Cloverleaf representations of simply connected $3$-manifolds
Edwin E.
Moise
1-30
Abstract: Let $M$ be a triangulated $3$-manifold satisfying the hypothesis of the Poincaré Conjecture. In the present paper it is shown that there is a finite linear graph $ {K_1}$ in the $ 3$-sphere, with exactly two components, and a finite linear graph ${K_2}$ in $M$, such that when the components of the graphs $ {K_i}$ are regarded as points, the resulting hyperspaces are homeomorphic. ${K_2}$ satisfies certain conditions which imply that each component of ${K_2}$ is contractible in $M$. Thus the conclusion of the theorem proved here is equivalent to the hypothesis of the Poincaré Conjecture.
Modular and distributive semilattices
Joe B.
Rhodes
31-41
Abstract: A modular semilattice is a semilattice $S$ in which $w \geq$ implies that there exist $x,y \in S$ such that $x \geq a,y \geq b$ and $x \wedge y = x \wedge w$. This is equivalent to modularity in a lattice and in the semilattice of ideals of the semilattice, and the condition implies the Kurosh-Ore replacement property for irreducible elements in a semilattice. The main results provide extensions of the classical characterizations of modular and distributive lattices by their sublattices: A semilattice $S$ is modular if and only if each pair of elements of $ S$ has an upper bound in $ S$ and there is no retract of $S$ isomorphic to the nonmodular five lattice. A semilattice is distributive if and only if it is modular and has no retract isomorphic to the nondistributive five lattice.
Associated and perspective simplexes
Leon
Gerber
43-55
Abstract: A set of $n + 1$ lines in $ n$-space such that any $({\text{n}} - 2)$-dimensional flat which meets $n$ of the lines also meets the remaining line is said to be an associated set of lines. Two Simplexes are associated if the joins of corresponding vertices are associated. A simple criterion is given for simplexes to be associated and an analogous one for Simplexes to be perspective. These are used to give a brief proof of the following generalization of the theorem of Pappus. Let $ \mathcal{A}$ and $\mathcal{B}$ be $n$-simplexes and let $p$ be a permutation on the vertices of $\mathcal{B}$. If $ \mathcal{A}$ and $\mathcal{B}$ are associated (respectively perspective) and $ \mathcal{A}$ and $\mathcal{B}p$ are associated (perspective) then $\mathcal{A}$ and $\mathcal{B}{p^k}$ are associated (perspective) for any integer $k$. Very short proofs are given of extensions to $ n$-dimensions of many theorems from Neuberg's famous Memoir sur le Tétraèdre, such as: the altitudes of a simplex are associated.
The structure of nested spaces
T. B.
Muenzenberger;
R. E.
Smithson
57-87
Abstract: The structure of nested spaces is studied in this paper using such tools as branches, chains, partial orders, and rays in the context of semitrees. A classification scheme for various kinds of acyclic spaces is delineated in terms of semitrees. Several families of order compatible topologies for semitrees are investigated, and these families are grouped in a spectrum (inclusion chain) of topologies compatible with the semitree structure. The chain, interval, and tree topologies are scrutinized in some detail. Several topological characterizations of semitrees with certain order compatible topologies are also derived.
A $5-r$ uniqueness theorem
Jessie Ann
Engle
89-104
Abstract: A Borel-regular Carathéodory outer measure $\Lambda$ on a separable metric space $X$, where $\Lambda$ is invariant with respect to a family $ H$ of homeomorphisms from $ X$ onto $X$, is unique if $\Lambda$ satisfies a $5$ - $r$ condition at one point in $X$ and if $H$ satisfies Condition I, a condition much weaker than, but related to, the invariance of distance under $ H$.
$k$-regular elements in semisimple algebraic groups
Peter P.
Andre
105-124
Abstract: In this paper, Steinberg's concept of a regular element in a semisimple algebraic group defined over an algebraically closed field is generalized to the concept of a $k$-regular element in a semisimple algebraic group defined over an arbitrary field of characteristic zero. The existence of semisimple and unipotent $ k$-regular elements in a semisimple algebraic group defined over a field of characteristic zero is proved. The structure of all $ k$-regular unipotent elements is given. The number of minimal parabolic subgroups containing a $k$-regular element is given. The number of conjugacy classes of $R$-regular unipotent elements is given, where $ R$ is the real field. The number of conjugacy classes of ${Q_p}$-regular unipotent elements is shown to be finite, where ${Q_p}$ is the field of $p$-adic numbers.
Theta constants and cusp forms
A. J.
Crisalli
125-132
Abstract: For principal congruence subgroups of levels 2 and 4 a basis for their cusp forms consisting of monomials of theta constants is displayed. Some conditions for the vanishing of Poincaré series of these groups are found.
On centralizers of generalized uniform subgroups of locally compact groups
Kwan-Yuk Law
Sit
133-146
Abstract: Let $G$ be a locally compact group and $ H$ a closed subgroup of $ G$ such that the homogeneous space $G/H$ admits a finite invariant measure. Let $ {Z_G}(H)$ be the centralizer of $H$ in $G$. It is shown that if $G$ is connected then ${Z_G}(H)$ modulo its center is compact. If $ G$ is only assumed to be locally connected it is shown that the commutator subgroup of ${Z_G}(H)$ has compact closure. Consequences of these results are found for special classes of groups, such as Lie groups. An example of a totally disconnected group $G$ is given to show that the results for $ {Z_G}(H)$ need not hold if $ G$ is not connected or locally connected.
