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Dans cet expos\'e, nous expliquons le r\'esultat suivant d\'emontr\'e r\'ecemment par Abramovich, Karu, Matsuki et W{\l}odarczyk : toute application birationnelle entre deux vari\'et\'es alg\'ebriques compl\`etes et lisses sur un corps alg\'ebriquement clos de caract\'eristique nulle se factorise en une suite finie d'\'eclatements et de contractions de centres lisses. Ce r\'esultat n'aurait sans doute pas vu le jour sans la d\'emonstration de son analogue dans le cadre de la g\'eom\'etrie torique par Morelli et W{\l}odarczyk au milieu des ann\'ees 90. La d\'emonstration donn\'ee par Abramovich, Karu, Matsuki et W{\l}odarczyk est une r\'eduction au cas torique, nous en donnons les grandes \'etapes. | Factorisation faible des applications birationnelles [d'après
Abramovich, Karu, Matsuki, Włodarczyk et Morelli] | 12,400 |
Monodromy groups, i.e. the groups of isometries of the intersection lattice L_X:=H_2/torsion generated by the monodromy action of all deformation families of a given surface, have been computed in math.AG/0006231 for any minimal elliptic surface with p_g>q=0. New and refined methods are now employed to address the cases of minimal elliptic surfaces with p_g+1>q>0. To this end we find explicit families such that any isometry is in the group generated by their monodromies or does not respect the invariance of the canonical class or the spinor norm. The monodromy action is moreover shown to act by the full symplectic group on the first homology modulo torsion. | Monodromy groups of irregular elliptic surfaces | 12,401 |
It is a classical result (apparently due to Tate) that all elliptic curves with a torsion point of order n ($4 \leq n \leq 10$, or n = 12) lie in a one-parameter family. However, this fact does not appear to have been used ever for computing the torsion of an elliptic curve. We present here a extremely down-to-earth algorithm using the existence of such a family. | Computing the rational torsion of an elliptic curve using Tate normal
form | 12,402 |
We describe smooth rational projective algebraic surfaces over an algebraically closed field of characteristic different from 2 which contain $n \ge \b_2-2$ disjoint smooth rational curves with self-intersection -2, where $\b_2$ is the second Betti number. In the last section this is applied to the study of minimal complex surfaces of general type with $p_g = 0$ and $K^2 = 8, 9$ which admit an automorphism of order 2. | Surfaces with many nodes | 12,403 |
Let $C$ be an isolated plane curve singularity, and $(C,l)$ be a decorated curve. In this article we compare the equisingular deformations of $C$ and the sandwiched singularity $X(C,l)$. We will prove that for $l \gg 0$ the functor of equisingular deformations of $C$ and $(C,l)$ are equivalent. From this we deduce a proof of a formula for the dimension of the equisingular stratum. Furthermore we will show how compute the equisingularity ideal of the curve singularity $C$, given the minimal (good) resolution of $C$. | Equisingular Deformations of Plane Curve and of Sandwiched Singularities | 12,404 |
Let R be a complete discrete valuation ring of mixed characteristics (0,p). Given an order p-automorphism of a formal disc (or annulus) over R, we describe the minimal semi-stable model for which the specialisations of fixed points are distincts and lie in the smooth locus of the special fiber. The description leads to a combinatorial object called Hurwitz tree. Our main result is a necessary and sufficient condition for a Hurwitz tree to arise from an order p-automorphism. | Arbres de Hurwitz et automorphismes d'ordre p des disques et des
couronnes p-adiques formels | 12,405 |
We prove that the dimension 0 part of the tautological ring of the moduli space of stable pointed curves is one-dimensional. This provides the first genus-free evidence for a conjecture of Faber and Pandharipande that the tautological ring of the moduli space is Gorenstein, answering in the affirmative a question of Hain and Looijenga. | On the tautological ring of $\bar{M}_{g,n}$ | 12,406 |
Homotopy Lie groups, recently invented by W.G. Dwyer and C.W. Wilkerson, represent the culmination of a long evolution. The basic philosophy behind the process was formulated almost 25 years ago by Rector in his vision of a homotopy theoretic incarnation of Lie group theory. What was then technically impossible has now become feasible thanks to modern advances such as Miller's proof of the Sullivan conjecture and Lannes's division functors. Today, with Dwyer and Wilkerson's implementation of Rector's vision, the tantalizing classification theorem seems to be within grasp. Supported by motivating examples and clarifying exercises, this guide quickly leads, without ignoring the context or the proof strategy, from classical finite loop spaces to the important definitions and striking results of this new theory. | Homotopy Lie groups | 12,407 |
The study of the action of the Steenrod algebra on the mod $p$ cohomology of spaces has many applications to the topological structure of those spaces. In this paper we present combinatorial formulas for the action of Steenrod operations on the cohomology of Grassmannians, both in the Borel and the Schubert picture. We consider integral lifts of Steenrod operations, which lie in a certain Hopf algebra of differential operators. The latter has been considered recently as a realization of the Landweber-Novikov algebra in complex cobordism theory; it also has connections with the action of the Virasoro algebra on the boson Fock space. Our formulas for Steenrod operations are based on combinatorial methods which have not been used before in this area, namely Hammond operators and the combinatorics of Schur functions. We also discuss several applications of our formulas to the geometry of Grassmannians. | The combinatorics of Steenrod operations on the cohomology of
Grassmannians | 12,408 |
The long hunt for a symmetric monoidal category of spectra finally ended in success with the simultaneous discovery of the third author's discovery of symmetric spectra and the Elmendorf-Kriz-Mandell-May category of S-modules. In this paper we define and study the model category of symmetric spectra, based on simplicial sets and topological spaces. We prove that the category of symmetric spectra is closed symmetric monoidal and that the symmetric monoidal structure is compatible with the model structure. We prove that the model category of symmetric spectra is Quillen equivalent to Bousfield and Friedlander's category of spectra. We show that the monoidal axiom holds, so that we get model categories of ring spectra and modules over a given ring spectrum. | Symmetric spectra | 12,409 |
Symmetric spectra were introduced by Jeff Smith as a symmetric monoidal category of spectra. In this paper, a detection functor is defined which detects stable equivalences of symmetric spectra. This detection functor is useful because the classic stable homotopy groups do not detect stable equivalences in symmetric spectra. One of the advantages of a symmetric monoidal category of spectra is that one can define topological Hochschild homology on ring spectra simply by mimicking the Hochschild complex from algebra. Using the detection functor mentioned above, this definition of topological Hochschild homology is shown to agree with Bokstedt's original definition. In particular, this shows that Bokstedt's definition is correct even for non-connective non-convergent symmetric ring spectra. | Symmetric ring spectra and topological Hochschild homology | 12,410 |
Using Ohkawa's theorem that the collection of Bousfield classes is a set, we perform a number of constructions with Bousfield classes. In particular, we describe a greatest lower bound operator; we also note that a certain subset DL of the Bousfield lattice is a frame, and we examine some consequences of this observation. We make several conjectures about the structure of the Bousfield lattice and DL. In particular, we conjecture that DL is obtained by killing "strange" spectra, such as the Brown-Comenetz dual of the sphere. We introduce a new "Boolean algebra of spectra" cBA, which contains Bousfield's BA and is complete. Our conjectures allow us to identify cBA as being isomorphic to the complete atomic Boolean algebra on {K(n) : n>= 0}, {A(n) : n>= 2}, and HF_p. Our conjectures imply that BA is the subBoolean algebra consisting of finite wedges of the K(n) and A(n), and their complements. | The structure of the Bousfield lattice | 12,411 |
