ends of maps ii versions of the finiteness obstruction and simple homotopy theory within e overx are developed this provides a setting for obstructions to the map analogs of the end ands cobordism theorems for manifolds these are applied to study equivariant mapping cylinder neighborhoods in topological group actions triangulations of locally triangulable spaces and block bundle structures on approximate fibrations

on the curvature of piecewise flat spaces we consider analogs of the lipschitz killing curvatures of smooth riemannian manifolds for piecewise flat spaces in the special case of scalar curvature the definition is due to t regge considerations in this spirit date back to j steiner we show that if a piecewise flat space approximates a smooth space in a suitable sense then the corresponding curvatures are close in the sense of measures

the periodicity in stable equivariant surgery the classical surgery theory see 5 and 23 computes the structure set sm m rel of manifolds homotopy equivalent to m relative to the boundary siebenmann showed that in topological category the structure set is 4 periodic sm m rel sm 4 m d4 rel up to a copy of see 12 cappell and weinberger gave a geometric interpretation of this periodicity in 8 by using weinberger s stratified surgery theory see 24 we extend this to an equivariant periodicity result for topological manifolds with homotopically stratified actions by compact lie groups with d4 replaced by the unit ball of certain group representations in particular if g is an odd order group acting on a topological manifold m then the equivariant stable structure sets satisfy s m rel s m d 4 g rel up to copies of 1993 john wiley sons inc

the borel conjecture for hyperbolic and cat 0 groups we prove the borel conjecture for a class of groups containing word hyperbolic groups and groups acting properly isometrically and cocompactly on a finite dimensional cat 0 space

the local structure of length spaces with curvature bounded above we show that a number of different notions of dimension coincide for length spaces with curvature bounded above we then apply this result showing that if x is a locally compact cat 0 space with cocompact isometry group then the dimension of the tits boundary and the asymptotic cone s of x are determined by the maximal dimension of a flat in x

stratified path spaces and fibrations the main objects of study are the homotopically stratified metric spaces introduced by quinn closed unions of strata are shown to be stratified forward tame stratified fibrations between spaces with stratifications are introduced paths which lie in a single stratum except possibly at their initial points form a space with a natural stratification and the evaluation map from that space of paths is shown to be a stratified fibration applications to mapping cylinders and to the geometry of manifold stratified spaces are expected in future papers

a categorification of morelli s theorem we prove a theorem relating torus equivariant coherent sheaves on toric varieties to polyhedrally constructible sheaves on a vector space at the level of k theory the theorem recovers morelli s description of the k theory of a smooth projective toric variety morelli in adv math 100 2 154 182 1993 specifically let x be a proper toric variety of dimension n and let m mathbb r mathrm lie t mathbb r vee cong mathbb r n be the lie algebra of the compact dual real torus t mathbb r vee cong u 1 n then there is a corresponding conical lagrangian t m and an equivalence of triangulated dg categories mathcal p mathrm erf t x cong mathit sh cc m mathbb r lambda where mathcal p mathrm erf t x is the triangulated dg category of perfect complexes of torus equivariant coherent sheaves on x and shcc m is the triangulated dg category of complex of sheaves on m with compactly supported constructible cohomology whose singular support lies in this equivalence is monoidal it intertwines the tensor product of coherent sheaves on x with the convolution product of constructible sheaves on m

constructing approximate fibrations in this paper two results concerning the construction of approximate fibrations are established the first shows that there are approximate fibrations p m s2 which are homotopic to bundle maps but which cannot be approximated by bundle maps here m can be a compact q manifold or some topological n manifold n 5 the second shows how to construct approximate fibrations p m b whose fibers do not have finite homotopy type for any b of euler characteristic zero here m can be a compact q manifold and b only has to be an anr or m can be an n manifold n 6 and b must then also be a topological manifold

