The equivariant Hurewicz map
L. Gaunce
Lewis
433-472
Abstract: Let $G$ be a compact Lie group, $ Y$ be a based $ G$-space, and $ V$ be a $ G$-representation. If $\pi _V^G(Y)$ is the equivariant homotopy group of $Y$ in dimension $V$ and $H_V^G(Y)$ is the equivariant ordinary homology group of $Y$ with Burnside ring coefficients in dimension $ V$, then there is an equivariant Hurewicz map $\displaystyle h:\pi _V^G(Y) \to H_V^G(Y).$ One should not expect this map to be an isomorphism, since $ H_V^G(Y)$ must be a module over the Burnside ring, but $ \pi _V^G(Y)$ need not be. However, here it is shown that, under the obvious connectivity conditions on $Y$, this map induces an isomorphism between $ H_V^G(Y)$ and an algebraically defined modification of $ \pi _V^G(Y)$. The equivariant Freudenthal Suspension Theorem contains a technical hypothesis that has no nonequivariant analog. Our results shed some light on the behavior of the suspension map when this rather undesirable technical hypothesis is not satisfied.
Global regularity on $3$-dimensional solvmanifolds
Jacek M.
Cygan;
Leonard F.
Richardson
473-488
Abstract: Let $M$ be any $3$-dimensional (nonabelian) compact solvmanifold. We apply the methods of representation theory to study the convergence of Fourier series of smooth global solutions to first order invariant partial differential equations $Df = g$ in $ {C^\infty }(M)$. We show that smooth infinite-dimensional irreducible solutions, when they exist, satisfy estimates strong enough to guarantee uniform convergence of the irreducible (or primary) Fourier series to a smooth global solution.
Characterizations of algebras arising from locally compact groups
Paul L.
Patterson
489-506
Abstract: Two Banach $ ^{\ast}$-algebras are naturally associated with a locally compact group, $ G$: the group algebra, $ {L^1}(G)$, and the measure algebra, $M(G)$. Either of these Banach algebras is a complete set of invariants for $G$. In any Banach $^{\ast}$-algebra, $A$, the norm one unitary elements form a group, $ S$. Using $S$ we characterize those Banach $ ^{\ast}$-algebras, $ A$, which are isometrically $^{\ast}$-isomorphic to $M(G)$. Our characterization assumes that $A$ is the dual of some Banach space and that its operations are continuous in the resulting weak $ ^{\ast}$ topology. The other most important condition is that the convex hull of $ S$ must be weak$ ^{\ast}$ dense in the unit ball of $A$. We characterize Banach $ ^{\ast}$-algebras which are isomerically isomorphic to ${L^1}(G)$ for some $G$ as those algebras, $A$, whose double centralizer algebra, $ D(A)$, satisfies our characterization for $M(G)$. In addition we require $A$ to consist of those elements of $ D(A)$ on which $ S$ (defined relative to $ D(A)$) acts continuously with its weak$^{\ast}$ topology. Using another characterization of $ {L^1}(G)$ we explicitly calculate the above isomorphism between $ A$ and ${L^1}(G)$.
On the singular cardinal hypothesis
W. J.
Mitchell
507-530
Abstract: We use core model theory to obtain the following lower bounds to the consistency strength for the failure of the Singular Cardinal Hypothesis: Suppose that $\kappa$ is a singular strong limit cardinal such that ${2^\kappa } > {\kappa ^ + }$. Then there is an inner model $K$ such that $o(\kappa ) = {\kappa ^{ + + }}$ in $K$ if $\kappa$ has uncountable cofinality, and $\forall \alpha < \kappa \exists \nu < \kappa o(\kappa ) \geqslant \nu$ in $K$ otherwise.
Adams' cobar equivalence
Yves
Félix;
Stephen
Halperin;
Jean-Claude
Thomas
531-549
Abstract: Let $F$ be the homotopy fibre of a continuous map $Y\xrightarrow{\omega }X$, with $X$ simply connected. We modify and extend a construction of Adams to obtain equivalences of DGA's and DGA modules, $\displaystyle \Omega {C_{\ast}}(X)\xrightarrow{ \simeq }C{U_{\ast}}(\Omega X),$ and $\displaystyle \Omega (C_{\ast}^\omega (Y);{C_{\ast}}(X))\xrightarrow{ \simeq }C{U_{\ast}}(F),$ where on the left-hand side $\Omega ( - )$ denotes the cobar construction. Our equivalences are natural in $X$ and $\omega$. Using this result we show how to read off the algebra ${H_{\ast}}(\Omega X;R)$ and the ${H_{\ast}}(\Omega X;R)$ module, ${H_{\ast}}(F;R)$, from free models for the singular cochain algebras $ C{S^{\ast}}(X)$ and $C{S^{\ast}}(Y)$; here we assume $R$ is a principal ideal domain and $ X$ and $Y$ are of finite $R$ type.
