on the topological 4 genus of torus knots we prove that the topological locally flat slice genus of large torus knots takes up less than three quarters of the ordinary genus as an application we derive the best possible linear estimate of the topological slice genus for torus knots with non maximal signature invariant

curvature diameter homotopy groups and cohomology rings we establish two topological results a if m is a 1 connected compact n manifold and q 2 then the minimal number of generators for the qth homotopy group q m mng q m is bounded above by a number depending only on mng h m and q where h m is the homology group c let mathscr m n y be the collection of compact orientable n manifolds whose oriented bundles admit so n invariant fibrations over y with fiber compact nilpotent manifolds such that the induced so n actions on y are equivalent then q m finitely generated m mathscr m n y contains only finite isomorphism classes depending only on n y q together with the results of cg and gr1 from a we conclude that i if m is a complete n manifold of nonnegative curvature then mng q m is bounded above by a number depending only on n and q 2 together with the results of ch and cfg from c we conclude that ii if m is a compact n manifold whose sectional curvature and diameter satisfy secm and diamm d then q m has a finite number of possible isomorphism classes depending on n d q 2 provided q m is finitely generated we also show that b if m is a compact n manifold with secm and diam m d then the cohomology ring h m may have infinitely many isomorphism classes in particular b answers some questions raised by k grove gro

a posteriori error control for distributed elliptic optimal control problems with control constraints discretized by hp finite elements abstract a distributed elliptic control problem with control constraints is considered which is formulated as a three field problem and consists of two variational equations for the state and the co state variables as well as of a variational inequality for the control variable two discretization approaches with h p finite elements are discussed in the first discretization approach all variables state co state and control are discretized a semi smooth newton method is introduced for solving the resulting algebraic system in the second discretization approach only the state and the co state variables are discretized whereas the control is determined by projection a simple fixed point scheme is presented for the iterative solution of this approach the main focus of the paper is on the derivation of reliable and efficient a posteriori error estimates which enables h p adaptive mesh refinements in particular the estimates can be applied to the iteration solutions so that they can be used as a stopping criterion of the iterative solution schemes in several numerical experiments the order of convergence of the adaptive discretization approaches and the efficiency as well as the reliability of the a posteriori error estimates are studied

a controlled plus construction for crumpled laminations given a closed n manifold m n 4 and a finitely generated perfect subgroup p of it1 m we previously developed a controlled version of quillen s plus construction namely a cobordism w m n with the inclusion j n f w a homotopy equivalence and kernel of i i1 m f it1 w equalling the smallest normal subgroup of xl1 m containing p together with a closed map p w 0 1 such that p l t is a closed n manifold for every t e 0 1 and in particular m p 1 o and n p l l we accomplished this by constructing an acyclic map of manifolds f m f n having the right fundamental groups and w arose as the mapping cylinder of f with a collar attached along n the main result here presents a condition under which the desired controlled plus construction can still be accomplished in many cases even when it1 m contains no finitely generated perfect subgroups by products of these results include a new method for constructing wild embeddings of codimension one manifolds and a better understanding of perfect subgroups of finitely presented groups

a survey of the theory of source coding with a fidelity criterion the purpose of this paper is threefold 1 to acquaint the reader with the types of problems that have been considered in the area of source coding with a fidelity criterion 2 to survey results that have been obtained on these problems and 3 to outline future research trends in the area

minimal entropy and collapsing with curvature bounded from below we show that if a closed manifold m admits an structure not necessarily polarized possibly of rank zero then its minimal entropy vanishes in particular this is the case if m admits a non trivial s1 action as a corollary we obtain that the simplicial volume of a manifold admitting an structure is zero we also show that if m admits an structure then it collapses with curvature bounded from below this in turn implies that m collapses with bounded scalar curvature or equivalently its yamabe invariant is non negative we show that structures of rank zero appear rather frequently every compact complex elliptic surface admits one as well as any simply connected closed 5 manifold we use these results to study the minimal entropy problem we show the following two theorems suppose that m is a closed manifold obtained by taking connected sums of copies of s4 p2 overline mathbb c p 2 s2 s2and the k3 surface then m has zero minimal entropy moreover m admits a smooth riemannian metric with zero topological entropy if and only if m is diffeomorphic to s4 p2 s2 s2 p2 overline mathbb c p 2 or p2 p2 finally suppose that m is a closed simply connected 5 manifold then m has zero minimal entropy moreover m admits a smooth riemannian metric with zero topological entropy if and only if m is diffeomorphic to s5 s3 s2 then on trivial s3 bundle over s2 or the wu manifold su 3 so 3

on h p adaptive bem for frictional contact problems in linear elasticity a mixed formulation for a tresca frictional contact problem in linear elasticity is considered in the context of boundary integral equations which is later extended to coulomb friction the discrete lagrange multiplier an approximation of the surface traction on the contact boundary part is a linear combination of biorthogonal basis functions in case of curved isoparametric elements these are the solutions of local problems in particular the biorthogonality allows to rewrite the variational inequality constraints as a simple set of complementarity problems thus enabling an efficient application of a semi smooth newton solver for the discrete mixed problem converging locally super linearly in the frictional case and quadratically in the frictionless case typically the solution of frictional contact problems is of reduced regularity at the interface between contact to non contact and from stick to slip to identify the a priori unknown locations of these interfaces two a posteriori error estimations are introduced in a first step the error is split into specific error contributions resulting from the contact and friction conditions and from the discretization error of a variational equation for the latter a residual and a bubble error estimation are considered explicitly the numerical experiments show the applicability of the derived error estimations and the superiority of h p adaptivity compared to low order uniform and adaptive approaches

