Free Products and 4-Dimensional Connected Sums

We show that a closed 4-manifold whose fundamental group isa nontrivial free product is stably homeomorphic to one whichadmits a corresponding decomposition into a connected sum, andthat connected sums of copies of S¹ x S³ and S¹ x S³ may becharacterized up to s-cobordism by their fundamental group,Euler characteristic and orientability. These results are appliedto obtain weak splitting and triviality theorems for links inS⁴.     

On 4-manifolds with zero-entropy metrics

We sharpen a recent result of Paternain and Petean by showingthat a closed 4-manifold which admits a Riemannian metric withzero topological entropy and infinite fundamental group eitheris aspherical or has a finite covering space homeomorphic toS³ x S¹ or S² x S¹ x S¹.     

The Canonical Decomposition of the Poset of a Hammock

In the Auslander-Reiten quiver of a representation-directedalgebra several hammocks occur naturally; they begin at theprojective cover of a simple module E and end in the correspondinginjective hull. It is known that hammocks are Auslander-Reitenquivers of posets, so there is a poset corresponding to eachsimple module; it describes the set of modules having E as acomposition factor. In this paper we show that this poset Sdecomposes canonically into a coideal S⁺ and an ideal S^–which can easily be described by vectorspace-categories correspondingto a one-point extension or a one-point coextension, respectively.In addition, we describe the simple modules for which S⁺ andS^– are not comparable, and also those for which S⁺ S^–. We also show how to use the results in order to prove for certainposets that they do not occur as posets corresponding to simplemodules.     

Perfect and Acyclic Subgroups of Finitely Presentable Groups

Acyclic groups of low dimension are considered. To indicatethe results simply, let G' be the nontrivial perfect commutatorsubgroup of a finitely presentable group G. Then def(G)1. Whendef(G)=1, G' is acyclic provided that it has no integral homologyin dimensions above 2 (a sufficient condition for this is thatG' be finitely generated); moreover, G/G' is then Z or Z². Naturalexamples are the groups of knots and links with Alexander polynomial1. A further construction is given, based on knots in S²x S¹.In these geometric examples, G' cannot be finitely generated;in general, it cannot be finitely presentable. When G is a 3-manifoldgroup it fails to be acyclic; on the other hand, if G' is finitelygenerated it has finite index in the group of a Q-homology 3-sphere.     

A pinching theorem for minimal hypersurfaces in a sphere

We study compact minimal hypersurfaces Mⁿ in Sⁿ⁺¹S^{n+1} with two distinct principal curvatures and prove that if the squared norm S of the second fundamental form of Mⁿ satisfies S \geqq nS \geqq n, then S o nS \equiv n and Mⁿ is a minimal Clifford torus.     

Totally Real Minimal Surfaces with Non-Circular Ellipse of Curvature in the Nearly Kahler S6

In [2] we discussed almost complex curves in the nearly KählerS⁶. These are surfaces with constant Kähler angle 0 or and, as a consequence of this, are also minimal and have circularellipse of curvature. We also considered minimal immersionswith constant Kähler angle not equal to 0 or , but withellipse of curvature a circle. We showed that these are linearlyfull in a totally geodesic S⁵ in S⁶ and that (in the simplyconnected case) each belongs to the S¹-family of horizontallifts of a totally real (non-totally geodesic) minimal surfacein CP². Indeed, every element of such an S¹-family has constantKähler angle and in each family all constant Kählerangles occur. In particular, every minimal immersion with constantKähler angle and ellipse of curvature a circle is obtainedby rotating an almost complex curve which is linearly full ina totally geodesic S⁵.     

A Multilinear Extension Inequality In Rn

Sharp decay estimates are provided in this paper for sphericalaverages of a certain multilinear extension operator on L² (S^n–1)x ... x L² (S^n–1). 2000 Mathematics Subject Classification42B10.     