We study phantom maps and homology theories in a stable homotopy category S via a certain Abelian category A. We express the group P(X,Y) of phantom maps X -> Y as an Ext group in A, and give conditions on X or Y which guarantee that it vanishes. We also determine P(X,HB). We show that any composite of two phantom maps is zero, and use this to reduce Margolis's axiomatisation conjecture to an extension problem. We show that a certain functor S -> A is the universal example of a homology theory with values in an AB 5 category and compare this with some results of Freyd. | Phantom Maps and Homology Theories | 12,412 |
A monoidal model category is a model category with a compatible closed monoidal structure. Such things abound in nature; simplicial sets and chain complexes of abelian groups are examples. Given a monoidal model category, one can consider monoids and modules over a given monoid. We would like to be able to study the homotopy theory of these monoids and modules. This question was first addressed by Stefan Schwede and Brooke Shipley in "Algebras and modules in monoidal model categories", who showed that under certain conditions, there are model categories of monoids and of modules over a given monoid. This paper is a follow-up to that one. We study what happens when the conditions of Schwede-Shipley do not hold. This will happen in any topological situation, and in particular, in topological symmetric spectra. We find that, with no conditions on our monoidal model category except that it be cofibrantly generated and that the unit be cofibrant, we still obtain a homotopy category of monoids, and that this homotopy category is homotopy invariant in an appropriate sense. | Monoidal model categories | 12,413 |
The question of whether a given H-space X is, up to homotopy, a loop space has been studied from a variety of viewpoints. Here we address this question from the aspect of homotopy operations, in the classical sense of operations on homotopy groups. First, we show how an H-space structure on X can be used to define the action of the primary homotopy operations on the shifted homotopy groups \pi_{*-1} X (which are isomorphic to \pi_* Y, if X=\Omega\Y. This action will behave properly with respect to composition of operations if X is homotopy-associative, and will lift to a topological action of the monoid of all maps between spheres if and only if X is a loop space. The obstructions to having such a topological action may be formulated in the framework of an obstruction theory for realizing \Pi-algebras, which is simplified here by showing that any (suitable) \Delta-simplicial space may be made into a full simplicial space (a result which may be useful in other contexts). | Loop spaces and homotopy operations | 12,414 |
We compute the 3-primary v1-periodic homotopy groups of the exceptional Lie group E7. Now E8 at the primes 3 and 5 is the only compact simple Lie group whose odd-primary v1-periodic homotopy groups remian to be computed. The main work is computing the unstable Novikov spectral sequence of \Omega E7/Sp(2). Showing that this converges to v1-periodic homotopy groups requires recent work of Bousfield and Bendersky-Thompson. | 3-primary v1-periodic homotopy groups of E7 | 12,415 |
A Lefschetz-type coincidence theorem for two maps f,g:X->Y from an arbitrary topological space X to a manifold Y is given: I(f,g)=L(f,g), the coincidence index is equal to the Lefschetz number. It follows that if L(f,g) is not equal to zero then there is an x in X such that f(x)=g(x). In particular, the theorem contains some well-known coincidence results for (i) X,Y manifolds and (ii) f with acyclic fibers. | A Lefschetz type coincidence theorem | 12,416 |
We describe an alternative approach to some results of Vassiliev on spaces of polynomials, by using the scanning method which was used by Segal in his investigation of spaces of rational functions. We explain how these two approaches are related by the Smale-Hirsch Principle or the h-Principle of Gromov. We obtain several generalizations, which may be of interest in their own right. | Spaces of polynomials with roots of bounded multiplicity | 12,417 |
We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal I of morphisms with certain properties, and that if I has these properties, then so does each of its powers. We show how a projective class leads to an Adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence. We use this to study various ideals. In the stable homotopy category we examine phantom maps, skeletal phantom maps, superphantom maps, and ghosts. (A ghost is a map which induces the zero map of homotopy groups.) We show that ghosts lead to a stable analogue of the Lusternik-Schnirelmann category of a space, and we calculate this stable analogue for low-dimensional real projective spaces. We also give a relation between ghosts and the Hopf and Kervaire invariant problems. In the case of A-infinity modules over an A-infinity ring spectrum, the ghost spectral sequence is a universal coefficient spectral sequence. From the phantom projective class we derive a generalized Milnor sequence for filtered diagrams of finite spectra, and from this it follows that the group of phantom maps from X to Y can always be described as a lim^1 group. The last two sections focus on algebraic examples. In the derived category of an abelian category we study the ideal of maps inducing the zero map of homology groups and find a natural setting for a result of Kelly on the vanishing of composites of such maps. We also explain how pure exact sequences relate to phantom maps in the derived category of a ring and give an example showing that phantoms can compose non-trivially. | Ideals in Triangulated Categories: Phantoms, Ghosts and Skeleta | 12,418 |
This paper discusses the question of how to recognize whether an operad is E_n (ie. equivalent to the little n-cubes operad). A construction is given which produces many new examples of E_n operads. This construction is developed in the context of an infinite family of right adjoint constructions for operads. Some other related constructions of E_n operads, so-called generalized tensor products, are also described. | Constructions of E_n Operads | 12,419 |
The aim of this brief note is mainly to advocate our approach to homotopy algebras based on the minimal model of an operad. Our exposition is motivated by two examples which we discuss very explicitly - the example of strongly homotopy associative algebras and the example of strongly homotopy Lie algebras. We then indicate what must be proved in order to show that these homotopy algebraic structures are really `stable under a homotopy.' The paper is based on a talk given by the author on June 16, 1998, at University of Osnabrueck, Germany. | Homotopy Algebras via Resolutions of Operads | 12,420 |
We construct splittings of some completions of the Z/(p)-Tate cohomology of E(n) and some related spectra. In particular, we split (a completion of) tE(n) as a (completion of) a wedge of E(n-1)'s as a spectrum, where t is shorthand for the fixed points of the Z/(p)-Tate cohomology spectrum (ie Mahowald's inverse limit of P_{-k} smash SE(n)). We also give a multiplicative splitting of tE(n) after a suitable base extension. | Completions of Z/(p)-Tate cohomology of periodic spectra | 12,421 |
This paper proves that the functor $C(*)$ that sends pointed, simply-connected CW-complexes to their chain-complexes equipped with diagonals and iterated higher diagonals, determines their integral homotopy type --- even inducing an equivalence of categories between the category of CW-complexes up to homotopy equivalence and a certain category of chain-complexes equipped with higher diagonals. Consequently, $C(*)$ is an algebraic model for integral homotopy types similar to Quillen's model of rational homotopy types. For finite CW complexes, our model is finitely generated. Our result implies that the geometrically induced diagonal map with all ``higher diagonal'' maps (like those used to define Steenrod operations) collectively determine integral homotopy type. | m-structures determine integral homotopy type | 12,422 |
