Differentiable group actions on homotopy spheres. II. Ultrasemifree actions
Reinhard
Schultz
255-297
Abstract: A conceptually simple but very useful class of topological or differentiable transformation groups is given by semifree actions, for which the group acts freely off the fixed point set. In this paper, the slightly more general notion of an ultrasemifree action is introduced, and it is shown that the existing machinery for studying semifree actions on spheres may be adapted to study ultrasemifree actions equally well. Some examples and applications are given to illustrate how ultrasemifree actions (i) may be used to study questions not answerable using semifree actions alone, and (ii) provide examples of unusual smooth group actions on spheres with no semifree counterparts.
A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves
Walter D.
Neumann
299-344
Abstract: Any graph-manifold can be obtained by plumbing according to some plumbing graph $\Gamma$. A calculus for plumbing which includes normal forms for such graphs is developed. This is applied to answer several questions about the topology of normal complex surface singularities and analytic families of complex curves. For instance it is shown that the topology of the minimal resolution of a normal complex surface singularity is determined by the link of the singularity and even by its fundamental group if the singularity is not a cyclic quotient singularity or a cusp singularity.
Les mod\`eles d\'enombrables d'une th\'eorie ayant des fonctions de Skolem
Daniel
Lascar
345-366
Abstract: Let $T$ be a countable complete theory having Skolem functions. We prove that if all the types over finitely generated models are definable (this is the case for example if $T$ is stable), then either $T$ has $ {2^{{\aleph _0}}}$ countable models or all its models are homogeneous. The proof makes heavy use of stability techniques.
On the weak behaviour of partial sums of Legendre series
S.
Chanillo
367-376
Abstract: We show that the partial sum operator associated with the Legendre series is restricted weak type, but not weak type, on the ${L^p}$ spaces when $p = 4$.
Involutions on Klein spaces $M(p,\,q)$
Paik Kee
Kim
377-409
Abstract: The Klein spaces $ M(p,\,q)$ are defined (up to homeomorphisms) to be the class of closed, orientable, irreducible $3$-manifolds with finite fundamental groups, in which a Klein bottle can be embedded. Their fundamental groups act freely on the $3$-sphere ${S^3}$ in the natural way. We obtain a complete classification of the PL involutions on Klein spaces $ M(p,\,q)$. It can be applied to the study of some transformation group actions on ${S^3}$ and double branched coverings of $ {S^3}$.
Geodesic rigidity in compact nonpositively curved manifolds
Patrick
Eberlein
411-443
Abstract: Our goal is to find geometric properties that are shared by homotopically equivalent compact Riemannian manifolds of sectional curvature $K \leqslant 0$. In this paper we consider mainly properties of free homotopy classes of closed curves. Each free homotopy class is represented by at least one smooth periodic geodesic, and the nonpositive curvature condition implies that any two periodic geodesic representatives are connected by a flat totally geodesic homotopy of periodic geodesic representatives. By imposing certain geometric conditions on these periodic geodesic representatives we define and study three types of free homotopy classes: Clifford, bounded and rank $1$. Let $M$, $M\prime$ denote compact Riemannian manifolds with $K \leqslant 0$, and let $ \theta :{\pi _1}(M,\,m) \to {\pi _1}(M\prime ,\,m\prime )$ be an isomorphism. Let $ \theta$ also denote the induced bijection on free homotopy classes. Theorem A. The free homotopy class $[\alpha ]$ in $M$ is, respectively, Clifford, bounded or rank $1$ if and only if the class $\theta [\alpha ]$ in $M\prime$ is of the same type. Theorem B. If $M$, $M\prime$ have dimension $3$ and do not have a rank $ 1$ free homotopy class then they have diffeomorphic finite covers of the form ${S^1} \times {M^2}$. The proofs of Theorems A and B use the fact that $\theta$ is induced by a homotopy equivalence $f:(M,\,m) \to (M\prime ,\,m\prime )$. Theorem C. The manifold $M$ satisfies the Visibility axiom if and only if $M\prime$ satisfies the Visibility axiom.
Quasicoherent sheaves over affine Hensel schemes
Silvio
Greco;
Rosario
Strano
445-465
Abstract: The following two theorems concerning affine Hensel schemes are proved. Theorem A. Every quasi-coherent sheaf over an affine Hensel scheme is generated by its global sections. Theorem B. ${H^p}(X,\,F) = 0$ for all positive $ p$ and all quasi-coherent sheaves $F$ over an affine Hensel scheme $ X$.
Fundamental solutions for differential equations associated with the number operator
Yuh Jia
Lee
467-476
Abstract: Let $(H,\,B)$ be an abstract Wiener space. If $ u$ is a twice $ H$-differentiable function on $B$ such that $ Du(x) \in {B^{\ast}}$ and $ {D^2}u(x)$ is of trace class, then we define $\mathfrak{N}u(x) = - \Delta u(x) + (x,\,Du(x))$, where $\Delta u(x) = {\operatorname{trace} _H}\,{D^2}u(x)$ is the Laplacian and $( \cdot ,\, \cdot )$ denotes the $B$- $ {B^{\ast}}$ pairing. The closure $\overline{\mathfrak{N}} $ of $\mathfrak{N}$ is known as the number operator. In this paper, we investigate the existence, uniqueness and regularity of solutions for the following two types of equations: (1) ${u_t} = - \mathfrak{N}u$ (initial value problem) and (2) ${\mathfrak{N}^k}u = f(k \geqslant 1)$. We show that the fundamental solutions of (1) and (2) exist in the sense of measures and we represent their solutions by integrals with respect to these measures.
Probability and interpolation
G. G.
Lorentz;
R. A.
Lorentz
477-486
Abstract: An $m \times n$ matrix $E$ with $n$ ones and $(m - 1)n$ zeros, which satisfies the Pólya condition, may be regular and singular for Birkhoff interpolation. We prove that for random distributed ones, $ E$ is singular with probability that converges to one if $m$, $n \to \infty $. Previously, this was known only if $m \geqslant (1 + \delta )n/\log n$. For constant $ m$ and $n \to \infty$, the probability is asymptotically at least $ \tfrac{1} {2}$.
The Dror-Whitehead theorem in prohomotopy and shape theories
S.
Singh
487-496
Abstract: Many analogues of the classical Whitehead theorem from homotopy theory are now available in pro-homotopy and shape theories. E. Dror has significantly extended the homology version of the Whitehead theorem from the well-known simply connected case to the more general, for instance, nilpotent case. We prove a full analogue of Dror's theorems in pro-homotopy and shape theories. More specifically, suppose $ \underline f :\underline X \to \underline Y$ is a morphism in the pro-homotopy category of pointed and connected topological spaces which induces isomorphisms of the integral homology pro-groups. Then $ \underline f$ induces isomorphisms of the homotopy pro-groups, for instance, when $\underline X$ and $\underline Y$ are simple, nilpotent, complete, or $\underline H $-objects; these notions are well known in homotopy theory and we have naturally extended them to pro-homotopy and shape theories.