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Let G be the fundamental group of the complement of a K(G,1) hyperplane arrangement (such as Artin's pure braid group) or more generally a homologically toroidal group (as defined in the paper). The subgroup of elements in the complex K-theory of BG which arises from complex unitary representations of G is shown to be trivial. In the case of real K-theory, this subgroup is an elementary abelian 2-group, which is characterized completely in terms of the first two Stiefel-Whitney classes of the representation. Furthermore, an orthogonal representation of G gives rise to a trivial bundle if and only if the representation factors through the spinor groups. In addition, quadratic relations in the cohomology algebra of the pure braid groups which correspond precisely to the Jacobi identity for certain choices of Poisson algebras are shown to give the existence of certain homomorphisms from the pure braid group to generalized Heisenberg groups. These cohomology relations correspond to non-trivial Spin representations of the pure braid groups which give rise to trivial bundles. | On representations and K-theory of the braid groups | 12,500 |
Paper written in French -- English abstract: This paper proves a particular case of a conjecture of N. Kuhn. This conjecture is as follows. Consider the Gabriel-Krull filtration of the category U of unstable modules. Let U_n, n>=0, be the n-th step of this filtration. The category U is the smallest thick sub-category that contains all sub-categories U_n and is stable under colimit [L. Schwartz, Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture, Chicago Lectures in Mathematics Series (1994)]. The category U_0 is the one of locally finite modules, i.e. the modules that are direct limit of finite modules. The conjecture is as follows, let X be a space then : * either H^*X is locally finite, * or H^*X does not belong to U_n, for all n. As an example the cohomology of a finite space, or of the loop space of a finite space are always locally finite. On the other side the cohomology of the classifying space of a finite group whose order is divisible by 2 does belong to any sub-category U_n. One proves this conjecture, modulo the additional hypothesis that all quotients of the nilpotent filtration are finitely generated. This condition is used when applying N. Kuhn's reduction of the problem. It is necessary to do it to be allowed to apply Lannes' theorem on the cohomology of mapping spaces.[N. Kuhn, On topologically realizing modules over the Steenrod algebra, Ann. of Math. 141 (1995) 321-347]. | La filtration de Krull de la categorie U et la cohomologie des espaces | 12,501 |
In this paper we develop homotopy theoretical methods for studying diagrams. In particular we explain how to construct homotopy colimits and limits in an arbitrary model category. The key concept we introduce is that of a model approximation. Our key result says that if a category admits a model approximation then so does any diagram category with values in this category. From the homotopy theoretical point of view categories with model approximations have similar properties to those of model categories. They admit homotopy categories (localizations with respect to weak equivalences). They also can be used to construct derived functors by taking the analogs of fibrant and cofibrant replacements. A category with weak equivalences can have several useful model approximations. We take advantage of this possibility and in each situation choose one that suits our needs. In this way we prove all the fundamental properties of the homotopy colimit and limit: Fubini Theorem (the homotopy colimit -respectively limit- commutes with itself), Thomason's theorem about diagrams indexed by Grothendieck constructions, and cofinality statements. Since the model approximations we present here consist of certain functors "indexed by spaces", the key role in all our arguments is played by the geometric nature of the indexing categories. | Homotopy theory of diagrams | 12,502 |
We show that if $K$ is a nilpotent finite complex, then $\Omega K$ can be built from spheres using fibrations and homotopy (inverse) limits. This is applied to show that if ${\mathrm {map}}_*(X,S^n)$ is weakly contractible for all $n$, then ${\mathrm {map}}_*(X,K)$ is weakly contractible for any nilpotent finite complex $K$. | Miller Spaces and Spherical Resolvability of Finite Complexes | 12,503 |
The classical ``computation'' methods in Algebraic Topology most often work by means of highly infinite objects and in fact +are_not+ constructive. Typical examples are shown to describe the nature of the problem. The Rubio-Sergeraert solution for Constructive Algebraic Topology is recalled. This is not only a theoretical solution: the concrete computer program +Kenzo+ has been written down which precisely follows this method. This program has been used in various cases, opening new research subjects and producing in several cases significant results unreachable by hand. In particular the Kenzo program can compute the first homotopy groups of a simply connected +arbitrary+ simplicial set. | Constructive Algebraic Topology | 12,504 |