\newcommand{\rhomi}[1]{\widetilde{H}_{#1}} \newcommand{\rbeti}[1]{\beta_{#1}} \newcommand{\kk}{\mathbf k} \newcommand{\dimk}{\dim_{\kk}} We show that algebraically shifting a pair of simplicial complexes weakly increases their relative homology Betti numbers in every dimension. More precisely, let $\Delta(K)$ denote the algebraically shifted complex of simplicial complex $K$, and let $\rbeti{j}(K,L)=\dimk \rhomi{j}(K,L;\kk)$ be the dimension of the $j$th reduced relative homology group over a field $\kk$ of a pair of simplicial complexes $L \subseteq K$. Then $\rbeti{j}(K,L) \leq \rbeti{j}(\Delta(K),\Delta(L))$ for all $j$. The theorem is motivated by somewhat similar results about Gr\"obner bases and generic initial ideals. Parts of the proof use Gr\"obner basis techniques. | Algebraic Shifting Increases Relative Homology | 12,423 |
By considering labeled configurations of ``bounded multiplicity'', one can construct a functor that fits between homology and stable homotopy. Based on previous work, we are able to give an equivalent description of this labeled construction in terms of loop space functors and symmetric products. This yields a direct generalization of the May-Milgram model for iterated loop spaces, and answers questions of Carlsson and Milgram posed in the handbook. We give a classifying space formulation of our results hence extending an older result of Segal. We finally relate our labeled construction to a theory of Lesh and give a generalization of a well-known theorem of Quillen, Barratt and Priddy. | An Interpolation between Homology and Stable Homotopy | 12,424 |
Let p be a prime. The Smith-Toda complex V(k) is a finite spectrum whose BP-homology is isomorphic to BP_*/(p,v_1,...,v_k). For example, V(-1) is the sphere spectrum and V(0) the mod p Moore spectrum. In this paper we show that if p > 5, then V((p+3)/2) does not exist and V((p+1)/2), if it exists, is not a ring spectrum. The proof uses the new homotopy fixed point spectral sequences of Hopkins and Miller. | On the nonexistence of Smith-Toda complexes | 12,425 |
In the first part, we determine conditions on spectra X and Y under which either every map from X to Y is phantom, or no nonzero maps are. We also address the question of whether such all or nothing behaviour is preserved when X is replaced with V smash X for V finite. In the second part, we introduce chromatic phantom maps. A map is n-phantom if it is null when restricted to finite spectra of type at least n. We define divisibility and finite type conditions which are suitable for studying n-phantom maps. We show that the duality functor W_{n-1} defined by Mahowald and Rezk is the analog of Brown-Comenetz duality for chromatic phantom maps, and give conditions under which the natural map Y --> W_{n-1}^2 Y is an isomorphism. | Phantom maps and chromatic phantom maps | 12,426 |
We define inductively a sequence of purely algebraic invariants - namely, classes in the Quillen cohomology of the Pi-algebra \pi_* X - for distinguishing between different homotopy types of spaces. Another sequence of such cohomology classes allows one to decide whether a given abstract Pi-algebra can be realized as the homotopy Pi-algebra of a space in the first place. The paper is written for a relatively general "resolution model category", so it also applies, for example, to rational homotopy types. | Algebraic invariants for homotopy types | 12,427 |
We consider partial liftings of maps at fibrations and compare the primary obstruction to extend the lifting with the obstruction to extend the lifting as a simple map into the total space. A relation between these two obstructions is proved for the case when the fiber is an Eilenberg-MacLane space. Furthermore it is shown that this result specialises to well known facts about secondary obstructions. | On the relation between lifting obstructions and ordinary obstructions | 12,428 |
Let M be an n-manifold, and let A be a space with a partial sum behaving as an n-fold loop sum. We define the space C(M;A) of configurations in M with summable labels in A via operad theory. Some examples are symmetric products, labelled configuration spaces, and spaces of rational curves. We show that C(I^n,dI^n;A) is an n-fold delooping of C(I^n;A), and for n=1 it is the classifying space by Stasheff. If M is compact, parallelizable, and A is path connected, then C(M;A) is homotopic to the mapping space Map(M,C(I^n,dI^n;A)). | Configuration spaces with summable labels | 12,429 |
In this note, we attack a question posed ten years ago by Tsukiyama about the injectivity of the so- called Forgetable map. We show that we can insert the Forgetable map in an exact sequence and that the problem can be reduced to the computation of the sequence which turns out unexpectedly to be related to the phantom map problem and the famous Halperin conjecture in rational homotopy theory. | Forgetable map and phantom maps | 12,430 |
We prove that strongly homotopy algebras (such as $A_\infty$, $C_\infty$, sh Lie, $B_\infty$, $G_\infty$,...) are homotopically invariant in the category of chain complexes. An important consequence is a rigorous proof that `strongly homotopy structures transfer over chain homotopy equivalences.' | Homotopy Algebras are Homotopy Algebras | 12,431 |
We refer to an action of the group Z/p (p here is an odd prime) on a stably complex manifold as simple if all its fixed submanifolds have the trivial normal bundle. The important particular case of a simple action is an action with only isolated fixed points. The problem of cobordism classification of manifolds with simple action of Z/p was posed by V.M.Buchstaber and S.P.Novikov in 1971. The analogous question of cobordism classification with stricter conditions on Z/p-action was answered by Conner and Floyd. Namely, Conner and Floyd solved the problem in the case of simple actions with identical sets of weights (eigenvalues of the differential of the map corresponding to the generator of Z/p) for all fixed submanifolds of same dimension. However, the general setting of the problem remained unsolved and is the subject of our present paper. We have obtained the description of the set of cobordism classes of stably complex manifolds with simple Z/p-action both in terms of the coefficients of universal formal group law and in terms of the characteristic numbers, which gives the complete solution to the above problem. In particular, this gives a purely cohomological obstruction to the existence of a simple Z/p-action (or an action with isolated fixed points) on a manifold. We also review connections with the Conner-Floyd results and with the well-known Stong-Hattori theorem. | On the cobordism classification of manifolds with Z/p-action | 12,432 |
For a given pair of maps f,g:X->M from an arbitrary topological space to an n-manifold, the Lefschetz homomorphism is a certain graded homomorphism L:H(X)->H(M) of degree (-n). We prove a Lefschetz-type coincidence theorem: if the Lefschetz homomorphism is nontrivial then there is an x in X such that f(x)=g(x). | Lefschetz Coincidence Theory for Maps Between Spaces of Different
Dimensions | 12,433 |
For X a simply-connected finite H-space, there is a Bousfield-Kan spectral sequence which converges to the homotopy of its K-completion. When X=Spin(2n+1), we expect that these homotopy groups equal the v1-periodic homotopy groups in dimension greater than n^2. In this paper, we accomplish two things. (1) We prove that, for any X, the 1-line of this spectral sequence is determined in an explicit way from K-theory and Adams operations. (2) For X=Spin(2n+1), we make an explicit computation of this 1-line. | The 1-line of the K-theory Bousfield-Kan spectral sequence for
Spin(2n+1) | 12,434 |
In this paper we compute homotopical bordism rings $MU^G_*$ for abelian compact Lie groups G, giving explicit generators and relations. The key constructions are operations on equivariant bordism which should play an important role in equivariant stable homotopy theory more generally. The main technique used is localization of the theory by inverting Euler classes. Applications to homotopy theory include analysis of the completion map from $MU^G_*$ to $MU^*(BG)$. Applications to geometry include classification up to cobordism of S^1 actions on stably complex four-manifolds with precisely three fixed points, answering a question of Bott. | Computations of Complex Equivariant Bordism Rings | 12,435 |
