which may be empty, such that 7r2(M) # 0, and which can be semi-linearly2 imbedded in a 3-manifold N, having the following property: the commutator quotient group of any non-trivial (but not necessarily proper) finitely generated subgroup of 7r,(N) has an element of infinite order (n.b. in particular this holds if 7r,(N) = 1). Then there exists a 2-sphere S semi-linearly imbedded in M, such that3 S X 0 in M. Dehn's lemma was included in a 1910 paper of M. Dehn [4] p. 147, but in 1928 H. Kneser [13] p. 260, observed that Dehn's proof contained a serious gap. In 1935 and 1938 appeared two papers by I. Johansson [11], [12], on Dehn's lemma. In the second one, p. 659, he proves that, if Dehn's lemma holds for all orientable 3-manifolds, it then holds for all non-orientable ones. We now prove in this paper that Dehn's lemma holds for all orientable 3-manifolds. Our proof makes use also of I. Johansson's first paper. As far as the sphere theorem is concerned we have to remark that, to the best knowledge of this author, the first one to attempt a theorem of this kind was H. Kneser in 1928, [13] p. 257; however his proof does not seem to be conclusive. In 1937 S. Eilenberg [5] p. 242, Remark 1, observed a relation between the nonvanishing of the second homotopy group and the existence of a non-contractible 2-sphere. Finally in 1939 J. H. C. Whitehead [25] p. 161, posed a problem which stimulated the author to prove the sphere theorem, stated above. We emphasize that, if 7r,(N) is a free group4 then the hypotheses of the sphere theorem are fulfilled, according to the following NIELSEN-SCHREIER THEOREM. Every subgroup of a free group is itself a free group.5

The geometric realization of a semi-simplicial complex

Abstract: homology and homotopy groups. The terminology for semi-simplicial complexes will follow John Moore [7]. In particular the face and degeneracy maps of K will be denoted by d: Kn Kn-1 and si:K. -> Kn+1 respectively. 1. The definition

On a Compact Lie Group Acting on a Manifold

Abstract: Let M be a manifold (= connected, separable, locally euclidean space) of dimension n + 1, and G a compact connected Lie group acting on M in such a way that there is at least one n-dimensional orbit.2 In this paper, we show that the space of orbits M/G is homeomorphic to one of (i) a circle, (ii) an open interval, (iii) a half-open interval, or (iv) a closed interval. Moreover, there exists a subgroup N C G such that in (i) and (ii) M is homeomorphic to (G/N) X (M/G). In (iii) there is a subgroup K D N such that K/N is an r-sphere for some 0 < r < n, and such that M is homeomorphic to (G/N) X (M/G) with G/N identified to G/K over the end point. In (iv), a similar situation holds for subgroups K1 and K2 so that K1/N and K2/N are spheres (not necessarily of the same dimension). This, then, completely determines all manifolds admitting such a transformation group, and, moreover, the group G must act on M in the obvious way. (Thus, the operation of G is equivalent to G acting differentiably). In particular, this will show that the only two-dimensional manifolds which admit a compact connected Lie group operating non-trivially are as follows: (1) the torus, (2) the infinite circular cylinder, (3) the plane, (4) the (open) Moebius strip, (5) the sphere, (6) the Klein bottle, and (7) the projective plane: that is, precisely the Klein spaces. Each of these admits a circle group of operators, and in fact only the circle group operates effectively and non-transitively (except, of course, the trivial-i.e., one point-Lie group). A further application determines all compact, connected and effective Lie groups which can operate on a 3-dimensional manifold. All those 3-dimensional manifolds which admit a compact Lie group acting in such a way that there is a 2-dimensional orbit are determined. There are just 15 such. The author wishes to express his thanks to Dr. P. Conner, Profs. D. Montgomery, M. Goto, and A. L. Shields for their interest and suggestions.