dissolving four manifolds and positive scalar curvature we prove that many simply connected symplectic four manifolds dissolve after connected sum with only one copy of s2 s2 for any finite group g that acts freely on the three sphere we construct closed smooth four manifolds with fundamental group g which do not admit metrics of positive scalar curvature but whose universal covers do admit such metrics

higher order fem for the obstacle problem of the p laplacian a variational inequality approach abstract we consider higher order finite element discretizations of a nonlinear variational inequality formulation arising from an obstacle problem with the p laplacian differential operator for p 1 we prove an a priori error estimate and convergence rates with respect to the mesh size h and in the polynomial degree q under assumed regularity moreover we derive a general a posteriori error estimate which is valid for any uniformly bounded sequence of finite element functions all our results contain the known results for the linear case of p 2 we present numerical results on the improved convergence rates of adaptive schemes mesh size adaptivity with and without polynomial degree adaptation for the singular case of p 1 5 and for the degenerated case of p 3

the full ward takahashi identity for colored tensor models colored tensor models ctm is a random geometrical approach to quantum gravity we scrutinize the structure of the connected correlation functions of general ctm interactions and organize them by boundaries of feynman graphs for rank d interactions including but not restricted to all melonic varphi 4 vertices to wit solely those quartic vertices that can lead to dominant spherical contributions in the large n expansion the aforementioned boundary graphs are shown to be precisely all possibly disconnected vertex bipartite regularly edge d colored graphs the concept of ctm compatible boundary graph automorphism is introduced and an auxiliary graph calculus is developed with the aid of these constructs certain u invariance of the path integral measure is fully exploited in order to derive a strong ward takahashi identity for ctms with a symmetry breaking kinetic term for the rank 3 varphi 4 theory we get the exact integral like equation for the 2 point function similarly exact equations for higher multipoint functions can be readily obtained departing from this full ward takahashi identity our results hold for some group field theories as well altogether our non perturbative approach trades some graph theoretical methods for analytical ones we believe that these tools can be extended to tensorial syk models

simply connected 4 manifolds with a given boundary let m be a closed oriented connected 3 manifold for each bilinear symmetric pairing zn l our goal is to calculate the set vl m of all oriented homeomorphism types of compact 1 connected oriented 4manifolds with boundary m and intersection pairing isomorphic to zn l for each pair zn l which presents h m we construct a double coset space bl m and a function ct vl m bl m the set bt m is the quotient of the group of all link pairing preserving isomorphisms of the torsion subgroup of h1 m by two naturally occuring subgroups when zn l is an even pairing we construct another double coset space bl m a function 6l vl m b fl m and a projection p2 bl m bl m such that p2 c l ct our main result states that when zn l is even the function c l is injective as is the function c x a vl m bl m x z 2 when z l is odd here a is a kirby siebenmann obstruction to smoothing it follows that the sets vl m are finite and of an order bounded above by a constant depending only on h1 m we also show that when hi m q 0 and z l is even ct p2 c 6l is injective it seems likely that via the functions cl x a and 6l the sets bl m x z 2 and bl m calculate vl m when zn l is respectively odd and even we verify this in several cases most notably when h1 m is free abelian the results above are based on a theorem which gives necessary and sufficient conditions for the existence of a homeomorphism between two 1connected 4 manifolds extending a given homeomorphism of their boundaries the theory developed is then applied to show that there is an m 0 depending only on h1 m such that for any self homeomorphism f of m fm extends to a self homeomorphism of any 1 connected compact 4 manifold with boundary m introduction in m freedman s fundamental paper fr he classified closed simply connected oriented 4 manifolds up to orientation preserving homeomorphism he showed that modulo the kirby siebenmann invariant these manifolds are in bijective correspondence through their intersection pairings with the set of unimodular bilinear symmetric pairings over z see theorem 1 5 of fr and corollary 2 2 3 of q in this paper we begin a classification of simply connected 4 manifolds with connected boundary our results with regard to this problem are based on a theorem which gives necessary and sufficient conditions for the existence of a homeomorphism between 1 connected 4 manifolds extending a given homeomorphism of their boundaries received by the editors september 19 1985 1980 mathematics subject classification 1985 revision primary 57n15 research supported by an nserc canada post doctoral fellowship 1986 american mathematical society 0002 9947 86 1 00 25 per page

vector bundles and regulous maps let x be a compact nonsingular affine real algebraic variety we prove that every pre algebraic vector bundle on x becomes algebraic after finitely many blowing ups using this theorem we then prove that the stiefel whitney classes of any pre algebraic mathbb r vector bundle on x are algebraic we also derive that the chern classes of any pre algebraic mathbb c vector bundles and the pontryagin classes of any pre algebraic mathbb r vector bundle are blow mathbb c algebraic we also provide several results on line bundles on x

asymmetric four dimensional manifolds il existe une inifite de varietes a 4 dimensions fermees simplement connexes qui n admettent pas d involution non triviale localement reguliere

stabilized mixed hp bem for frictional contact problems in linear elasticity we analyze stabilized mixed hp boundary element methods for frictional contact problems for the lame equation the stabilization technique circumvents the discrete inf sup condition for the mixed problem and thus allows us to use the same mesh and polynomial degree for the primal and dual variables we prove a priori convergence rates in the case of tresca friction using gauss legendre lagrange polynomials as test and trial functions for the lagrange multiplier additionally a residual based a posteriori error estimate for a more general class of discretizations is derived it in particular applies to discretizations based on bernstein polynomials for the discrete lagrange multiplier which we also analyze the discretization and the a posteriori error estimate are extended to the case of coulomb friction several numerical experiments underline our theoretical results demonstrate the behavior of the method and its insensitivity to the scaling and perturbations of the stabilization term

