some remarks on the segal and sullivan conjectures introduction the recurrent theme of this paper is the concept of invariance introduced in 1 let g be a finite group and let hg q denote the homotopy category of g cw complexes given a set f of morphisms of our category a functor h defined on hg q is said to be g invariant if it carries each element of p to an isomorphism maps in hg q are determined by fixed point data accordingly we obtain interesting sets of morphisms by considering the so called sc equivalences here 3c is a collection of subgroups of g and a map f x y is an sc equivalence if h xh yh is a homotopy equivalence for each h e sc we then say that h is sc invariant if it inverts all the sc equivalences the smaller sc is the stronger the statement becomes since less and less data is required of a map to ensure that it is inverted by h the generalization of the segal conjecture proved in 1 produces the smallest class of subgroups sc such that the functor s i7r is sc invariant s 17r i denotes pro group valued equivariant stable cohomotopy localized at a multiplicatively closed subset s of the burnside ring a g and completed at an ideal i c a g the class sc admits a description given entirely in terms of s and i but does nonetheless depend on the cohomology theory 7 rg indeed while equivariant k theory enjoys a similar even stronger property 2 this is not so for an arbitrary a g module valued theory h g our first objective will be to produce a collection 3c s i of subgroups such that s h is 3c s i invariant for a large class of cohomology theories h namely those representable by g spectra families i e collections i of subgroups that are closed under taking subconjugates play a privileged role in our context this is due to the existence of a space el which is i equivalent to a point and yet is i isotropic namely ejk 0 unless k e i it follows that there exists a universal

collocation approximation to eigenvalues of an ordinary differential equation numerical illustrations we display the numerical results associated with the collocation of three eigenvalue problems using from one to four gauss points per partition interval in order to document the sharpness of the error bounds we have previously obtained the ordinary differential operators involved are real with constant coefficients two of the problems have an eigenvalue whose ascent exceeds one we propose an explanation for the observed manner in which a set of simple approximate eigenvalues can approach a single multiple eigenvalue

correction to the sullivan conjecture on maps from classifying space on page 49 i assert that given an unstable coalgebra c over the mod p steenrod algebra a c e ca the module of primitives pc is the suspension of an unstable a module e pc e u this is true for p 2 but certainly false for p 2 consider for instance the a coalgebra h bzp which plays such a large role in this paper this error makes nonsense of the spectral sequence 2 5 ext m 1rp cp ext s e ma c which is central to the proof of the sullivan conjecture however we may proceed as follows consider instead of the category u the category v of a modules satisfying the modified unstable condition xpt ofor lxi 2pt

multi quadric quasi interpolation method coupled with fdm for the degasperis procesi equation in this paper the degasperis procesi dp equation which contains nonlinear high order derivatives and discontinuous solutions is recast to its equivalent form u p based on the equivalent formulation of dp equation a new numerical method is presented the multi quadric mq quasi interpolation method coupled with finite difference method is applied to approximate the spatial derivatives and the third order tvd method is used to approximate the time derivative several examples are presented to demonstrate the effectiveness of the proposed method

classifying spaces of kac moody groups we study the structure of classifying spaces of kac moody groups from a homotopy theoretic point of view they behave in many respects as in the compact lie group case the mod p cohomology algebra is noetherian and lannes t functor computes the mod p cohomology of classifying spaces of centralizers of elementary abelian p subgroups also spaces of maps from classifying spaces of finite p groups to classifying spaces of kac moody groups are described in terms of classifying spaces of centralizers while the classifying space of a kac moody group itself can be described as a homotopy colimit of classifying spaces of centralizers of elementary abelian p subgroups up to p completion we show that these properties are common to a larger class of groups also including parabolic subgroups of kac moody groups and centralizers of finite p subgroups