On the Lusternik-Schnirelmann Category of Abstract Groups

Abstract: Let H be an abstract group. The cohomology groups H'(1l, A) may then be considered for any integer q > 0 and any abelian group A with II as a group of operators. The least integer n such that H'(II, A) = 0 for all A and all q > n is called the dimension of I. If no such integer exists then dim II = 00. For given HI, one can construct a connected aspherical CW-complex K11 with H as fundamental group.' Any two such complexes have the same homotopy type. The least dimension of such a complex Ku is called the geometric dimension of H. It is always infinite if H contains elements of finite order. The (Lusternik-Schnirelmann) category of a topological space X is the least integer n such that X can be covered by open sets UO, * * *, Un such that each Ui is contractible in X. If no such integer exists then cat X = oo. If we replace the phrase "UE is contractible in X" by "each closed path in Us is contractible in X" we obtain the 1-dimensional category of X (notation: cat, X).2 Both cat X and cat1 X are homotopy type invariants. The category of the aspherical complex Ku (which is independent of the choice the complex K11) is called the category of I. Since K11 is aspherical we have cat Ku = cat1 Ku1, there is therefore no need to define cat IH. The main result of this note is THEOREM 1. For any group HI

Intersection theory of manifolds with operators with applications to knot theory

Abstract: Let 9N be an oriented combinatorial manifold with boundary, let 9)1, be any of its covering complexes, and let G be any free abelian group of covering transformations. The homology groups of 9N* are R-modules, where R denotes the integral group ring of the (multiplicative) group G. Each such homology module H has a well-defined torsion sub-module T, and the corresponding Betti module is B = H/T. The automorphism y -> -1 of the group G extends to a unique automorphism a -a & of the ring R = R. Following Reidemeister [4] there is defined an intersection S which is a pairing of the homology modules of dual dimension to the ring R, and also a linking V which is a pairing of dual torsion sub-modules to Ro/R, where Ro is the quotient field of R. Two duality theorems are proved: (1) S is a primitive pairing to R/rm of dual Betti modules with coefficients modulo tm and Tm respectively, where or is zero or a prime element of R and m is any positive integer. (2) V is a primitive pairing of dual torsion modules to Ro/R. These theorems are analogous to the Burger duality theorems [1]; in case R is the ring of integers, they specialize to the Poincar6-Lefschetz duality theorems for manifolds with boundary. Although dual modules are not, in general, isomorphic, it is demonstrated that if one torsion module has elementary divisors A0 c Al c ... then its dual has elementary divisors Ao C Al C * ... . (The numbering differs from that of [2] in that we begin with the first non-zero ideal.) This result is applied to the maximal abelian covering of a link in a closed 3-manifold. It is proved that the elementary divisors Ai of the 1-dimensional torsion module are symmetric in the sense that Ai = At . For the case of a knot or link in Euclidean 3-space, the ideal A0 is generated by the Alexander polynomial, and the symmetry of A0 has been proved previously by Seifert [6] for knots and Torres [7] for links. The "symmetry" of the Alexander polynomial was proved by Seifert and Torres in a slightly more precise form [8, Cors. 2 and 3]. The problem of similar extra precision in the more general case of a link in an arbitrary closed 3-manifold remains open.

On Boundaries and Lateral Conditions for the Kolmogorov Differential Equations

Abstract: Kolmogorov [13] has shown that under appropriate regularity conditions such a P(t) will satisfy a pair of matrix differential equations (adjoint to each other). Since then a great many investigations have been concerned with problems of non-uniqueness, with solutions which fail to satisfy the adjoint equation, and with other "pathological" situations. It is the purpose of the present paper to show that the unbounded matrix operator Q on which the Kolmogorov equations depend shares the essential properties of second order elliptic differential operators, such as the SturmLiouville operator in one dimension or the Laplacian A in two dimensions. The Kolmogorov equations are then the exact counterpart of the heat equation ut = Au, except that due to the essential lack of symmetry of Q the adjoint equation bears little resemblance to the original. Our solutions will depend on boundary conditions in which one easily recognizes the classical boundary conditions of, say, the theory of harmonic functions. Of course, the latter depend on normal derivatives and no analogue to partial derivatives exists in our case. However, our boundary conditions are expressed in terms of functionals which remain meaningful under the most general circumstances, and can be applied to harmonic functions. There they reduce to the normal derivatives whenever the boundary is sufficiently smooth, but they give the proper expression for an arbitrary boundary. The perfect analogy of the Kolmogorov equations with diffusion equations explains the many phenomena and problems connected with the non-uniqueness of the solutions, and leads to new insights. Consider, for simplicity, an isolated point X of the boundary. It is possible to extend Q (that is, the Kolmogorov equations) and the matrix P(t) to the countable set E + a. This extension is not uniquely determined, but it involves very little arbitrariness. It adds a new row and a new column to Q, and these bear no similarity to the remaining rows and columns of Q. This leads in the most natural manner to generalizations of the