Deformations of group actions
Allan L.
Edmonds
147-160
Abstract: Let $G$ be a finite group and $M$ be a compact piecewise linear (PL) manifold. Define a PL $G$-isotopy to be a level-preserving PL action of $G$ on $M \times [0,1]$. In this paper PL $ G$-isotopies are studied and PL $G$-isotopic actions (which need not be equivalent) are characterized.
A homotopy theory of pro-spaces
Jerrold W.
Grossman
161-176
Abstract: The category of towers of spaces, $\ldots \to {X_{s + 1}} \to {X_s} \to \ldots \to {X_0}$, viewed as pro-spaces, appears to be useful in the study of the relation between homology and homotopy of nonsimply connected spaces. We show that this category admits the structure of a closed model category, in the sense of Quillen; notions of fibration, cofibration, and weak equivalence are defined and shown to satisfy fundamental properties that the corresponding notions satisfy in the category of spaces. This enables one to develop a ``homotopy theory'' for pro-spaces.
On the decomposition of tensor products of principal series representations for real-rank one semisimple groups
Robert Paul
Martin
177-211
Abstract: Let $G$ be a connected semisimple real-rank one Lie group with finite center. It is shown that the decomposition of the tensor product of two representations from the principal series of $G$ consists of two pieces, ${T_c}$ and ${T_d}$, where ${T_c}$ is a continuous direct sum with respect to Plancherel measure on $\hat G$ of representations from the principal series only, occurring with explicitly determined multiplicities, and ${T_d}$ is a discrete sum of representations from the discrete series of $G$, occurring with multiplicities which are, for the present, undetermined.
Teichm\"uller spaces and representability of functors
Michael
Engber
213-226
Abstract: The Teichmëller space of compact Riemann surfaces with punctures is exhibited as the object representing a certain functor. This extends the work of Grothendieck, who treated the unpunctured case. The relationship between the two cases is exploited to obtain specific results on the connection between the two Teichmüller spaces.
Large models of countable height
Harvey
Friedman
227-239
Abstract: Eery countable transitive model $M$ of ZF (without choice) has an ordinal preserving extension satisfying ZF, of power ${ \sqsupset _{M \cap On}}$. An application to infinitary logic is given.
On the analytic continuation of the Minakshisundaram-Pleijel zeta function for compact Riemann surfaces
Burton
Randol
241-246
Abstract: A formula is derived for the Minakshisundaram-Pleijel zeta function in the half-plane Re $s < 0$.
Identities involving the coefficients of a class of Dirichlet series. VII
Bruce C.
Berndt
247-261
Abstract: Let $a(n)$ be an arithmetical function, and consider the Riesz sum $ {A_\rho }(x) = {\Sigma _{n \leq x}}a(n){(x - n)^\rho }$. For $a(n)$ belonging to a certain class of arithmetical functions, $ {A_\rho }(x)$ can be expressed in terms of an infinite series of Bessel functions. K. Chandrasekharan and R. Narasimhan have established this identity for the widest known range of $ \rho$. Their proof depends upon equi-convergence theory of trigonometric series. An alternate proof is given here which uses only the classical theory of Bessel functions.
${\rm PI}$-algebras satisfying identities of degree $3$
Abraham A.
Klein
263-277
Abstract: A method of classification of PI-algebras over fields of characteristic 0 is described and applied to algebras satisfying polynomial identities of degree 3. Two algebras satisfying the same identities of degree 3 are considered in the same class. For the degree 3 all the possible classes are obtained. In each case the identities of degree 4 that can be deduced from those of degree 3 have been obtained by means of a computer. These computations have made it possible to obtain-except for three cases-all the identities of higher degrees. It turns out that except for a finite number of cases an algebra satisfying an identity of degree 3 is either nilpotent of order 4, or commutative of order 4, namely the product of 4 elements of the algebra is a symmetric function of its factors.
$K$-groups generated by $K$-spaces
Eric C.
Nummela
279-289
Abstract: A $K$-group $G$ with identity $e$ is said to be generated by the $ K$-space $X$ if $X$ is a subspace of $G$ containing $e,X$ algebraically generates $G$, and the canonical morphism from the Graev free $K$-group over $(X,e)$ on-to $G$ is a quotient morphism. An internal characterization of the topology of such a group $ G$ is obtained, as well as a sufficient condition that a subgroup $H$ of $G$ be generated by a subspace $Y$ of $H$. Several illuminating examples are provided.
Relationships between $\lambda $-nuclearity and pseudo-$\mu $-nuclearity
William B.
Robinson
291-303
Abstract: It is shown that for any Köthe space $\lambda$, $\lambda$-nuclearity coincides with pseudo- $\lambda {\lambda ^ \times }$-nuclearity. More particular results, including a Grothendieck-Pietsch criterion for $\lambda$-nuclearity of sequence spaces, are given for Köthe spaces which are regular.