towers and injective cohomology algebras since carlsson s paper 5 and the work of lannes and zarati 9 topo1ogists have learned to recognize many spaces y so h y is injective in u for example if g is a finite group with p sylow subgroup z p kn the h bg is injective further infinite examples include eb z p k and any of the many wedge summands of these spaces a complete classification of u injectives is given in 18 and from this it can be seen that the only such injectives which are not a unbounded are the brown gitler modules these have been adequately discussed elsewhere see 6 among many an immediate consequence of theorem a will be the following result suppose y satisfies the same hypotheses as in theorem a

deconstructing hopf spaces we characterize hopf spaces with finitely generated cohomology as an algebra over the steenrod algebra we deconstruct the original space into an h space y with finite mod p cohomology and a finite number of p torsion eilenberg mac lane spaces we give a precise description of homotopy commutative h spaces in this setting

numerical solution of the degasperis procesi equation by the cubic b spline quasi interpolation method abstract in this paper a numerical scheme is presented to solve the non dissipative degasperis procesi equation based on the u p formulation the cubic b spline quasi interpolation coupled with the finite difference method is applied to approximate the spatial derivatives and an optimal third order tvd runge kutta method to estimate the time derivative of the dependent variable the accuracy and effectiveness of the proposed method are validated by six classical problems numerical results indicate that the proposed scheme is simple easy to implement with high accuracy

homotopy and homology of simplicial abelian hopf algebras let a be a simplicial bicommutative hopf algebra over the field mathbb f 2 with the property that pi 0a cong mathbb f 2 we show that pi ast a is a functor of the andre quillen homology of a where a is regarded as an mathbb f 2 algebra then we give a method for calculating that andre quillen homology independent of knowledge of pi ast a

the classification of p compact groups for p odd a p compact group as defined by dwyer and wilkerson is a purely homotopically defined p local analog of a compact lie group it has long been the hope and later the conjecture that these objects should have a classification similar to the classification of compact lie groups in this paper we finish the proof of this conjecture for p an odd prime proving that there is a one to one correspondence between connected p compact groups and finite reflection groups over the p adic integers we do this by providing the last and rather intricate piece namely that the exceptional compact lie groups are uniquely determined as p compact groups by their weyl groups seen as finite reflection groups over the p adic integers our approach in fact gives a largely self contained proof of the entire classification theorem for p odd

localization and periodicity in unstable homotopy theory in this paper we develop a hierarchy of natural localizations of spaces called the vn periodizations for n 0 which may be used to expose and study periodic phenomena in unstable homotopy theory these vn periodizations act less radically than the corresponding homological localizations 2 and respect fibrations to a very considerable extent a major part of this paper is devoted to developing the general theory of periodizations of spaces thereby providing a foundation for the study of the vn periodization and many others some of this general theory has been developed independently by dror farjoun 13 14 and we refer the reader to his work for an alternative approach with other interesting general results during the past decade remarkable progress was made by ravenel hopkins devinatz and smith 12 16 17 33 toward a global understanding of stable periodic phenomena and we hope that the present paper will help to prepare the way for a similar understanding of unstable periodic phenomena an excellent exposition of localization and periodicity in stable homotopy theory is now available in ravenel s book 35 major features of stable homotopy are understood chromatically as manifestations of more basic periodic phenomena these phenomena belong to a hierarchy starting with those detected rationally followed by those detected in classical k theory and in the successive morava k theories most fundamentally each finite cw spectrum has an intrinsic periodicity given by a vn self map which becomes a selfequivalence after suitable localization for simplicity we describe our results in the pointed homotopy category hoo of connected cw complexes for spaces w y e hoo we call y w periodic or w local when the pointed mapping space map w y is contractible or equivalently when xt w y for t 0 as shown more generally in 3 corollary 7 2 10 or 13 there is a natural initial example x pwx of a map from x to a w periodic space pwx in hoo and we call this the w periodization or w localization of x our notation is derived from the classical example where w sn l and pwx is the n th postnikov section