We show that if U is a hypercover of a topological space X then the natural map from hocolim U to X is a weak equivalence. This fact is used to construct topological realization functors for the A^1-homotopy theory of schemes over real and complex fields. | Hypercovers in topology | 12,505 |
We study natural subalgebras Ch_E(G) of group cohomology defined in terms of infinite loop spaces E and give representation theoretic descriptions of those based on QS^0 and the Johnson-Wilson theories E(n). We describe the subalgebras arising from the Brown-Peterson spectra BP and as a result give a simple reproof of Yagita's theorem that the image of BP^*(BG) in H^*(BG;F_p) is F-isomorphic to the whole cohomology ring; the same result is shown to hold with BP replaced by any complex oriented theory E with a map of ring spectra from E to HF_p which is non-trivial in homotopy. We also extend the constructions to define subalgebras of H^*(X;F_p) for any space X; when X is finite we show that the subalgebras Ch_{E(n)}(X) give a natural unstable chromatic filtration of H^*(X;F_p). | Subalgebras of group cohomology defined by infinite loop spaces | 12,506 |
We study co-H-maps from a suspension to the suspension of the projective plane and provide examples of non-suspension 3-cell co-H-spaces. These (infinitely many) examples are related to the homotopy groups of the 3-sphere. For each element of order 2 in $\pi_n(S^3)$, there is a corresponding non-suspension co-H-space of cells in dimensions 2, 3 and n+2. Our ideas are to study Hopf invariants in combinatorial way by using the Cohen groups. | On Co-H-maps to the Suspension of the Projective Plane | 12,507 |
We construct a canonical Thom isomorphism in Grojnowski's equivariant elliptic cohomology, for virtual T-oriented T-equivariant spin bundles with vanishing Borel-equivariant second Chern class, which is natural under pull-back of vector bundles and exponential under Whitney sum. It extends in the complex-analytic case the non-equivariant sigma orientation of Hopkins, Strickland, and the author. The construction relates the sigma orientation to the representation theory of loop groups and Looijenga's weighted projective space, and sheds light even on the non-equivariant case. Rigidity theorems of Witten-Bott-Taubes including generalizations by Kefeng Liu follow. | The sigma orientation for analytic circle-equivariant elliptic
cohomology | 12,508 |
Let f:E-->B be a fibration of fiber F. Eilenberg and Moore have proved that there is a natural isomorphism of vector spaces between H^*(F;F_p) and Tor^{C^*(B)}(C^*(E),F_p). Generalizing the rational case proved by Sullivan, Anick [Hopf algebras up to homotopy, J. Amer. Math. Soc. 2 (1989) 417--453] proved that if X is a finite r-connected CW-complex of dimension < rp+1 then the algebra of singular cochains C^*(X;F_p) can be replaced by a commutative differential graded algebra A(X) with the same cohomology. Therefore if we suppose that f:E-->B is an inclusion of finite r-connected CW-complexes of dimension < rp+1, we obtain an isomorphism of vector spaces between the algebra H^*(F;F_p) and Tor^{A(B)}(A(E),F_p) which has also a natural structure of algebra. Extending the rational case proved by Grivel-Thomas-Halperin [PP Grivel, Formes differentielles et suites spectrales, Ann. Inst. Fourier 29 (1979) 17--37] and [S Halperin, Lectures on minimal models, Soc. Math. France 9-10 (1983)] we prove that this isomorphism is in fact an isomorphism of algebras. In particular, $H^*(F;F_p) is a divided powers algebra and p-th powers vanish in the reduced cohomology \mathaccent "707E {H}^*(F;F_p). | On the cohomology algebra of a fiber | 12,509 |
Let X --> B be an orientable sphere bundle. Its Gysin sequence exhibits H^*(X) as an extension of H^*(B)-modules. We prove that the class of this extension is the image of a canonical class that we define in the Hochschild 3-cohomology of H^*(B), corresponding to a component of its A_infty-structure, and generalizing the Massey triple product. We identify two cases where this class vanishes, so that the Gysin extension is split. The first, with rational coefficients, is that where B is a formal space; the second, with integer coefficients, is where B is a torus. | Splitting of Gysin extensions | 12,510 |
We define and discuss G-formality for certain spaces endowed with an action by a compact Lie group. This concept is essentially formality of the Borel construction of the space in a category of commutative differential graded algebras over R, where R is the cohomology of the classifying space BG. These results may be applied in computing the equivariant cohomology of their loop spaces. | Formality in an Equivariant Setting | 12,511 |
