Rational LS category and its applications
Yves
Félix;
Stephen
Halperin
1-37
Abstract: Let $S$ be a $1$-connected CW-complex of finite type and put $ {\text{ca}}{{\text{t}}_0}(S) =$ Lusternik-Schnirelmann category of the localization $ {S_{\mathbf{Q}}}$. This invariant is characterized in terms of the minimal model of $S$. It is shown that if $\phi :S \to T$ is injective on $ {\pi _ \ast } \otimes {\mathbf{Q}}$ then $ {\text{ca}}{{\text{t}}_0}(S) \leqslant {\text{ca}}{{\text{t}}_0}(T)$, and this result is strengthened when $ \phi$ is the fibre inclusion of a fibration. It is also shown that if $ {H^ \ast }(S;{\mathbf{Q}}) < \infty$ then either ${\pi _ \ast }(S) \otimes {\mathbf{Q}} < \infty$ or the groups ${\pi _k}(S) \otimes {\mathbf{Q}}$ grow exponentially with $k$.
Cohomology of nilmanifolds and torsion-free, nilpotent groups
Larry A.
Lambe;
Stewart B.
Priddy
39-55
Abstract: Let $M$ be a nilmanifold, i.e. $M = G/D$ where $ G$ is a simply connected, nilpotent Lie group and $D$ is a discrete uniform, nilpotent subgroup. Then $M \simeq K(D,1)$. Now $D$ has the structure of an algebraic group and so has an associated algebraic group Lie algebra $ L(D)$. The integral cohomology of $M$ is shown to be isomorphic to the Lie algebra cohomology of $L(D)$ except for some small primes depending on $ D$. This gives an effective procedure for computing the cohomology of $ M$ and therefore the group cohomology of $D$. The proof uses a version of form cohomology defined for subrings of $ {\mathbf{Q}}$ and a type of Hirsch Lemma. Examples, including the important unipotent case, are also discussed.
Contractive projections on $C\sb{0}(K)$
Yaakov
Friedman;
Bernard
Russo
57-73
Abstract: We show that the range of a norm one projection on a commutative $ {C^\ast}$-algebra has a ternary product structure (Theorem 2). We describe and characterize all such projections in terms of extreme points in the unit ball of the image of the dual (Theorem 1). We give necessary and sufficient conditions for the range to be isometric to a $ {C^\ast}$-algebra (Theorem 4) and we show that the range is a ${C_\sigma }$-space (Theorem 5).
Simple knots in compact, orientable $3$-manifolds
Robert
Myers
75-91
Abstract: A simple closed curve $J$ in the interior of a compact, orientable $ 3$-manifold $M$ is called a simple knot if the closure of the complement of a regular neighborhood of $ J$ in $M$ is irreducible and boundary-irreducible and contains no non-boundary-parallel, properly embedded, incompressible annuli or tori. In this paper it is shown that every compact, orientable $ 3$-manifold $M$ such that $ \partial M$ contains no $ 2$-spheres contains a simple knot (and thus, from work of Thurston, a knot whose complement is hyperbolic). This result is used to prove that such a $3$-manifold is completely determined by its set $ \mathcal{K}(M)$ of knot groups, i.e, the set of groups ${\pi _1}(M - J)$ as $J$ ranges over all the simple closed curves in $ M$. In addition, it is proven that a closed $3$-manifold $M$ is homeomorphic to ${S^3}$ if and only if every simple closed curve in $ M$ shrinks to a point inside a connected sum of graph manifolds and $3$-cells.
The asymptotic expansion for the trace of the heat kernel on a generalized surface of revolution
Ping Charng
Lue
93-110
Abstract: Let $M$ be a smooth compact Riemannian manifold without boundary. Let $I$ be an open interval. Let $h(r)$ be a smooth positive function. Let $g$ be the metric on $M$. Consider the fundamental solution $ E(x,y,{r_1},{r_2};t)$ of the heat equation on $M \times I$ with metric $ {h^2}(r)g + dr \otimes dr$ (when $E$ exists globally we call it the heat kernel on $M \times I$). The coefficients of the asymptotic expansion of the trace $E$ are studied and expressed in terms of corresponding coefficients on the basis $M$. It is fulfilled by means of constructing a parametrix for $E$ which is different from a parametrix in the standard form. One important result is that each of the former coefficients is a linear combination of the latter coefficients.