In this paper we compute homotopical equivariant bordism for the group ${\bf Z/2}$, namely $MO^{\bf Z/2}$, geometric equivariant bordism $\Omega^{\bf Z/2}_*$, and their quotient as modules over geometric bordism. This quotient is a module of stable transversality obstructions. In doing these computations, we use the techniques the author developed in the complex setting. Because we are working in the real setting only with Z/2, these techniques simplify greatly. | Transversality Obstructions and Equivariant Bordism for G=Z/2 | 12,436 |
Equivariant elliptic cohomology with complex coefficients was defined axiomatically by Ginzburg, Kapranov and Vasserot and constructed by Grojnowski. We give an invariant definition of S^1-equivariant elliptic cohomology, and use it to give an entirely cohomological proof of the rigidity theorem of Witten for the elliptic genus. We also state and prove a rigidity theorem for families of elliptic genera. | Equivariant Elliptic Cohomology and Rigidity | 12,437 |
We describe algebraic obstruction theories for realizing an abstract coalgebra K_* over the mod p Steenrod algebra as the homology of a topological space, and for distinguishing between the p-homotopy types of different realizations. The theories are expressed in terms of the Quillen cohomology of K_*. | Realizing coalgebras over the Steenrod algebra | 12,438 |
A smooth closed 3-manifold $M$ fibered by tori $T^2$ is characterized by an element $\phi \in GL(2,\mathbb{Z})$. We show that $M$ is the boundary of a 4-manifold fibered by tori over a surface such that the bundle structure on $M$ is the restriction of the bundle structure on the 4-manifold if and only if $\phi$ is from the commutator subgroup $(GL(2,\mathbb{Z}))'$. The notions of oriented and unoriented cobordisms in the class of closed 3-manifolds fibered by tori are introduced. It turns out that in this case the cobordisms form a group, namely $\mathbb{Z}_{12}$ in the oriented case and $\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ in the unoriented one. When the surface on the base of oriented cobordism is orientable, it is shown that its minimal genus can be calculated by Culler's algorithm. | The toric cobordisms | 12,439 |
We explain the essence of perturbation problems. The key to understanding is the structure of chain homotopy equivalence -- the standard one must be replaced by a finer notion which we call a strong chain homotopy equivalence. We prove an Ideal Perturbation Lemma and show how both new and classical results follow from this ideal statement. | Ideal Perturbation Lemma | 12,440 |
The theory of indices of Morse--Bott vector fields on a manifold is constructed and the famous localization problem for the transfer map is solved on its base in the present paper. As a consequence, we obtained addition theorems for the universal Pontrjagin characteristic classes in cobordisms. These results gave us a possibility to complete the construction, which was begun more than twenty years ago, of the universal characteristic classes' theory. | An Index of an Equivariant Vector Field and Addition Theorems for
Pontrjagin Characteristic Classes | 12,441 |
The complex Lie group GO(2n,C) by definition consists of all complex matrices A of size 2n, such that A times transpose(A) is a non-zero scalar. In this paper we determine explicitly the singular cohomology ring of the classifying space BGO(2n,C) with mod 2 coefficients, in terms of generators and relations. The method consists of analysing a certain derivation on the cohomology ring of BO(2n) (which is a polynomial ring in the Stiefel-Whitney classes) via a Koszul complex, and using this to `solve' the Gysin sequence for the bundle BO(2n) over BGO(2n,C). | Characteristic Classes for GO(2n,C) | 12,442 |
This paper proves that the homotopy type of a pointed, simply-connected, 2-reduced simplicial set is determined by the chain-complex augmented by functorial diagonal and higher diagonal maps (a simple generalization of the ones used to define Steenrod operations). The treatment of this problem is completely self-contained, and includes material that simplifies, extends, and corrects material from the authors AMS Memoir, "Iterating the cobar construction". | Operads and algebraic homotopy | 12,443 |
Let D_n be the Dickson invariant ring of F_2[X_1,...,X_n] acted by the general linear group GL(n,\F_2). In this paper, we provide an elementary proof of the conjecture by [Hung]: each element in D_n is in the image of the Steenrod square in F_2[X_1,...,X_n], where n>3. | Dickson Invariants in the image of the Steenrod Square | 12,444 |
This work is devoted to the investigation of the problem about inverse mapping systems expansions of ultrauniform spaces $X$ using polyhedra over non-Archimedean locally compact fields $\bf L$. Theorems about expansions of complete ultrametric and ultrauniform spaces are proved. Absolute polyhedral expansions and inverse mapping systems of expansions for non-complete spaces are investigated. This article also contains results about a relation of $dim (X)$ and dimensions of polyhedra over $\bf L$. | Absolute non-Archimedean polyhedral expansions of ultrauniform spaces | 12,445 |
In this paper we study various simplicial complexes associated to the commutative structure of a finite group G. We define NC(G) (resp. C(G)) as the complex associated to the poset of pairwise non-commuting (resp. commuting) sets in G. We observe that NC(G) has only one positive dimensional connected component, which we call BNC(G), and we prove that BNC(G) is simply connected. Our main result is a simplicial decomposition formula for BNC(G) which follows from a result of A. Bjorner, M. Wachs and V. Welker on inflated simplicial complexes. As a corollary, we obtain that if G has a nontrivial center or if G has odd order, then the homology group H_{n-1}(BNC(G)) is nontrivial for every n such that G has a maximal noncommuting set of order n. | On commuting and non-commuting complexes | 12,446 |
We show that every combinatorial model category can be obtained, up to Quillen equivalence, by localizing a model category of diagrams of simplicial sets. This says that any combinatorial model category can be built up from a category of `generators' and a set of `relations'---that is, any combinatorial model category has a presentation. | Combinatorial model categories have presentations | 12,447 |
We construct a Thom class in complex equivariant elliptic cohomology extending the equivariant Witten genus. This gives a new proof of the rigidity of the Witten genus, which exhibits a close relationship to recent work on non-equivariant orientations of elliptic spectra. | The Witten genus and equivariant elliptic cohomology | 12,448 |
Let X be a simply connected space and k a commutative ring. Goodwillie, Burghelea and Fiedorowiscz proved that the Hochschild cohomology of the singular chains on the pointed loop space HH^{*}S_*(\Omega X) is isomorphic to the free loop space cohomology H^{*}(X^{S^{1}}). We proved that this isomorphism is compatible with both the cup product on HH^{*}S_*(\Omega X) and on H^{*}(X^{S^{1}}). In particular, we explicit the algebra H^{*}(X^{S^{1}}) when X is a suspended space, a complex projective space or a finite CW-complex of dimension p such that \frac {1}{(p-1)!}\in k. | The cohomology ring of free loop spaces | 12,449 |
The singular chain complex of the iterated loop space is expressed in terms of the cobar construction. After that we consider the spectral sequence of the cobar construction and calculate its first term over Z/p-coefficients and over a field of characteristic zero. Finally we apply these results to calculate the homology of the iterated loop spaces of the stunted real and complex projective spaces. In the Appendix, written by F.Sergeraert there are considered computer methods for calculations of the homology of iterated loop spaces. | The homology of iterated loop spaces | 12,450 |