on hyperbolic groups with spheres as boundary let g be a torsion free hyperbolic group and let n 6 be an integer we prove that g is the fundamental group of a closed aspherical manifold if the boundary of g is homeomorphic to an n 1 dimensional sphere

the polytope algebra let f be an ordered field and let p denote the family of all convex polytopes in the d dimensional vector space v over f the universal abelian group corresponding to the translation invariant valuations on p has generators p for p p with 0 satisfying the relations v p q p p q whenever p q p q p and t p t p for p p and t v with multiplication induced by m p q p q is almost a graded commutative algebra over f in that dr 0 r with 0 z r a vector space over f r 1 and r s r s r s 0 r 0 forr d the dilatation d p p for p p and f is such that x rx for x r and 0 negative dilatations arise from the euler map e p p f 1 dimf f the sum extending over all faces f of p since x rx for x r and 0 separating group homomorphisms for are the frame functionals which give the volumes of the faces of polytopes determined by successive support hyperplanes in sequences of directions two isomorphisms on are described one related to cones of outer normal vectors and the other to the polytope groups obtained from by discarding polytopes of dimension less than d various applications of the polytope algebra are given including a theory of mixed polytopes which has implications for mixed valuations

the k theoretic farrell jones conjecture for hyperbolic groups we prove the k theoretic farrell jones conjecture for hyperbolic groups with twisted coefficients in any associative ring with unit

anti de sitter strictly ghc regular groups which are not lattices for d 4 5 6 7 8 we exhibit examples of mathrm ads d 1 strictly ghc regular groups which are not quasi isometric to the hyperbolic space mathbb h d nor to any symmetric space this provides a negative answer to question 5 2 in 9a12 and disproves conjecture 8 11 of barbot merigot bm12 we construct those examples using the tits representation of well chosen coxeter groups on the way we give an alternative proof of moussong s hyperbolicity criterion mou88 for coxeter groups built on danciger gueritaud kassel dgk17 and find examples of coxeter groups w such that the space of strictly ghc regular representations of w into mathrm po d 2 mathbb r up to conjugation is disconnected

rigidity and crystallographic groups i this is the first in a series of articles about rigidity theorems for crystallographic groups the main theorem is 1 1 it is an h cobordism rigidity statement it implies see 1 6 that any crystallographic manifold with odd order holonomy which is h cobordant and simply homotopically equivalent to the standard one is actually homeomorphic to it in the second part of this work 9 we will exploit this result to prove that one can remove the word h cobordant from the previous sentence the main theorem concerns an involution on the group wh p mr this is the group of g h cobordisms of the flat torus mr of a crystallographic group f with holonomy group g definitions of these terms appear in 1 4 we prove a vanishing result of the tate cohomology

approximation by equivariant homeomorphisms i locally linear locally smoothable actions of finite groups on finite dimensional manifolds are considered in which two incident components of fixed point sets of subgroups either coincide or one has codimension at least three in the other for these actions an equivariant o approximation theorem is proved using engulfing techniques as corollaries are obtained equivariant fibrations are bundles and controlled h cobordism theorems as well as an equivariant version of edwards cell like mapping theorem and the vanishing of the set of transfer invariant g homotopy topological structures rel bound ary on tn x dp when tn is the n torus with trivial g action and dp is a representation disc here we consider locally linear locally smooth br pl and topological actions of a finite group g on n manifolds in addition we require that a g manifold m have gaps of codimension 3 i e for h c g if mh is a component of the h fixed point set mh and m h c mh is the subspace of points x for which the isotropy subgroup gx strictly contains h then either m h mh or m h has codimension 3 in mh we shall use the term not necessarily locally linear g manifold when we wish to drop the local linearity hypothesis but retain the property that if mh c m but mh m then mh is a locally flat inequivariantly submanifold of mk of codimension at least three this while self contained is the second in a series of papers sw2 8 s in which we analyze the extent of failure of the topological invariance of equivari ant whitehead torsion and the consequent failure of subgroups of the equivariant whitehead group whgl m of illman ill cf rothenberg r to classify pl or smooth g h cobordisms up to topological equivalence under our gap hypotheses the arguments of browder and quinn bq and rothenberg r show that g h cobordisms are classified up to pl or smooth equivalence by such subgroups we eventually conclude in sw7 s cf sw2 that the topological equivalence classes of g h cobordisms on a g manifold m that are products over the union of fixed point components of m having dimension 5 are classised by a group whg p m to which whgl m maps homomorphically the image in whg p m of the per tinent subgroup of whgl m classifies up to equivariant homeomorphism rel m those g h cobordisms on m that admit equivariant handlebody decompositions relative to m this homomorphism is not generally surjective and there are g h cobordisms that do not admit such handle decompositions cf sw5 s in order

