Extending closed plane curves to immersions of the disk with $n$ handles
Keith D.
Bailey
1-24
Abstract: Let $f:S \to E$ be a normal curve in the plane. The extensions of $f$ to immersions of the disk with $n$ handles $({T_n})$ can be determined as follows. A word for $f$ is constructed using the definitions of Blank and Marx and a combinatorial structure, called a $ {T_n}$-assemblage, is defined for such words. There is an immersion extending $ f$ to ${T_n}$ iff the tangent winding number of $f$ is $1 - 2n$ and $f$ has a ${T_n}$-assemblage. For each $n$, a canonical curve $ {f_n}$ with a topologically unique extension to ${T_n}$ is described (${f_0}$ = Jordan curve). Any extendible curve with the minimum number $(2n + 2\;{\text{for}}\;n > 0)$ of self-intersections is equivalent to ${f_n}$.
Interpolation properties of generalized perfect splines and the solutions of certain extremal problems. I
Samuel
Karlin
25-66
Abstract: The existence of generalized perfect splines satisfying certain interpolation and/or moment conditions are established. In particular, the existence of ordinary perfect splines obeying boundary and interpolation conditions is demonstrated; precise criteria for the uniqueness of such interpolatory perfect splines are indicated. These are shown to solve a host of variational problems in certain Sobolev spaces.
One-sided congruences on inverse semigroups
John
Meakin
67-82
Abstract: By the kernel of a one-sided (left or right) congruence $\rho$ on an inverse semigroup $ S$, we mean the set of $ \rho$-classes which contain idempotents of $S$. We provide a set of independent axioms characterizing the kernel of a one-sided congruence on an inverse semigroup and show how to reconstruct the one-sided congruence from its kernel. Next we show how to characterize those partitions of the idempotents of an inverse semigroup $S$ which are induced by a one-sided congruence on $ S$ and provide a characterization of the maximum and minimum one-sided congruences on $S$ inducing a given such partition. The final two sections are devoted to a study of indempotent-separating one-sided congruences and a characterization of all inverse semigroups with only trivial full inverse subsemigroups. A Green-Lagrange-type theorem for finite inverse semigroups is discussed in the fourth section.
Iterated integrals, fundamental groups and covering spaces
Kuo Tsai
Chen
83-98
Abstract: Differential $ 1$-forms are integrated iteratedly along paths in a differentiable manifold $ X$. The purpose of this article is to consider those iterated integrals whose value along each path depends only on the homotopy class of the path. The totality of such integrals is shown to be dual, in an appropriate sense, to the ``maximal'' residually torsion free nilpotent quotient of the fundamental group $ {\pi _1}(X)$. Taken as functions on the universal covering space $ \tilde X$, these integrals separate points of $\tilde X$ if and only if $ {\pi _1}(X)$ is residually torsion free nilpotent.
Groups of free involutions of homotopy $S\sp{[n/2]}\times S\sp{[(n+1)/2]}$'s
H. W.
Schneider
99-136
Abstract: Let $M$ be an oriented $ n$-dimensional manifold which is homotopy equivalent to ${S^l} \times {S^{n - l}}$, where $l$ is the greatest integer in $ n/2$. Let $Q$ be the quotient manifold of $ M$ by a fixed point free involution. Associated to each such $Q$ are a unique integer $k\bmod {2^{\varphi (l)}}$, called the type of $ Q$, and a cohomology class $ \omega$ in ${H^1}(Q;{Z_2})$ which is the image of the generator of the first cohomology group of the classifying space for the double cover of $Q$ by $M$. Let ${I_n}(k)$ be the set of equivalence classes of such manifolds $Q$ of type $k$ for which $ {\omega ^{l + 1}} = 0$, where two such manifolds are equivalent if there is a diffeomorphism, orientation preserving if $k$ is even, between them. It is shown in this paper that if $n \geq 6$, then ${I_n}(k)$ can be given the structure of an abelian group. The groups ${I_8}(k)$ are partially calculated for $ k$ even.
On the fixed point set of a compact transformation group with some applications to compact monoids
Karl Heinrich
Hofmann;
Michael
Mislove
137-162
Abstract: Under various special additional hypotheses we prove that the fixed point set of the group of inner automorphisms of a compact connected monoid with zero is connected.