Denote by E(Y) the group of homotopy classes of self-homotopy equivalences of a finite-dimensional complex Y. We give a selection of results about certain subgroups of E(Y). We establish a connection between the Gottlieb groups of Y and the subgroup of E(Y) consisting of homotopy classes of self-homotopy equivalences that fix homotopy groups through the dimension of Y, denoted by E_#(Y). We give an upper bound for the solvability class of E_#(Y) in terms of a cone decomposition of Y. We dualize the latter result to obtain an upper bound for the solvability class of the subgroup of E(Y) consisting of homotopy classes of self-homotopy equivalences that fix cohomology groups with various coefficients. We also show that with integer coefficients, the latter group is nilpotent. | Subgroups of the group of self-homotopy equivalences | 12,451 |
Halperin has conjectured that the Serre spectral sequence of any fibration that has fibre space a certain kind of elliptic space should collapse at the E_2-term. In this paper we obtain an equivalent phrasing of this conjecture, in terms of formality relations between base and total spaces in such a fibration (Theorem 3.4). Also, we obtain results on relations between various numericalinvariants of the base, total and fibre spaces in these fibrations. Some of our results give weak versions of Halperin's conjecture (Remark 4.4 and Corollary 4.5). We go on to establish some of these weakened forms of the conjecture (Theorem 4.7). In the last section, we discuss extensions of our results and suggest some possibilities for future work. | Variations on a conjecture of Halperin | 12,452 |
We develop an obstruction theory for homotopy of homomorphisms f,g : M -> N between minimal differential graded algebras. We assume that M = Lambda V has an obstruction decomposition given by V = V_0 oplus V_1 and that f and g are homotopic on Lambda V_0. An obstruction is then obtained as a vector space homomorphism V_1 -> H^*(N). We investigate the relationship between the condition that f and g are homotopic and the condition that the obstruction is zero. The obstruction theory is then applied to study the set of homotopy classes [M, N]. This enables us to give a fairly complete answer to a conjecture of Copeland-Shar on the size of the homotopy set [A,B] when A and B are rational spaces. In addition, we give examples of minimal algebras (and hence of rational spaces) that have few homotopy classes of self-maps. | Rational obstruction theory and rational homotopy sets | 12,453 |
The Adams spectral sequence was invented by J.F.Adams almost fifty years ago for calculations of stable homotopy groups of topological spaces and in particular of spheres. The calculation of differentials of this spectral sequence is one of the most difficult problem of Algebraic Topology. Here we consider an approach to solve this problem in the case of Z/2 coefficients and find inductive formulas for the differentials. It is based on the Stasheff algebra structures, operad methods and functional homology operations. | Stasheff structures and differentials of the Adams spectral sequence | 12,454 |
Let C_*(K) denote the cellular chains on the Stasheff associahedra. We construct an explicit combinatorial diagonal \Delta : C_*(K) --> C_*(K) \otimes C_*(K); consequently, we obtain an explicit diagonal on the A_\infty-operad. We apply the diagonal \Delta to define the tensor product of A_\infty-(co)algebras in maximal generality. | A Diagonal on the Associahedra | 12,455 |
We define an inner product (suitably interpreted) on the K(n)-local spectrum LG := L_{K(n)}BG_+, where G is a finite group or groupoid. This gives an inner product on E^*BG_+ for suitable K(n)-local ring spectra E. We relate this to the usual inner product on the representation ring when n=1, and to the Hopkins-Kuhn-Ravenel generalised character theory. We show that LG is a Frobenius algebra object in the K(n)-local stable category, and we recall the connection between Frobenius algebras and topological quantum field theories to help analyse this structure. In many places we find it convenient to use groupoids rather than groups, and to assist with this we include a detailed treatment of the homotopy theory of groupoids. We also explain some striking formal similarities between our duality and Atiyah-Poincare duality for manifolds. | K(n)-local duality for finite groups and groupoids | 12,456 |
In this paper we study E^*BV_k, where E=BP<m,n> is a cohomology theory with coefficient ring F_p[v_m,...,v_n] (if m>0) or Z_(p)[v_1,...,v_n] (if m=0). We use ideas from the theory of multiple level structures, developed in earlier work of the author with John Greenlees. Our results apply when k is less than or equal to w=n+1-m. If k<w we find that E^*BV_k has no v_m-torsion. When k=w, we show that the v_m-torsion is annihilated by the ideal I_{n+1}=(v_m,...,v_n), and that it is a free module on one generator over the ring F_p[[x_0,...,x_{w-1}]]. We give three very different formulae for this generator; it is not at all obvious that these give the same element, and we only have a rather indirect proof of this. | The BP<n> cohomology of elementary abelian groups | 12,457 |
We set up a framework for using algebraic geometry to study the generalised cohomology rings that occur in algebraic topology. This idea was probably first introduced by Quillen and it underlies much of our understanding of complex oriented cohomology theories, exemplified by the work of Morava. Most of the results have close and well-known analogues in the algebro-geometric literature, but with different definitions or technical assumptions that are often inconvenient for topological applications. We merely put everything together in a systematic and convenient way. | Formal schemes and formal groups | 12,458 |
Let V_0 and V_1 be complex vector bundles over a space X. We use the theory of divisors on formal groups to give obstructions in generalised cohomology that vanish when V_0 and V_1 can be embedded in a bundle U in such a way that V_0\cap V_1 has dimension at least k everywhere. We study various algebraic universal examples related to this question, and show that they arise from the generalised cohomology of corresponding topological universal examples. This extends and reinterprets earlier work on degeneracy classes in ordinary cohomology or intersection theory. | Common subbundles and intersections of divisors | 12,459 |
We compute the mod two homology Hopf rings of the spectra KO and KU. The spaces in these spectra are the infinite classical groups and their coset spaces, and their homology was first calculated in the Cartan seminars, but the Hopf ring structure was first determined in the second author's unpublished PhD thesis. The presentation given here serves as an introduction to the first author's much more intricate work on the connective spectrum bo. The Hopf ring viewpoint turns out to be very convenient for understanding the homological effect of various maps between classical groups and fibrations of their connective covers. | The Hopf Rings for KO and KU | 12,460 |
We relate the brace products of a fibration with section to the differentials in its serre spectral sequence. In the particular case of free loop fibrations, we establish a link between these differentials and browder operations in the fiber. Applications and several calculations (for the particular case of spheres and wedges of spheres) are given. | On the Topology of Fibrations with Section and Free Loop Spaces | 12,461 |
In this paper we study the global structure of the stable homotopy theory of spectra. We establish criteria for when the homotopy theory associated to a given stable model category agrees with the classical stable homotopy theory of spectra. One sufficient condition is that the associated homotopy category is equivalent to the stable homotopy category as a triangulated category with an action of the ring of stable homotopy groups of spheres. In other words, the classical stable homotopy theory, with all of its higher order information, is determined by the homotopy category as a triangulated category with an action of the stable homotopy groups of spheres. Another sufficient condition is the existence of a small generating object (corresponding to the sphere spectrum) for which a specific `unit map' from the infinite loop space QS^0 to the endomorphism space is a weak equivalence. | A uniqueness theorem for stable homotopy theory | 12,462 |