stratified fiber bundles quinn homology and brane stability of hyperbolic orbifolds we revisit the problem of stability of string vacua involving hyperbolic orbifolds using methods from homotopy theory and k homology we propose a definition of type ii string theory on such backgrounds that further carry stratified systems of fiber bundles which generalize the more conventional orbifold and symmetric string backgrounds together with a classification of wrapped branes by a suitable generalized homology theory for spaces stratified fibered over hyperbolic orbifolds we use the algebraic k theory of their fundamental groups and quinn homology to derive criteria for brane stability in terms of an atiyah hirzebruch type spectral sequence with its lift to k homology stable d branes in this setting carry stratified charges which induce new additive structures on the corresponding k homology groups we extend these considerations to backgrounds which support h flux where we use k groups of twisted group algebras of the fundamental groups to analyze stability of locally symmetric spaces with k amenable isometry groups and derive stability conditions for branes wrapping the fibers of an eilenberg maclane spectrum functor

coefficients for the farrell jones conjecture abstract we introduce the farrell jones conjecture with coefficients in an additive category with g action this is a variant of the farrell jones conjecture about the algebraic k or l theory of a group ring rg it allows to treat twisted group rings and crossed product rings the conjecture with coefficients is stronger than the original conjecture but it has better inheritance properties since known proofs using controlled algebra carry over to the set up with coefficients we obtain new results about the original farrell jones conjecture the conjecture with coefficients implies the fibered version of the farrell jones conjecture

4 dimensional locally cat 0 manifolds with no riemannian smoothings we construct examples of 4 dimensional manifolds m supporting a locally cat 0 metric whose universal covers q m satisfy hruska s isolated flats condition and contain 2 dimensional flats f with the property that 1f s s s s 1 q m are nontrivial knots as a consequence we obtain that the group 1 m cannot be isomorphic to the fundamental group of any compact riemannian manifold of nonpositive sectional curvature in particular if k is any compact locally cat 0 manifold then m k is a locally cat 0 manifold which does not support any riemannian metric of nonpositive sectional curvature

an end theorem for stratified spaces we show that a manifold stratified space x is the interior of a compact manifold stratified space with boundary if and only if x is tame ended and a k theoretic obstruction x vanishes the obstruction x is a localization of quinn s mapping cylinder neighborhood obstruction the main results are theorem 1 6 and theorem 1 7 below in particular this explains when a g manifold is the interior of a compact g manifold with boundary one of our methods is a new transversality theorem theorem 1 16

groups generated by reflections and aspherical manifolds not covered by euclidean space a coxeter system r v is a group r a coxeter group together with a set of generators v such that each element of v has order two and such that all relations in r are consequences of relations of the form vw m v w 1 where v w e v and m v w denotes the order of vw the m v w s which are positive integers or oo obviously determine the coxeter system up to isomorphism the list of finite coxeter groups is short and well known however general coxeter groups are much more flexible in fact the set v and the m v w s can be specified arbitrarily at least if v is finite subject only to the conditions m v v 1 and m v w m w v 2 if v w suppose that r v is a coxeter system that x is a hausdorff space and that xv v v is a locally finite family of closed subspaces indexed by v the xv are called the panels of x 2 there is a classical method for constructing a transformation group from these data for each x e x let v x denote the set of v in v such that x e xv for each subset s of v let rs be the subgroup generated by s and let xs be the face of x defined by