On bounded functions satisfying averaging conditions. I
Rotraut Goubau
Cahill
163-174
Abstract: Let $R(T)$ be the space of real valued ${L^\infty }$ functions defined on the unit circle $ C$ consisting of those functions $f$ for which $li{m_{h \to 0}}(1/h)\int_\theta ^{\theta + h} {f({e^{it}})dt = f({e^{i\theta }})}$ for every ${e^{i\theta }}$ in $C$. The extreme points of the unit ball of $R(T)$ are found and the extreme points of the unit ball of the space of all bounded harmonic functions in the unit disc which have non-tangential limit at each point of the unit circle are characterized. We show that if $g$ is a real valued function in ${L^\infty }(C)$ and if $K$ is a closed subset of $ \{ {e^{i\theta }}\vert li{m_{h \to 0}}(1/h)\int_\theta ^{\theta + h} {g({e^{it}})dt = g({e^{i\theta }})\} } $, then there is a function in $R(T)$ whose restriction to $K$ is $g$. If $E$ is a $ {G_\delta }$ subset of $ C$ of measure 0 and if $ F$ is a closed subset of $ C$ disjoint from $ E$, there is a function of norm 1 in $R(T)$ which is on $E$ and 1 on $F$. Finally, we show that if $E$ and $F$ are as in the preceding result, then there is a function of norm 1 in $ {H^\infty }$ (unit disc) the modulus of which has radial limit along every radius, which has radial limit of modulus 1 at each point of $F$ and radial limit 0 at each point of $E$.
Weakly almost periodic functions and almost convergent functions on a group
Ching
Chou
175-200
Abstract: Let $G$ be a locally compact group, $ UC(G)$ the space of bounded uniformly continuous complex functions on $G,{C_0}(G)$ the subspace of $UC(G)$ consisting of functions vanishing at infinity. Let $W(G)$ be the space of weakly almost periodic functions on $G$ and ${W_0}(G)$ the space of functions in $W(G)$ such that their absolute values have zero invariant mean. If $G$ is amenable let $F(G)$ be the space of almost convergent functions in $UC(G)$ and ${F_0}(G)$ the space of functions in $F(G)$ such that their absolute values are almost convergent to zero. The inclusive relations among the above-mentioned spaces are studied. It is shown that if $G$ is noncompact and satisfies certain conditions, e.g. $G$ is nilpotent, then each of the quotient Banach spaces $ UC(G)/W(G),{W_0}(G)/{C_0}(G),{F_0}(G)/{W_0}(G)$ contains a linear isometric copy of ${l^\infty }$. On the other hand, an example of a noncompact group $G$ is given which satisfies the condition that $ {C_0}(G) = {W_0}(G)$.
On $h$-local integral domains
Willy
Brandal
201-212
Abstract: Related to the question of determining the integral domains with the property that finitely generated modules are a direct sum of cyclic submodules is the question of determining when an integral domain is $h$-local, especially for Bezout domains. Presented are ten equivalent conditions for a Prüfer domain with two maximal ideals not to be $h$-local. If $R$ is an integral domain with quotient field $ Q$, if every maximal ideal of $R$ is not contained in the union of the rest of the maximal ideals of $R$, and if $Q/R$ is an injective $R$-module, then $R$ is $h$-local; and if in addition $R$ is a Bezout domain, then every finitely generated $R$-module is a direct sum of cyclic submodules. In particular if $R$ is a semilocal Prüfer domain with $ Q/R$ an injective $ R$-module, then every finitely generated $R$-module is a direct sum of cyclic submodules.
The Radon-Nikodym property in conjugate Banach spaces
Charles
Stegall
213-223
Abstract: We characterize conjugate Banach spaces ${X^\ast }$ having the Radon-Nikodym Property as those spaces such that any separable subspace of $ X$ has a separable conjugate. Several applications are given.
Toeplitz matrices generated by the Laurent series expansion of an arbitrary rational function
K. Michael
Day
224-245
Abstract: Let ${T_n}(f) = ({a_{i - j}})_{i,j = 0}^n$ be the finite Toeplitz matrices generated by the Laurent expansion of an arbitrary rational function. An identity is developed for $\det ({T_n}(f) - \lambda )$ which may be used to prove that the limit set of the eigenvalues of the $ {T_n}(f)$ is a point or consists of a finite number of analytic arcs.
