Geodesics in piecewise linear manifolds
David A.
Stone
1-44
Abstract: A simplicial complex M is metrized by assigning to each simplex $a \in {\mathbf{M}}$ a linear simplex ${a^\ast}$ in some Euclidean space ${{\mathbf{R}}^k}$ so that face relations correspond to isometries. An equivalence class of metrized complexes under the relation generated by subdivisions and isometries is called a metric complex; it consists primarily of a polyhedron M with an intrinsic metric $ {\rho _{\mathbf{M}}}$. This paper studies geodesics in metric complexes. Let $P \in {\mathbf{M}}$; then the tangent space ${T_P}({\mathbf{M}})$ is canonically isometric to an orthogonal product of cones from $ P,{{\mathbf{R}}^k} \times {\nu _P}({\mathbf{M}})$; once k is as large as possible. $ {\nu _P}({\mathbf{M}})$ is called the normal geometry at P in M. Let $P\bar X$ be a tangent direction at P in $ {\nu _P}({\mathbf{M}})$. I define numbers $ {\kappa _ + }(P\bar X)$ and $ {\kappa _ - }(P\bar X)$, called the maximum and minimum curvatures at P in the direction $P\bar X$. THEOREM. Let M be a complete, simply-connected metric complex which is a p.l. n-manifold without boundary. Assume $ {\kappa _ + }(P\bar X) \leqslant 0$ for all $P \in {\mathbf{M}}$ and all $ P\bar X \subseteq {\nu _P}({\mathbf{M}})$. Then M is p.l. isomorphic to $ {{\mathbf{R}}^n}$. This is analogous to a well-known theorem for smooth manifolds by E. Cartan and J. Hadamard. THEOREM (ROUGHLY). Let M be a complete metric complex which is a p.l. n-manifold without boundary. Assume (1) there is a number $\kappa \geqslant 0$ such that $ {\kappa _ - }(P\bar X) \geqslant \kappa$ whenever P is in the $ (n - 2)$-skeleton of M and whenever $ P\bar X \subseteq {\nu _P}({\mathbf{M}})$; (2) the simplexes of M are bounded in size and shape. Then M is compact. This is analogous to a weak form of a well-known theorem of S. B. Myers for smooth manifolds.
Espaces fibr\'es lin\'eaires faiblement n\'egatifs sur un espace complexe
Vincenzo
Ancona
45-61
Abstract: Let F be a coherent sheaf over a compact reduced complex space $X,L($F$)$ the linear fibre space associated with F, ${S^k}($F$)$ the kth symmetric power of F. We show that if the zero-section of $L($F$)$ is exceptional, then ${H^r}(X,$E${ \otimes _{{O_X}}}{S^k}($F$ )) = 0$ for every coherent sheaf E on X and for $r \geqslant 1$ and sufficiently large k. Using this result, we deduce moreover that Supp F is a Moišezon space.
Evolution system approximations of solutions to closed linear operator equations
Seaton D.
Purdom
63-79
Abstract: With S a linearly ordered set with the least upper bound property, with g a nonincreasing real-valued function on S, and with A a densely defined dissipative linear operator, an evolution system M is developed to solve the modified Stieljes integral equation $M(s,t)x = x + A((L)\smallint _S^t dgM( \cdot ,t)x)$. An affine version of this equation is also considered. Under the hypothesis that the evolution system associated with the linear equation is strongly (resp. weakly) asymptotically convergent, an evolution system is used to approximate strongly (resp. weakly) solutions to the closed operator equation $ Ay = - z$.
Balanced subgroups of abelian groups
Roger H.
Hunter
81-98
Abstract: The balanced subgroups of Fuchs are generalised to arbitrary abelian groups. Projectives and injectives with respect to general balanced exact sequences are classified; a new class of groups is introduced in order to classify these projectives.
The sets that are scissor congruent to an unbounded convex subset of the plane
Sydell Perlmutter
Gold
99-117
Abstract: It is shown that an unbounded convex plane body is scissor congruent to the union of a congruent body with a finite number of arbitrary topological discs. It is proved that 'is scissor congruent to' is an equivalence relation. Thus two unbounded convex plane bodies are scissor congruent if and only if the union of one with a finite number of topological discs is scissor congruent to the other.
Universally torsionless and trace modules
Gerald S.