To clarify the method behind the paper "Ganea's conjecture on Lusternik-Schnirelman category" by the author, a generalisation of Berstein-Hilton Hopf invariants is defined as `higher Hopf invariants'. They detect the higher homotopy associativity of Hopf spaces and are studied as obstructions not to increase the LS category by one by attaching a cone. Under a condition between dimension and LS category, a criterion for Ganea's conjecture on LS category is obtained using the generalised higher Hopf invariants, which yields the main result of "Ganea's ..." for all the cases except the case when $p=2$. As an application, conditions in terms of homotopy invariants of the characteristic maps are given to determine the LS category of sphere-bundles-over-spheres. Consequently, a closed manifold $M$ is found not to satisfy Ganea's conjecture on LS category and another closed manifold $N$ is found to have the same LS category as its `punctured submanifold' $N-\{P\}$, $P \in N$. But all examples obtained here support the conjecture in "Ganea's ...". | A_{\infty}-method in Lusternik-Schnirelmann category | 12,512 |
A criterion to determine the L-S category of a total space of a sphere-bundle over a sphere is given in terms of homotopy invariants of its characteristic map, and thus providing a complete answer to Ganea's Problem 4. As a result, we obtain a necessary and sufficient condition for such a total space $N$ to have the same L-S category as its `once punctured submanifold' $N\smallsetminus\{P\}$, $P \in N$. Also a necessary condition for such a total space $M$ to satisfy Ganea's conjecture is obtained. | Lusternik-Schnirelmann category of a sphere-bundle over a sphere | 12,513 |
We determine the L-S category of Sp(3) by showing that the 5-fold reduced diagonal $\widebar{\Delta}_5$ is given by $\nu^2$, using a Toda bracket and a generalised cohomology theory $h^*$ given by $h^*(X,A) = \{X/A,{\mathbb S}[0,2]\}$, where ${\mathbb S}[0,2]$ is the 3-stage Postnikov piece of the sphere spectrum ${\mathbb S}$. This method also yields a general result that $\cat(Sp(n)) \geq n+2$ for $n \geq 3$, which improves the result of Singhof. | L-S categories of simply-connected compact simple Lie groups of low rank | 12,514 |
On construit des foncteurs de formes differentielles generalisees. Ceux-ci, dans le cas d'espaces nilpotents de type fini, determinent le type d'homotopie faible des espaces. Ils sont munis, d'une maniere elementaire et naturelle, de l'action de cup-i produits. Pour les algebres commutatives a homotopit pres (algebres sur une resolution cofibrante de l'operade des algebres commutatives), on demontre en utilisant les formes differentielles generalisees que le modele de la fibre d'une application simpliciale est la cofibre du modele de ce morphisme. We construct functors of generalized differential forms. In the case of nilpotent spaces of finite type, they determine the weak homotopy type of the spaces. Moreover they are equipped, in an elementary and natural way, with the action of cup-i products. Working with commutative algebras up to homotopy (viewed as algebras over a cofibrant resolution of the operad of commutative algebras), we show using these functors that the model of the fiber of a simplicial map is the cofiber of the algebraic model of this map. | Formes differentielles generalisees sur une operade et modeles
algebriques des fibrations | 12,515 |
The loop homology of a closed orientable manifold $M$ of dimension $d$ is the ordinary homology of the free loop space $M^{S^1}$ with degrees shifted by $d$, i.e. $\mathbb H_*(M^{S^1}) = H_{*+d}(M^{S^1})$. Chas and Sullivan have defined a loop product on $\mathbb H_*(M^{S^1})$ and an intersection morphism $I : \mathbb H_*(M^{S^1}) \to H_*(\Omega M)$. The algebra $\mathbb H_*(M^{S^1})$ is commutative and $I$ is a morphism of algebras. In this paper we produce a model that computes the algebra $\mathbb H_*(M^{S^1})$ and the morphism $I$. We show that the kernel of $I$ is nilpotent and that the image is contained in the center of $H_*(\Omega M)$, which is in general quite small. | Loop homology algebra of a closed manifold | 12,516 |
We study the so-called Gray filtration on the set of phantom maps between two spaces. Using both its algebraic characterization and the Sullivan completion approach to phantom maps, we generalize some of the recent results of Le, McGibbon and Strom. We particularly emphasize on the set of phantom maps with infinite Gray index, describing it in an original algebraic way. We furthermore introduce and study a natural filtration on SNT-sets (that is sets of homotopy types of spaces having the same n-type for all n), which appears to have the same algebraic characterization of the Gray one on phantom maps. For spaces whose rational homotopy type is that of an H-space or a co-H-space, we establish criteria permitting to determinate those subsets of this filtration which are non trivial, generalizing work of McGibbon and Moller. We finally describe algebraically the natural connection between phantom maps and SNT-theory, associating to a phantom map its homotopy fiber or cofiber. We use this description to show that this connection respect filtrations, and to find generic examples of spaces for which the filtration on the corresponding SNT-set consists of infinitely many strict inclusions. | Phantom maps, SNT-theory, and natural filtrations on lim^1 sets | 12,517 |