We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work of this author with Schwede on the uniqueness of models of the stable homotopy theory of spectra. As an application we show that with an added assumption about underlying model structures Margolis' axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence. Also, the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra (S-modules, \W-spaces, orthogonal spectra, simplicial functors) with symmetric spectra. The equivalences of modules, algebras and commutative algebras in these categories are also considered. | Monoidal uniqueness theorems for stable homotopy theory | 12,463 |
The Bass-Heller-Swan-Farrell-Hsiang-Siebenmann decomposition of the Whitehead group $K_1(A_{\rho}[z,z^{-1}])$ of a twisted Laurent polynomial extension $A_{\rho}[z,z^{-1}]$ of a ring $A$ is generalized to a decomposition of the Whitehead group $K_1(A_{\rho}((z)))$ of a twisted Novikov ring of power series $A_{\rho}((z))=A_{\rho}[[z]][z^{-1}]$. The decomposition involves a summand $W_1(A,\rho)$ which is an abelian quotient of the multiplicative group $W(A,\rho)$ of Witt vectors $1+a_1z+a_2z^2+... \in A_{\rho}[[z]]$. An example is constructed to show that in general the natural surjection $W(A,\rho)^{ab} \to W_1(A,\rho)$ is not an isomorphism. | The Whitehead group of the Novikov ring | 12,464 |
This is a survey paper on spaces of automorphisms of manifolds and spaces of manifolds in a fixed homotopy type. It describes the main theorems of traditional surgery theory, but also the main theorems of pseudoisotopy theory, alias concordance theory, Waldhausen style. It culminates in (an outline of) a synthesis of these two theories, producing algebraic models, valid in a stable range, for spaces of manifolds in a fixed homotopy type. This is inspired by earlier work of Burghelea-Lashof and Hatcher. The algebraic models are a mix of algebraic L-theory and algebraic K-theory. | Automorphisms of manifolds | 12,465 |
In this paper we show the existence of non minimal critical points of the Yang-Mills functional over a certain family of 4-manifolds with generic SU(2)-invariant metrics using Morse and homotopy theoretic methods. These manifolds are acted on fixed point freely by the Lie group SU(2) with quotient a compact Riemann surface of even genus. We use a version of invariant Morse theory for the Yang-Mills functional used by Parker and by Rade. | Morse theory for the Yang-Mills functional via equivariant homotopy
theory | 12,466 |
We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits. We also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial, or `continuous', functors. This is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial Quillen equivalence. | Simplicial structures on model categories and functors | 12,467 |
Let $F\hookrightarrow E\twoheadrightarrow B$ be a fibration whose base space $B$ is a finite simply-connected CW-complex of dimension $\leq p$ and whose total space $E$ is a path-connected CW-complex of dimension $\leq p-1$. If $\alpha\in H^{+}(F;\mathbb{F}_p)$ then $\alpha ^{p}=0$. | P-th powers in mod p cohomology of fibers | 12,468 |
This paper is the supplement to the section 2 of the paper "Floating bundles and their applications" (math.AT/0102054). Below we construct the denumerable set of extensions of the formal group of geometric cobordisms $F(x\otimes 1,1\otimes x)$ by the Hopf algebra $H=\Omega_U^*(Gr).$ | Supplement to the paper "Floating bundles and their applications" | 12,469 |
In [math.AT/9907138] we proved that strongly homotopy algebras are homotopy invariant concepts in the category of chain complexes. Our arguments were based on the fact that strongly homotopy algebras are algebras over minimal cofibrant operads and on the principle that algebras over cofibrant operads are homotopy invariant. In our approach, algebraic models for colored operads describing diagrams of homomorphisms played an important role. The aim of this paper is to give an explicit description of these models. A possible application is an appropriate formulation of the `ideal' homological perturbation lemma for chain complexes with algebraic structures. Our results also provide a conceptual approach to `homotopies through homomorphism' for strongly homotopy algebras. We also argue that strongly homotopy algebras form a honest (not only weak Kan) category. The paper is a continuation of our program to translate the famous book "M. Boardman, R. Vogt: Homotopy Invariant Algebraic Structures on Topological Spaces" to algebra. | Homotopy Diagrams of Algebras | 12,470 |
Let \Omega X be the space of Moore loops on a finite, q-connected, n-dimensional CW complex X, and let R be a subring of Q containing 1/2. Let p(R) be the least non-invertible prime in R. For a graded R-module M of finite type, let FM = M / Torsion M. We show that the inclusion of the sub-Lie algebra P of primitive elements of FH_*(\Omega X;R) induces an isomorphism of Hopf algebras UP = FH_*(\Omega X;R), provided p(R) > n/q - 1. Furthermore, the Hurewicz homomorphism induces an embedding of F(\pi_*(\Omega X)\otimes R) in P, with torsion cokernel. As a corollary, if X is elliptic, then FH_*(\Omega X;R) is a finitely-generated R-algebra. | A Torsion-Free Milnor-Moore Theorem | 12,471 |
We compute the Brown-Peterson cohomology of QX, the free infinite loop-space on X, when X is a space whose Morava K-theory is flat over its BP-cohomology, in particular a space whose Morava K-theory is concentrated in even degrees. Our computation is in terms of a destabilization functor for BP-cohomology. We also show that for such X, the Morava K-homology of QX is a free commutative algebra. | On Brown-Peterson cohomology of QX | 12,472 |
This paper is the supplement to the section 2 of the paper "Floating bundles and their applications" (math.AT/0102054). Below we study some properties of category, connected with cobordism rings of FBSP. In particular, we shall show that it is the tensor category. | Supplement 2 to the paper "Floating bundles and their applications" | 12,473 |
The aim of this paper is to prove the following result. For any commutative formal group ${\frak F}(x\otimes 1,1\otimes x),$ which is considered as a formal group over $H_\mathbb{Q},$ there exists a homomorphism to a formal group of the form ${\frak c}+x\otimes 1+1\otimes x,$ where $\frak c\in H_\mathbb{Q}{\mathop{\hat{\otimes}} \limits_{R_\mathbb{Q}}}H_\mathbb{Q}$ such that $(\id \otimes \epsilon){\frak c}=0= (\epsilon \otimes \id){\frak c}.$ | Logarithms of formal groups over Hopf algebras | 12,474 |
In this paper we examine certain filtrations of topological Hochschild homology and topological cyclic homology. As an example we show how the filtration with respect to a nilpotent ideal gives rise to an analog of a theorem of Goodwillie saying that rationally relative K-theory and relative cyclic homology agree. Our variation says that the p-torsion parts agree in a range of degrees. We use it to compute K_i(Z/p^m) for i < p-2. | Filtered Topological Cyclic Homology and relative K-theory of nilpotent
ideals | 12,475 |
We introduce Clapp-Puppe type generalized Lusternik-Schnirelmann (co)category in a Quillen model category. We establish some of their basic properties and give various characterizations of them. As the first application of these characterizations, we show that our generalized (co)category is invariant under Quillen modelization equivalences. In particular, generalized (co)category of spaces and simplicial sets coincide. Another application of these characterizations is to define and study rational cocategory. Various other applications are also given. | Clapp-Puppe Type Lusternik-Schnirelmann (Co)category in a Model Category | 12,476 |