integral action in output feedback for multi input multi output nonlinear systems we address a particular problem of output regulation for multi input multi output nonlinear systems specifically we are interested in making the stability of an equilibrium point and the regulation to zero of an output robust to small unmodelled discrepancies between design model and actual system in particular those introducing an offset we propose a novel procedure which is intended to be relevant to realistic scenarios as illustrated by a non academic example

the topology of real algebraic sets with isolated singularities in this paper we give a topological classification of real algebraic sets with isolated singularities showing that they are exactly smooth closed manifolds with smooth subpolyhedra crushed to points the question of which topological spaces are homeomorphic to real algebraic sets solutions of polynomial equations in euclidean space has been long studied in 1936 seifert showed that any smooth compact stably parallelizable manifold is diffeomorphic to a component of an algebraic set 121 and in 1952 nash extended this result to all smooth compact manifolds 11 in 1973 tognoli showed that any smooth compact manifold is diffeomorphic to a nonsingular algebraic set 13 so at least compact nonsingular algebraic sets are classified little has been done with singular algebraic sets however since the transversality arguments used by seifert nash and tognoli no longer apply except in some special cases one could use stability of singularities such as kuiper 71 and akbulut 1 used to show certain nonsmoothable pl manifolds are algebraic sets or one could use the projective version of seifert nash tognoli as king did 6 but one could still not hope these techniques would allow even a characterization of isolated singularities to get around this problem we take a cue from hironaka s resolution of singularities 4 the idea is to take a topological resolution of a space if it exists we can apply transversality techniques seifertnash tognoli to the resolved space and then blow down algebraically and end up with the original space as an algebraic set it seems likely that this technique allows one to classify all algebraic sets but in any case we show that it classifies all algebraic sets with isolated singularities in future papers we will use this technique to show for instance that all compact pl manifolds are homeomorphic to real algebraic sets 17 and that 2 dimensional real algebraic sets are topologically characterized as polyhedra satisfying sullivan s even local euler characteristic condition 16

dirac operators coupled to instantons on positive definite four manifolds we study the moduli space of instantons on a simply connected positive definite closed four manifold by analyzing the classifying map of the index bundle of a family of dirac operators parameterized by the moduli space as an application we compute the cohomology rings for the charge 1 and 2 moduli spaces in the rank stable limit

a note on the design of hp adaptive finite element methods for elliptic partial differential equations we introduce an hp adaptive finite element algorithm based on a combination of reliable and efficient residual error indicators and a new hp extension control technique which assesses the local regularity of the underlying analytical solution on the basis of its local legendre series expansion numerical experiments confirm the robustness and reliability of the proposed algorithm

real algebraic morphisms represent few homotopy classes we study the problem of representing homotopy classes of maps between real algebraic varieties of regular maps

on real algebraic morphisms into even dimensional spheres given affine real algebraic varieties x and y let us denote by r x y the set of regular mappings real algebraic morphisms from x into y for definitions and notions of real algebraic geometry see 2 where the theory of real algebraic varieties is treated systematically our aim in this paper is to study the set m x s of regular mappings from affine nonsingular real algebraic varieties x into s the unit sphere ezn 1x2 1 earlier we obtained several results in this direction 3 4 cf also 2 chap 13 in particular it was shown in 4 that given a compact connected nonsingular orientable real algebraic subset x of rp of odd dimension k either each smooth mapping from x to sk is homotopic to a regular mapping or precisely those mappings of even topological degree have this property now we shall study the case of regular mappings into even dimensional spheres strangely enough the situation then is radically different from that mentioned above for odd dimensional spheres for a very large class of compact smooth manifolds of even dimension 2k most algebraic models x of these manifolds have the property that every regular mapping from x into s2k is null homotopic the meaning of most will be made precise later cf remark 1 6 theorem 2 1 example 2 3 this class of manifolds contains all compact co hypersurfaces of r2k 1

coupling regularization and adaptive hp bem for the solution of a delamination problem in this paper we couple regularization techniques with the adaptive hp version of the boundary element method hp bem for the efficient numerical solution of linear elastic problems with nonmonotone contact boundary conditions as a model example we treat the delamination of composite structures with a contaminated interface layer this problem has a weak formulation in terms of a nonsmooth variational inequality the resulting hemivariational inequality is first regularized and then discretized by an adaptive hp bem we give conditions for the uniqueness of the solution and provide an a priori error estimate furthermore we prove the very first a posteriori error estimate for the nonsmooth variational problem utilizing a novel mixed regularized formulation thus enabling hp adaptivity various numerical experiments illustrate the behavior strengths and limitations of the proposed high order approximation scheme

fem and bem coupling for a nonlinear transmission problem with signorini contact this paper is concerned with the interface problem consisting of a nonlinear uniformly monotone partial differential equation in some bounded lipschitz domain omega in reals n n 2 or n geq 3 and the laplace equation with some radiation condition in the unbounded exterior domain omega c reals n setminus overline omega the two problems are coupled by transmission conditions and signorini contact conditions on the interface gamma partial omega the exterior part of the interface problem is rewritten in terms of boundary integral operators this leads to a variational inequality with a nonlinear monotone operator by this approach existence and uniqueness of a solution in appropriate sobolev spaces are obtained its approximation is performed by coupling the finite element method fem in omega and the boundary element method bem the latter living on the interface gamma yielding a discrete monotone variational inequality we present an abstract cea type error estimate and derive asymptotic error estimates