Garfinkel
119-144
Abstract: We study, over an arbitrary ring R, a class of right modules intermediate between the projective and the flat content modules. Over the ring of rational integers these modules are the locally free abelian groups. Over any commutative ring they are the modules which remain torsionless under all scalar extensions. They each possess a certain separability property exactly when R is left semihereditary. We define M to be universally torsionless if the natural map $M \otimes A \to {\operatorname{Hom}}({M^\ast},A)$ is monic for all left modules A. We give various equivalent conditions for M to be universally torsionless, one of which is that M is a trace module, i.e. that $x \in M \cdot {M^\ast}(x)$ for all $x \in M$. We show the countably generated such modules are projective. Chase showed that rings over which products of projective or flat modules are also, respectively, projective or flat have other interesting properties and that they are characterized by certain left ideal theoretical conditions. We show similar results hold when the trace or content properties are preserved by products.
Automorphisms of ${\rm GL}\sb{n}(R)$
Bernard R.
McDonald
145-159
Abstract: Let R be a commutative ring and S a multiplicatively closed subset of R having no zero divisors. The pair $\langle R,S\rangle$ is said to be stable if the ring of fractions of R, ${S^{ - 1}}R$, defined by S is a ring for which all finitely generated projective modules are free. For a stable pair $ \langle R,S\rangle$ assume 2 is a unit in R and V is a free R-module of dimension $ \geqslant 3$. This paper examines the action of a group automorphism of $ GL(V)$ (the general linear group) on the elementary matrices relative to a basis B of V. In the case that R is a local ring, a Euclidean domain, a connected semilocal ring or a Dedekind domain whose quotient field is a finite extension of the rationals, we obtain a description of the action of the automorphism on all elements of $GL(V)$.
Geometry of Banach spaces of functions associated with concave functions
Paul
Hlavac;
K.
Sundaresan
161-189
Abstract: Let $(X,\Sigma ,\mu )$ be a positive measure space, and $ \phi$ be a concave nondecreasing function on ${R^ + } \to {R^ + }$ with $\phi (0) = 0$. Let $ {N_\phi }(R)$ be the Lorentz space associated with the function $\phi$. In this paper a complete characterization of the extreme points of the unit ball of ${N_\phi }(R)$ is provided. It is also shown that the space $ {N_\phi }(R)$ is not reflexive in all nontrivial cases, thus generalizing a result of Lorentz. Several analytical properties of spaces ${N_\phi }(R)$, and their abstract analogues ${N_\phi }(E)$, are obtained when E is a Banach space.
Cell-like closed-$0$-dimensional decompositions of $R\sp{3}$ are $R\sp{4}$ factors
Robert D.
Edwards;
Richard T.
Miller
191-203
Abstract: It is proved that the product of a cell-like closed-0-dimensional upper semicontinuous decomposition of ${R^3}$ with a line is ${R^4}$. This establishes at once this feature for all the various dogbone-inspired decompositions of ${R^3}$. The proof makes use of an observation of L. Rubin that the universal cover of a wedge of circles admits a 1-1 immersion into the wedge crossed with $ {R^1}$.
Some $C\sp{\ast} $-alegebras with a single generator
Catherine L.
Olsen;
William R.
Zame
205-217
Abstract: This paper grew out of the following question: If X is a compact subset of ${C^n}$, is $C(X) \otimes {{\mathbf{M}}_n}$ (the $ {C^\ast}$-algebra of $n \times n$ matrices with entries from $ C(X)$) singly generated? It is shown that the answer is affirmative; in fact, $ A \otimes {{\mathbf{M}}_n}$ is singly generated whenever A is a $ {C^\ast}$-algebra with identity, generated by a set of $ n(n + 1)/2$ elements of which $n(n - 1)/2$ are selfadjoint. If A is a separable ${C^\ast}$-algebra with identity, then $A \otimes K$ and $ A \otimes U$ are shown to be singly generated, where K is the algebra of compact operators in a separable, infinite-dimensional Hilbert space, and U is any UHF algebra. In all these cases, the generator is explicitly constructed.
Essential embeddings of annuli and M\"obius bands in $3$-manifolds
James W.
Cannon;
C. D.
Feustel
219-239
Abstract: In this paper we give conditions when the existence of an ``essential'' map of an annulus or Möbius band into a 3-manifold implies the existence of an ``essential'' embedding of an annulus or Möbius band into that 3-manifold. Let $ {\lambda _1}$ and ${\lambda _2}$ be disjoint simple ``orientation reversing'' loops in the boundary of a 3-manifold M and A an annulus. Let $f:(A,\partial A) \to (M,\partial M)$ be a map such that ${f_\ast}:{\pi _1}(A) \to {\pi _1}(M)$ is monic and $f(\partial A) = {\lambda _1} \cup {\lambda _2}$. Then we show that there is an embedding $g:(A,\partial A) \to (M,\partial M)$ such that $g(\partial A) = {\lambda _1} \cup {\lambda _2}$.