postnikov pieces and bz p homotopy theory we present a constructive method to compute the cellularization with respect to b m z p for any integer m 1 of a large class of h spaces namely all those which have a finite number of non trivial b m z p homotopy groups the pointed mapping space map b m z p x is a postnikov piece we prove in particular that the b m z p cellularization of an h space having a finite number of b m z p homotopy groups is ap torsion postnikov piece along the way we characterize the bz p r cellular classifying spaces of nilpotent groups

the bar complex of an e infinity algebra the standard reduced bar complex b a of a differential graded algebra a inherits a natural commutative algebra structure if a is a commutative algebra we address an extension of this construction in the context of e infinity algebras we prove that the bar complex of any e infinity algebra can be equipped with the structure of an e infinity algebra so that the bar construction defines a functor from e infinity algebras to e infinity algebras we prove the homotopy uniqueness of such natural e infinity structures on the bar construction we apply our construction to cochain complexes of topological spaces which are instances of e infinity algebras we prove that the n th iterated bar complexes of the cochain algebra of a space x is equivalent to the cochain complex of the n fold iterated loop space of x under reasonable connectedness completeness and finiteness assumptions on x

vector bundles over classifying spaces of compact lie groups the completion theorem of atiyah and segal as says that the complex k theory group k bg of the classifying space of any compact lie group g is isomorphic to r g the representation ring completed with respect to its augmentation ideal however the group k bg bg z bu does not directly contain information about vector bundles over the infinite dimensional complex bg itself the purpose of this paper is to compare the grothendieck group of vector bundles over bg which we denote k bg with both k bg and r g the main result is an algebraic description of k bg in terms of the representation rings of certain subgroups of g as one consequence we show that of the natural maps

the performance of the collocation and galerkin methods with hermite bi cubics this paper presents a study of the performance of the collocation and galerkin methods using hermite bi cubic basis functions it is a sequel to the studies of houstis et al 6 and weiser et al

2 nilpotent real section conjecture we show a 2 nilpotent section conjecture over mathbb r for a geometrically connected curve x over mathbb r such that each irreducible component of its normalization has mathbb r points pi 0 x mathbb r is determined by the maximal 2 nilpotent quotient of the fundamental group with its galois action as the kernel of an obstruction of jordan ellenberg this implies that for x smooth and proper x mathbb r pm is determined by the maximal 2 nilpotent quotient of mathrm gal mathbb c x with its mathrm gal mathbb r action where x mathbb r pm denotes the set of real points equipped with a real tangent direction showing a 2 nilpotent birational real section conjecture

fibrations of classifying spaces we investigate fibrations of the form z y x where two of the three spaces are classifying spaces of compact connected lie groups we obtain certain finiteness conditions on the third space which make it also a classifying space our results allow to express some of the basic notions in group theory in terms of homotopy theory i e in terms of classifying spaces as an application we prove that every retract of the classifying space of a compact connected lie group is again a classifying space

higher comparison maps for the spectrum of a tensor triangulated category abstract for each object in a tensor triangulated category we construct a natural continuous map from the object s support a closed subset of the category s triangular spectrum to the zariski spectrum of a certain commutative ring of endomorphisms when applied to the unit object this recovers a construction of p balmer these maps provide an iterative approach for understanding the spectrum of a tensor triangulated category by starting with the comparison map for the unit object and iteratively analyzing the fibers of this map via higher comparison maps we illustrate this approach for the stable homotopy category of finite spectra in fact the same underlying construction produces a whole collection of new comparison maps including maps associated to and defined on each closed subset of the triangular spectrum these latter maps provide an alternative strategy for analyzing the spectrum by iteratively building a filtration of closed subsets by pulling back filtrations of affine schemes

equivariant stable homotopy and sullivan s conjecture let g be a p group and let x be a g complex let eg denote a contractible space on which g acts freely by the homotopy fixed point set of x we mean the fixed point set f eg x g where f eg x denotes the function space of all maps eg x equipped with a g action by 9f 9f9 1 if we let denote the one point space with trivial g action we may also consider the g space f x it is canonically g homeomorphic to x the g map e g induces a map x g f x g in 19 d sullivan proposed the following