In this note we introduce an action of cyclohedra on the free loop space. We will then discuss how this action can be used for an appropriate recognition principle in the same manner as the action of Stasheff's associahedra can be used to recognize based loop spaces. We will also interpret one result of R.L. Cohen as an approximation theorem, in the spirit of Beck and May, for free loop spaces. | Free Loop Spaces and Cyclohedra | 12,518 |
In "Elliptic spectra, the Witten genus, and the Theorem of the cube" (Invent. Math. 146 (2001)), the authors constructed a natural map from the Thom spectrum MU<6> to any elliptic spectrum, called the "sigma orientation". MU<6> is an H-infinity ring spectrum, and in this paper we show that if E is a K(2)-local H-infinity elliptic spectrum, then the sigma orientation is a map of H-infinity spectra. | The sigma orientation is an H-infinity map | 12,519 |
Let M be a simply-connected closed oriented N-dimensional manifold. We prove that for any field of coefficients there exists a natural homomorphism of commutative graded algebras $\Psi : H_\ast (\Omega {aut}_1 M) \to H_{\ast +N}(M^{S^1})$ where $H_\ast (M^{S^1})$ is the loop algebra defined by Chas-Sullivan. As usual ${aut}_1 X$ (resp. $\Omega X$) denotes the monoid of the self-equivalences homotopic to the identity map (resp. the space of based loops) of the space X. Moreover, if $\bk$ is of characteristic zero, $\Psi$ yields isomorphisms $\pi_n(\Omega {aut}_1 M) \otimes \bk \cong \hH^{n+N}_{(1)}$ where $\displaystyle \oplus_{l=1}^\infty \hH^n_{(l)}$ denotes the Hodge decomposition on $H^\ast (M ^{S^1})$. | Spaces of self-equivalences and free loops spaces | 12,520 |
The Barratt-Eccles operad is a simplicial operad formed by the classical homogeneous bar construction of the symmetric groups. We prove that these simplicial sets decompose as unions of prisms indexed by surjections. We observe that the cellular complexes given by this prismatic structure are nothing but the components of the surjection operad (the operad introduced by J. McClure and J. Smith in their work on the Deligne conjecture). | A prismatic decomposition of the Barratt-Eccles operad | 12,521 |
The usual way of defining weak equivalences for simplicial presheaves is to require an isomorphism on all sheaves of homotopy groups. We unravel some of the machinery here, and give a more concrete description in terms of local homotopy lifting properties. This characterization is used to prove some basic facts about the local homotopy theory of simplicial presheaves. | Weak equivalences of simplicial presheaves | 12,522 |
We prove that Jardine's model category of simplicial presheaves can be obtained by localizing the `discrete' version at the collection of all hypercovers. One consequence is that the fibrant objects can be explicitly identified in terms of a hypercover descent condition. Another is a very simple approach to change-of-site functors. In an appendix, we discuss how this hypercover localization compares to the more naive process of localizing at the Cech complexes; the two are not the same in general, but agree in some cases of interest. | Hypercovers and simplicial presheaves | 12,523 |
In this note we show that the positivity property of the equivariant signature of the loop space, first observed in [MS1] in the case of the even-dimensional projective spaces, is valid for Picard number 2 toric varieties. A new formula for the equivariant signature of the loop space in the case of a toric spin variety is derived. | The Chiral de Rham Complex and Positivity of the Equivariant Signature
of the Loop Space | 12,524 |
Let V be a mod 2 vector space of rank k. W. Singer defined a transfer homomorphism from the GL(k,2) coinvariants of the primitives in the homology of BV to the cohomology of the Steenrod algebra, as an algebraic version of the geometric transfer from the stable homotopy of BV to the stable homotopy of spheres. It has been shown that the algebraic transfer is highly nontrivial and, more precisely, that it is an isomorphism for k=1, 2, or 3. However, Singer showed that it is not an epimorphism for k=5. In this paper, we prove that it also fails to be an epimorphism when k=4. Precisely, it does not detect the non zero elements in the g family, in stems 20, 44, 92, and in general, 12*2^s - 4, for each s > 0. The transfer still fails to be an epimorphism even after inverting Sq^0, thereby giving a negative answer to a prediction by Minami. | On behavior of the algebraic transfer | 12,525 |