We discuss the Adams Spectral Sequence for R-modules based on commutative localized regular quotient ring spectra over a commutative S-algebra R in the sense of Elmendorf, Kriz, Mandell, May and Strickland. The formulation of this spectral sequence is similar to the classical case and the calculation of its E_2-term involves the cohomology of certain `brave new Hopf algebroids' E^R_*E. In working out the details we resurrect Adams' original approach to Universal Coefficient Spectral Sequences for modules over an R ring spectrum. We show that the Adams Spectral Sequence for S_R based on a commutative localized regular quotient R ring spectrum E=R/I[X^{-1}] converges to the homotopy of the E-nilpotent completion pi_*hat{L}^R_ES_R=R_*[X^{-1}]^hat_{I_*}. We also show that when the generating regular sequence of I_* is finite, hatL^R_ES_R is equivalent to L^R_ES_R, the Bousfield localization of S_R with respect to E-theory. The spectral sequence here collapses at its E_2-term but it does not have a vanishing line because of the presence of polynomial generators of positive cohomological degree. Thus only one of Bousfield's two standard convergence criteria applies here even though we have this equivalence. The details involve the construction of an I-adic tower R/I <-- R/I^2 <-- ... <-- R/I^s <-- R/I^{s+1} <-- ... whose homotopy limit is hatL^R_ES_R. We describe some examples for the motivating case R=MU. | On the Adams Spectral Sequence for R-modules | 12,477 |
Sufficient conditions on a space are given which guarantee that the $K$-theory ring and the ordinary cohomology ring with coefficients over a principal ideal domain are invariants of, respectively, the adic genus and the SNT set. An independent proof of Notbohm's theorem on the classification of the adic genus of $BS^3$ by $KO$-theory $\lambda$-rings is given. An immediate consequence of these results about adic genus is that the power series ring $\mathbf{Z} \lbrack \lbrack x \rbrack \rbrack$ admits uncountably many pairwise non-isomorphic $\lambda$-ring structures. | On adic genus, Postnikov conjugates, and lambda-rings | 12,478 |
We study the category pro-SSet of pro-simplicial sets, which arises in etale homotopy theory, shape theory, and pro-finite completion. We establish a model structure on pro-SSet so that it is possible to do homotopy theory in this category. This model structure is closely related to the strict structure of Edwards and Hastings. In order to understand the notion of homotopy groups for pro-spaces we use local systems on pro-spaces. We also give several alternative descriptions of weak equivalences, including a cohomological characterization. We outline dual constructions for ind-spaces. | A model structure on the category of pro-simplicial sets | 12,479 |
In this paper we develop an axiomatic approach to coarse homology theories. We prove a uniqueness result concerning coarse homology theories on the category of `coarse CW-complexes'. This uniqueness result is used to prove a version of the coarse Baum-Connes conjecture for such spaces. | Coarse homology theories | 12,480 |
If F is a collection of topological spaces, then a homotopy class \alpha in [X,Y] is called F-trivial if \alpha_* = 0: [A,X] --> [A,Y] for all A in F. In this paper we study the collection Z_F(X,Y) of all F-trivial homotopy classes in [X,Y] when F = S, the collection of spheres, F = M, the collection of Moore spaces, and F = \Sigma, the collection of suspensions. Clearly Z_\Sigma (X,Y) \subset Z_M(X,Y) \subset Z_S(X,Y), and we find examples of finite complexes X and Y for which these inclusions are strict. We are also interested in Z_F(X) = Z_F(X,X), which under composition has the structure of a semigroup with zero. We show that if X is a finite dimensional complex and F = S, M or \Sigma, then the semigroup Z_F(X) is nilpotent. More precisely, the nilpotency of Z_F(X) is bounded above by the F-killing length of X, a new numerical invariant which equals the number of steps it takes to make X contractible by successively attaching cones on wedges of spaces in F, and this in turn is bounded above by the F-cone length of X. We then calculate or estimate the nilpotency of Z_F(X) when F = S, M or \Sigma for the following classes of spaces: (1) projective spaces (2) certain Lie groups such as SU(n) and Sp(n). The paper concludes with several open problems. | Homotopy classes that are trivial mod F | 12,481 |
It is shown that only countably many spaces in the genus of $\hpinfty$, the infinite quaternionic projective space, can admit essential maps from $\cpinfty$, the infinite complex projective space. Examples of countably many homotopically distinct spaces in the genus of $\hpinfty$ which admit essential maps from $\cpinfty$ are constructed. These results strengthen a theorem of McGibbon and Rector which states that among the uncountably many homotopy types in its genus, $\hpinfty$ is the only one which admits a maximal torus. | A remark on the genus of the infinite quaternionic projective space | 12,482 |
In this paper we define a secondary Brown-Kervaire quadratic forms. Among the applications we obtain a complete classification of (n-2)-connected 2n-dimensional framed manifolds up to homeomorphism and homotopy equivalence, . In particular, we prove that the homotopy type of such manifolds determine their homeomorphism type. | Secondary Brown-Kervaire Quadratic forms and $π$-manifolds | 12,483 |
A successful generalization of phantom map theory to the equivariant case for all compact Lie groups is obtained in this paper. One of the key observations is the discovery of the fact that homotopy fiber of equivariant completion splits as product of equivariant Eilenberg-Maclane spaces which seems impossible at first sight by the example of Triantafillou. | Equivariant Phantom maps | 12,484 |
In our previous work, a relation between Tsukiyama problem about self homotopy equivalence was found by using a generalization of phantom map. In this note, fundamental result is established for such a generalization. This is the first time one can deal with phantom maps to space not satisfying finite type condition. Application to Forgetful map is also discussed briefly. | Phantom elements and its Applications | 12,485 |
The paper containes a proof that the mapping class group of the manifold $S^3\times S^3$ is isomorphic to a central extension of the (full) Jacobi group $\Gamma^J$ by the group of 7-dimensional homotopy spheres. Using a presentation of the group $\Gamma^J$ and the $\mu$-invariant of the homotopy spheres, we give a presentation of the mapping class group of $S^3\times S^3$ with generators and defining relations. We also compute cohomology of the group $\Gamma^J$ and determine a 2-cocycle that corresponds to the mapping class group of $S^3\times S^3$. | On the Jacobi Group and the Mapping Class Group of $S^3\times S^3$ | 12,486 |
The Adams spectral sequence was invented by J.F.Adams fifty years ago for calculations of stable homotopy groups of topological spaces and in particular of spheres. The calculation of differentials of this spectral sequence is one of the most difficult problem of Algebraic Topology. Here we consider an approach to find inductive formulas for the differentials. It is based on the A_\infty-structures, E_\infty-structures and functional homology operations. As This approach it will be applied to the Kervaire invariant problem. | E_\infty-structure and differentials of the Adams spectral sequence | 12,487 |
This paper explicitely constructs cofree coalgebras over operads in the category of DG-modules. Special cases are considered in which the general expression simplifies (such as the pointed, irreducible case). It is shown that the existence of an operad-action on a coalgebra implies a ``generalized coassociativity'' that facilitates the construction. Some areas of application are discussed. | Cofree coalgebras over operads | 12,488 |
In this paper the Hochschild-cochain-complex of an A-infinity-algebra A with values in an A-infinity-bimodule M over A and maps between them is defined. Then, an infinity-inner-product on A is defined to be an A-infinity-bimodule-map between A and its dual A*. There is a graph-complex associated to A-infinity-algebras with infinity-inner-product. | Infinity-Inner-Products on A-Infinity-Algebras | 12,489 |