Fixed point theorems for mappings satisfying inwardness conditions
James
Caristi
241-251
Abstract: Let X be a normed linear space and let K be a convex subset of X. The inward set, ${I_K}(x)$, of x relative to K is defined as follows: ${I_K}(x) = \{ x + c(u - x):c \geqslant 1,u \in K\}$. A mapping $T:K \to X$ is said to be inward if $Tx \in {I_K}(x)$ for each $x \in K$, and weakly inward if Tx belongs to the closure of ${I_K}(x)$ for each $x \in K$. In this paper a characterization of weakly inward mappings is given in terms of a condition arising in the study of ordinary differential equations. A general fixed point theorem is proved and applied to derive a generalization of the Contraction Mapping Principle in a complete metric space, and then applied together with the characterization of weakly inward mappings to obtain some fixed point theorems in Banach spaces.
On the topology of a compact inverse Clifford semigroup
D. P.
Yeager
253-267
Abstract: A description of the topology of a compact inverse Clifford semigroup S is given in terms of the topologies of its subgroups and that of the semilattice X of idempotents. It is further shown that the category of compact inverse Clifford semigroups is equivalent to a full subcategory of the category whose objects are inverse limit preserving functors $F:X \to G$, where X is a compact semilattice and G is the category of compact groups and continuous homomorphisms, and where a morphism from $ F:X \to G$ to $ G:Y \to G$ is a pair $(\varepsilon ,w)$ such that $\varepsilon$ is a continuous homomorphism of X into Y and w is a natural transformation from F to $ G\varepsilon$. Simpler descriptions of the topology of S are given in case the topology of X is first countable and in case the bonding maps between the maximal subgroups of S are open mappings.
Composition series and intertwining operators for the spherical principal series. II
Kenneth D.
Johnson
269-283
Abstract: In this paper, we consider the connected split rank one Lie group of real type ${F_4}$ which we denote by $F_4^1$. We first exhibit $F_4^1$ as a group of operators on the complexification of A. A. Albert's exceptional simple Jordan algebra. This enables us to explicitly realize the symmetric space $F_4^1/{\text{Spin}}(9)$ as the unit ball in ${{\mathbf{R}}^{16}}$ with boundary ${S^{15}}$. After decomposing the space of spherical harmonics under the action of ${\text{Spin}}(9)$, we obtain the matrix of a transvection operator of $F_4^1{\text{/Spin}}(9)$ acting on a spherical principal series representation. We are then able to completely determine the Jordan Holder series of any spherical principal series representation of $ F_4^1$.
A property for inverses in a partially ordered linear algebra
Taen Yu
Dai;
Ralph
DeMarr
285-292
Abstract: We consider a Dedekind $\sigma$-complete partially ordered linear algebra A which has the following property: if $x \in A$ and $1 \leqslant x$, then $- u \leqslant {x^{ - 1}}$, where $u = {u^2}$. This property is used to show that A must be commutative. We also show that A is the direct sum of two algebras, each of which behaves like an algebra of real-valued functions.
Generalized inverses and spectral theory
Constantin
Apostol;
Kevin
Clancey
293-300
Abstract: The concept of a generalized spectral projection associated with a subset in the semi-Fredholm domain of a bounded operator on a Hilbert space is introduced. These generalized spectral projections possess many of the desirable properties of spectral projections associated with spectral sets. In particular, generalized spectral projections are used to separate finite sets of singular points from the semi-Fredholm domain.
On the Stone-\v Cech compactification of the space of closed sets
John
Ginsburg
301-311
Abstract: For a topological space X, we denote by ${2^X}$ the space of closed subsets of X with the finite topology. If X is normal and $ {T_1}$, the map $F \to {\text{cl}_{\beta X}}F$ is an embedding of ${2^X}$ onto a dense subspace of ${2^{\beta X}}$, and, in this way, we regard ${2^{\beta X}}$ as a compactification of $ {2^X}$. This paper is motivated by the following question. When can ${2^{\beta X}}$ be identified as the Stone-Čech compactification of ${2^X}$? In [11], J. Keesling states that $ \beta ({2^X}) = {2^{\beta X}}$ implies ${2^X}$ is pseudocompact. We give a proof of this result and establish the following partial converse. If ${2^X} \times {2^X}$ is pseudocompact, then $ \beta ({2^X}) = {2^{\beta X}}$. A corollary of this theorem is that $\beta ({2^X}) = {2^{\beta X}}$ when X is ${\aleph _0}$-bounded.