heegaard floer homology of matsumoto s manifolds abstract we consider a homology sphere m n k 1 k 2 presented by two knots k 1 k 2 with linking number 1 and framing 0 n we call the manifold matsumoto s manifold we show that m n t 2 3 k 2 never bounds any contractible 4 manifold if n 2 k 2 holds we also give a formula of ozsvath szabo s invariant as the total sum of the euler numbers of the reduced filtration we compute the invariants of the twisted whitehead doubles of torus knots and correction terms of the branched covers of the whitehead doubles by using owens and strle s obstruction we show that the 12 twisted whitehead double of the 2 7 torus knot and the 20 twisted whitehead double of the 3 7 torus knot are not slice but the double branched covers bound rational homology 4 balls these are new examples having a gap between what a knot is slice and what a double branched cover bounds a rational homology 4 ball

the weierstrass approximation theorem for maps between real algebraic varieties p n r for some n endowed with the zariski topology and the sheaf of rvalued regular functions morphisms between real algebraic varieties will be called regular maps every real algebraic variety carries also the euclidean topology that is the topology induced by the usual metric topology on r unless explicitly stated otherwise all topological notions related to real algebraic varieties will refer to the euclidean topology given two real algebraic varieties x and y we regard the set r x y of all regular maps from x into y as a subset of the space c x y of all continuous maps fromx into y endowed with the compact open topology if x and y are nonsingular then we also regard r x y as a subset of the space c x y of all c maps from x into y endowed with the c compact open topology the weak c topology in the terminology used in 6

nonlocally smoothable topological symmetries of four manifolds il existe une infinite de varietes a 4 dimensions simplement connexes closes qui n admettent pas d involutions localement regularisables mais qui en admet des topologiques

correction to source coding for average rate and average distortion new variable length coding theorems it has been pointed out to the author of the above tilted paper see ibid vol it 29 p 785 92 nov 1983 that a crucial inequality does not hold in general he presents a modification of this work that remedies the defect

block source coding theory for asymptotically mean stationary sources several properties of operational and information theoretic distortion rate functions and of mutual information rate are developed for asymptotically mean stationary ams processes with standard alphabets and used to prove a block source coding theorem for ergodic ams sources in the development we prove an ergodic decomposition of the mutual information rate of ams sources we show that the operational distortion rate function of an ams source equals that of its stationary mean and we provide a new and direct proof of the equality of the ergodic and stationary process distortion rate functions the latter result yields a proof of the block source coding theorem not requiring the nedoma decomposition and complicated code construction used by gallager and berger

4 manifold topology ii dwyer s filtration and surgery kernels even when the fundamental group is intractable i e not good many interesting 4 dimensional surgery problems have topological solutions we unify and extend the known examples and show how they compare to the presumed counterexamples by reference to dwyer s filtration on second homology the development brings together many basic results on the nilpotent theory of links as a special case a class of links only slightly smaller than homotopically trivial links is shown to have free slices on their whitehead doubles

coupling of finite and boundary element methods for an elastoplastic interface problem a class of transmission problems is considered in which a nonlinear variational problem in one domain is coupled with a linear elliptic problem in a second domain a typical example is a problem from three dimensional elasticity theory where an elastoplastic material is embedded into a linear elastic material the nonlinear problem is given in variational form with a strictly convex functional the linear elliptic problem is described by boundary integral equations on the coupling boundary the typical saddle point structure of such problems is analyzed galerkin approximations are studied which consist of a finite element approximation in the first domain coupled with a boundary element method on the coupling boundary the convergence of the galerkin approximation is based on the saddle point structure which is shown to hold for the exact as well as the discretized problems

surgery in dimension four and noncompact 5 manifolds this paper describes a precise relationship between the problems of completing surgery in dimension four and finding boundaries for noncompact 5 manifolds two outstanding problems of low dimensional topology are completing 4 dimensional surgery and finding boundaries for noncompact 5 dimensional manifolds although no complete theory is known for either problem some interesting results have been found see 1 4 5 10 and addendum in this paper we describe a complete relationship between these two problems consequently knowledge of 4 dimensional surgery yields information about noncompact 5 manifolds and conversely all manifolds and maps in this paper are assumed to be smooth each manifold is provided with a basepoint and when appropriate basepaths will be chosen to given 2 spheres without explicit mention i wish to thank gerald a anderson and frank quinn for helpful discussions also i wish to thank patrick gilmer for his advice and encouragement during this project 1 statement of results let f b m am y x be a degree one normal map in the sense of wall 11 where m is a compact connected 4 manifold and y x is a poincare pair not necessarily simple if x 0 then we require that f jam am x induces an isomorphism of homology groups with a z vi y coefficients also we assume that f is 2 connected k2 m is a free a module and the surgery obstruction a f b e lh 7 c is zero if h c k2 m is a subkernel we will say that h is representable in m if we can find disjoint framed embedded 2 spheres si sr c int m such that the inclusion m ui si m induces an isomorphism of fundamental groups and the classes of sl sr in h2 m a 7t2 m form a basis over a for h the subkernel h is stably representable if for some integer k 0 h ed kei ek is representable in m k s2 x s2 here ei denotes the class of s2 x 1 c ith summand s2 x s2 and e1 ek is the submodule generated by el ek we regard h ed ke ek as received by the editors august 10 1982 1980 mathematics subject classification primary 57r65 secondary 57m99 1985 american mathematical society 0002 9947 85 1 00 25 per page