whitehead groups and the bass conjecture this paper will be concerned with proving that certain whitehead groups of torsion free elementary amenable groups are torsion groups and related results and then applying these results to the bass conjecture in particular we shall establish the strong bass conjecture for an arbitrary elementary amenable group

geometric transfer and the homotopy type of the automorphism groups of a manifold introduction for m a compact manifold with or without boundary in one of the categories diff pl or top we denote by a m the semisimplicial ss group of automorphisms of m i e the group of diffeomorphisms pl homeomorphisms or homeomorphisms resp in this paper we describe in the stability range and localized away from two the homotopy structure of a m we do this in terms of a m the ss group of block automorphisms whose homotopy structure is a sophisticated formulation of surgery theory abk and some new homotopy functors 5 theorem c the functors 5 and 5 l here considered have already been studied in the pl category by hatcher h from a different point of view the present geometric description is appropriate to the description of the homotopy structure of a m and enables us to produce a natural map b sc x 5 x from the associative monoid of simple homotopy equivalence of x to 5 x which will be important for our study besides the geometric topology results blr two ideas partially developed in bli are basic the first is the transfer map and stability for concordances described in the pl context by hatcher and in the differentiable context in bli the second is the splitting of a z 1 weak commutative topological group with involution as a product up to homotopy of two pieces the symmetric respectively antisymmetric parts applied to our situation those parts manifest very different behavior the results of this paper have been sketched by the first author in the survey paper b there is an overlap between some but not all results of a and results of hatcher h2 published in the same volume as b the results of this paper are stated in the next section statement of results and the proofs are given in subsequent sections we continue this section with a description of the notations used

the boundary of negatively curved groups gromov s article gr contains fundamental properties of negatively curved groups several sets of seminar notes are available frn swn usn that contain more detailed accounts of and further expand on gromov s article a sequence xi e r i 1 2 is convergent at infinity provided xi x t 00 as i j t 00 two sequences xi yj convergent at infinityare equivalent if xi yi t 00 as i t 00 a point in the boundary ar of r is an equivalence class of sequences convergent at infinity now r u ar has a natural topology in which r is a discrete subspace and a typical not necessarily open neighborhood of a point a e ar represented by a sequence xj is given by y e r u arj a y n where a y lim xi y if y e r and a y lim xi yi if year is represented by a sequence yj ar is a compact metrizable finite dimensional space gr swn the important tool that relates ar with the cohomological properties of r is the rips complex pd r for every d 0 pd r is the simplicial complex whose vertices are elements of r and a collection xi xk e r spans a simplex if d xi x d for all i j the natural group action of r on itself by left translations gives rise to an action on pd r the key observation of rips is that when d is sufficiently large pd r is contractible and therefore when r is torsion free provides a model for er more precisely we have proposition 1 1 gr lemma 1 7 a swn 4 2 proposition 9 suppose that r is a negatively curved group and let t5 be as in the definition choose an integer

algebraic k theory of virtually free groups we provide a general procedure for computing the algebraic k theory of finitely generated virtually free groups the procedure describes these groups in terms of the algebraic k theory of various finite subgroups and various farrell nil groups we illustrate this process by carrying out the computation for several interesting classes of examples the first two classes serve as a check on the method and show that our algorithm recovers results that already exist in the literature the last two classes of examples yield new computations