Approximating Surfaces With Polyhedral Ones

Abstract: If J is a simple closed curve in the plane E2, the Schoenflies Theorem says that there is a homeomorphism of E2 onto itself that takes J onto a circle. The theorem does not generalize directly to E3 because there is a simple surface S in E3 such that there is no homeomorphism of E3 onto itself that takes S onto the surface of a sphere. However, if S is polyhedral, there is a homeomorphism of E3 onto itself that takes S onto the surface of a sphere [1]. Hence, polyhedral surfaces are imbedded in E3 in a simpler fashion than are some other surfaces. The intersection of two polyhedral surfaces is much less complicated than the intersection of some other surfaces. For this reason, mathematicians sometimes impose the additional condition of being polyhedral on some of the surfaces they use. Applications of the results of this paper show that this extra condition is not always necessary. Harrold gives several applications of the approximation theorem for surfaces (Theorem 7) of this paper in [7]. I make extensive use of it in showing that there is a simple closed curve in E3 which pierces no disk [4]. In a subsequent paper I shall show that an extension of this theorem to topological 2-complexes can be used to give alternate proofs of the theorems due to Moise to the effect that each 3-manifold can be triangulated [9] (also a 3-manifold with boundary can be triangulated [2]) and that each homeomorphism of one 3-manifold with boundary into another can be approximated arbitrarily closely by a piecewise linear homeomorphism [8]. In this paper we consider the extent to which arbitrary surfaces can be approximated by polyhedral ones. We show that each surface can be homeomorphically approximated arbitrarily closely by one that is locally polyhedral. Therefore, it can be approximated by a polyhedron if it is compact [Lemma 1 of 2]. A simple surface is a continuum topologically equivalent to the surface of a sphere. We use the term "surface" synonomously with "2-manifold with boundary. " An n-manifold is a separable metric space K such that each point of K lies in an open set that is topologically equivalent to En (Euclidean n-space). An n-manifold with boundary is a separable metric space K such that each point of K lies in an open set N such that I7 is topologically equivalent to In (cube in En). We note that an n-manifold is an n-manifold with boundary but not conversely. The term boundary is used in two senses. In the point set sense we say that the boundary of a set X is the intersection of X (the closure of X) and the closure

The existence of a slice

Abstract: Throughout this paper, G is assumed to be a compact Lie group which acts as a topological transformation group on a space M. The symbol G, denotes the closed subgroup of G consisting of all elements of G which leave fixed the point p of M. It has been shown by Gleason that under certain conditions [1] there exists a local cross-section of the orbits at a point p. However a local cross-section does not always exist. In the case where G acts differentiably on a differentiable manifold, it is known that there exists at every point a somewhat more general object which might be called a slice [2, 5]. By using an invariant Riemannian metric a slice could be roughly described as a cell K orthogonal to G(p) at p, with G,(K) = K, and dim K the complementary dimension of G(p). In case p is fixed under G, K is merely an invariant neighborhood of p, so that in this case the definition has little content. We shall now define a slice without differentiability and then go on to show that it exists in the topological case for finite-dimensional spaces which may or may not be manifolds. The proof makes use of a recent theorem of Michael [3]. DEFINITION. If a compact Lie group G acts on a space M containing a point p then a slice at p is a closed set K which satisfies the following conditions: (1) p EK (2) Gp(K) = K (3) if y E K, then G, C G, and Gp(y) = K n G(y) (4) there is a compact cell R in G which is a local cross-section of the cosets of Gp in G at e and for which the map

