The genus of a map
Sara
Hurvitz
1-28
Abstract: The elements $[f'](f':X' \to Y')$ of the genus $- G(f)$ of a map $f:X \to Y$ are equivalence classes of homotopy classes of maps $f'$ which satisfy: For every prime $p$ there exist homotopy equivalences $X - {G^X}(f)$ and the genus of $f$ over $ Y - {G_Y}(f)$ are defined similarly. In this paper we prove that under certain conditions on $f$, the sets $G(f)$, ${G^X}(f)$ and ${G_Y}(f)$ are finite and admit an abelian group structure. We also compare the genus of $f$ to those of $X$ and $Y$, calculate it for some principal fibrations of the form $ K(G,n - 1) \to X \to Y$, and deal with the noncancellation phenomenon.
Orientation-reversing Morse-Smale diffeomorphisms on the torus
Steve
Batterson
29-37
Abstract: For orientation-reversing diffeomorphisms on the torus necessary and sufficient conditions are given for an isotopy class to admit a Morse-Smale diffeomorphism with a specified periodic behavior.
Schur products of operators and the essential numerical range
Quentin F.
Stout
39-47
Abstract: Let $ \mathcal{E} = \{ {e_n}\} _{n = 1}^\infty$ be an orthonormal basis for a Hilbert space $ \mathcal{H}$. For operators $A$ and $B$ having matrices $({a_{ij}})_{i,\;j = 1}^\infty$ and $ ({b_{ij}})_{i,\;j}^\infty = 1$, their Schur product is defined to be $ ({a_{ij}}{b_{ij}})_{i,\:j}^\infty = 1$. This gives $ \mathcal{B}(\mathcal{H})$ a new Banach algebra structure, denoted $ {\mathcal{P}_\mathcal{E}}$. For any operator $T$ it is shown that $T$ is in the kernel (hull(compact operators)) in some $ {\mathcal{B}_\mathcal{E}}$ iff 0 is in the essential numerical range of $ T$. These conditions are also equivalent to the property that there is a basis such that Schur multiplication by $ T$ is a compact operator mapping Schatten classes into smaller Schatten classes. Thus we provide new results linking $ \mathcal{B}(\mathcal{H})$, $ {\mathcal{B}_\mathcal{E}}$ and $ \mathcal{B}(\mathcal{B}(\mathcal{H}))$.
Some general theorems on the cohomology of classifying spaces of compact Lie groups
Mark
Feshbach
49-58
Abstract: This paper is divided into two parts. The first part proves a number of general theorems on the cohomology of the classifying spaces of compact Lie groups. These theorems are proved by transfer methods, relying heavily on the double coset theorem [F$_{1}$]. Several of these results are well known while others are quite new. For the most part the proofs of the theorems are independent of each other and are quite short. Nevertheless they are true in great generality. Several are proven for arbitrary compact Lie groups and arbitrary cohomology theories. Perhaps the most interesting of the new results relates the cohomology of the classifying space of an arbitrary compact Lie group with that of the normalizer of a maximal torus. The second part of the paper generalizes many theorems to certain equivariant cohomology theories. Some of these theorems appear in [F$ _{2}$].
On spectral theory and convexity
C. K.
Fong;
Louisa
Lam
59-75
Abstract: A compact convex set $K$ in a locally convex algebra is said to be a spectral carrier if, for all $x$, $y \in K$, we have $ xy = yx \in K$ and $x + y - xy \in K$. We show that if a compact convex set $K$ is a spectral carrier, then the idempotents in $ K$ are exactly the extreme points of $K$ and form a complete lattice. Conversely, if a compact set $K$ is a closed convex hull of a lattice of commuting idempotents, then $K$ is a spectral carrier. Furthermore, a metrizable spectral carrier is a Choquet simplex if and only if its extreme points form a chain of idempotents.
Submonotone subdifferentials of Lipschitz functions
Jonathan E.
Spingarn
77-89
Abstract: The class of "lowwer-${C^1}$" functions, that is functions which arise by taking the maximum of a compactly indexed family of $ {C^1}$ functions, is characterized in terms of properties of the generalized subdifferential. A locally Lipschitz function is shown to be lower-${C^1}$ if and only if its subdifferential is "strictly submonotone". Other properties of functions with "submonotone" subdifferentials are investigated.
Invariance of solutions to invariant nonparametric variational problems
John E.