Equivariant bordism and Smith theory. IV
R. E.
Stong
313-321
Abstract: This paper analyzes two types of characteristic numbers defined for manifolds with ${Z_4}$ action, showing their relation and that neither suffices to detect ${Z_4}$ equivariant bordism. This extends work of Bix who had given examples not detected by one type of number.
Homogeneous manifolds with negative curvature. I
Robert
Azencott;
Edward N.
Wilson
323-362
Abstract: This paper solves the problem of determining which Lie groups act simply transitively on a Riemannian manifold with negative curvature. The results obtained extend those of Heintze for the case of strictly negative curvature. Using results of Wolf and Heintze, it is established that every connected, simply connected, homogeneous manifold M with negative curvature admits a Lie group S acting simply transitively by isometries and every group with this property must be solvable. Formulas for the curvature tensor on M are established and used to show that the Lie algebra of any such group S must satisfy a number of structural conditions. Conversely, given a Lie algebra $\mathfrak{s}$ satisfying these conditions and any member of an easily constructed family of inner products on $\mathfrak{s}$, a metric deformation argument is used to obtain a modified inner product which gives rise to a left invariant Riemannian structure with negative curvature on the associated simply connected Lie group.
Involutions on homotopy spheres and their gluing diffeomorphisms
Chao Chu
Liang
363-391
Abstract: Let $hS({P^{2n + 1}})$ denote the set of equivalence classes of smooth fixed-point free involutions on $ (2n + 1)$-dimensional homotopy spheres. Browder and Livesay defined an invariant $ \sigma ({\Sigma ^{2n + 1}},T)$ for each $({\Sigma ^{2n + 1}},T) \in hS({P^{2n + 1}})$, where $\sigma \in Z$ if n is odd, $\sigma \in {Z_2}$ if n is even. They showed that for $n \geqslant 3,\sigma ({\Sigma ^{2n + 1}},T) = 0$ if and only if $ ({\Sigma ^{2n + 1}},T)$ admits a codim 1 invariant sphere. For any $({\Sigma ^{2n + 1}},T)$, there exists an A-equivariant diffeomorphism f of ${S^n} \times {S^n}$ such that $({\Sigma ^{2n + 1}},T) = ({S^n} \times {D^{n + 1}},A){ \cup _f}({D^{n + 1}} \times {S^n},A)$, where A denotes the antipodal map. Let $ \beta (f) = \sigma ({\Sigma ^{2n + 1}},T)$. In the case n is odd, we can show that the Browder-Livesay invariant is additive: $\beta (fg) = \beta (f) + \beta (g)$. But if n is even, then there exists f and g such that $\beta (gf) = \beta (g) + \beta (f) \ne \beta (fg)$. Let $ {D_0}({S^n} \times {S^n},A)$ be the group of concordance classes of A-equivariant diffeomorphisms which are homotopic to the identity map of ${S^n} \times {S^n}$. We can prove that ``For $ n \equiv 0,1,2 \bmod 4, hS({P^{2n + 1}})$ is in 1-1 correspondence with a subgroup of $ {D_0}({S^n} \times {S^n},A)$. As an application of these theorems, we demonstrated that ``Let $ \Sigma _0^{8k + 3}$ denote the generator of $ b{P_{8k + 4}}$. Then the number of $ (\Sigma _0^{8k + 3},T)$'s with $\sigma (\Sigma _0^{8k + 3},T) = 0$ is either 0 or equal to the number of $ ({S^{8k + 3}},T)$'s with $ \sigma ({S^{8k + 3}},T) = 0$, where ${S^{8k + 3}}$ denotes the standard sphere".
An asymptotic formula for an integral in starlike function theory
R. R.
London;
D. K.
Thomas
393-406
Abstract: The paper is concerned with the integral $\displaystyle H = \int _0^{2\pi }\vert f{\vert^\sigma }\vert F{\vert^\tau }{(\operatorname{Re} F)^\kappa }\;d\theta$ in which f is a function regular and starlike in the unit disc, $F = zf'/f$, and the parameters $\sigma ,\tau ,\kappa$ are real. A study of H is of interest since various well-known integrals in the theory, such as the length of $f(\vert z\vert = r)$, the area of $f(\vert z\vert \leqslant r)$, and the integral means of f, are essentially obtained from it by suitably choosing the parameters. An asymptotic formula, valid as $r \to 1$, is obtained for H when f is a starlike function of positive order $ \alpha$, and the parameters satisfy $\alpha \sigma + \tau + \kappa > 1,\tau + \kappa \geqslant 0,\kappa \geqslant 0,\sigma > 0$. Several easy applications of this result are made; some to obtaining old results, two others in proving conjectures of Holland and Thomas.