isomorphism conjectures in algebraic k theory duff in this paper we are concerned with the four functors y y3 if and which map from the category of topological spaces x to the category diff of q spectra the functor 3 or if maps the space x to the qspectrum of stable topological or smooth pseudoisotopies of x the spaces diff dif 3i x id9 x i 0 are hatcher s deloopings of 970 x yd x cf 29 the functor j maps a path connected space x to the algebraic ktheoretic s2 spectrum for the integral group ring z7r1x the spaces x x i 0 are the gersten wagoner deloopings of z o x cf 27 52 5 x is also defined on a nonpath connected space x to be x1 where xi i e i are the path components of x and ii xi indicates the direct limit of all finite products of the i xi i e i if x is path connected then the functor maps x to the l surgery classifying spaces for oriented surgery problems with fundamental group r i1x identified by their fourfold periodicity k 90 x k 4j x cf 35 39 44 47 in addition if x is not path connected then we set y77 x tiei5 fj xi where xi i e i are the path components of x results obtained by the authors over the past five years cf 15 25 and in particular 19 16 a 1 a 18 together with many earlier results and conjectures which are reviewed in 1 6 1 1 6 6 suggest that for any connected cwcomplex x and for 5 denoting any of the above functors the q spectrum x should be computable in a simple way from the values taken by 9 on covering spaces of x which have very simple fundamental groups in more detail we let 9 x denote the collection of all subgroups h c ir ix which are either finite or virtually infinite cyclic a group h is virtually infinite cyclic if there is a short exact sequence of groups 1 z h f 1 with f equal to a finite group and z equal to the infinite cyclic group for each such group we let xh x denote the covering space projection corresponding to h then the q spectrum 5 x should be computable in a simple way from the q spectra 5 xh h e 9 x the purpose of this paper is to formulate precisely a conjecture along these lines and to verify the conjecture for any x

a pl geometric study of algebraic k theory this paper manages to apply the farrell jones theory on algebraic k groups of closed negatively curved riemannian manifolds to gromov s hyperbolic group theory the paper reaches the conclusion that for any finite polyhedron k with negative curvature wh 1 k 0

a characterization of certain shimura curves in the moduli stack of abelian varieties let f x y be a semistable family of complex abelian varieties over a curve y of genus g y and smooth over the com

bounded homotopy equivalences of hilbert cube manifolds let m and f be hubert cube manifolds with f compact the purpose of this paper is to study homotopy equivalences m r x f which have bounded control in the redirection roughly these homotopy equivalences form a semi sim plicial complex 4 rm x f the controlled whitehead space using results about approximate fibrations rm x f is related to the semi simplicial complex of bounded concordances on r x f then the homotopy groups of rm x f are computed in terms of the lower algebraic at theoretic functors k i 1 introduction let f be a compact hilbert cube manifold we are interested in homotopy equivalences m rm x f which are controlled in the redirection where m is also a hilbert cube manifold to say is controlled in the redirection means that pf m rm is an approximate fibration where p rm x f rm is projection the collection of all such homotopy equivalences which are additionally given to be retractions onto the collared submanifold rm x f of m form the vertices of a semi simplicial complex ta rm x f see 2 for the precise definition the main result of this paper is the computation of the homotopy groups of wa rm x f see corollary 1 below in order to do this we relate wa rm x f to the semi simplicial complex h rm x f of bounded concordances on rm x f an simplex of eb rm x f is a homeomorphism

higher dimensional manifolds with s1 category 2 a closed topological n manifold mn is of s1 category 2 if it can be covered by two open subsets w1 w2 such that the inclusions wi mn factor homotopically through maps wi s1 we show that for n 3 if rm cat s 1 m n 2 then mn is homeomorphic to sn or sn 1 s1 or the non orientable sn 1 bundle over s1 we also obtain an unknotting theorem for locally flat knots of sn 2 in sn and a characterization of s1 sn 1

