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 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

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

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

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

proper actions fixed point algebras and naturality in nonabelian duality abstract suppose a locally compact group g acts freely and properly on a locally compact hausdorff space x and let be the induced action on c 0 x we consider a category in which the objects are c dynamical systems a g for which there is an equivariant homomorphism of c 0 x into the multiplier algebra m a rieffel has shown that such systems are proper and saturated and hence have a generalized fixed point algebra a which is morita equivalent to a r g we show that the assignment a a is functorial and that rieffel s morita equivalence is natural in a suitable sense we then use our results to prove a categorical version of landstad duality which characterizes crossed products by coactions and to prove that mansfield imprimitivity for crossed products by homogeneous spaces is natural

induced representations of crossed products by coactions abstract let a m a c r g be a coaction of a locally compact group g on a c algebra a then for any closed normal amenable subgroup h of g we define a coaction a m a c r g h of g h on a we present dense subalgebras of the crossed products a g and a g h and use these to induce representations of a g h to representations of a g we then formulate an imprimitivity theorem for the induction process that is we classify the induced representations we also investigate the continuity of the induction process and are able to prove rieffel s conjecture that the dual action to a coaction is proper and saturated

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

weakly proper group actions mansfield s imprimitivity and twisted landstad duality alcides buss and siegfried echterhoffabstract using the theory of weakly proper actions of locally compactgroups recently developed by the authors we give a uni ed proof of both re duced and maximal versions of mans eld s imprimitivity theorem and obtaina general version of landstad s duality theorem for twisted group coactions as one application we obtain the stabilization trick for arbitrary twisted coac tions showing that every twisted coaction is morita equivalent an in atedcoaction

generalized almost periodic and ergodic solutions of linear differential equations on the half line in banach spaces abstract the bohl bohr amerio kadets theorem states that the indefinite integral y p of an almost periodic ap r x is again ap if y is bounded and the banach space x does not contain a subspace isomorphic to c0 this is here generalized in several directions instead of r it holds also for defined only on a half line j instead of ap functions abstract classes a with suitable properties are admissible a can be weakened to in some mean class m q 1 a then p m q a here ma contains all f l1loc with 1 h 0 h f s ds in a for all h 0 usually a ma m 2 a strictly furthermore instead of boundedness of y mean boundedness y in some m k l or in m k e e ergodic functions suffices the loomis doss result on the almost periodicity of a bounded for which all differences t h t are ap for h 0 is extended analogously also to higher order differences studying difference spaces a in this connection we obtain decompositions of the form any bounded measurable function is the sum of a bounded ergodic function and the indefinite integral of a bounded ergodic function the bohr neugebauer result on the almost periodicity of bounded solutions y of linear differential equations p d y of degree m with ap is extended similarly for m q m a then y m q a provided for example y is in some m k u with u l or is totally ergodic and for the half line re 0 for all eigenvalues p 0 analogous results hold for systems of linear differential equations special case bounded and p ergodic implies p bounded if all re 0 there exists a unique solution y growing not too fast this y is in m q a if m q m a for quite general a

asymptotic behavior of solutions of systems of neutral and convolution equations suppose j for some r or j r and let x be a banach space we study asymptotic behavior of solutions on j of neutral system of equations with values in x this reduces to questions concerning the behavior of solutions of convolution equations h v 0 b where h hj k i s anr r matrix hj k d 1 b bj and bj d r x for 1 j kr we prove that if 0 is a bounded uniformly continuous solution of with b from some translation invariant suitably closed class a then 0 belongs to a provided for example that det h has countably many zeros on r and c0 3 x in particular if b is asymptotically almost periodic almost automorphic or recurrent 0 is too our results extend theorems of bohr neugebauer bochner doss basit and zhikov and also certain theorems of fink madych staffans and others also we investigate bounded solutions of this leads to an extension of the known classes of almost periodicity to larger classes called mean classes we explore mean classes and prove that bounded solutions of belong to mean classes provided certain conditions hold these results seem new even for the simplest difference equation 0 t 1 0 t b t with j x r and b stepanoff almost periodic 1998 academic press

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

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

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

bauer furuta invariants under mathbb z _2 actions s bauer and m furuta defined a stable cohomotopy refinement of the seiberg witten invariants in this paper we prove a vanishing theorem of bauer furuta invariants for 4 manifolds with smooth mathbb z 2 actions as an application we give a constraint on smooth mathbb z 2 actions on homotopy k3 k3 and construct a nonsmoothable locally linear mathbb z 2 action on k3 k3 we also construct a nonsmoothable locally linear mathbb z 2 action on k3

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

smooth group actions on definite 4 manifolds and moduli spaces in this paper we give an application of equivariant moduli spaces to the study of smooth group actions on certain 4 manifolds a rich source of examples for such actions is the collection of algebraic surfaces compact and nonsingular together with their groups of algebraic automorphisms from this collection further examples of smooth but generally nonalgebraic actions can be constructed by an equivariant connected sum along an orbit of isolated points given a smooth oriented 4 manifold x which is diffeomorphic to a connected sum of algebraic surfaces we can ask i which finite groups can act smoothly on x preserving the orientation and ii how closely does a smooth action on x resemble some equivariant connected sum of algebraic actions on the algebraic surface factors of x for the purposes of this paper we will restrict our attention to the simplest case namely x p2 c p2 c a connected sum of n copies of the complex projective plane arranged so thatx is simply connected furthermore assumption all actions will be assumed to induce the identity on h x z

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

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

mansfield s imprimitivity theorem for full crossed products for any maximal coaction a g s and any closed normal subgroup n of g there exists an imprimitivity bimodule y g g n a between the full crossed product a g n and a g n together with inf dec compatible coaction sy of g the assignment a y g g n a y implements a natural equivalence between the crossed product functors x g x n and x g n in the category whose objects are maximal coactions of g and whose morphisms are isomorphism classes of right hilbert bimodule coactions of g

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

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

induced c algebras and landstad duality for twisted coactions suppose n is a closed normal subgroup of a locally compact group g a coaction e a m a c n of n on a c algebra a can be inflated to a coaction s of g on a and the crossed product a x g is then isomorphic to the induced c algebra ind a xe n we prove this and a natural generalization in which a xf n is replaced by a twisted crossed product a x g n g in case g is abelian we recover a theorem of olesen and pedersen we then use this to extend the landstad duality of the first author to twisted crossed products and give several applications in particular we prove that if l n d g n 1 is topologically trivial but not necessarily split as a group extension then every twisted crossed product a x g n g is isomorphic to a crossed product of the form a x n

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 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

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