On the integrable and square-integrable representations of ${\rm Spin}(1, 2m)$
Ernest
Thieleker
1-40
Abstract: All the unitary equivalence classes of irreducible integrable and square-integrable representations of the groups ${\text{Spin}}(1,2m),m \geqslant 2$, are determined. The method makes use of some elementary results on differential equations and the classification of irreducible unitary representations of these groups. In the latter classification, certain ambiguities resulting from possible equivalences not taken into account in a previous paper, are cleared up here.
Inverting a cylinder through isometric immersions and isometric embeddings
B.
Halpern;
C.
Weaver
41-70
Abstract: It is shown that a right circular cylinder can be turned inside out through immersions which preserve its flat Riemannian metric if and only if its diameter is greater than its height. Such a cylinder can be turned inside out through embeddings which preserve its flat Riemannian metric provided its diameter is greater than $(\pi + 2)/\pi$ times its height. A flat Möbius strip has an immersion into Euclidean three dimensional space which preserves its Riemannian metric if and only if its length is greater than $\pi /2$ times its height.
The Blumberg problem
William A. R.
Weiss
71-85
Abstract: A compact Hausdorff space and a real-valued function on this space are constructed such that the function is not continuous on any dense subspace. This solves the Blumberg problem. Some related results are established.
Decompositions of linear maps
Sze Kai J.
Tsui
87-112
Abstract: In the first part we show that the decomposition of a bounded selfadjoint linear map from a ${C^\ast}$-algebra into a given von Neumann algebra as a difference of two bounded positive linear maps is always possible if and only if that range algebra is a ``strictly finite'' von Neumann algebra of type I. In the second part we define a ``polar decomposition'' for some bounded linear maps and show that polar decomposition is possible if and only if the map satisfies a certain ``norm condition". We combine the concepts of polar and positive decompositions to show that polar decomposition for a selfadjoint map is equivalent to a strict Hahn-Jordan decomposition (see Theorems 2.2.4 and 2.2.8).
Semigroups of unbounded linear operators in Banach space
Rhonda Jo
Hughes
113-145
Abstract: One-parameter families of unbounded linear operators acting in a Banach space X, and satisfying the semigroup and strong continuity properties on a suitable subspace of X, are discussed; the notion of infinitesimal generator is generalized to this unbounded setting, and a Hille-Yosida-type theorem is proved. The theory is illustrated by several examples, which include fractional integrals and derivatives acting in $ {L^p}(0,\infty )$.
Circle actions on simply connected $4$-manifolds
Ronald
Fintushel
147-171
Abstract: Locally smooth $ {S^1}$-actions on simply connected 4-manifolds are studied in terms of their weighted orbit spaces. An equivariant classification theorem is proved, and the weighted orbit space is used to compute the quadratic form of a given simply connected 4-manifold with ${S^1}$-action. This is used to show that a simply connected 4-manifold which admits a locally smooth $ {S^1}$-action must be homotopy equivalent to a connected sum of copies of ${S^4},C{P^2}, - C{P^2}$, and ${S^2} \times {S^2}$.
Finiteness in the minimal models of Sullivan
Stephen
Halperin
173-199
Abstract: Let X be a 1-connected topological space such that the vector spaces $ {\Pi _ \ast }(X) \otimes {\mathbf{Q}}$ and ${H^\ast}(X;{\mathbf{Q}})$ are finite dimensional. Then ${H^\ast}(X;{\mathbf{Q}})$ satisfies Poincaré duality. Set $ {\chi _\Pi } = \sum {( - 1)^p}\dim {\Pi _p}(X) \otimes {\mathbf{Q}}$ and ${\chi _c} =$ $\sum {( - 1)^p}\dim {H^p}(X;{\mathbf{Q}})$. Then $ {\chi _\Pi } \leqslant 0$ and $ {\chi _c} \geqslant 0$. Moreover the conditions: (1) ${\chi _\Pi } = 0$, (2) ${\chi _c} > 0,{H^\ast}(X;{\mathbf{Q}})$ evenly graded, are equivalent. In this case $ {H^\ast}(X;{\mathbf{Q}})$ is a polynomial algebra truncated by a Borel ideal. Finally, if X is a finite 1-connected C.W. complex, and an r-torus acts continuously on X with only finite isotropy, then ${\chi _\Pi } \leqslant - r$.
Topological examples of projective modules
Richard G.