on the diffeomorphism groups of elliptic surfaces in this paper we determine for relatively minimal elliptic surfaces with positive euler number the image of the natural representation of the group of orientation preserving self diffeomorphisms on h the second homology group reduced modulo torsion to this end we construct as many embedded spheres of square 2 such that an isometry not induced from any combination of reflections at such spheres or from complex conjugation can be shown not to be induced from some diffeomorphism at all this is done with the help of seiberg witten invariants

theory of ordered spaces this is the first of a planned series of investigations on the theory of ordered spaces based upon four axioms two of these the order i 1 1 and the local structure ii 5 1 axioms provide the structure of the theory and the other two the identification i 1 11 and cone i 2 7 axioms eliminate pathologies or excessive generality in the present paper the axioms are supplemented by the nontriviality conditions i 1 9 and a regularity property ii 4 2

the global structure of simple space times according to a standard definition of penrose a space time admitting well defined future and past null infinitiesi andi is asymptotically simple if it has no closed timelike curves and all its endless null geodesics originate fromi and terminate ati the global structure of such space times has previously been successfully investigated only in the presence of additional constraints the present paper deals with the general case it is shown thati is diffeomorphic to the complement of a point in some contractible open 3 manifold the strongly causal regioni0 ofi is diffeomorphic to mathbb s 2 times mathbb r and every compact connected spacelike 2 surface ini is contained ini0 and is a strong deformation retract of bothi0 andi moreover the space time must be globally hyperbolic with cauchy surfaces which subject to the truth of the poincare conjecture are diffeomorphic to 3

stein structures and holomorphic mappings we prove that every continuous map from a stein manifold x to a complex manifold y can be made holomorphic by a homotopic deformation of both the map and the stein structure on x in the absence of topological obstructions the holomorphic map may be chosen to have pointwise maximal rank the analogous result holds for any compact hausdorff family of maps but it fails in general for a noncompact family our main results are actually proved for smooth almost complex source manifolds x j with the correct handlebody structure the paper contains another proof of eliashberg s int j math 1 29 46 1990 homotopy characterization of stein manifolds and a slightly different explanation of the construction of exotic stein surfaces due to gompf ann math 148 2 619 693 1998 j symplectic geom 3 565 587 2005

fundamental groups of positively curved manifolds with symmetry we obtain a classification for the fundamental groups of closed n manifolds of positive sectional curvature which admit an isometric t k action with k ge frac n 6 1 n ne 11 15

self affine manifolds abstract this paper studies closed 3 manifolds which are the attractors of a system of finitely many affine contractions that tile r 3 such attractors are called self affine tiles effective characterization and recognition theorems for these 3 manifolds as well as theoretical generalizations of these results to higher dimensions are established the methods developed build a bridge linking geometric topology with iterated function systems and their attractors a method to model self affine tiles by simple iterative systems is developed in order to study their topology the model is functorial in the sense that there is an easily computable map that induces isomorphisms between the natural subdivisions of the attractor of the model and the self affine tile it has many beneficial qualities including ease of computation allowing one to determine topological properties of the attractor of the model such as connectedness and whether it is a manifold the induced map between the attractor of the model and the self affine tile is a quotient map and can be checked in certain cases to be monotone or cell like deep theorems from geometric topology are applied to characterize and develop algorithms to recognize when a self affine tile is a topological or generalized manifold in all dimensions these new tools are used to check that several self affine tiles in the literature are 3 balls an example of a wild 3 dimensional self affine tile is given whose boundary is a topological 2 sphere but which is not itself a 3 ball the paper describes how any 3 dimensional handlebody can be given the structure of a self affine 3 manifold it is conjectured that every self affine tile which is a manifold is a handlebody

minimal two spheres and the topology of manifolds with positive curvature on totally isotropic two planes there has been much interest among differential geometers in finding relationships between curvature and topology of riemannian manifolds for the most part efforts have been directed towards understanding the implications of positive or negative sectional curvature ricci curvature or scalar curvature but there are other hypotheses on the curvature which also deserve investigation in this article we will consider the topological implications of a new curvature assumption positive curvature on totally isotropic two planes and we will prove via the sacks uhlenbeck theory of minimal two spheres that a compact simply connected riemannian manifold of dimension at least four with positive curvature on totally isotropic two planes is homeomorphic to a sphere as corollaries we will obtain a proof via minimal two spheres for riemannian manifolds of dimension at least four of the berger klingenberg rauch toponogov sphere theorem strengthened to pointwise quarter pinching as well as a proof that every simply connected compact riemannian manifold with positive curvature operators is homeomorphic to a sphere let m be an n dimensional riemannian manifold with tangent space tpm at the point p e m recall that the curvature operator at p is the self adjoint

a decomposition theorem for h cobordant smooth simply connected compact 4 manifolds theorem let m and n be smooth h cobordant compact 1 connected 4 manifolds there exist decompositions m m0 m1 and n n0 n1 where m0 and n0 are smooth compact contractible 4 manifolds with boundary so that m1 and n1 are dieomorphic if m and n are closed then we may further arrange that m1 and n1 are 1 connected in fact if w is an h cobordism connecting m and n then w can be written w w0 i w1 where w0 m0 n0 is an often nontrivial h cobordism and w1 m1 n1 is smoothly the product h cobordism m1 i m1 f0 g m1 f1 g this theorem lies in the mainstream of combinatorial 4 manifold theory circa 1975 but oddly the result was overlooked it now ts into a rich context donaldson theory and raises new questions when compared with the known examples in 1988 akbulut a showed that a manifold x homotopy equivalent to a k3 surface could after blowing up a point be transformed into a rational surface q by cutting out and regluing a certain contractible mazur manifold by an involution of its boundary at the time a rather delicate argument of fintushel and stern relying on the observed presence of the homology sphere 2 3 7 x showed that certain donaldson invariants of x and its blow up x were nontrivial this distinguished x from the h cobordant manifold q in retrospect this was the rst nontrivial example of our theorem the result was a surprise the donaldson invariant which derives from the second homology had been localized on a contractible piece recently bizaca and gompf bg have shown that x is in fact dieomorphic to k3 allowing a more direct analysis of its donaldson invariant see e g qk or km they also nd an innite family of examples

