A general Stone-Gel'fand duality
J.
Lambek;
B. A.
Rattray
1-35
Abstract: We give a simple characterization of full subcategories of equational categories. If $ \mathcal{a}$ is one such and $\mathcal{B}$ is the category of topological spaces, we consider a pair of adjoint functors $ {\mathcal{a}^{op}}\underset{F}{\overset{U}{\longleftrightarrow}}\mathcal{B}$ which are represented by objects I and J in the sense that the underlying sets of $U(A)$ and $F(B)$ are $ \mathcal{a}(A,I)$ and $\mathcal{B}(B,J)$. (One may take I and J to have the same underlying set.) Such functors always establish a duality between Fix FU and Fix UF. We study conditions under which one can conclude that FU and UF are reflectors into Fix FU and Fix UF, that Fix FU = Image F = the limit closure of I in $ \mathcal{a}$ and that Fix UF = Image U = the limit closure of J in $\mathcal{B}$. For example, this happens if (1) $\mathcal{a}$ is a limit closed subcategory of an equational category, (2) J is compact Hausdorff and has a basis of open sets of the form $\{ x \in J\vert\alpha (I)(x) \ne \beta (I)(x)\}$, where $\alpha$ and $\beta$ are unary $ \mathcal{a}$-operations, and (3) there are quaternary operations $\xi$ and $\eta$ such that, for all $x \in {J^4},\xi (I)(x) = \eta (I)(x)$ if and only if ${x_1} = {x_2}$ or ${x_3} = {x_4}$. (The compactness of J may be dropped, but then one loses the conclusion that Fix FU is the limit closure of I.) We also obtain a quite different set of conditions, a crucial one being that J is compact and that every f in $ \mathcal{B}({J^n},J)$, n finite, can be uniformly approximated arbitrarily closely by $ \mathcal{a}$-operations on I. This generalizes the notion of functional completeness in universal algebra. The well-known dualities of Stone and Gelfand are special cases of both situations and the generalization of Stone duality by Hu is also subsumed.
Higher order Massey products and links
Edward J.
O’Neill
37-66
Abstract: In this paper we generalize Steenrod's functional cup products, calling the generalizations functional triple products, and relate them with Massey 4-products. We then study certain links using this machinery and a new relation that is satisfied by 4-products under conditions on X which permit applications to links. Finally, many examples illustrating the connection between Massey higher products, both ordinary and matrix, and links are presented. This material constituted a portion of the author's doctoral dissertation. The author would like to thank his thesis advisor Professor W. S. Massey for his encouragement and guidance.
Circle-preserving functions of spheres
Joel
Gibbons;
Cary
Webb
67-83
Abstract: Suppose a function of the standard sphere ${S^2}$ into the standard sphere ${S^{2 + m}}$, $m \geqslant 0$, sends every circle into a circle but is not a circlepreserving bijection of ${S^2}$. Then the image of the function must lie in a five-point set or, if it contains more than five points, it must lie in a circle together with at most one other point. We prove the local version of this theorem together with a generalization to n dimensions. In the generalization, the significance of 5 is replaced by $2n + 1$. There is also proved a 3-dimensional result in which, compared to the n-dimensional theorem, we are allowed to weaken the structure assumed on the image set of the function.
Uniquely arcwise connected plane continua have the fixed-point property
Charles L.
Hagopian
85-104
Abstract: This paper contains a solution to a fixed-point problem of G. S. Young [17, p. 884] and R. H. Bing [4, Question 4, p. 124]. Let M be an arcwise connected plane continuum that does not contain a simple closed curve. We prove that every continuous function of M into M has a fixed point.
On tails and domains of attraction of stable measures in Banach spaces
Aloisio
Araujo;
Evarist
Giné
105-119
Abstract: The exact tail behavior of stable measures in Banach spaces and measures in their domains of attraction is given. Conditions for a p.m. to be in the domain of attraction of a stable p.m. of order $\alpha$ are derived which are sufficient in type p spaces, $p > \alpha $, and necessary in general. This paper also contains a short proof of the Lévy-Khinchin formula in Banach spaces.
Small zeros of additive forms in many variables
Wolfgang M.