On the indecomposable representations of algebras

Abstract: Let A be a finite dimensional associative algebra and d an integer, let gA(d) be the number of inequivalent indecomposable representations of A of degree d. We shall say that A is of strongly unbounded (representation) type if gA(d) is infinite for an infinite number of integers d. In this paper, we are concerned with the structure of algebras of strongly unbounded type. Theorem 2.1 states that an algebra is of strongly unbounded type if it has an infinite ideal lattice. This is a generalization of a theorem of R. M. Thrall which generalized the condition given by Nakayaina in [4] for an algebra to be of strongly unbounded type. In Section 3, we associate a graph with each ideal in the radical. Theorem 3.2 states that if any of these graphs has a cycle, a vertex of order four, or a chain that branches at each end then the algebra is of strongly unbounded type. The cycle and vertex of order four conditions are generalizations of conditions given by R. Brauer [2]. The branching chain condition is a generalization of a condition used by Thrall [6]. Two other conditions were recently obtained by T. Yoshii [9]. The function gA(d) can be used to define other classes of algebras. We say A is of finite (representation) type if Ed gA(d) is finite. Also, A is of bounded (representation) type if there exists d0 such that gA(d) = 0 for d ? d0 ; A is of unbounded type if not of bounded type. Nakayama first noted the existence of algebras of unbounded type in [4]. Concerning these classes of algebras, R. Brauer and R. M. Thrall have conjectured that algebras of bounded type are actually of finite type and that (over infinite fields) algebras of unbounded type are actually of strongly unbounded type. D. G. Higman [3] has shown that a group algebra is of unbounded type if and only if it has a non cyclic Sylow pgroup, p the characteristic of the field. T. Yoshii has confirmed the first conjecture in the case that the field is algebraically closed and the radical squared is zero [10]. Using the same assumptions, in [8] he obtains necessary and sufficient conditions that an algebra be of bounded type. Section 1 contains preliminary results. Sections 2 and 3 contain main theorems. Throughout this paper, fields are all infinite, modules are finite dimensional over these fields. Representation means a matrix representation describing the action of the algebra on a module with respect to a fixed basis.

Wild Cantor attractors exist

Abstract: T I lHE scientific community dealing with earth characteristics, marine and air navigation, and modern concepts of national defense is vitally concerned with longitudinal ties between continents. Expanded interest in modern geodesy has resulted particularly from research programs in developing intercontinental guided missiles and earth-circling satellites. The art of navigation has always been the most practical application of longitude, but in this the art does not require any high degree of precision. Concomitant to the age of exploration and rapid expansion of water-borne traffic was the urgent demand for a means of determining longitude at sea. Beginning with Harrison's chronometer, steady refinements were made until modern times, when continental ties are possible with the aid of telegraph and radio. Aside from navigational considerations, several purely scientific requirements involve the precise determination of longitude with reference to a standard meridian, such as Greenwich. The astronomer requires an accurate position for the time coordination of world-wide astronomical observations; the geophysicist needs longitudes of the highest precision in his studies of the drift of continents; and, finally, the geodesist desires to place all triangulation datums as closely as possible in their proper relation one with the other. Longitudinal studies, no matter how localized in place or how distant in time, have an important bearing on the subject. In celebrating the sesquicentennial of the United States Coast and Geodetic Survey during 1957, it is fitting to recall the Survey's contribution to the establishment of precise longitude in the United States. Early in the history of this bureau the need became apparent for a precise tie between some main-scheme triangulation stations in the United States and the meridians of any and all of the European observatories. The Harvard Observatory at Cambridge was adopted for the point of origin in the United States and in due time was tied to Greenwich by the method discussed in this paper. The second superintendent of the Coast Survey, Alexander Dallas Bache, was well aware of the difficulties other nations had encountered in fixing the