Greenlees defined an abelian category A whose derived category is equivalent to the rational S^1-equivariant stable homotopy category whose objects represent rational S^1-equivariant cohomology theories. We show that in fact the model category of differential graded objects in A (dgA) models the whole rational S^1-equivariant stable homotopy theory. That is, we show that there is a Quillen equivalence between dgA and the model category of rational S^1-equivariant spectra, before the quasi-isomorphisms or stable equivalences have been inverted. This implies that all of the higher order structures such as mapping spaces, function spectra and homotopy (co)limits are reflected in the algebraic model. The new ingredients here are certain Massey product calculations and the work on rational stable model categories from "Classification of stable model categories" and "Equivalences of monoidal model categories" with Schwede. In an appendix we show that Toda brackets, and hence also Massey products, are determined by the triangulated derived category. | An algebraic model for rational S^1-equivariant stable homotopy theory | 12,490 |
A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent `the same homotopy theory'. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a `ring spectrum with several objects', i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard's work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the Eilenberg-Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for a ring R. | Classification of stable model categories | 12,491 |
We show that C if is a proper model category, then the pro-category pro-C has a strict model structure in which the weak equivalences are the levelwise weak equivalences. The strict model structure is the starting point for many homotopy theories of pro-objects. | Strict model structures for pro-categories | 12,492 |
We study a family of subrings, indexed by the natural numbers, of the mod-p cohomology of a finite group G. These subrings are based on a family of v_n-periodic complex oriented cohomology theories and are constructed as rings of generalised characteristic classes. We identify the varieties associated to these subrings in terms of colimits over categories of elementary abelian subgroups of G, naturally interpolating between the work of Quillen on var(H^*(BG)), the variety of the whole cohomology ring, and that of Green and Leary on the variety of the Chern subring, var(Ch(G)). Our subrings give rise to a "chromatic" (co)filtration, which has both topological and algebraic definitions, of var(H^*(BG)) whose final quotient is the variety var(Ch(G)). | Chromatic characteristic classes in ordinary group cohomology | 12,493 |
Let Map_T(K,X) denote the mapping space of continuous based functions between two based spaces K and X. If K is a fixed finite complex, Greg Arone has recently given an explicit model for the Goodwillie tower of the functor sending a space X to the suspension spectrum \Sigma^\infty Map_T(K,X). Applying a generalized homology theory h_* to this tower yields a spectral sequence, and this will converge strongly to h_*(Map_T(K,X)) under suitable conditions, e.g. if h_* is connective and X is at least dim K connected. Even when the convergence is more problematic, it appears the spectral sequence can still shed considerable light on h_*(Map_T(K,X)). Similar comments hold when a cohomology theory is applied. In this paper we study how various important natural constructions on mapping spaces induce extra structure on the towers. This leads to useful interesting additional structure in the associated spectral sequences. For example, the diagonal on Map_T(K,X) induces a `diagonal' on the associated tower. After applying any cohomology theory with products h^*, the resulting spectral sequence is then a spectral sequence of differential graded algebras. The product on the E_\infty -term corresponds to the cup product in h^*(Map_T(K,X)) in the usual way, and the product on the E_1-term is described in terms of group theoretic transfers. We use explicit equivariant S-duality maps to show that, when K is the sphere S^n, our constructions at the fiber level have descriptions in terms of the Boardman-Vogt little n-cubes spaces. We are then able to identify, in a computationally useful way, the Goodwillie tower of the functor from spectra to spectra sending a spectrum X to \Sigma ^\infty \Omega ^\infty X. | Product and other fine structure in polynomial resolutions of mapping
spaces | 12,494 |
The space of unordered configurations of distinct points in the plane is aspherical, with Artin's braid group as its fundamental group. Remarkably enough, the space of ordered configurations of distinct points on the real projective line, modulo projective equivalence, has a natural compactification (as a space of equivalence classes of trees) which is also (by a theorem of Davis, Januszkiewicz, and Scott) aspherical. The classical braid groups are ubiquitous in modern mathematics, with applications from the theory of operads to the study of the Galois group of the rationals. The fundamental groups of these new configuration spaces are not braid groups, but they have many similar formal properties. | Braids, trees, and operads | 12,495 |
The aim of this paper is to explain how, through the work of a number of people, some algebraic structures related to groupoids have yielded algebraic descriptions of homotopy n-types. Further, these descriptions are explicit, and in some cases completely computable, in a way not possible with the traditional Postnikov systems, or with other models, such as simplicial groups. | Computing Homotopy Types Using Crossed N-Cubes of Groups | 12,496 |
If C=C_\phi denotes the mapping cone of an essential phantom map \phi from the suspension of the Eilenberg-Mac Lane complex K=K(Z,5) to the 4-sphere S=S^4 we derive the following properties: (1) The LS category of the product of C with any n-sphere S^n is equal to 3; (2) The LS category of the product of C with itself is equal to 3, hence is strictly less than twice the LS category of C. These properties came to light in the course of an unsuccessful attempt to find, for each positive integer m, an example of a pair of 1-connected CW-complexes of finite type in the same Mislin (localization) genus with LS categories m and 2m. If \phi is such that its p-localizations are inessential for all primes p, then by the main result of [J. Roitberg, The Lusternik-Schnirelmann category of certain infinite CW-complexes, Topology 39 (2000), 95-101], the pair C_*, C where C_*= S wedge \Sigma ^2 K, provides such an example in the case m=1. | The product formula for Lusternik-Schnirelmann category | 12,497 |
A classical E-infinity operad is formed by the bar construction of the symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the context of simplicial sets in order to have an analogue of the Milnor FK-construction for infinite loop spaces. The purpose of this article is to prove that the associative algebra structure on the normalized cochain complex of a simplicial set extends to the structure of an algebra over the Barratt-Eccles operad. We also prove that differential graded algebras over the Barratt-Eccles operad form a closed model category. Similar results hold for the normalized Hochschild cochain complex of an associative algebra. More precisely, the Hochschild cochain complex is acted on by a suboperad of the Barratt-Eccles operad which is equivalent to the classical little squares operad. | Combinatorial operad actions on cochains | 12,498 |
An algebraic theory $T$ is a category with objects $t_0,t_2...$ such that for each $n$ the object $t_n$ is an $n$-fold categorical product of $t_1$. A strict $T$-algebra is a product preserving functor $A: T\to Spaces$. Lawvere showed that for a suitable choice of T giving such an algebra amounts to providing the space $A(t_1)$ with a familiar structure of a monoid group, ring, Lie algebra... Given a functor $X: T\to Spaces$ which preserves products up to a weak equivalence we show that $X$ is more or less canonically weakly equivalent to a strict $T$-algebra $LX$. Thus any `homotopy' algebraic structure on the space $X(t_1)$ can be rigidified to a strict algebraic structure on a space weakly equivalent to $X(t_1)$. This fact can be interpreted as a generalization of the results establishing equivalence of homotopy theories of loop spaces and simplicial groups, products of Eilenberg-Mac Lane spaces and abelian monoids etc. | Algebraic theories in homotopy theory | 12,499 |
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