The third fundamental form of minimal surfaces in a sphere

We study minimal surfaces in a sphere Sⁿ with regard to the following question: to what extent minimal surfaces in Sⁿ are determined by restrictions on the Gaussian curvature of the Gaussian image in the sense of Obata?     

A Note on a Maximal Function Over Arbitrary Sets of Directions

Let M_S be the universal maximal operator over unit vectors ofarbitrary directions. This operator is not bounded in L²(R²).We consider a sequence of operators over sets of finite equidistributeddirections converging to M_S. We provide a new proof of N. Katz'sbound for such operators. As a corollary, we deduce that M_Sis bounded from some subsets of L² to L². These subsets arecomposed of positive functions whose Fourier transforms havea logarithmic decay or which are supported on a disc. 1991 MathematicsSubject Classification 42B25.     

On Closed Orbits of Morse-Smale Flows on 3-Manifolds

Any link in a 3-manifold is the closed orbits of a non-singularMorse-Smale flow after taking the split sum with the unlinkand the connected sum with S² x S¹s. Current address: Department of Mathematics, University of Toronto,Toronto, Ontario M5S 1A1, Canada     

Hypersurfaces in a Unit Sphere Sn+1(1) with Constant Scalar Curvature

The paper considers n-dimensional hypersurfaces with constantscalar curvature of a unit sphere S^n–1(1). The hypersurfaceS^k(c₁)xS^n–k(c₂) in a unit sphere Sⁿ⁺¹(1) is characterized,and it is shown that there exist many compact hypersurfaceswith constant scalar curvature in a unit sphere Sⁿ⁺¹(1) whichare not congruent to each other in it. In particular, it isproved that if M is an n-dimensional (n > 3) complete locallyconformally flat hypersurface with constant scalar curvaturen(n–1)r in a unit sphere Sⁿ⁺¹(1), then r > 1–2/n,and (1) when r (n–2)/(n–1), if then M is isometric to S¹xS^n–1(c),where S is the squared norm of the second fundamental form ofM; (2) there are no complete hypersurfaces in Sⁿ⁺¹(1) with constantscalar curvature n(n–1)r and with two distinct principalcurvatures, one of which is simple, such that r = (n–2)/(n–1)and      

On Hardy, BMO and Lipschitz spaces of invariantly harmonic functions in the unit ball

The invariantly harmonic functions in the unit ball Bⁿ in Cⁿare those annihilated by the Bergman Laplacian . The Poisson-Szegökernel P(z,) solves the Dirichlet problem for : if f C(Sⁿ),the Poisson-Szegö transform of f, where d is the normalized Lebesgue measure on Sⁿ,is the unique invariantly harmonic function u in Bⁿ, continuousup to the boundary, such that u=f on Sⁿ. The Poisson-Szegötransform establishes, loosely speaking, a one-to-one correspondencebetween function theory in Sⁿ and invariantly harmonic functiontheory in Bⁿ. When n 2, it is natural to consider on Sⁿ functionspaces related to its natural non-isotropic metric, for theseare the spaces arising from complex analysis. In the paper,different characterizations of such spaces of smooth functionsare given in terms of their invariantly harmonic extensions,using maximal functions and area integrals, as in the correspondingEuclidean theory. Particular attention is given to characterizationin terms of purely radial or purely tangential derivatives.The smoothness is measured in two different scales: that ofSobolev spaces and that of Lipschitz spaces, including BMO andBesov spaces. 1991 Mathematics Subject Classification: 32A35,32A37, 32M15, 42B25.     

Average Linking Numbers of Closed Orbits of Hyperbolic Flows

The existence of an asymptotic average of linking numbers ofperiodic orbits of a basic set of an Axiom A flow in S³ or R³is proved.     

Spin Structures on Lefschetz Fibrations

Necessary and sufficient conditions for Lefschetz fibrationsover D² and S² to carry spin structures are presented. The conditionsare given in terms of vanishing cycles and homology sections.     