Unitary Invariants for Representations of Operator Algebras

Abstract: This paper will be concerned with certain developments in the spectral multiplicity (unitary invariants) theory of self-adjoint families of operators. The subject has its roots in the classical multiplicity solution of Hahn [12] and Hellinger [14] of the problem of describing those self-adjoint operators which are unitarily equivalent to a given self-adjoint operator. In their basic form, the results we obtain will provide a solution to this problem when the operators in question are not necessarily normal. These results were outlined in [19] (though, as stated there, they are incorrectly applied in the case of non-separable Hilbert spaceswe have made the revisions necessary to include the general case in this paper). Since the publication of [12, 14], the question has been re-examined and several variations and improvements made on the original solution. One may mention in this connection Wecken [49], Nakano [31, 32], Plessner-Rohlin [39] and Halmos [13]. These improvements have brought into focus the critical r6le played by measure-theoretic constructs in the unitary determination of self-adjoint operators. Moreover, the theory developed in [31, 13] applies, almost without change, to the unitary determination of commutative C*-algebras. (The following section contains a precise description of the terminology and notation we use.) Concurrently with these later improvements in the spectral multiplicity theory of a single self-adjoint operator, Murray and von Neumann undertook an investigation of rings of operators, more particularly, factors [28, 29, 30, 33, 34], while Nakano carried out a multiplicity decomposition of abelian rings of operators in terms of maximal abelian algebras [32]. For all but the type III case, the Murray and von Neumann results reduced the unitary equivalence problem for factors to an algebraic problem and the determination of a \"coupling constant\" [30]. In recent years, Dixmier [6] and Kaplansky [22] have developed techniques for dealing with general rings of operators in much the same way as Murray and von Neumann dealt with factors. Making use of these techniques, Dye [7, 8], Griffin [10] and Pallu de la Barriere [51] carried the unitary equivalence theory developed by Murray and von Neumann to general rings of operators having no part of type III. Slightly before this, Segal [40, 41] put the results of [32] in a

On the Holonomy Group of Locally Euclidean Spaces

Abstract: Part I of this paper is devoted to giving an algebraic characterization of the fundamental group of a compact locally euclidean manifold (i.e. a compact riemann manifold with curvature and torsion tensors identically zero). If M is a compact locally euclidean manifold then the fundamental group ir of M may be considered as a subgroup of the group of rigid motions, R(n), of n-dimensional euclidean space, En, and the orbit space of xr acting on En, E /Xr, is homeomorphic to M. Bieberbach in [3] showed that if N denotes the subgroup of ir consisting of pure translations then 1. N is a free abelian group generated by n linearly independent translations. 2. FIN is a finite group. Now it is clear from the discussion in [1], that in/N is isomorphic to the holonomy group, H, of M as a riemann manifold. Also, since E'/I is a manifold, we must have 'r operating on En without fixed points. This is equivalent, assuming 1 and 2, to 3. in has no finite subgroups. To see that 7r can have no finite subgroups, we may use the P. A. Smith Theory or an elementary argument on convex sets, since we have linear transformations. The converse will be established later. In Theorem 1 we prove that a finitely generated group in is isomorphic to the fundamental group of some compact locally euclidean manifold if and only if there exists a normal subgroup N c ir which is maximal abelian (contained in no larger abelian group), where N is free on n generators, and satisfies conditions 2 and 3 above. In the second part of this paper, we will show that any finite group can be the holonomy group of a compact locally euclidean manifold. We will actually show the following: Let G be a finite group and let F/R = G, where F is a finitely generated free group and F and R are non-abelian. Then if Fo = F/[R, R] and Ro = R/[R, R], where [R, R] denotes the commutator subgroup of R, then Fo/Ro = G and Fo is isomorphic to the fundamental group of a compact locally euclidean manifold. We give an example to show that the converse is not true. The authors would like to take this opportunity to thank M. Auslander and A. Rosenberg for their suggestions and advice during the writing of this paper.