an efficient method for subtracting off singularities at corners for laplace s equation on considere l approximation de la solution d un probleme aux valeurs limites pour l equation de laplace avec singularite sur la frontiere la methode est basee sur un algorithme permettant de calculer les coefficients du developpement asymptotique de la solution a la singularite

a dispersively accurate compact finite difference method for the degasperis procesi equation in this paper we are aimed to solve the non dissipative degasperis procesi equation based on the u p formulation to resolve the computational difficulty at the wave crest where the first order derivative may diverge and the shockpeakon solution may form the first order spatial derivative term in the two step equations will be approximated in a conservative form the resulting equations will be approximated by the symplecticity preserving time stepping scheme and the spatial discretization scheme that can optimize the numerical wavenumber for the first order spatial derivative term this scheme will be developed in a three point grid stencil with the accuracy order of seventh within the combined compact finite difference framework besides the validation of numerical accuracy we will in particular address the discrete conservation of hamiltonians even when peakon collides with antipeakon and generates as a result a shockpeakon we will also demonstrate the capability of applying the proposed numerical method to sharply resolve some important features of the third order dispersive dp equation

conservative finite difference schemes for the degasperis procesi equation we consider the numerical integration of the degasperis procesi equation which was recently introduced as a completely integrable shallow water equation for the equation we propose nonlinear and linear finite difference schemes that preserve two invariants associated with the bi hamiltonian form of the equation at the same time we also prove the unique solvability of the schemes and show some numerical examples

global weak solutions for a new periodic integrable equation with peakon solutions abstract we prove the existence of global weak solutions for a new periodic integrable equation with peakon solutions

on simplicial commutative algebras with vanishing andre quillen homology in this paper we study the andre quillen homology of simplicial commutative l algebras l a field having certain vanishing properties when l has non zero characteristic we obtain an algebraic version of a theorem of j p serre and y umeda that characterizes such simplicial algebras having bounded homotopy groups we further discuss how this theorem fails in the rational case and as an application indicate how the algebraic serre theorem can be used to resolve a conjecture of d quillen for algebras of finite type over noetherian rings having non zero characteristic

homotopy groups of highly connected manifolds abstract in this paper we give a formula for the homotopy groups of n 1 connected 2n manifolds as a direct sum of homotopy groups of spheres in the case the n th betti number is larger than 1 we demonstrate that when the n th betti number is 1 the homotopy groups might not have such a decomposition the techniques used in this computation also yield formulas for homotopy groups of connected sums of sphere products and cw complexes of a similar type in all the families of spaces considered here we establish a conjecture of j c moore

characterizations of spectra with injective cohomology which satisfy the brown gitler property we work in the stable homotopy category ofp complete connective spectra having mod p homology of nite type h x means cohomology with z p coecients and is a left module over the steenrod algebraa a spectrum z is called spacelike if it is a wedge summand of a suspension spectrum and a spectrum x satises the brown gitler property if the natural map x z homa h z h x is onto for all spacelike z it is known that there exist spectra t n satisfying the brown gitler prop erty and with h t n isomorphic to the injective envelope of h s n i n the categoryu of unstablea modules call a spectrum x standard if it is a wedge of spectra of the form l t n where l is a stable wedge summand of the classifying space of some elementary abelian p group such spectra haveu injective cohomology and allu injectives appear in this way working directly with the two properties of t n stated above we clarify and extend earlier work by many people on brown gitler spectra our main theorem is that if x is a spectrum withu injective cohomology the following conditions are equivalent a there exists a spectrum y whose cohomology is a reducedu injective and a map x y that is epic in cohomology b there exists a spacelike spectrum z a nd am apx z that is epic in cohomology c 11 x x is monic in cohomology d x satises the brown gitler property e x is spacelike f x is standard m2uis reduced if it has no nontrivial submodule which is a suspension as an application we prove that the snaith summands of 2 s 3 are brown gitler spectra a new result for the most interesting summands at odd primes another application combines the theorem with the second author s work on the whitehead conjecture of independent interest enroute to proving that b implies c we prove that the homology suspension has the following property if an n connected space x admits a map to an n fold suspension that is monic in mod p homol ogy then nn x x is onto in mod p homology