hyperbolization of polyhedra hyperbolization is a process for converting a simplicial complex into a metric space with nonpositive curvature in the sense of gromov several such processes are described in 19 3 4 one of the purposes of this paper is to elaborate this idea of gromov another purpose is to use it to construct the three examples described below our approach to hyperbolization is based on the following construction of williams 32 suppose that x is a space and that x is a map onto the standard z simplex suppose also that k is an dimensional simplicial complex to these data williams associates a space xak constructed by replacing each n simplex in the barycentric subdivision of k by a copy of x the pair x is a hyperbolized simplex if x is a nonpositively curved manifold with boundary and has appropriate properties it is proved in 4 that hyperbolized simplices exist if x is a hyperbolized simplex then xak is nonpositively curved it is called a hyperbolization of k in all three examples we begin with a polyhedral homology manifold having a desired feature a hyperbolization then has the added feature of nonpositive curvature the first example is a closed aspherical fourmanifold which cannot be triangulated taking the product of this example with a a2 torus we obtain an aspherical manifold of any dimension 4 which is not homotopy equivalent to a pl manifold the second example is a closed smooth manifold of dimension n 5 which carries a topological metric of nonpositive curvature while its universal cover though contractible is not homeomorphic to a euclidean n space r as we shall see such a manifold cannot carry a pl or smooth metric of nonpositive curvature the third example is a further refinement m carries a topological metric of strict negative curvature and its universal

prym covers theta functions and kobayashi curves in hilbert modular surfaces algebraic curves in hilbert modular surfaces that are totally geodesic for the kobayashi metric have very interesting geometric and arithmetic properties e g they are rigid there are very few methods known to construct such algebraic geodesics that we call kobayashi curves we give an explicit way of constructing kobayashi curves using determinants of derivatives of theta functions this construction also allows to calculate the euler characteristics of the teich m uller curves constructed by mcmullen using prym covers

stability of hodge bundles and a numerical characterization of shimura varieties let u be a connected non singular quasi projective variety and f a u a family of abelian varieties of dimension g suppose that the induced map u ag is generically finite and there is a compactification y with complement s y u a normal crossing divisor such that y log s is nef and y s is ample with respect to u we characterize whether u is a shimura variety by numerical data attached to the variation of hodge structures rather than by properties of the map u ag or by the existence of cm points more precisely we show that f a u is a kuga fibre space if and only if two conditions hold first each irreducible local subsystem v of rf ca is either unitary or satisfies the arakelov equality second for each factor m in the universal cover of u whose tangent bundle behaves like the one of a complex ball an iterated kodaira spencer map associated with v has minimal possible length in the direction of m if in addition f a u is rigid it is a connected shimura subvariety of ag of hodge type

group actions and equivariant lipschitz analysis the idea that one could prove by analytic methods the topological invariance of characteristic classes was suggested in an influential lecture by singer the concrete realization of this in the case of rational pontrjagin classes was the result of works by sullivan who showed that topological manifolds have essentially unique lipschitz structures and teleman who showed that lipschitz manifolds are the type of object for which one can analyze the signature operator a refinement of the same method proves the ko l 2 orientability of topological bundles much of this can be done for manifolds with group actions the main implication is that for certain sorts of topological group actions including all smooth and pl actions one can construct an equivariant signature operator whose class in equivariant if homology is a topological invariant this class satisfies the g signature theorem the topological method has implications for various precise classification problems e g semifree topological actions on the sphere but here we shall present only the consequences of these lipschitzian ideas a topologically locally linear g manifolds for odd order g have kog 1 2 orientations madsen rothenberg for all finite g there are lipschitz structures but the signature class is a zero divisor for even order groups the same method produces upon application of the localization theorem atiyah singer classes for some actions with nonmanifold fixed set weinberger extending earlier joint work with cappell b for odd order groups topological and linear conjugacy of linear representations are equivalent hsiang pardon madsen rothenberg this is a consequence of a alternatively one can return to the original argument of atiyah bott milnor for semifree representations a stronger result is true the atiyah bott local numbers associated to representations agree for elements that have discrete fixed sets in topologically equivalent representations this distinguishes many pairs of representations of even order groups c r top g r lip g for all finite g where r cat is the grothendieck group of representations up to cat conjugacy weinberger after inverting 2 this implies that there is no analytic definition of reidemeister torsion that works well in the lipschitz setting the reason is that following