a family of freely slice good boundary links we show that every good boundary link with a pair of derivative links on a seifert surface satisfying a homotopically trivial plus assumption is freely slice this subsumes all previously known methods for freely slicing good boundary links with two or more components and provides new freely slice links

source coding for average rate and average distortion new variable length coding theorems although variable length codes are known to be an efficient class of codes in noiseless coding little is known about these codes for sources subjected to a fidelity constraint variable length source coding theorems with a fidelity criterion are proved and variable length coding is shown to be applicable to sources without any regularity requirements such as stationarity also unlike fixed length coding it is shown that the convergence of average performances to the distortion rate function does not depend on the sample path properties of the source

the heat flow in nonlinear hodge theory we study the nonlinear hodge systemd 0 and 2 0 for an exterior form on a compact oriented riemannian manifold m where q is a given positive function the solutions are called harmonic forms they are the stationary points on cohomology classes of the functional e me 2 dm with e q q 2 the codifferential of a form is defined as 1 with 2 we evolve a given closed form 0 by the nonlinear heat flow system d for a time dependent exterior form x t on m this system is the differential of the normalized gradient flow u for e with 0 du under a technical assumption on the function 2 q q q we show that the nonlinear heat flow system d with initial condition 0 0 has a unique solution for all times which converges to a harmonic form in the cohomology class of 0 this yields a nonlinear hodge theorem that every cohomology class of m has a unique harmonic representative

piecewise regular maps let v w be real algebraic varieties that is up to isomorphism real algebraic sets and x subseteq v some subset a map from x into w is said to be regular if it can be extended to a regular map defined on some zariski locally closed subvariety of v that contains x furthermore a continuous map f x rightarrow w is said to be piecewise regular if there exists a stratification mathscr s of v such that for every stratum s in mathscr s the restriction of f to each connected component of x cap s is a regular map by a stratification of v we mean a finite collection of pairwise disjoint zariski locally closed subvarieties whose union is equal to v assuming that the subset x of v is compact we prove that every continuous map from x into a grassmann variety or a unit sphere can be approximated by piecewise regular maps as an application we obtain a variant of the algebraization theorem for topological vector bundles if the variety v is compact and nonsingular we prove that each continuous map from v into a unit sphere is homotopic to a piecewise regular map of class mathscr c k where k is an arbitrary nonnegative integer

on the adaptive coupling of fem and bem in 2 d elasticity this paper concerns the combination of the finite element method fem and the boundary element method bem using the symmetric coupling as a model problem in two dimensions we consider the hencky material a certain nonlinear elastic material in a bounded domain with navier lame differential equation in the unbounded complementary domain using some boundary integral operators the problem is rewritten such that the galerkin procedure leads to a fem bem coupling and quasi optimally convergent discrete solutions beside this a priori information we derive an a posteriori error estimate which allows up to a constant factor the error control in the energy norm since information about the singularities of the solution is not available a priori in many situation and having in mind the goal of an automatic mesh refinement we state adaptive algorithms for the h version of the fem bem coupling illustrating numerical results are included

higher order alexander invariants and filtrations of the knot concordance group we establish certain nontriviality results for several filtrations of the smooth and topological knot concordance groups first as regards the n solvable filtration of the topological knot concordance group c defined by k orr p teichner and the first author 0 f n 5 f n f 1 5 f 1 0 f 0 5 f 0 c we refine the recent nontriviality results of cochran and teichner by including information on the alexander modules these results also extend those of c livingston and the second author we exhibit similar structure in the closely related symmetric grope filtration ofc we also show that the grope filtration of the smooth concordance group is nontrivial using examples that cannot be distinguished by the ozsvath szabo invariant nor by j rasmussen s s invariant our broader contribution is to establish in the relative case the key homological results whose analogues cochran orr teichner established in the absolute case we say two knots k 0 and k 1 are concordant modulo n solvability if k o k 1 f n our main result is that for any knot k whose classical alexander polynomial has degree greater than 2 and for any positive integer n there exist infinitely many knots k i that are concordant to k modulo n solvability but are all distinct modulo n 5 solvability moreover the k i and k share the same classical seifert matrix and alexander module as well as sharing the same higher order alexander modules and seifert presentations up to order n 1

a fem bem coupling method for a nonlinear transmission problem modelling coulomb friction contact a nonlinear transmission problem modelling elastoplastic interface problems with contact and coulomb friction is reduced to a boundary domain variational inequality we present a corresponding fem bem coupling procedure and derive for its galerkin solution an a priori error estimate furthermore we reformulate the problem as an equivalent saddle point problem whose discretization can be solved by the uzawa algorithm convergence of the fem bem coupling method is proved and numerical results are given

a direct proof of the equality between the block definition and the process definition of distortion rate functions for stationary ergodic sources a direct proof is given for the equivalence of the process definition and the block definition of distortion rate functions for stationary ergodic sources with finite alphabets and bounded distortions