Swan
201-234
Abstract: A new and more elementary proof is given for LØnsted's theorem that vector bundles over a finite complex can be represented by projective modules over a noetherian ring. The rings obtained are considerably smaller than those of LØnsted. In certain cases, methods associated with Hilbert's 17th problem can be used to give a purely algebraic description of the rings. In particular, one obtains a purely algebraic characterization of the homotopy groups of the classical Lie groups. Several examples are given of rings such that all projective modules of low rank are free. If $m \equiv 2 \bmod 4$, there is a noetherian ring of dimension m with nontrivial projective modules of rank m such that all projective modules of ${\text{rank}} \ne m$ are free.
On the cohomology groups of a polarisation and diagonal quantisation
J. H.
Rawnsley
235-255
Abstract: The sheaf ${\mathcal{S}_F}(L)$ of germs of sections of a line bundle L on a manifold X covariant constant with respect to a flat connection defined for vectors in a complex subbundle F of the tangent bundle has a resolution by differential forms defined on F with values in L provided F satisfies the integrability conditions of the complex Frobenius theorem. This includes as special cases the de Rham and Dolbeault resolutions.
The reduced Witt ring of a formally real field
Ron
Brown
257-292
Abstract: The reduced Witt rings of certain formally real fields are computed here in terms of some basic arithmetic invariants of the fields. For some fields, including the rational function field in one variable over the rational numbers and the rational function field in two variables over the real numbers, this is done by computing the image of the total signature map on the Witt ring. For a wider class of fields, including all those with only finitely many square classes, it is done by computing the Witt rings of certain ultracompletions of the field and representing the reduced Witt ring as an appropriate subdirect product of the Witt rings of the ultracompletions. The reduced Witt rings of a still wider class of fields, including for example the fields of transcendence degree one and the rational function field in three variables over the real numbers, are computed similarly, except that the description of the subdirect product no longer involves only local conditions.
Coherence in nonmonoidal closed categories
Miguel L.
Laplaza
293-311
Abstract: A Trionmonoidal closed category is a category with an internal homomorphism functor, left Yoneda natural arrows, unity object and some natural transformations and coherence axioms. The object of this paper is to give a complete solution of the coherence problem in this structure: we use a cut-elimination theorem as basic tool to prove that the elementary natural transformations are characterized by their graph (roughly speaking the graph is the type of identification imposed by a natural transformation on the arguments of its domain and codomain).
Invariant-free representations of augmented rings
Peter M.
Curran
313-319
Abstract: Let $\Gamma$ be an augmented ring in the sense of Cartan-Eilenberg, and let there be given a representation of $\Gamma$ in $ {\text{End}_k}\;A$, where A is a finite dimensional vector space over the field k. We show that all cohomology of $\Gamma$ in A is trivial if there are no invariants in A under the action of a suitable commutative subring of $\Gamma$. This generalizes a previous result of the author for group cohomology, and is applied to obtain sufficient conditions for the vanishing of the cohomology of Lie algebras and associative algebras.
Continua whose cone and hyperspace are homeomorphic
Sam B.
Nadler
321-345
Abstract: Let X be a (nonempty) metric continuum. By the hyperspace of X we mean $C(X) = \{ A$: A is a nonempty subcontinuum of $X\}$ with the Hausdorff metric H. It is determined that there are exactly eight hereditarily decomposable continua X such that the cone over X is homeomorphic to $C(X)$. Information about cone-to-hyperspace homeomorphisms, and about arc components for general classes of continua whose cone and hyperspace are homeomorphic is obtained. It is determined that indecomposable continua whose cone and hyperspace are homeomorphic have arcwise connected composants and, if finite-dimensional, have a strong form of the cone = hyperspace property.
Growth problems for subharmonic functions of finite order in space
N. V.
Rao;
Daniel F.
Shea
347-370
Abstract: For a function $ u(x)$ subharmonic (or $ {C^2}$) in ${{\mathbf{R}}^m}$, we compare the ``harmonics'' (defined in §1) of u with those of a related subharmonic function whose total Riesz mass in $\vert x\vert \leqslant r$ is the same as that of u, but whose ${L^2}$ norm on $\vert x\vert = r$ is maximal, for all $0 < r < \infty$. We deduce estimates on the growth of the Riesz mass of u in $\vert x\vert \leqslant r$, as $r \to \infty$.