Our main motivation for the work presented in this paper is to construct a localization functor, in a certain sense dual to the f-localization of Bousfield and Farjoun, and to study some of its properties. We succeed in a case which is related to the Sullivan profinite completion. As a corollary we prove the existence of certain cohomological localizations. | Homotopical localizations at a space | 12,526 |
Let G=S^1, G=Z/p or more generally G be a finite p group, where p is an odd prime number. If G acts on a space whose cohomology ring satisfies Poincare duality (with appropriate coefficients k), we prove a mod 4 congruence between the total Betti number of X^G and a number which depends only on the k[G]-module structure of H^*(X;k). This improves the well known mod 2 congruences that hold for actions on general spaces. | Poincare duality in P.A. Smith theory | 12,527 |
We obtain explicit formulas for the rational homotopy groups of generalised symmetric spaces, i.e., the homogeneous spaces for which the isotropy subgroup appears as the fixed point group of some finite order automorphism of the group. In particular, this gives explicit formulas for the rational homotopy groups of all classical compact symmetric spaces. | Rational homotopy groups of generalised symmetric spaces | 12,528 |
Hu, Kriz and May recently reexamined ideas implicit in Priddy's elegant homotopy theoretic construction of the Brown-Peterson spectrum at a prime p. They discussed May's notions of nuclear complexes and of cores of spaces, spectra, and commutative S-algebras. Their most striking conclusions, due to Hu and Kriz, were negative: cores are not unique up to equivalence, and BP is not a core of MU considered as a commutative S-algebra, although it is a core of MU considered as a p-local spectrum. We investigate these ideas further, obtaining much more positive conclusions. We show that nuclear complexes have several non-obviously equivalent characterizations. Up to equivalence, they are precisely the irreducible complexes, the minimal atomic complexes, and the Hurewicz complexes with trivial mod p Hurewicz homomorphism above the Hurewicz dimension, which we call complexes with no mod p detectable homotopy. Unlike the notion of a nuclear complex, these other notions are all invariant under equivalence. This simple and conceptual criterion for a complex to be minimal atomic allows us to prove that many familiar spectra, such as ko, eo_2, and BoP at the prime 2, all BP<n> at any prime p, and the indecomposable wedge summands of the suspension spectra of $CP^\infty$ and $HP^\infty$ at any prime p are minimal atomic. | Minimal Atomic Complexes | 12,529 |
We show how the formal Wirthmuller isomorphism theorem proven in "Isomorphisms between left and right adjoints", by Fausk, Hu, and May, simplifies the proof of the Wirthmuller isomorphism in equivariant stable homotopy theory. Other examples from equivariant stable homotopy theory show that the hypotheses of the formal Wirthmuller and formal Grothendieck isomorphism theorems in the cited paper cannot be weakened. | The Wirthmuller isomorphism revisited | 12,530 |
We establish a weak form of Carlson's conjecture on the depth of the mod-p cohomology ring of a p-group. In particular, Duflot's lower bound for the depth is tight if and only if the cohomology ring is not detected on a certain family of subgroups. The proofs use the structure of the cohomology ring as a comodule over the cohomology of the centre via the multiplication map. We demonstrate the existence of systems of parameters (so-called polarised systems) which are particularly well adapted to this comodule structure. | On Carlson's depth conjecture in group cohomology | 12,531 |
The category of crossed complexes gives an algebraic model of CW-complexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the problem of calculating non-abelian extensions. We show how the strong properties of this category allow for the computation of free crossed resolutions for amalgamated sums and HNN-extensions of groups, and so obtain computations of higher homotopical syzygies in these cases. | Crossed complexes, and free crossed resolutions for amalgamated sums and
HNN-extensions of groups | 12,532 |
We provide a general definition of higher homotopy operations, encompassing most known cases, including higher Massey and Whitehead products, and long Toda brackets. These operations are defined in terms of the W-construction of Boardman and Vogt, applied to the appropriate diagram category; we also show how some classical families of polyhedra (including simplices, cubes, associahedra, and permutahedra) arise in this way. | Higher Homotopy Operations | 12,533 |