Polynomial solutions to constant coefficient differential equations
S. Paul
Smith
551-569
Abstract: Let $ {D_1}, \ldots ,{D_r} \in \mathbb{C}[\partial /\partial {x_1}, \ldots ,\partial /\partial {x_n}]$ be constant coefficient differential operators with zero constant term. Let $\displaystyle S = \{ f \in \mathbb{C}[{x_1}, \ldots ,{x_n}]\vert{D_j}(f) = 0\;{\text{for all }}1 \leqslant j \leqslant r\}$ be the space of polynomial solutions to the system of simultaneous differential equations ${D_j}(f) = 0$. It is proved that $S$ is a module over $\mathcal{D}(V)$, the ring of differential operators on the affine scheme $V$ with coordinate ring $ \mathbb{C}[\partial /\partial {x_1}, \ldots ,\partial /\partial {x_n}]/\left\langle {{D_1}, \ldots ,{D_r}} \right\rangle$. If $ V$ is smooth and irreducible, then $S$ is a simple $ \mathcal{D}(V)$-module, $ S = 1.\mathcal{D}(V)$, and the generators for $ \mathcal{D}(V)$ yield an algorithm for obtaining a basis for $S$. If $V$ is singular, then $S$ need not be simple. However, $S$ is still a simple $\mathcal{D}(V)$-module for certain curves $ V$, and certain homogeneous spaces $V$, and this allows one to obtain a basis for $ S$, through knowledge of $ \mathcal{D}(V)$.
A direct geometric proof of the Lefschetz fixed point formulas
John D.
Lafferty;
Yan Lin
Yu;
Zhang
Weiping
571-583
Abstract: In this paper we prove the Lefschetz fixed point formulas of Atiyah, Singer, Segal, and Bott for isometries by using the direct geometric method initiated by Patodi.
Studying links via closed braids. V. The unlink
Joan S.
Birman;
William W.
Menasco
585-606
Abstract: The main result is a version of Markov's Theorem which does not involve stabilization, in the special case of the $ r$-component link. As a corollary, it is proved that the stabilization index of a closed braid representative of the unlink is at most $ 1$. To state the result, we need the concept of an "exchange move", which modifies a closed braid without changing its link type or its braid index. For generic closed braids exchange moves change conjugacy class. Theorem $1$ shows that exchange moves are the only obstruction to reducing a closed $n$-braid representative of the $ r$-component unlink to the standard closed $r$-braid representative, through a sequence of braids of nonincreasing braid index.
Locally constant cohomology
E.
Spanier
607-624
Abstract: In this paper we study locally constant cohomology theories on a space $X$. We prove that for cohomology theories on a category of paracompact spaces the homotopy axiom of Eilenberg-Steenrod is a consequence of the other Eilenberg-Steenrod axioms together with continuity and either additivity or weak additivity. We also prove that if $H$ is a cohomology theory on the space of a simplicial complex $K$ which is locally constant on every open simplex of $K$ there is a spectral sequence converging to $ H(\vert K\vert)$ whose $ {E_2}$-term is the usual simplicial cohomology of $K$ with coefficients in various stacks on $ K$ defined by $ H$. This generalizes some known spectral sequences.
An intrinsic description of shape
José M. R.
Sanjurjo
625-636
Abstract: We give in this paper a description of the shape category of compacta in terms of multivalued maps. We introduce the notion of a multi-net and prove that the shape category of compacta is isomorphic to the category $HN$ whose objects are metric compacta and whose morphisms are homotopy classes of multi-nets. This description is intrinsic in the sense that it does not make use of external elements such as ANR-expansions or embeddings in appropriate AR-spaces. We present many applications of this new formulation of shape.
The connectivity of multicurves determined by integral weight train tracks
Andrew
Haas;
Perry
Susskind
637-652
Abstract: An integral weighted train track on a surface determines the isotopy class of an embedded closed $1$-manifold. We are interested in the connectivity of the resulting $1$-manifold. In general there is an algorithm for determining connectivity, and in the simplest case of a $ 2$-parameter train track on a surface of genus one there is an explicit formula. We derive a formula for the connectivity of the closed $1$-manifold determined by a $4$-parameter train track on a surface of genus two which is computable in polynomial time. We also give necessary and sufficient conditions on the parameters for the resulting $1$-manifold to be connected.