numerical solution of burgers fisher equation by cubic b spline quasi interpolation abstract in this paper numerical solution of the burgers equation is presented based on the cubic b spline quasi interpolation at first the cubic b spline quasi interpolation is introduced moreover the numerical scheme is presented by using the derivative of the quasi interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the time derivative of the dependent variable the accuracy of the proposed method is demonstrated by some test problems the numerical results are found in good agreement with exact solutions the advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement

an operator splitting method for the degasperis procesi equation an operator splitting method is proposed for the degasperis procesi dp equation by which the dp equation is decomposed into the burgers equation and the benjamin bona mahony bbm equation then a second order tvd scheme is applied for the burgers equation and a linearized implicit finite difference method is used for the bbm equation furthermore the strang splitting approach is used to construct the solution in one time step the numerical solutions of the dp equation agree with exact solutions e g the multipeakon solutions very well the proposed method also captures the formation and propagation of shockpeakon solutions and reveals wave breaking phenomena with good accuracy

autour de la cohomologie de maclane des corps finis nous decrivons une nouvelle methode de calcul de la cohomologie de maclane des corps finis cette theorie est intimement reliee aux extensions du groupe additif deja etudiees par l breen et a l homologie de hochschild topologique de m bokstedt et donc a la k theorie stable notre approche utilise de maniere cruciale l annulation de la cohomologie de maclane du corpsf p avec pour coefficients l algebre symetrique ou l on a inverse le frobenius nous recourons alors a l analyse des complexes de koszul et de de rham en caracteristique non nulle

indecomposablea module summands inh v which are unstable algebras letp be a prime and denote bya the modp steenrod algebra we determine the indecomposablea module summands ofh p d f p which admit the structure of an unstablea algebra in fact it turns out that this is equivalent to the problem of determining those indecomposablea module summands which arise as the modp cohomology of a space or even a classifying space of a finite group we reduce this problem to one in modular representation theory namely for whichd andp is the projective cover of the trivial one dimensional gl d f p representationf p a permutation module our solution of this latter problem makes use of the classification of subgroups of gl d f p acting transitively on f p d 0 and hence depends on the classification of finite simple groups on feit thompson s odd order theorem ifp 2

lannes t functor on summands of h b z p s let h be the mod p cohomology of the classifying space b z p thought of as an object in the category u of unstable modules over the steenrod algebra lannes constructed a functor t u u which is left adjoint to the functor a a h in this paper we evaluate t on the indecomposable u summands of h s the tensor product of s copies of h our formula involves the composition factors of certain tensor products of irreducible representations of the semigroup ring f p m s s z p

nullification and cellularization of classifying spaces of finite groups in this note we discuss the effect of the bz p nullification p bz p and the bz p cellularization cw bz p over classifying spaces of finite groups and we relate them with the corresponding functors with respect to moore spaces that have been intensively studied in the last years we describe p bz p bg by means of a covering fibration and we classify all finite groups g for which bg is bz p cellular we also carefully study the analogous functors in the category of groups and their relationship with the fundamental groups of p bz p bg and cw bz p bg

b spline quasi interpolation based numerical methods for some sobolev type equations in this article we intend to use quadratic and cubic b spline quasi interpolants to develop higher order numerical methods for some sobolev type equations in one space dimension our aim is also to compare the performance of the proposed methods in terms of the accuracy and the rate of convergence we also discuss another approach to the cubic b spline quasi interpolation based method where we achieve fourth order of accuracy in space we theoretically establish the order of accuracy for the three proposed methods and also establish the l 2 stability in the linear case using von neumann analysis as a particular case of the sobolev type equations we take the equal width and the benjamin bona mahony burgers equations and perform several numerical experiments to support our theoretical results