$q$-extension of the $p$-adic gamma function. II
Neal
Koblitz
111-129
Abstract: Taylor series and asymptotic expansions are developed for $q$-extensions of the $p$-adic psi (derivative of log-gamma) function "twisted" by roots of unity. Connections with $ p$-adic $L$-functions and $q$-expansions of Eisenstein series are discussed. The $p$-adic series are compared with the analogous classical expansions.
Trace-like functions on rings with no nilpotent elements
M.
Cohen;
Susan
Montgomery
131-145
Abstract: Let $R$ be a ring with no nilpotent elements, with extended center $C$, and let $E$ be the set of idempotents in $ C$. Our first main result is that for any finite group $G$ acting as automorphisms of $R $, there exist a finite set $L \subseteq E$ and an $ {R^G}$-bimodule homomorphism $ \tau :R \to {(RL)^G}$ such that $\tau (R)$ is an essential ideal of $ {(RE)^G}$. This theorem is applied to show the following: if $R$ is a Noetherian, affine $ PI$-algebra (with no nilpotent elements) over the commutative Noetherian ring $ A$, and $G$ is a finite group of $ A$-automorphisms of $ R$ such that $ {R^G}$ is Noetherian, then $ {R^G}$ is affine over $ A$.
When is the natural map $X\rightarrow \Omega \Sigma X$ a cofibration?
L. Gaunce
Lewis
147-155
Abstract: It is shown that a map $f:X \to F(A,W)$ is a cofibration if its adjoint $ f:X \wedge A \to W$ is a cofibration and $X$ and $A$ are locally equiconnected (LEC) based spaces with $A$ compact and nontrivial. Thus, the suspension map $ \eta :X \to \Omega \sum X$ is a cofibration if $X$ is LEC. Also included is a new, simpler proof that C.W. complexes are LEC. Equivariant generalizations of these results are described.
Branching degrees above low degrees
Peter A.
Fejer
157-180
Abstract: In this paper, we investigate the location of the branching degrees within the recursively enumerable (r.e.) degrees. We show that there is a branching degree below any given nonzero r.e. degree and, using a new branching degree construction and a technique of Robinson, that there is a branching degree above any given low r.e. degree. Our results extend work of Shoenfield and Soare and Lachlan on the generalized nondiamond question and show that the branching degrees form an automorphism base for the r.e. degrees.
Algebraic and geometric models for $H\sb{0}$-spaces
J.
Aguadé;
A.
Zabrodsky
181-190
Abstract: For every $ {H_0}$-space (i.e. a space whose rationalization is an $H$-space) we construct a space $J$ depending only on ${H^\ast}(X;{\mathbf{Z}})$ and a rational homotopy equivalence $J \to X$ (i.e. $J$ is a universal space to the left of all $ {H_0}$-spaces having the same integral cohomology ring as $X$ is constructed generalizing the James reduced product. We study also the integral cohomology of $ {H_0}$-spaces and we prove that under certain conditions it contains an algebra with divided powers.
No division implies chaos
Tien Yien
Li;
Michał
Misiurewicz;
Giulio
Pianigiani;
James A.
Yorke
191-199
Abstract: Let $I$ be a closed interval in ${R^1}$ and $f:I \to I$ be continuous. Let ${x_0} \in I$ and $\displaystyle {x_{i + 1}} = f({x_i})\quad {\text{for}}\;i > 0.$ We say there is no division for $ ({x_1},{x_2}, \ldots ,{x_n})$ if there is no $a \in I$ such that ${x_j} < a$ for all $j$ even and ${x_j} < a$ for all $j$ odd. The main result of this paper proves the simple statement: no division implies chaos. Also given here are some converse theorems, detailed estimates of the existing periods, and examples which show that, under our conditions, one cannot do any better.
The Catalan equation over function fields
Joseph H.