Brothers
91-111
Abstract: Let $f$ be a weak solution to the Euler-Lagrange equation of a convex nonparametric variational integral in a bounded open subset $D$ of $ {{\mathbf{R}}^n}$. Assume the boundary $B$ of $D$ to be rectifiable. Let $D$ be a compact connected Lie group of diffeomorphisms of a neighborhood of $D \cup B$ which leave $D$ invariant and assume the variational integral to be $G$-invariant. Conditions are formulated which imply that if $f$ is continuous on $D \cup B$ and $ f \circ g\vert B = f\vert B$ for $g \in G$ then $ f \circ g = f$ for every $ g \in G$. If the integrand $ L$ is strictly convex then $ f$ can be shown to have a local uniqueness property which implies invariance. In case $L$ is not strictly convex the graph $ {T_f}$ of $f$ in ${{\mathbf{R}}^n} \times {\mathbf{R}}$ is interpreted as the solution to an invariant parametric variational problem, and invariance of ${T_f}$, hence of $f$, follows from previous results of the author. For this purpose a characterization is obtained of those nonparametric integrands on ${{\mathbf{R}}^n}$ which correspond to a convex positive even parametric integrand on $ {{\mathbf{R}}^n} \times {\mathbf{R}}$ in the same way that the nonparametric area integrand corresponds to the parametric area integrand.
A representation-theoretic criterion for local solvability of left invariant differential operators on nilpotent Lie groups
Lawrence
Corwin
113-120
Abstract: Let $L$ be a left invariant differential operator on the nilpotent Lie group $N$. It is shown that if $\pi (L)$ is invertible for all irreducible representations $\pi$ in general position (and if the inverses satisfy some mild technical conditions), then $ L$ is locally solvable. This result generalizes a theorem of ${\text{L}}$. Rothschild.
Arithmetic of elliptic curves upon quadratic extension
Kenneth
Kramer
121-135
Abstract: This paper is a study of variations in the rank of the Mordell-Weil group of an elliptic curve $E$ defined over a number field $F$ as one passes to quadratic extensions $K$ of $F$. Let $S(K)$ be the Selmer group for multiplication by $ 2$ on $E(K)$. In analogy with genus theory, we describe $S(K)$ in terms of various objects defined over $ F$ and the local norm indices $ {i_\upsilon } = {\dim _{{{\mathbf{F}}_2}}}E({F_\upsilon })/$Norm$\{ E({K_w})\}$ for each completion ${F_\upsilon }$ of $F$. In particular we show that $ \dim S(K) + \dim E{(K)_2}$ has the same parity as $\Sigma {i_\upsilon }$. We compute ${i_\upsilon }$ when $E$ has good or multiplicative reduction modulo $\upsilon$. Assuming that the $2$-primary component of the Tate-Shafarevitch group $ \mathcyr{SH}(K)$ is finite, as conjectured, we obtain the parity of rank $ E(K)$. For semistable elliptic curves defined over $ {\mathbf{Q}}$ and parametrized by modular functions our parity results agree with those predicted analytically by the conjectures of Birch and Swinnerton-Dyer.
The cohomology algebras of finite-dimensional Hopf algebras
Clarence
Wilkerson
137-150
Abstract: The cohomology algebra of a finite dimensional graded connected cocommutative biassociative Hopf algebra over a field $ K$ is shown to be a finitely generated $K$-algebra. Counterexamples to the analogue of a result of Quillen (that nonnilpotent cohomology classes should have nonzero restriction to some abelian sub-Hopf algebra) are constructed, but an elementary proof of the validity of this "detection principle" for the special case of finite sub-Hopf algebras of the $ \operatorname{mod} 2$ Steenrod algebra is given. As an application, an explicit formula for the Krull dimension of the cohomology algebras of the finite skeletons of the $\operatorname{mod} 2$ Steenrod algebra is given.
Homotopy groups of the space of self-homotopy-equivalences
Darryl
McCullough
151-163
Abstract: Let $M$ be a connected sum of $ r$ closed aspherical manifolds of dimension $n \geqslant 3$, and let $ EM$ denote the space of self-homotopy-equivalences of $M$, with basepoint the identity map of $ M$. Using obstruction theory, we calculate $ {\pi _q}(EM)$ for $1 \leqslant q \leqslant n - 3$ and show that ${\pi _{n - 1}}(EM)$ is not finitely-generated. As an application, for the case $n = 3$ and $r \geqslant 3$ we show that infinitely many generators of ${\pi _1}(E{M^3},{\text{i}}{{\text{d}}_M})$ can be realized by isotopies, to conclude that ${\pi _1}({\text{Homeo}}({M^3}),{\text{i}}{{\text{d}}_M})$ is not finitely-generated.