Subcategories of uniform spaces
Michael D.
Rice
305-314
Abstract: The problem of embedding a topological space as a closed subspace of a product of members from a given family has received considerable attention in the past twenty years, while the corresponding problem in uniform spaces has been largely ignored. In this paper we initiate the study of the closed uniform subspaces of products of metric spaces. In §1 we introduce the functor $m$, which is used in §2 to characterize the closed subspaces of products of metric spaces and separable metric spaces, and the closed subspaces of powers of the open unit interval $(0, 1)$. In §3 we obtain various descriptions of the functor $d$ which associates to each uniform space a closed subspace of a product of metric spaces and establish the equation $md = dm$. This leads to a characterzation of the completeness of $euX$, the uniform space generated by the countable $ u$-uniform covers, in terms of the completeness of $uX$ and a countable intersection property on Cauchy filters.
A sufficient condition for hyperbolicity of partial differential operators with constant coefficient principal part
Joseph L.
Dunn
315-327
Abstract: Let $P$ be a differential operator with principal part ${P_m}$, and suppose that ${P_m}$ has constant coefficients and is hyperbolic. It is shown that the condition for hyperbolicity of $P$ when $P$ has constant coefficients, namely, that $ P$ is weaker than $ {P_m}$ is also a sufficient condition for hyperbolicity in the case where $P$ does not have constant coefficients. Some generalizations are also made to the case where $P$ is a square matrix of differential operators.
Solvable groups and quadratic forms
Richard
Tolimieri
329-345
Abstract: A solvable Lie group and a unitary representation are constructed from a given binary quadratic form. The multiplicity of this representation is related to the arithmetic of the form.
Polynomials and the limit point condition
Robert M.
Kauffman
347-366
Abstract: An $n$th order, possibly nonselfadjoint, ordinary differential expression $L$ is said to be in the limit point condition if the maximal operator ${L_M}$ in $ {L_2}[0,\infty )$ is an $ n$-dimensional extension of the minimal operator ${L_0}$. If range ${L_0}$ is closed, this definition is equivalent to the assertion that nullity ${L_M} +$ nullity${({L^ + })_M} = n$, where ${L^ + }$ is the formal adjoint of $ L$. It also implies that any operator $T$ such that ${L_0} \subseteq T \subseteq {L_M}$ is the restriction of ${L_M}$ to a set of functions described by a boundary condition at zero. In this paper, we discuss the question of when differential expressions involving complex polynomials in selfadjoint expressions are in the limit point condition.
The algebraic EHP sequence
William M.
Singer
367-382
Abstract: Let $A$ be the dual of the $\bmod - 2$ Steenrod algebra. If $ M,N$, are graded unstable $ A$-comodules, one can define and compute the derived functors ${\text{Coext} _A}(M,N)$ using unstable injective resolutions of $N$. Bousfield and Curtis have shown that these unstable Coext groups can be fit into a long exact ``EHP sequence", an algebraic analogue of the EHP sequence of homotopy theory. Our object in the present paper is to study the relationship between the $E,H$, and $P$ homomorphisms and the composition pairing $ {\text{Coext} _A}(N,R) \otimes {\text{Coext} _A}(M,N) \to {\text{Coext} _A}(M,R)$. Among our results is a formula that measures the failure of the composition product to commute.
Bounds in piecewise linear topology
L. B.
Treybig
383-405
Abstract: The following types of results are obtained: Given a polyhedral $ 2$-sphere $P$ with rectilinear triangulation $ T$ lying in the interior of a solid tetrahedron $G$ in ${E^3}$, then there is a simplicial isotopy $f:G \times [0,1] \to G$ taking $ P$ onto a tetrahedron so that for $t$ in $ [0,1],f(x,t) = x$ on Bd$(G)$ and ${f_t}$ is affine on each element of the triangulation $ S$ of $G$, where card $(S)$ is a known function of card $ (T)$. Also, given (1) $ P$ as above, (2) polyhedral disks ${D_1}$ and ${D_2}$, where Bd$({D_1}) =$ Bd$({D_2}) \subset P$ and $ \operatorname{Int} ({D_1}) \cup \operatorname{Int} ({D_2}) \subset \operatorname{Int} (P)$ and (3) a triangulation $T$ of ${D_1} \cup {D_2} \cup P$, then analogous results are found for a simplicial isotopy $f$ which is fixed on $P$ and takes ${D_1}$ onto ${D_2}$. Given $G$ as above and a piecewise linear homeomorphism $h:G \to G$ which is fixed on ${\text{Bd(G)}}$ and affine on each $ r \in R$, then analogous bounds are found for a simplicial isotopy $f:G \times [0,1] \to G$ so that ${f_0}(x) = x$ and $ {f_1}(r) = h(r)$ for all $ r$ in $R$. In the second half of this paper the normal surface and normal equation theory of Haken is briefly explained and extended slightly. Bounds are found in connection with nontrivial integer entried solutions of normal equations. Also bounds are found for the number of Simplexes used in triangulating normal surfaces associated with certain solutions of the extended normal equations.