Weak maps of combinatorial geometries
Dean
Lucas
247-279
Abstract: Weak maps of combinatorial geometries are studied, with particular emphasis on rank preserving weak bijections. Equivalent conditions for maps to be reversed under duality are given. It is shown that each simple image (on the same rank) of a binary geometry $G$ is of the form $ G/F \oplus F$ for some subgeometry $F$ of $G$. The behavior of invariants under mappings is studied. The Tutte polynomial, Whitney numbers of both kinds, and the Möbius function are shown to behave systematically under rank preserving weak maps. A weak map lattice is presented and, through it, the lattices of elementary images and preimages of a fixed geometry are studied.
Partitions of unity and a closed embedding theorem for $(C\sp{p},b\sp*)$-manifolds
Richard E.
Heisey
281-294
Abstract: Many manifolds of fiber bundle sections possess a natural atlas $\{ ({U_\alpha },{\phi _\alpha })\}$ such that the transition maps ${\phi _\beta }\phi _\alpha ^{ - 1}$, in addition to being smooth, are continuous with respect to the bounded weak topology of the model. In this paper we formalize the idea of such manifolds by defining $({C^p},{b^\ast })$-manifolds, $({C^p},{b^\ast })$-morphisms, etc. We then show that these manifolds admit $({C^p},{b^\ast })$-partitions of unity subordinate to certain open covers and that they can be embedded as closed $ ({C^p},{b^\ast })$-submanifolds of their model. A corollary of our work is that for any Banach space $B$, the conjugate space ${B^\ast }$ admits smooth partitions of unity subordinate to covers by sets open in the bounded weak-$ \ast$ topology.
Manifolds modelled on $R\sp{\infty }$ or bounded weak-* topologies
Richard E.
Heisey
295-312
Abstract: Let $ {R^\infty } = \mathop {\lim {R^n}}\limits_ \to$, and let ${B^ \ast }({b^ \ast })$ denote the conjugate, ${B^ \ast }$, of a separable, infinite-dimensional Banach space with its bounded weak-$\ast$ topology. We investigate properties of paracompact, topological manifolds $ M,N$ modelled on $ F$, where $F$ is either $ {R^\infty }$ or ${B^ \ast }({b^ \ast })$. Included among our results are that locally trivial bundles and microbundles over $M$ with fiber $F$ are trivial; there is an open embedding $M \to M \times F$; and if $M$ and $N$ have the same homotopy type, then $M \times F$ and $ N \times F$ are homeomorphic. Also, if $U$ is an open subset of ${B^ \ast }({b^ \ast })$, then $U \times {B^ \ast }({b^ \ast })$ is homeomorphic to $U$. Thus, two open subsets of ${B^ \ast }({b^ \ast })$ are homeomorphic if and only if they have the same homotopy type. Our theorems about $ {B^ \ast }({b^ \ast })$-manifolds, $ {B^ \ast }({b^ \ast })$ as above, immediately yield analogous theorems about $ B(b)$-manifolds, where $ B(b)$ is a separable, reflexive, infinite-dimensional Banach space with its bounded weak topology.
Spaces of vector measures
A.
Katsaras
313-328
Abstract: Let ${C_{rc}} = {C_{rc}}(X,E)$ denote the space of all continuous functions $f$, from a completely regular Hausdorff space $ X$ into a locally convex space $E$, for which $f(X)$ is relatively compact. As it is shown in [8], the uniform dual ${C'_{rc}}$ of ${C_{rc}}$ can be identified with a space $ M(B,E')$ of $E'$-valued measures defined on the algebra of subsets of $X$ generated by the zero sets. In this paper the subspaces of all $\sigma$-additive and all $\tau $-additive members of $ M(B,E')$ are studied. Two locally convex topologies $\beta$ and $ {\beta _1}$ are considered on ${C_{rc}}$. They yield as dual spaces the spaces of all $\tau$-additive and all $\sigma$-additive members of $M(B,E')$ respectively. In case $ E$ is a locally convex lattice, the $\sigma$-additive and $\tau$-additive members of $M(B,E')$ correspond to the $\sigma $-additive and $ \tau$-additive members of $ {C_{rc}}$ respectively.
Absolutely continuous functions on idempotent semigroups in the locally convex setting
A.
