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

nonlinear displacement based finite element analyses of composite shells a new total lagrangian formulation presented here is a new total lagrangian finite element formulation for general laminated composite shells undergoing large displacement large rotation and large strain motion the theory fully accounts for geometric nonlinearities large rotations large in plane strains general initial curvatures transverse shear deformations and interlaminar normal stresses by using jaumann stress and strain measures an exact coordinate transformation and a new concept of orthogonal virtual rotations in addition with the inclusion of transverse normal and shear stresses the theory accounts for three dimensional stress effects and gives accurate effective structural stiffnesses because of the inclusion of all possible initial curvatures the developed strain displacement expressions can be used for any surface structures and are fully nonlinear which contain von karman strains as a special case moreover the theory accounts for the continuity of interlaminar shear and peeling stresses and the normal and shear stress conditions on the bonding surfaces by using a new layer wise local displacement field which results in a higher order shear theory that contains most shear deformation theories as special cases in this paper we develop a fully nonlinear displacement based finite element formulation based on the derived shell theory the verification and accuracy of the theory will be presented in a subsequent paper by comparing obtained numerical results with some exact solutions

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

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

numerical minimization of dirichlet laplacian eigenvalues of four dimensional geometries we develop the first numerical study in four dimensions of optimal eigenmodes associated with the dirichlet laplacian we describe an extension of the method of fundamental solutions adapted to the four dimensional context based on our numerical simulation and a postprocessing adapted to the identification of relevant symmetries we provide and discuss the numerical description of the eighth first optimal domains

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

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

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

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

black box multigrid one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in this paper we investigate such a situation for the multigrid method the end result is code boxmg in which one need only specify the logically rectangular positive definite matrix problem boxmg does everything else necessary to set up the auxilliary coarser problems to achieve a multigrid solution

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

robust parallel smoothing for multigrid via sparse approximate inverses sparse approximate inverses are considered as smoothers for multigrid they are based on the spai algorithm m j grote and t huckle siam j sci comput 18 1997 pp 838 853 which constructs a sparse approximate inverse m of a matrix a by minimizing i ma in the frobenius norm this yields a new hierarchy of smoothers spai 0 spai 1 spai varepsilon advantages of spai smoothers over classical smoothers are inherent parallelism possible local adaptivity and improved robustness the simplest smoother spai 0 is based on a diagonal matrix m it is shown to satisfy the smoothing property for symmetric positive definite problems numerical experiments show that spai 0 smoothing is usually preferable to damped jacobi smoothing for the spai 1 smoother the sparsity pattern of m is that of a its performance is typically comparable to that of gauss seidel smoothing however both the computation and the application of the smoother remain inherently parallel in more difficult situations where the simpler spai 0 and spai 1 smoothers are not adequate the spai varepsilon smoother provides a natural procedure for improvement where needed numerical examples illustrate the usefulness of spai smoothing

on red black sor smoothing in multigrid optimal relaxation parameters are obtained for red black gauss seidel relaxation in multigrid solvers of a family of elliptic equations the resulting relaxation schemes are found to retain very high efficiency over an appreciable range of coefficients of the elliptic differential operator yielding simple inexpensive and fully parallelizable smoothers in many situations where less cost effective block and alternating direction schemes are commonly used

accelerated multigrid convergence and high reynolds recirculating flows techniques are developed for accelerating multigrid convergence in general and for advection diffusion and incompressible flow problems with small viscosity in particular it is shown by analysis that the slowing down of convergence is due mainly to poor coarse grid correction to certain error components and means for dealing with this problem are suggested analyzed and tested by numerical experiments showing very significant improvement in convergence rates at little cost

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

multigrid smoothing for symmetric nine point stencils analytical formulae are obtained for the smoothing factors yielded by damped jacobi relaxation and by red black relaxation applied to symmetric nine point stencil discretizations of elliptic partial differential operators in 2d the results include point and line relaxation and full and partial coarsening several unusual results are implied by the formulae numerical results of multigrid cycles match the analytical predictions well

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

4 manifold topology i subexponential groups the technical lemma underlying the 5 dimensional topologicals cobordism conjecture and the 4 dimensional topological surgery conjecture is a purely smooth category statement about locating 1 null immersions of disks these conjectures are theorems precisely for those fundamental groups good groups where the 1 null disk lemma ndl holds we expand the class of known good groups to all groups of subexponential growth and those that can be formed from these by a finite number of application of two operations 1 extension and 2 direct limit the finitely generated groups in this class are amenable and no amenable group is known to lie outside this class

collapsed 5 manifolds with pinched positive sectional curvature abstract let m be a closed 5 manifold of pinched curvature 0 sec m 1 we prove that m is homeomorphic to a spherical space form if one of the following conditions holds i the center of the fundamental group has index w a constant depending on ii 1 4 1 10 6 2 and the fundamental group is a non cyclic group of order c a constant iii the volume is less than and the fundamental group is either isomorphic to a spherical 5 space group or has an odd order and it has a center of index w a constant