The duality between subsemigroups of Lie groups and monotone functions
Karl-Hermann
Neeb
653-677
Abstract: In this paper we give a characterization of those convex cones $ W$ in the Lie algebra ${\mathbf{L}}(G)$ of a connected Lie group $ G$ which are global in $G$, i.e. for which there exists a closed subsemigroup $ S$ in $G$ having $W$ as its tangent wedge ${\mathbf{L}}(S)$. The main result is the Characterization Theorem II.12. We also prove in Corollary II.6 that each germ of a strictly $W$-positive function belongs to a global function if there exists at least one strictly $ W$-positive function. We apply the Characterization Theorem to obtain some general conditions for globality and to give a complete description of the global cones in compact Lie algebras.
Almost periodic potentials in higher dimensions
Vassilis G.
Papanicolaou
679-696
Abstract: This work was motivated by the paper of R. Johnson and J. Moser (see [J-M] in the references) on the one-dimensional almost periodic potentials. Here we study the operator $L = - \Delta /2 - q$, where $q$ is an almost periodic function in $ {R^d}$. It is shown that some of the results of [J-M] extend to the multidimensional case (our approach includes the one-dimensional case as well). We start with the kernel $ k(t,x,y)$ of the semigroup ${e^{ - tL}}$. For fixed $t > 0$ and $u \in {R^d}$, it is known (we review the proof) that $k(t,x,x + u)$ is almost periodic in $x$ with frequency module not bigger than the one of $q$. We show that $k(t,x,y)$ is, also, uniformly continuous on $ [a,b] \times {R^d} \times {R^d}$. These results imply that, if we set $ y = x + u$ in the kernel ${G^m}(x,y;z)$ of $ {(L - z)^{ - m}}$ it becomes almost periodic in $x$ (for the case $u = 0$ we must assume that $m > d/2$), which is a generalization of an old one-dimensional result of Scharf (see [S.G]). After this, we are able to define ${w_m}(z) = {M_x}[{G^m}(x,x;z)]$, and, by integrating this $m$ times, an analog of the complex rotation number $ w(z)$ of [J-M]. We also show that, if $ e(x,y;\lambda )$ is the kernel of the projection operator ${E_\lambda }$ associated to $L$, then the mean value $\alpha (\lambda ) = {M_x}[e(x,x;\lambda )]$ exists. In one dimension, this (times $\pi$) is the rotation number. In higher dimensions ($d = 1$ included), we show that $d\alpha (\lambda )$ is the density of states measure of [A-S] and it is related to ${w_m}(z)$ in a nice way. Finally, we derive a formula for the functional derivative of ${w_m}(z;q)$ with respect to $q$, which extends a result of [J-M].
Rigidity of $p$-completed classifying spaces of alternating groups and classical groups over a finite field
Kenshi
Ishiguro
697-713
Abstract: A $p$-adic rigid structure of the classifying spaces of certain finite groups $\pi$, including alternating groups $ {A_n}$ and finite classical groups, is shown in terms of the maps into the $ p$-completed classifying spaces of compact Lie groups. The spaces $(B\pi )_p^ \wedge$ have no nontrivial retracts. As an application, it is shown that $ (B{A_n})_p^ \wedge \simeq (B{\Sigma _n})_p^ \wedge$ if and only if $n\not \equiv 0,1,\;\bmod \,p$. It is also shown that $(BSL(n,{\mathbb{F}_q}))_p^ \wedge \simeq (BGL(n,{\mathbb{F}_q}))_p^ \wedge$ where $q$ is a power of $p$ if and only if $ (n,q - 1) = 1$.
Symmetric local algebras with $5$-dimensional center
M.
Chlebowitz;
B.
Külshammer
715-731
Abstract: We prove that a symmetric split local algebra whose center is $ 5$-dimensional has dimension $5$ or $8$. This implies that the defect groups of a block of a finite group containing exactly five irreducible Frobenius characters and exactly one irreducible Brauer character have order $5$ or are nonabelian of order $8$.
Prinjective modules, reflection functors, quadratic forms, and Auslander-Reiten sequences
J. A.
de la Peña;
D.