Katsaras
329-337
Abstract: Let $E$ be a locally convex space and let $ T$ be a semigroup of semicharacters on an idempotent semigroup. It is shown that there exists an isomorphism between the space of $ E$-valued functions on $ T$ and the space of all $ E$-valued finitely additive measures on a certain algebra of sets. The space of all $E$-valued functions on $T$ which are absolutely continuous with respect to a positive definite function $F$ is identified with the space of all $E$-valued measures which are absolutely continuous with respect to the measure ${m_F}$ corresponding to $ F$. Finally a representation is given for the operators on the set of all $ E$-valued finitely additive measures on an algebra of sets which are absolutely continuous with respect to a positive measure.
On the action of $\Theta \sp{n}$. I
H. E.
Winkelnkemper
339-346
Abstract: We prove two theorems about the inertia groups of closed, smooth, simply-connected $n$-manifolds. Theorem A shows that, in certain dimensions, the special inertia group, unlike the full inertia group, can never be equal to ${\Theta ^n}$; Theorem B shows, in $ \operatorname{dimensions} \equiv 3\bmod 4$, how to construct explicit closed $ n$-manifolds $ {M^n}$ such that $\Theta (\partial \pi )$ is contained in the inertia group of ${M^n}$.
Wild spheres in $E\sp{n}$ that are locally flat modulo tame Cantor sets
Robert J.
Daverman
347-359
Abstract: Kirby has given an elementary geometric proof showing that if an $ (n - 1)$-sphere $ \Sigma$ in Euclidean $ n$-space ${E^n}$ is locally flat modulo a Cantor set that is tame relative to both $\Sigma$ and ${E^n}$, then $\Sigma$ is locally flat. In this paper we illustrate the sharpness of the result by describing a wild $ (n - 1)$-sphere $ \Sigma$ in ${E^n}$ such that $\Sigma$ is locally flat modulo a Cantor set $C$ and $C$ is tame relative to ${E^n}$. These examples then are used to contrast certain properties of embedded spheres in higher dimensions with related properties of spheres in ${E^3}$. Rather obviously, as Kirby points out in [11], his result cannot be weak-ened by dismissing the restriction that the Cantor set be tame relative to ${E^n}$. It is well known that a sphere in $ {E^n}$ containing a wild (relative to ${E^n}$) Cantor set must be wild. Consequently the only variation on his work that merits consideration is the one mentioned above. The phenomenon we intend to describe also occurs in $3$-space. Alexander's horned sphere [1] is wild but is locally flat modulo a tame Cantor set. In fact, at one spot methods used here parallel those used to construct that example. However, other properties of $ 3$-space are strikingly dissimilar to what can be derived from the higher dimensional examples constructed here, for, as discussed in §2, natural analogues to some important results concerning locally flat embeddings in ${E^3}$ are false.
Simultaneous approximation of additive forms
Ming Chit
Liu
361-373
Abstract: Let $X = ({x_1}, \cdots ,{x_s})$ be a vector of $ s$ real components and ${f_i}(X) = \sum\nolimits_{j = 1}^s {{\theta _{ij}}x_j^k} (k = 2,3, \cdots ;i = 1, \cdots ,R) R$ additive forms, where ${\theta _{ij}}$ are arbitrary real numbers. The author obtains some results on the simultaneous approximation of $ \vert\vert{f_i}(X)\vert\vert$, where $\vert\vert t\vert\vert$ means the distance from $ t$ to the nearest integer.
Amalgamated products of semigroups: the embedding problem
Gérard
Lallement
375-394
Abstract: A necessary and sufficient condition for a semigroup amalgam to be embeddable is given. It is in the form of a countable set of equational implications with existential quantifiers. Furthermore it is shown that no finite set of equational implications can serve as a necessary and sufficient condition. Howie's sufficient condition (see [5]) is derived as a consequence of our main theorem.
Holomorphic functions with growth conditions
Bent E.
Petersen
395-406
Abstract: Let $P$ be a $p \times q$ matrix of polynomials in $n$ complex variables. If $\Omega$ is a domain of holomorphy in ${{\mathbf{C}}^n}$ and $u$ is a $q$-tuple of holomorphic functions we show that the equation $Pv = Pu$ has a solution $v$ which is a holomorphic $q$-tuple in $\Omega$ and which satisfies an ${L^2}$ estimate in terms of $ Pu$. Similar results have been obtained by Y.-T. Siu and R. Narasimhan for bounded domains and by L. Höormander for the case $ \Omega = {{\mathbf{C}}^n}$.