Silverman
201-205
Abstract: Let $K$ be the function field of a projective variety. Fix $ a,b,c \in {K^ \ast }$. We show that if $ \max \{ m,n\}$ is sufficiently large, then the Catalan equation $a{x^m} + b{y^n} = c$ has no nonconstant solutions $ x,y \in K$.
Topological invariant means on the von Neumann algebra ${\rm VN}(G)$
Ching
Chou
207-229
Abstract: Let $VN(G)$ be the von Neumann algebra generated by the left regular representation of a locally compact group $G$, $A(G)$ the Fourier algebra of $G$ and $ TIM(\hat G)$ the set of topological invariant means on $VN(G)$. Let ${\mathcal{F}_1} = \{ \mathcal{O} \in {({l^\infty })^ \ast }\} :\mathcal{O} \geqslant 0,\;\vert\vert\mathcal{O}\vert\vert = 1$ and $\mathcal{O}(f) = 0\;{\text{if}}\;f \in {l^\infty }$ and $f(n) \to 0\} $. We show that if $ G$ is nondiscrete then there exists a linear isometry $\Lambda$ of $ {({l^\infty })^ \ast }$ into $VN{(G)^ \ast }$ such that $ \Lambda ({\mathcal{F}_1}) \subset TIM(\hat G)$. When $G$ is further assumed to be second countable then ${\mathcal{F}_1}$ can be embedded into some predescribed subsets of $ TIM(\hat G)$. To prove these embedding theorems for second countable groups we need the existence of a sequence of means $\{ {u_n}\}$ in $A(G)$ such that their supports in $VN(G)$ are mutually orthogonal and $ \vert\vert u{u_n} - {u_n}\vert\vert \to 0\;{\text{if}}\;u$ is a mean in $A(G)$. Let $F(\hat G)$ be the space of all $T \in VN(G)$ such that $m(T)$ is a constant as $m$ runs through $ TIM(\hat G)$ and let $ W(\hat G)$ be the space of weakly almost periodic elements in $VN(G)$. We show that the following conditions are equivalent: (i) $G$ is discrete, (ii) $F(\hat G)$ is an algebra and (iii) $ (A(G) \cdot VN(G)) \cap F(\hat G) \subset W(\hat G)$.
An almost sure invariance principle for Hilbert space valued martingales
Gregory
Morrow;
Walter
Philipp
231-251
Abstract: We obtain an almost sure approximation of a martingale with values in a real separable Hilbert space $H$ by a suitable $H$-valued Brownian motion.
Local blow-up of stratified sets up to bordism
S.
Buoncristiano;
M.
Dedò
253-280
Abstract: Homological obstructions are given, whose vanishing is a necessary and sufficient condition for the existence of a blow-up of an abstract prestratification $ V$ along a 'locally top-dimensional' substratification $V'$.
Operator-self-similar processes in a finite-dimensional space
William N.
Hudson;
J. David
Mason
281-297
Abstract: A general representation for an operator-self-similar process is obtained and its class of exponents is characterized. It is shown that such a process is the limit in a certain sense of an operator-normed process and any limit of an operator-normed process is operator-self-similar.
Products of $k$-spaces and spaces of countable tightness
G.
Gruenhage;
Y.
Tanaka
299-308
Abstract: In this paper, we obtain results of the following type: if $ f:X \to Y$ is a closed map and $X$ is some "nice" space, and ${Y^2}$ is a $k$-space or has countable tightness, then the boundary of the inverse image of each point of $ Y$ is "small" in some sense, e.g., Lindelöf or $ {\omega _1}$-compact. We then apply these results to more special cases. Most of these applications combine the "smallness" of the boundaries of the point-inverses obtained from the earlier results with "nice" properties of the domain to yield "nice" properties on the range.
Global solvability on compact Heisenberg manifolds
Leonard F.
Richardson
309-317
Abstract: We apply the methods of primary and irreducible Fourier series on compact nilmanifolds to determine the ranges of all first order invariant operators on the compact Heisenberg manifolds. We show that the sums of primary solutions behave better on these manifolds than on any multidimensional torus.