Schmidt
121-133
Abstract: It is shown that if s is large as a function of k and of $\varepsilon > 0$, then the diophantine equation $ {a_1}{x_1}^k + \cdots + {a_s}x_s^k = {b_1}y_1^k + \cdots + {b_s}y_s^k$ with positive coefficients $ {a_1}, \ldots ,{a_s}$, ${b_1}, \ldots ,{b_s}$ has a nontrivial solution in nonnegative integers ${x_1}, \ldots ,{x_s}$, ${y_1}, \ldots ,{y_s}$ not exceeding $ {m^{\left( {1/k} \right) + \varepsilon }}$, where m is the maximum of the coefficients.
Some infinite free boundary problems
David E.
Tepper;
Gerald
Wildenberg
135-144
Abstract: Let $\Gamma$ be the boundary of an unbounded simply connected region $ \mathcal{D}$, and let $ \mathcal{C}(\Gamma )$ denote the family of all simply connected regions $\Delta \subset \mathcal{D}$ such that $ \partial \Delta = \Gamma \cup \gamma$ where $ \gamma \cap \Gamma$ contains only the infinite point. For $\Delta \in \mathcal{C}(\Gamma )$ we call $\gamma$ the free boundary of $\Delta$. Given a positive constant $ \lambda$, we seek to find a region ${\Delta _\lambda } \in \mathcal{C}(\Gamma )$ with free boundary $ {\gamma _\lambda }$ such that there is a bounded harmonic function V in $ {\Delta _\lambda }$ with the properties that (i) $V = 0$ on $\Gamma$, (ii) $V = 1$ on $\gamma$, (iii) $\left\vert {{\text{grad }}V(z)} \right\vert = \lambda$ for $ z \in {\gamma _\lambda }$. We give sufficient conditions for existence and uniqueness of $ {\Delta _\lambda }$. We also give quantitative properties of ${\gamma _\lambda }$.
The periodic behavior of Morse-Smale diffeomorphisms on compact surfaces
Carolyn C.
Narasimhan
145-169
Abstract: Necessary and sufficient conditions are given for the existence of Morse-Smale diffeomorphisms homotopic to the identity with prescribed periodic characteristics on any compact 2-manifold.
On a case of extensions of group schemes
B.
Weisfeiler
171-189
Abstract: The extensions of a smooth connected commutative group scheme whose generic fiber is ${G_m}$ by the additive group scheme are studied. The results are most explicit in the case when the basic scheme is the spectrum of an integral domain containing a field.
The PL Grassmannian and PL curvature
Norman
Levitt
191-205
Abstract: A space ${\mathcal{G}_{n,k}}$ is constructed, together with a block bundle over it, which is analogous to the Grassmannian ${G_{n,k}}$ in that, given a PL manifold ${M^n}$ as a subcomplex of an affine triangulation of $ {R^{n + k}}$, there is a natural ``Gauss map'' ${M^n} \to {\mathcal{G}_{n,k}}$ covered by a block-bundle map of the PL tubular neighborhood of ${M^n}$ to the block bundle over ${G_{n,k}}$. Certain subcomplexes of $ {G_{n,k}}$ are then studied in connection with immersion problems, the chief result being that a connected manifold $ {M^n}$ (nonclosed) PL immerses in ${R^{n + k}}$ satisfying certain ``local'' conditions if and only if its stable normal bundle is represented by a map to the subcomplex of ${G_{n,k}}$ corresponding to the condition. An important example of such a condition is a restriction on PL curvature, e.g., nonnegative or nonpositive, PL curvature having been defined by D. Stone.
Automorphisms of the deformation space of a Kleinian group
James A.
Gentilesco
207-220
Abstract: In the following paper, we determine the biholomorphic automorphisms of a cross-product of Teichmüller spaces. This in turn helps us to determine the biholomorphic automorphisms of the deformation space of a Kleinian group using the fact that its holomorphic universal covering space is a cross-product of Teichmüller spaces. Among other interesting results, we show that in general the deformation space is not a homogeneous space.
Erratum to: ``Torsion in the bordism of oriented involutions'' (Trans. Amer. Math. {\bf 231} (1977), no. 2, 541--548)
Russell J.
Rowlett
221-221