On f-injective modules

In this paper, the notions of f-injective and f^*-injective modules are introduced. Elementary properties of these modules are given. For instance, a ring R is coherent iff any ultraproduct of f-injective modules is absolutely pure. We prove that the class S^* \Sigma^* of f^*-injective modules is closed under ultraproducts. On the other hand, S^* \Sigma^* is not axiomatisable. For coherent rings R, S^* \Sigma^* is axiomatisable iff every c₀ \chi_0 -injective module is f^*-injective. Further, it is shown that the class S \Sigma of f-injective modules is axiomatisable iff R is coherent and every c₀ \chi_0 -injective module is f-injective. Finally, an f-injective module H, such that every module embeds in an ultraprower of H, is given.     

Symmetric Groups as Products of Abelian Subgroups

A proof is given that the full symmetric group over any infiniteset is the product of finitely many Abelian subgroups. In fact,289 subgroups suffice. Sharp bounds are also obtained on theminimal number k, such that the finite symmetric group S_n isthe product of k Abelian subgroups. Using this, S_n is provedto be the product of 72n^1/2(log n)^3/2 cyclic subgroups. 2000Mathematics Subject Classification 20B30, 20D40.     

UNIVERSAL SPACES OF TWO-CELL COMPLEXES AND THEIR EXPONENT BOUNDS

Let P²ⁿ⁺¹ be a two-cell complex which is formed by attachinga (2n + 1)-cell to a 2m-sphere by a suspension map. We constructa universal space U for P²ⁿ⁺¹ in the category of homotopy associative,homotopy commutative H-spaces. By universal, we mean that Uis homotopy associative, homotopy commutative and has the propertythat any map f: P²ⁿ⁺¹ Y to a homotopy associative, homotopycommutative H-space Y extends to a uniquely determined H-map: U Y. We then prove upper and lowerbounds of the H-homotopy exponent of U. In the case of a modp^r, Moore space U is the homotopy fibre S²ⁿ⁺¹{p^r} of the p^r-powermap on S²ⁿ⁺¹, and we reproduce Neisendorfer's result that S²ⁿ⁺¹{p^r}is homotopy associative, homotopy commutative and that the p^r-powermap on S²ⁿ⁺¹{p^r} is null homotopic.     

Dickson-Mui Invariants

Let V be a finite-dimensional vector space over a finite field.The general and special linear groups, GL(V) and SL(V), acton the exterior algebras ^*V and ^*V^* of V and its dual V^*, andon the symmetric algebra S^*V. The subring of SL(V)-invariantsof ^*VS^*V was determined by Dickson and Mui. This paper describesthe equivalent, but simpler, calculation of the invariant subringof ^*VS^*V as a representation of GL(V)/SL(V). 2000 MathematicsSubject Classification 13A50.     

Hilbert空间上线性算子广义逆A(2)T,S的存在性及其表示式

设H₁和H₂是两个Hilbert空间, B(H₁,H₂)表示从H₁到H₂的所有有界线性算子的集合, T和S分别是H₁和H₂的两个闭子空间. 如果存在线性算子X ∈ B(H₂,H₁)满足XAX=X, R(X)=T, N(X)=S,则称X为线性算子$A$的具有指定像空间T和零空间S的外逆,记为A⁽²⁾_T,S. 该文进一步研究了线性算子广义逆A⁽²⁾_T,S存在的若干等价条件及其性质,建立了算子广义逆A⁽²⁾_T,S的表示形式.     

Strong Jordan Separation and Applications to Rigidity

We prove that simple, thick hyperbolic P-manifolds of dimensionat least three exhibit Mostow rigidity. We also prove a quasi-isometryrigidity result for the fundamental groups of simple, thickhyperbolic P-manifolds of dimension at least three. The keytool in the proof of these rigidity results is a strong formof the Jordan separation theorem, for maps from Sⁿ Sⁿ⁺¹ whichare not necessarily injective.