On the double suspension homomorphism at odd primes
J. R.
Harper;
H. R.
Miller
319-331
Abstract: We work with the $ {E_1}$-term for spheres and the stable Moore space, given by the $\Lambda $-algebra at odd primes. Writing $W(n) = \Lambda (2n + 1)/\Lambda (2n - 1)$ and $M(0) = {H_ \ast }({S^0}{ \cup _p}{e^1})$, we construct compatible maps ${f_n} \cdot W(n) \to M(0)\tilde \otimes \Lambda$ and prove the Metastability Theorem: in homology ${f_n}$ induces an isomorphism for $ \sigma < 2({p^2} - 1)(s - 2) + pqn - 2p - 2$ where $\sigma =$ stem degree$ $, $s =$ homological degree resulting from the bigrading of $\Lambda$ and $ q = 2p - 2$. There is an operator ${\upsilon _1}$ corresponding to the Adams stable self-map of the Moore space and ${\upsilon _1}$ extends to $W(n)$. A corollary of the Metastability Theorem and the Localization Theorem of the second author is that the map ${f_n}$ induces an isomorphism on homology after inverting $ {\upsilon _1}$.
Resolvent operators for integral equations in a Banach space
R. C.
Grimmer
333-349
Abstract: Conditions are given which ensure the existence of a resolvent operator for an integrodifferential equation in a Banach space. The resolvent operator is similar to an evolution operator for nonautonomous differential equations in a Banach space. As in the finite dimensional case, this operator is used to obtain a variation of parameters formula which can be used to obtain results concerning the asymptotic behaviour of solutions and weak solutions.
On the oscillation theory of $f\sp{\prime\prime}+Af=0$ where $A$ is entire
Steven B.
Bank;
Ilpo
Laine
351-363
Abstract: In this paper, we investigate the distribution of zeros of solutions of $f'' + A(z)f = 0$. More specifically, results are obtained concerning the exponent of convergence of the zero-sequence of a solution in both the case where $A(z)$ is a polynomial, and the case where $ A(z)$ is transcendental.
If all normal Moore spaces are metrizable, then there is an inner model with a measurable cardinal
William G.
Fleissner
365-373
Abstract: We formulate an axiom, HYP, and from it construct a normal, metacompact, nonmetrizable Moore space. HYP unifies two better known axioms. The Continuum Hypothesis implies HYP; the nonexistence of an inner model with a measurable cardinal implies HYP. As a consequence, it is impossible to replace Nyikos' "provisional" solution to the normal Moore space problem with a solution not involving large cardinals. Finally, we discuss how this space continues a process of lowering the character for normal, not collectionwise normal spaces.
Exact dynamical systems and the Frobenius-Perron operator
A.
Lasota;
James A.
Yorke
375-384
Abstract: Conditions are investigated which guarantee exactness for measurable maps on measure spaces. The main application is to certain piecewise continuous maps $T$ on $[0,1]$ for which $T'(0) > 1$. We assume $[0,1]$ can be broken into intervals on which $ T$ is continuous and convex and at the left end of these intervals $T = 0$ and $dt/dx > 0$. Such maps have an invariant absolutely continuous density which is exact.
A symplectic Banach space with no Lagrangian subspaces
N. J.
Kalton;
R. C.
Swanson
385-392
Abstract: In this paper we construct a symplectic Banach space $(X,\Omega )$ which does not split as a direct sum of closed isotropic subspaces. Thus, the question of whether every symplectic Banach space is isomorphic to one of the canonical form $Y \times {Y^ \ast }$ is settled in the negative. The proof also shows that $ \mathfrak{L}(X)$ admits a nontrivial continuous homomorphism into $\mathfrak{L}(H)$ where $H$ is a Hilbert space.
Circle actions and fundamental groups for homology $4$-spheres
Steven
Plotnick
393-404
Abstract: We generalize work of Fintushel and Pao to give a topological classification of smooth circle actions on oriented $4$-manifolds $\Sigma$ satisfying ${H_1}(\Sigma ) = 0$. We then use these ideas to construct infinite families of homology $4$-spheres that do not admit effective circle actions, and whose fundamental groups cannot be $ 3$-manifold groups.