enriched collocation finite element method a model problem abstract the standard collocation finite element method requires a fine mesh in the vicinity of singularities the concept of enriched elements is introduced to rectify this limitation the additional degree of freedom is handled by introducing a constraint equation relating the stress intensity factor to the nodal values in the interior the numerical results indicate that the recommended strategy is viable and leads to a significant improvement in the accuracy

rational equivalences between classifying spaces the paper contains a homotopy classification of rational equivalences between classifying spaces of compact connected lie groups with an application to genus sets of such spaces

group cohomology and control of p fusion we show that if an inclusion of finite groups h g of index prime to p induces a homeomorphism of mod p cohomology varieties or equivalently an f isomorphism in mod p cohomology then h controls p fusion in g if p is odd this generalizes classical results of quillen who proved this when h is a sylow p subgroup and furthermore implies a hitherto difficult result of mislin about cohomology isomorphisms for p 2 we give analogous results at the cost of replacing mod p cohomology with higher chromatic cohomology theories

homotopy fixed point methods for lie groups and finite loop spaces a loop space x is by definition a triple x bx e in which x is a space bx is a connected pointed space and e x qbx is a homotopy equivalence from x to the space qbx of based loops in bx we will say that a loop space x is finite if the integral homology h x z is finitely generated as a graded abelian group i e if x appears at least homologically to be a finite complex in this paper we prove the following theorem

degenerate fibres in the stone cech compactification of the universal bundle of a finite group applied to a continuous surjection e b of completely regular hausdorff spaces e and b the stone cech compactification functor yields a surjection e b for an n fold covering map we show that the fibres of while never containing more than n points may degenerate to sets of cardinality properly dividing n in the special case of the universal bundle eg bg of a p group g we show more precisely that every possible type of g orbit occurs among the fibres of to prove this we use a weak form of the so called generalized sullivan conjecture

homotopy classification of self maps of bg via g actions to complete the classification of self maps of bg theorem 4 2 it remains to prove theorem 4 1 i e to show that the higher limits of certain functors n 1h 2 f p g z p wall vanish and that h0 mp g f i has the expected value in this section these limits h mp g hi will be calculated in the easy cases when g is a simply connected classical group i e one of the groups su n sp n or spin n or when i 2 the computation of h mp g hi for the simply connected exceptional lie groups will be put off until section 6 we begin with some general properties of higher limits let 6 be any small discrete category it is convenient when dealing with higher limits of functors on to let 4 moldenote the category of contravariant functors from v to ad then for any f e w muok we can identify

on the homotopy type of p completions of infra nilmanifolds infra nilmanifolds are compact k g 1 manifolds with g a torsion free finitely generated virtually nilpotent group motivated by previous results of various authors on p completions of k g 1 spaces with g a finite or a nilpotent group we study the homotopy type of p completions of infra nilmanifolds for any prime p we prove that the p completion of an infra nilmanifold is a virtually nilpotent space which is either aspherical or has infinitely many nonzero homotopy groups the same is true for p localization moreover we show by means of examples that rationalizations of infra nilmanifolds may be elliptic or hyperbolic

derived representation theory of lie algebras and stable homotopy categorification of slk abstract we set up foundations of representation theory over s the sphere spectrum which is the initial ring of stable homotopy theory in particular we treat s lie algebras and their representations characters g l n s verma modules and their duals harish chandra pairs and zuckermann functors as an application we construct a khovanov s l k stable homotopy type with a large prime hypothesis which is a new link invariant using a stable homotopy analogue of the method of j sussan

global weak solutions and blow up structure for the degasperis procesi equation in this paper we study several qualitative properties of the degasperis procesi equation we first established the precise blow up rate and then determine the blow up set of blow up strong solutions to this equation for a large class of initial data we finally prove the existence and uniqueness of global weak solutions to the equation provided the initial data satisfies appropriate conditions