on the farrell jones conjecture for higher algebraic k theory here b is the classifying space of the group and we denote by k r the non connective algebraic k theory spectrum of the ring r the homotopy groups of this spectrum are denoted kn r and coincide with quillen s algebraick groups of r qui73 in positive dimensions and with the negative k groups of bass bas68 in negative dimensions the homotopy groups of the spectrum x k r are denotedhn x k r they yield a generalized homology theory and in particular standard computational tools such as the atiyah hirzebruch spectral sequence apply to the left hand side of the assembly map above as a corollary of the main result of this paper we prove conjecture 1 1 in the case where is the fundamental group of a closed riemannian manifold with strictly negative sectional curvature in fact our result is more general and applies to group rings r where r is a completely arbitrary coefficient ring note that if one replaces in conjecture 1 1 the coefficient ring z by an arbitrary coefficient ringr the corresponding conjecture would be false already in the simplest non trivial case if c is the infinite cyclic group the bass heller swan formula bhs64 gra76 p 236 for kn rc kn r t 1 yields that kn rc kn 1 r kn r nkn r nkn r where nkn r is defined as the cokernel of the split inclusion kn r kn r t and does not vanish in general but since s is a model for bc one obtains on the left hand side of the assembly map only hn bc k r kn r kn 1 r

identifying assembly maps in k and l theory in this paper we prove the equivalence of various algebraically or geometrically defined assembly maps used in formulating the main conjectures in k and l theory and c theory

the farrell jones isomorphism conjecture for finite covolume hyperbolic actions and the algebraic theory of bianchi groups we prove the farrell jones isomorphism conjecture for groups acting properly discontinuously via isometries on real hyperbolic n space ihin with finite volume orbit space we then apply this result to show that for any bianchi group f wh f ko zr and ki zr vanish for i 1

the dualizing spectrum of a topological group to a topological group g we assign a naive g spectrum d g called the dualizing spectrum of g when the classifying space bg is finitely dominated we show that d g detects poincare duality in the sense that bg is a poincare duality space if and only if d g is a homotopy finite spectrum secondly we show that the dualizing spectrum behaves multiplicatively on certain topological group extensions in proving these results we introduce a new tool a norm map which is defined for any g and for any naive g spectrum e applications of the dualizing spectrum come in two flavors i applications in the theory of poincare duality spaces and ii applications in the theory of group cohomology on the poincare duality space side we derive a homotopy theoretic solution to a problem posed by wall which says that in a fibration sequence of fini the total space satisfies poincare duality if and only if the base and fiber do the dualizing spectrum can also be used to give an entirely homotopy theoretic construction of the spivak fibration of a finitely dominated poincare duality space we also include a new proof of browder s theorem that every finite h space satisfies poincare duality in connection with group cohomology we show how to define a variant of farrell tate cohomology for any topological or discrete group g with coefficients in any naive equivariant cohomology theory e when e is connective and when g admits a subgroup h of finite index such that bh is finitely dominated we show that this cohomology coincides with the ordinary cohomology of g with coefficients in e in degrees greater than the cohomological dimension of h in an appendix we identify the homotopy type of d g for certain kinds of groups the class includes all compact lie groups torsion free arithmetic groups and bieri eckmann duality groups

approximately fibering a manifold over an aspherical one the paper is devoted to the problem when a map from some closed connected manifold to an aspherical closed manifold approximately fibers i e is homotopic to manifold approximate fibration we define obstructions in algebraic k theory their vanishing is necessary and under certain conditions sufficient basic ingredients are quinn s thin h cobordism theorem and end theorem and knowledge about the farrell jones conjectures in algebraic k and l theory and the maf rigidity conjecture by hughes taylor williams

on the k theory of crystallographic groups for any crystallographic group f we show that the groups ki t are isomorphic via the forget control map to the controlled a groups ki r c for all 0