Uniqueness of product and coproduct decompositions in rational homotopy theory
Roy
Douglas;
Lex
Renner
165-180
Abstract: Let $X$ be a nilpotent rational homotopy type such that (1) $S(X)$, the image of the Hurewicz map has finite total rank, and (2) the basepoint map of $ M$, a minimal algebra for $ X$, is an element of the Zariski closure of $ {\text{Aut}}(M)$ in ${\text{End}}(M)$ (i.e. $X$ has "positive weight"). Then (A) any retract of $X$ satisfies the two properties above, (B) any two irreducible product decompositions of $ X$ are equivalent, and (C) any two irreducible coproduct decompositions of $ X$ are equivalent.
Which curves over ${\bf Z}$ have points with coordinates in a discrete ordered ring?
Lou
van den Dries
181-189
Abstract: A criterion is given for curves defined over $ {\mathbf{Z}}$ to have an infinite point in a discrete ordered ring. Using this, one can decide effectively whether a given polynomial in $ {\mathbf{Z}}[X,Y]$ has a zero in a model for the axioms of open induction. Riemann-Roch for curves over ${\mathbf{Q}}$ is the main tool used.
Quasisymmetric embeddings in Euclidean spaces
Jussi
Väisälä
191-204
Abstract: We consider quasi-symmetric embeddings $ f:G \to {R^n}$, $ G$ open in ${R^p}$, $ p \leqslant n$. If $ p = n$, quasi-symmetry implies quasi-conformality. The converse is true if $ G$ has a sufficiently smooth boundary. If $p < n$, the Hausdorff dimension of $fG$ is less than $n$. If $fG$ has a finite $p$-measure, $f$ preserves the property of being of $ p$-measure zero. If $ p < n$ and $n \geqslant 3$, ${R^n}$ contains a quasi-symmetric $ p$-cell which is topologically wild. We also prove auxiliary results on the relations between Hausdorff measure and Čech cohomology.
Characterization of some zero-dimensional separable metric spaces
Jan
van Mill
205-215
Abstract: Let $X$ be a separable metric zero-dimensional space for which all nonempty clopen subsets are homeomorphic. We show that, up to homeomorphism, there is at most one space $Y$ which can be written as an increasing union $\cup _{n = 1}^\infty {F_n}$ of closed sets so that for all $n \in {\mathbf{N}}$, ${F_n}$ is a copy of $X$ which is nowhere dense in ${F_{n + 1}}$. If moreover $X$ contains a closed nowhere dense copy of itself, then $Y$ is homeomorphic to ${\mathbf{Q}} \times X$ where ${\mathbf{Q}}$ denotes the space of rational numbers. This gives us topological characterizations of spaces such as ${\mathbf{Q}} \times {\mathbf{C}}$ and $ {\mathbf{Q}} \times {\mathbf{P}}$.
Some countability conditions on commutative ring extensions
Robert
Gilmer;
William
Heinzer
217-234
Abstract: If $S$ is a finitely generated unitary extension ring of the commutative ring $ R$, then $S$ cannot be expressed as the union of a strictly ascending sequence $\{ {R_n}\} _{n = 1}^\infty $ of intermediate subrings. A primary concern of this paper is that of determining the class of commutative rings $T$ for which the converse holds--that is, each unitary extension of $T$ not expressible as $\cup _1^\infty {T_i}$ is finitely generated over $ T$.
Localizable analytically uniform spaces and the fundamental principle
Sönke
Hansen
235-250
Abstract: The Fundamental Principle of Ehrenpreis states that the solutions of homogeneous linear partial differential equations with constant coefficients have natural integral representations. Using the Oka-Cartan procedure Ehrenpreis derived this theorem for spaces of functions and distributions which he called localizable analytically uniform (LAU-spaces). With a new definition of LAU-spaces we explain how Hörmander's results on cohomology with bounds fit into Ehrenpreis' method of proof of the Fundamental Principle. Furthermore, we show that many of the common Fréchet-Montel spaces of functions are LAU-spaces.
Class groups of cyclic groups of square-free order
Andrew
Matchett
251-254
Abstract: Let $G$ be a finite cyclic group of square free order. Let ${\text{Cl(}}ZG{\text{)}}$ denote the projective class group of the integral group ring $ZG$. Our main theorem describes explicitly the quotients of a certain filtration of $ {\text{Cl(}}ZG{\text{)}}$. The description is in terms of class groups and unit groups of the rings of cyclotomic integers involved in $ZG$. The proof is based on a Mayer-Vietoris sequence.