on the cohomology of highly connected covers of finite hopf spaces relying on the computation of the andre quillen homology groups for unstable hopf algebras we prove that if the mod p cohomology of both the fiber and the base in an h fibration is finitely generated as algebra over the steenrod algebra then so is the mod p cohomology of the total space in particular the mod p cohomology of the n connected cover of a finite h space is always finitely generated as algebra over the steenrod algebra

on the p adic completions of nonnilpotent spaces this paper deals with the p adic completion f p x developed by bousfield kan for a space x and prime p a space x is called f p good when the map x f p x is a mod p homology equivalence and called f p bad otherwise general examples of f p good spaces are established beyond the usual nilpotent or virtually nilpotent ones these include the polycyclic by finite spaces however the wedge of a circle with a sphere of positive dimension is shown to be f p bad this provides the first example of an f p bad space of finite type and implies that the p profinite completion of a free group on two generators must have nontrivial higher mod p homology as a discrete group

rational homotopy of the homotopy fixed point sets of circle actions in this paper when g is the circle s 1 and m is a g space we study the rational homotopy type of the fixed point set m g the homotopy fixed point set m hg and the natural injection m g m hg c 2009 elsevier inc all rights reserved

homology decompositions for p compact groups we construct a homotopy theoretic setup for homology decompositions of classifying spaces of p compact groups this setup is then used to obtain a subgroup decomposition for p compact groups which generalizes the subgroup decomposition with respect to p stubborn subgroups for a compact lie group constructed by jackowski mcclure and oliver

a propos de conjectures de serre et sullivan we first give a new proof of a conjecture of j p serre on the homotopy of finite complexes which was recently proved by c mcgibbon and j neisendorfer the emphasis is on a property of the mod 2 homology of certain spaces their quasi boundedness as right modules over the steenrod algebra this property is preserved when one goes from a simply connected space to its loop space and also when one takes a covering of anh space then we show that this notion of quasi boundedness simplifies h miller s proof of d sullivan s conjecture on the contractibility of the space of pointed maps from the classifying space of the groupe 2 into a finite complex

derived koszul duality and tq homology completion of structured ring spectra abstract working in the context of symmetric spectra we consider algebraic structures that can be described as algebras over an operad o topological quillen homology or tq homology is the spectral algebra analog of quillen homology and stabilization working in a framework for homotopical descent the associated tq homology completion map appears as the unit map of an adjunction comparing the homotopy categories of o algebra spectra with coalgebra spectra over the associated comonad via tq homology we prove that this adjunction between homotopy categories can be turned into an equivalence by replacing o algebras with the full subcategory of 0 connected o algebras we also construct the spectral algebra analog of the unstable adams spectral sequence that starts from the tq homology groups tq x of an o algebra x and prove that it converges strongly to x when x is 0 connected

on stable maps from eilenberg maclane spaces to classifying spaces 0 introduction in recent years there has been a general interest in the problem of understanding the homotopy type of function spaces map x y where x is an infinite complex the special case where x is the classifying space of a finite group gives the celebrated segal s conjecture 4 when y lim fqnn qs and sullivan s conjecture 10 when y is a finite complex one would like to know whether the affirmative solution to these conjectures can also be applied to the study of map x y for other infinite complexes x the philosophy behind this approach to the problem has already manifested itself in 5 6 7 and 8 among others to give interesting results in this note we will utilize the solution to segal s conjecture to compute stable maps from eilenberg maclane spaces to classifying spaces as an application we will also describe the homotopy type of map x y at a prime p where x k 1t n wt finite n 2 and y q g g 0 u sp the original motivation behind this paper was due to anderson hodgkin 1 there they proved vanishing results on the topological k theory of higher eilenberg maclane spaces using atyiah s theorem on the k theory of the classifying spaces of compact lie groups we shall prove that the same result holds for stable cohomotopy groups the author is especially grateful to professor gunnar carlsson for pointing out 1 and how segal s conjecture comes into play