This paper provides analogues of the results of [G.Walker and R.M.W. Wood, Linking first occurrence polynomials over F_2 by Steenrod operations, J. Algebra 246 (2001), 739--760] for odd primes p. It is proved that for certain irreducible representations L(lambda) of the full matrix semigroup M_n(F_p), the first occurrence of L(lambda) as a composition factor in the polynomial algebra P=F_p[x_1,...,x_n] is linked by a Steenrod operation to the first occurrence of L(lambda) as a submodule in P. This operation is given explicitly as the image of an admissible monomial in the Steenrod algebra A_p under the canonical anti-automorphism chi . The first occurrences of both kinds are also linked to higher degree occurrences of L(lambda) by elements of the Milnor basis of A_p. | Linking first occurrence polynomials over F_p by Steenrod operations | 12,534 |
The normalized cochain complex of a simplicial set N^*(Y) is endowed with the structure of an E_{infinity} algebra. More specifically, we prove in a previous article that N^*(Y) is an algebra over the Barratt-Eccles operad. According to M. Mandell, under reasonable completeness assumptions, this algebra structure determines the homotopy type of Y. In this article, we construct a model of the mapping space Map(X,Y). For that purpose, we extend the formalism of Lannes' T functor in the framework of E_{infinity} algebras. Precisely, in the category of algebras over the Barratt-Eccles operad, we have a division functor -oslash N_(X) which is left adjoint to the functor Hom_F(N_*(X),-). We prove that the associated left derived functor -oslash^L N_*(X) is endowed with a quasi-isomorphism N^*(Y) oslash^L N_*(X) --> N^* Map(X,Y). | Derived division functors and mapping spaces | 12,535 |
We study fibred spaces with fibres in a structure category $\V$ and we show that cellular approximation, Blakers--Massey theorem, Whitehead theorems, obstruction theory, Hurewicz homomorphism, Wall finiteness obstruction, and Whitehead torsion theorem hold for fibred spaces. For this we introduce the cohomology of fibred spaces. | Homotopy and homology of fibred spaces | 12,536 |
We study the group of homotopy classes of self maps of compact Lie groups which induce the trivial homomorphism on homotopy groups. We completely determine the groups for SU(3) and Sp(2). We investigate these groups for simple Lie groups in the cases when Lie groups are p-regular or quasi p-regular. We apply our results to the groups of self homotopy equivalences. | pi_*-kernels of Lie groups | 12,537 |
We explore two constructions in homotopy category with algebraic precursors in the theory of noncommutative rings and homological algebra, namely the Hochschild cohomology of ring spectra and Morita theory. The present paper provides an extension of the algebraic theory to include the case when $M$ is not necessarily a progenerator. Our approach is complementary to recent work of Dwyer and Greenlees and of Schwede and Shipley. A central notion of noncommutative ring theory related to Morita equivalence is that of central separable or Azumaya algebras. For such an Azumaya algebra A, its Hochschild cohomology HH^*(A,A) is concentrated in degree 0 and is equal to the center of A. We introduce a notion of topological Azumaya algebra and show that in the case when the ground S-algebra R is an Eilenberg-Mac Lane spectrum of a commutative ring this notion specializes to classical Azumaya algebras. A canonical example of a topological Azumaya R-algebra is the endomorphism R-algebra F_R(M,M) of a finite cell R-module. We show that the spectrum of mod 2 topological K-theory KU/2 is a nontrivial topological Azumaya algebra over the 2-adic completion of the K-theory spectrum widehat{KU}_2. This leads to the determination of THH(KU/2,KU/2), the topological Hochschild cohomology of KU/2. As far as we know this is the first calculation of THH(A,A) for a noncommutative S-algebra A. | Topological Hochschild cohomology and generalized Morita equivalence | 12,538 |
Spaces in the genus of infinite quaternionic projective space which admit essential maps from infinite complex projective space are classified. In these cases the sets of homotopy classes of maps are described explicitly. These results strengthen the classical theorem of McGibbon and Rector on maximal torus admissibility for spaces in the genus of infinite quaternionic projective space. An interpretation of these results in the context of Adams-Wilkerson embedding in integral $K$-theory is also given. | Maps to spaces in the genus of infinite quaternionic projective space | 12,539 |