on the optimal control of variational hemivariational inequalities abstract the present paper represents a continuation of 26 there a continuous dependence result for the solution of an elliptic variational hemivariational inequality was obtained and then used to prove the existence of optimal pairs for two associated optimal control problems in the current paper we complete this study with more general results indeed we prove the continuous dependence of the solution with respect to a parameter which appears in all the data of the problem including the set of constraints the nonlinear operator and the two functionals which govern the variational hemivariational inequality this allows us to consider a general associated optimal control problem for which we prove the existence of optimal pairs together with a new convergence result the mathematical tools developed in this paper are useful in the analysis and control of a large class of boundary value problems which in a weak formulation lead to elliptic variational hemivariational inequalities to provide an example we illustrate our results in the study of an inequality which describes the equilibrium of an elastic body in frictional contact with a foundation made of a rigid body covered by a layer of soft material

stable mapping class groups of 4 manifolds with boundary we give a complete algebraic description of the mapping class groups of compact simply connected 4 manifolds with boundary up to connected sum with copies of s 2 s 2

hp fem convergence for unilateral contact problems with tresca friction in plane linear elastostatics abstract this paper is concerned with the convergence of the h p version of the finite element method h p fem for some nonsmooth unilateral problems in linear elastostatics we consider in particular the deformation of an elastic body unilaterally supported by a rigid foundation admitting tresca friction given friction along the rigid foundation solely subjected to body forces and surface tractions without being fixed along some part of its boundary for the discretization of the unilateral constraint and the nonsmooth friction functional we employ gauss lobatto quadrature we show convergence of the h p fem approximations for mechanically definite problems without imposing any regularity assumption moreover we treat the coercive case when the body is fixed along some part of the boundary based on an abstract cea falk estimate and operator interpolation arguments we establish an a priori error estimate in the energy norm under a reasonable regularity assumption

a direct boundary element method for signorini problems in this paper a signorini problem is reduced to a variational inequality on the boundary and a direct boundary element method is presented for its solution furthermore error estimates for the approximate solutions of signorini problems are given in addition we show that the signorini problem may be formulated as a saddle point problem on the boundary

locally flat 2 knots in with the same fundamental group we consider a locally flat 2 sphere in s x s2 representing a primitive homology class 4 which is referred to as a 2 knot in s x s2 representing 4 then for any given primitive class there exists a 2 knot in s2 x s2 representing 4 with simply connected complement in this paper we consider the classification of 2 knots in s2 x s2 whose complements have a fixed fundamental group we show that if the complement of a 2 knot s in s2 x s2 is simply connected then the ambient isotopy type of s is determined in the case of nontrivial 7r1 however we show that the ambient isotopy type of a 2 knot in s2 x s2 with nontrivial 71 is not always determined by 71

metabelian representations twisted alexander polynomials knot slicing and mutation given a knot complement x and its p fold cyclic cover x p x we identify twisted polynomials associated to gl 1 left bf f t pm 1 right representations of 1 x p with twisted polynomials associated to related gl p left bf f t pm 1 right representations of 1 x which factor through metabelian representations this provides a simpler and faster algorithm to compute these polynomials allowing us to prove that 16 of 18 previously unknown algebraically slice knots of 12 or fewer crossings are not slice we also use this improved algorithm to prove that the 24 mutants of the pretzel knot p 3 7 9 11 15 corresponding to permutations of 7 9 11 15 represent distinct concordance classes

rigidity theorems for right angled reflection groups let t be a right angled reflection group let m and m be coxeter manifolds then any t map f m fm is t homotopic to a homeo morphism introduction this paper contains the results given in the author s ph d the sis davis in 9 showed that in every dimension 4 there exists a cocompact reflection group on a contractible manifold not homeomorphic to an euclidean space also he gave the first example of a closed aspherical manifold not covered by euclidean space davis construction provides an infinite coxeter group acting locally smoothly effectively and properly discontinuously on a contractible manifold with compact quotient

topologically invariant chern numbers of projective varieties abstract we prove that a rational linear combination of chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the euler and pontryagin numbers in dimension at least three we prove that only multiples of the top chern number which is the euler characteristic are invariant under diffeomorphisms that are not necessarily orientation preserving these results solve a long standing problem of hirzebruch s we also determine the linear combinations of chern numbers that can be bounded in terms of betti numbers

a posteriori error estimators for obstacle problems another look we show that a posteriori estimators for the obstacle problem are easily obtained from the theory for linear equations the theory would be even simpler if the lagrange multiplier does not have a nonconforming contribution as it has in actual finite element computations