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

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

a fake compact contractible 4 manifold here we construct a fake smooth structure on a compact contractible 4 manifold w where w is a well known mazur manifold obtained by attaching in two handle to s x b along its boundary as in figure 1 here we use the conventions of 2 the results of this paper imply theorem 1 there is a smooth contractible 4 manifold v with dv d w such that v is homeomorphic but not diffeomorphic to w relative to the boundary let a be the loop in dw given by s x p c s x s d s x b as in figure 2 zeeman raised the question whether a is slice in w 12 i e if a bounds an imbedded smooth d in w even though it turned out that a is slice in another smooth contractible manifold with the same boundary 2 the original question has remained open let d w d w be the diffeomorphism obtained by first surgering s x b to b xs in the interior of w then surgering the other imbedded b x s back to s x b i e replacing the dots in figure 2 clearly this diffeomorphism extends to a self homotopy equivalence of w in fact by 9 extends to a homeomorphism f w w in 2 p 279 the question of whether extends to a diffeomorphism of w was posed if it did a would be slice in w since f a is clearly slice in w here we answer these questions negatively theorem 2 is not slice in w in particular f does not extend to a self diffeomorphism of w theorem 1 follows from theorem 2 as follows let f w w be a homeomorphism extending let v be the smooth structure on w obtained by pulling back the smooth structure of w by f this gives a diffeomorphism f v w extending on the boundary if g w v

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

group actions corks and exotic smoothings of mathbb r 4 r 4 we provide the first information on diffeotopy groups of exotic smoothings of mathbb r 4 for each of uncountably many smoothings there are uncountably many isotopy classes of self diffeomorphisms we realize these by various explicit group actions there are also actions at infinity by nonfinitely generated groups for which no nontrivial element extends over the whole manifold in contrast every diffeomorphism of the end of the universal mathbb r 4 extends our techniques apply to many other open 4 manifolds and are related to cork theory we show that under broad hypotheses cork twisting is equivalent up to blowups to twisting on an exotic mathbb r 4 and give applications

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

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

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

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

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

on first and second order moderate rotation theories of laminated plates the first and second order small strain and moderate rotation theories of anisotropic laminated plates are developed and numerically evaluated beginning with an assumed displacement field and introducing various order of magnitude assumptions the governing equilibrium equations of laminated plates are derived from the principle of virtual displacements the finite element formulation for the second order moderate rotation theory is developed and numerical results are presented to evaluate the new plate theories in comparison with the von karman plate theory with shear deformation for comparison purposes the 2 d elasticity theory with full non linearity is also analysed using the finite element method it is found that the second order theory with moderate rotations is closest to the 2 d finite elasticity theory

two color fourier analysis of the multigrid method with red black gauss seidel smoothing a two color fourier analytical approach is proposed to analyze the multigrid method which employs the red black gauss seidel smoothing iteration for solving the poisson equation in this approach fourier components in the high frequency region are folded into the low frequency region so that the coupling between the low and high fourier components is transformed into a coupling between components of red and black computational waves in the low frequency region we show that the two color two grid method asymptotically reduces to a one color two grid method whose physical mechanism is more transparent than for its original two color form the two color fourier analysis is also used to design variants of the standard multigrid algorithm

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

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

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

the geometry of the moduli space of cp2 instantons the moduli space of self dual connections instantons of instanton numberk 1 overcp2 is topologically a cone oncp2 the complement of the vertex of this cone carries a natural riemannian metric the l2 metric we find an explicit formula for this metric and deduce certain consequences in particular i the boundary is a totally geodesic submanifold of the completion ii is positively curved at and iii is negatively curved in certain directions near the vertex of the cone

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

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

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

coarse grid correction for nonelliptic and singular perturbation problems the suitability of coarse grid correction operators for multigrid solution of discretized nonelliptic partial differential equations is studied it is found that the requirements for the orders of intergrid transfers and the order of approximation of the coarse grid operators differ in an important way from the well known rules pertaining to elliptic operators both galerkin coarsening and coarsening schemes based on the differential operator are examined and new coarse grid correction strategies are suggested in light of this analysis

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

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 theory of shells with small strain accompanied by moderate rotation this paper is concerned with a constrained theory of shells in the presence of small strain accompanied by moderate rotation the constrained theory accounts for the effect of transverse normal strain and includes of course the special case corresponding to the kirchhoff love theory of shells in which the effect of transverse normal strain is absent after precise estimates for local moderate rotation and relative displacement gradients in terms of infinitesimal strain have been effected a complete theory is formulated with the use of linear constitutive equations the nature of the complete theory is further examined when initially the shell like body is a plate and it is shown that our kinematical formulae strain displacement relations as well as the relevant differential equations of the theory in the absence of the effect of transverse normal strain systematically reduce to those used in the von karman plate equations also in the light of the present results an assessment of kinematical aspects of previously developed theories of shells undergoing small strain and moderate rotation is indicated

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

dual decompositions of 4 manifolds iii s cobordisms the main result is that an s cobordism topological or smooth of 4 manifolds has a product structure outside a core sub s cobordism these cores are arranged to have quite a bit of structure for example they are smooth and abstractly forgetting boundary structure diffeomorphic to a standard neighborhood of a 1 complex the decomposition is highly nonunique so cannot be used to define an invariant but it shows that the topological s cobordism question reduces to the core case the simply connected version of the decomposition with 1 complex a point is due to curtis freedman hsiang and stong controlled surgery is used to reduce topological triviality of core s cobordisms to a question about controlled homotopy equivalence of 4 manifolds there are speculations about further reductions the decompositions on the ends of the s cobordism are dual decompositions with homotopically controlled handle structures and the main result is an application of earlier papers in the series

multigrid smoothing factors for red black gauss seidel relaxation applied to a class of elliptic operators analytic formulae are obtained for the smoothing factors yielded by gauss seidel relaxation in two color ordering for a class of scalar elliptic operators block and point relaxations in conjunction with full or partial coarsening are encompassed for operators with general constant positive coefficients in general dimensions and for an arbitrary number of relaxation sweeps it is found that there is no direct dependence of the smoothing factors on the dimension and that the effect of the number of relaxation sweeps on the smoothing factor is usually independent of the operator coefficients and of the relaxation scheme the results are compared with computed results of two level analyses smoothing strategies implied by the formulae are discussed

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