Simson
733-753
Abstract: Let $A,\;B$ be artinian rings and let $ _A{M_B}$ be an $ (A - B)$-bimodule which is a finitely generated left $A$-module and a finitely generated right $ B$-module. A right $ _A{M_B}$-prinjective module is a finitely generated module ${X_R} = (X_A', X_B'', \varphi :X_A' \otimes_A M_B \to X''_B)$ over the triangular matrix ring $\displaystyle R = \left( {\begin{array}{*{20}{c}} A & {_A{M_B}} 0 & B \end{array} } \right)$ such that $X_A'$ is a projective $A$-module, $X''_B$ is an injective $B$-module, and $\varphi$ is a $B$-homomorphism. We study the category $ \operatorname{prin} (R)_B^A$ of right $_A{M_B}$-prinjective modules. It is an additive Krull-Schmidt subcategory of $\bmod (R)$ closed under extensions. For every $ X,\;Y$ in $\operatorname{prin} (R)_B^A,\;\operatorname{Ext} _R^2(X,\,Y) = 0$. When $R$ is an Artin algebra, the category $\operatorname{prin} (R)_B^A$ has Auslander-Reiten sequences and they can be computed in terms of reflection functors. In the case that $R$ is an algebra over an algebraically closed field we give conditions for $ \operatorname{prin} (R)_B^A$ to be representation-finite or representation-tame in terms of a Tits form. In some cases we calculate the coordinates of the Auslander-Reiten translation of a module using a Coxeter linear transformation.
Determinant expression of Selberg zeta functions. II
Shin-ya
Koyama
755-772
Abstract: This paper is the $ \operatorname{PSL} (2,{\mathbf{C}})$-version of Part I. We show that for $ \operatorname{PSL} (2,{\mathbf{C}})$ and its subgroup $\operatorname{PSL} (2,O)$, the Selberg zeta function with its gamma factors is expressed as the determinant of the Laplacians, where $ O$ is the integer ring of an imaginary quadratic field. All the gamma factors are calculated explicitly. We also give an explicit computation to the contribution of the continuous spectrum to the determinant of the Laplacian.
Toeplitz and Hankel operators on Bergman spaces
Karel
Stroethoff;
De Chao
Zheng
773-794
Abstract: In this paper we consider Toeplitz and Hankel operators on the Bergman spaces of the unit ball and the polydisk in ${\mathbb{C}^n}$ whose symbols are bounded measurable functions. We give necessary and sufficient conditions on the symbols for these operators to be compact. We study the Fredholm theory of Toeplitz operators for which the corresponding Hankel operator is compact. For these Toeplitz operators the essential spectrum is computed and shown to be connected. We also consider symbols that extend to continuous functions on the maximal ideal space of ${H^\infty }(\Omega )$; for these symbols we describe when the Toeplitz or Hankel operators are compact.
Log concave sequences of symmetric functions and analogs of the Jacobi-Trudi determinants
Bruce E.
Sagan
795-811
Abstract: We prove that various sequences of elementary and complete homogeneous symmetric functions are log concave or PF. As corollaries we show that certain sequences of $q$-binomial coefficients and $ q$-Stirling numbers have these properties. The principal technique used is a combinatorial interpretation of determinants using lattice paths due to Gessel and Viennot [G-V 85].
A global \L ojasiewicz inequality for algebraic varieties
Shanyu
Ji;
János
Kollár;
Bernard
Shiffman
813-818
Abstract: Let $X$ be the locus of common zeros of polynomials $ {f_1}, \ldots ,{f_k}$ in $ n$ complex variables. A global upper bound for the distance to $X$ is given in the form of a Lojasiewicz inequality. The exponent in this inequality is bounded by ${d^{\min (n,k)}}$ where $d = \max (3,\deg {f_i})$. The estimates are also valid over an algebraically closed field of any characteristic.
On explosions of solutions to a system of partial differential equations modelling chemotaxis
W.
Jäger;
S.
Luckhaus
819-824
Abstract: A system of partial differential equations modelling chemotactic aggregation is analysed (Keller-Segel model). Conditions on the system of parameters are given implying global existence of smooth solutions. In two space dimensions and radially symmetric situations, explosion of the bacteria concentration in finite time is shown for a class of initial values.
Holomorphic flows in ${\bf C}\sp 3,0$ with resonances
Júlio Cesar
Canille Martins
825-837
Abstract: The topological classification, by conjugacy, of the germs of holomorphic diffeomorphisms $f:\,{{\mathbf{C}}^2},0 \to {{\mathbf{C}}^2},0$ with $df(0) = \operatorname{diag} ({\lambda _1},{\lambda _2})$, where $ {\lambda _1}$ is a root of unity and $ \vert{\lambda _2}\vert \ne 1$ is given. This type of diffeomorphism appears as holonomies of singular foliations ${\mathcal{F}_X}$ induced by holomorphic vector fields $X:{{\mathbf{C}}^3},0 \to {{\mathbf{C}}^3},0$ normally hyperbolic and resonant. An explicit example of a such vector field without holomorphic invariant center manifold is presented. We prove that there are no obstructions in the holonomies for ${\mathcal{F}_X}$ to be topologically equivalent to a product type foliation.