construction of group actions on four manifolds it is shown that any cyclic group of odd prime order acts on any closed simply connected topological 4 manifold inducing the identity on integral homology the action is locally linear except perhaps at one isolated fixed point in the case of primes greater than three a more careful argument is used to show that the action can be constructed to be locally linear introduction in this paper we shall show that every closed simply connected topological 4 manifold m4 admits an action of any cyclic group zp of odd prime order p the action will be homologically trivial that is induce the identity on integral homology be pseudofree that is have only isolated fixed points except in certain cases when p 3 and be locally linear except perhaps at one isolated fixed point when p 2 and the intersection form of m4 has even type then the same conclusion holds this is also due to s kwasik and p vogel 1985 by a somewhat different proof if p 2 and the intersection form of m has odd type however then an action of z2 which is homologically trivial must have a fixed point set containing two dimensional components see 7 further it is harder to control the kirby siebenmann triangulation obstruction which is then not determined by the intersection form it is interesting to ask whether these actions can be locally linear or smooth when m4 is smooth the existence of a locally linear z2 action implies the vanishing of the kirby siebenmann obstruction see kwasik and vogel 1984 we shall show however that for p 3 in general these actions can be constructed to be locally linear in a recent preprint kwasik has shown that this is the case for the fake cp2 when p is odd by a rather different proof these actions are homologically trivial and have only isolated fixed points except sometimes when p 3 the g signature theorem shows that not every simply connected 4 manifold admits pseudofree locally linear actions when p 3 we shall see for example that neither the e8 manifold the kummer surface nor a nontrivial connected sum of copies of cp2 admits such an action for p 3 in broad outline our construction goes as follows let a closed oriented simply connected 4 manifold m be given by studying equivariant framed links we construct a compact smooth 4 manifold with boundary having the intersection form of m and admitting a smooth zp action which is pseudofree and homologically trivial the action is free on the boundary homology sphere e in a paper primarily focused on equivariant plumbing diagrams for high dimensional highly connected manifolds with even intersection forms weintraub 1975 carried out most of this we give an independent development because we need to be able to better control received by the editors october 16 1985 and in revised form january 7 1986 1980 mathematics subject classification 1985 revision primary 57s17 57s25 57n15 research supported in part by a grant from the national science foundation 1987 american mathematical society 0002 9947 87 1 00 25 per page

surgery in cusp neighborhoods and the geography of irreducible 4 manifolds since its inception the theory of smooth 4 manifolds has relied upon complex surface theory to provide its basic examples this is especially true for simply connected 4 manifolds a longstanding conjecture held that every smooth simply connected closed 4 manifold is the connected sum of manifolds each admitting a complex structure with one of its orientations for this conjecture the 4 sphere was considered as the empty connected sum quite recently this was disproved by gompf and mrowka gm1 who produced infinite families of examples of 4 manifolds homeomorphic to the k3 surface and which admit no complex structure with either orientation the discovery of these dramatic counterexamples to the h cobordism theorem in dimension 4 was the most recent success in a long line of applications of donaldson invariants to the study of smooth structures on 4 manifolds beginning with donaldson s d4 and followed by fm1 ko ov1 ov2 which depended on donaldson s f invariant or a suitable generalization and fmm dk e2 eo1 mn2 sa using donaldson s polynomial invariant a common thread running through many of these counterexamples to the h cobordism conjecture is the process of logarithmic transformation which introduces multiple fibres in an elliptic surface in controlled situations this process will change the diffeomorphism type of a given elliptic surface while keeping its homotopy type fixed taking advantage of the complex structure on an elliptic surface this change can often be detected by using the relationship between anti self dual connections over a complex surface and stable holomorphic bundles d2 an alternative and more differential geometric

a surgery sequence in dimension four the relations with knot concordance we present a systematic treatment of the classification problem for compact smooth 4 manifolds m it is modeled on the surgery exact sequence the central theorem in the classification of n manifolds n 5 the price for the extension to dimension 4 is a hole in m where a homotopy 1 skeleton should be there is no homotopy theoretic or surgical obstruction to complet ing m with a wedge of circles so that the completion has the topology of a compact smooth manifold this point set problem is all that stands between 4 manifolds and the tranquility that prevails in higher dimensions when m is simply connected only a point is missing from the model the applications of this are discussed in f and fq 1 the general theory has application to knot and link theory in particular knots with alexander polynomial 1 are characterized geometrically as knots admitting a certain

a p version finite element method for nonlinear elliptic variational inequalities in 2d this article introduces and analyzes a p version fem for variational inequalities resulting from obstacle problems for some quasi linear elliptic partial differential operators we approximate the solution by controlling the obstacle condition in images of the gauss lobatto points we show existence and uniqueness for the discrete solution up from the p version for the obstacle problem we prove the convergence of up towards the solution with respect to the energy norm and assuming some additional regularity for the solution we derive an a priori error estimate in numerical experiments the p version turns out to be superior to the h version concerning the convergence rate and the number of unknowns needed to achieve a certain exactness of the approximation

linking and causality in globally hyperbolic space times the classical linking number lk is defined when link components are zero homologous in 15 we constructed the affine linking invariant alk generalizing lk to the case of linked submanifolds with arbitrary homology classes here we apply alk to the study of causality in lorentzian manifolds

a diffeomorphism classification of manifolds which are like projective planes we give a complete diffeomorphism classification of 1 connected closed manifolds m with integral homology h m z z z provided that dim m 6 4

block definition approaches to source coding with a fidelity criterion we give a new variable rate and variable distortion coding scheme and show a coding theorem for time discrete stationary sources with abstract alphabets and a single letter fidelity criterion without assuming reference letters the approach used to prove the coding theorem is then specialized to maximum distortion coding and to fixed rate coding and the corresponding coding theorems are proved for stationary sources we also consider the possibility of extending the results to time continuous sources we still need a reference letter for fixed rate coding

the rate distortion dimension of sets and measures data compression of independent samples drawn from a fractal set is considered the asymptotic ratio of rate to magnitude log distortion characterizes the effective dimension occupied by the underlying distribution this quantity is shown to be identical to renyi s 1959 information dimension for self similar fractal sets this dimension is distribution dependent in sharp contrast with the behavior of absolutely continuous measures the rate distortion dimension of a set is defined as the maximal rate distortion dimension for distributions supported on this set kolmogorov s metric dimension is an upper bound on the rate distortion dimension while the hausdorff dimension is a lower bound examples of sets for which the rate distortion dimension differs from these bounds are provided

gauge theory on asymptotically periodic 4 manifolds il existe une famille non denombrable de classes de diffeomorphismes des 4 varietes orientees homeomorphes a r 4