In this paper we propose and partially carry out a program to use $K$-theory to refine the topological realization problem of unstable algebras over the Steenrod algebra. In particular, we establish a suitable form of algebraic models for $K$-theory of spaces, called $\psi^p$-algebras, which give rise to unstable algebras by taking associated graded algebras mod $p$. The aforementioned problem is then split into (i) the \emph{algebraic} problem of realizing unstable algebras as mod $p$ associated graded of $\psi^p$-algebras and (ii) the \emph{topological} problem of realizing $\psi^p$-algebras as $K$-theory of spaces. Regarding the algebraic problem, a theorem shows that every connected and even unstable algebra can be realized. We tackle the topological problem by obtaining a $K$-theoretic analogue of a theorem of Kuhn and Schwartz on the so-called Realization Conjecture. | A K-theoretic refinement of topological realization of unstable algebras | 12,540 |
Boardman, Johnson, and Wilson gave a precise formulation for an unstable algebra over a generalized cohomology theory. Modifying their definition slightly in the case of complex K-theory by taking into account its periodicity, we prove that an unstable algebra for complex $K$-theory is precisely a filtered $\lambda$-ring, and vice versa. | Unstable $K$-cohomology algebra is filtered lambda-ring | 12,541 |
It is an old conjecture, that finite $H$-spaces are homotopy equivalent to manifolds. Here we prove that this conjecture is true for loop spaces. Actually, we show that every quasi finite loop space is equivalent to a stably parallelizable manifold. The proof is conceptual and relies on the theory of p-compact groups. On the way we also give a complete classification of all simple 2-compact groups of rank 2. | Quasi finite loop spaces are manifolds | 12,542 |
We construct an explicit diagonal \Delta_P on the permutahedra P. Related diagonals on the multiplihedra J and the associahedra K are induced by Tonks' projection P --> K and its factorization through J. We introduce the notion of a permutahedral set Z and lift \Delta_P to a diagonal on Z. We show that the double cobar construction \Omega^2(C_*(X)) is a permutahedral set; consequently \Delta_P lifts to a diagonal on \Omega^2(C_*(X)). Finally, we apply the diagonal on K to define the tensor product of A_\infty-(co)algebras in maximal generality. | Diagonals on the Permutahedra, Multiplihedra and Associahedra | 12,543 |
We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZ-algebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZ-algebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that bascially any rational stable model category is Quillen equivalent to modules over a differential graded Q-algebra (with many objects). | HZ-algebra spectra are differential graded algebras | 12,544 |
We construct Quillen equivalences between the model categories of monoids (rings), modules and algebras over two Quillen equivalent model categories under certain conditions. This is a continuation of our earlier work where we established model categories of monoids, modules and algebras [Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 (2000), 491-511]. As an application we extend the Dold-Kan equivalence to show that the model categories of simplicial rings, modules and algebras are Quillen equivalent to the associated model categories of connected differential graded rings, modules and algebras. We also show that our classification results from [Stable model categories are categories of modules, Topology, 42 (2003) 103-153] concerning stable model categories translate to any one of the known symmetric monoidal model categories of spectra. | Equivalences of monoidal model categories | 12,545 |
In the paper the notion of truncating twisting function $\tau :X\to Q$ from a simplicial set $X$ to a cubical set $Q$ and the corresponding notion of twisted Cartesian product of these sets $X\times_{\tau}Q$ are introduced. The latter becomes a cubical set whose chain complex coincides with the standard twisted tensor product $C_*(X)\otimes_{\tau_*}C_*(Q)$. This construction together with the theory of twisted tensor products for homotopy G-algebras allows to obtain multiplicative models for fibrations. | A cubical model for a fibration | 12,546 |
In the paper the notion of truncating twisting function from a cubical set to a permutahedral set and the corresponding notion of twisted Cartesian product of these sets are introduced. The latter becomes a permutocubical set that models in particular the path fibration on a loop space. The chain complex of this twisted Cartesian product in fact is a comultiplicative twisted tensor product of cubical chains of base and permutahedral chains of fibre. This construction is formalized as a theory of twisted tensor products for Hirsch algebras. | The twisted Cartesian model for the double path fibration | 12,547 |
For a 1-connected spectrum E, we study the moduli space of suspension spectra which come equipped with a weak equivalence to E. We construct a spectral sequence converging to the homotopy of the moduli space in positive degrees. In the metastable range, we get a complete homotopical classification of the path components of the moduli space. Our main tool is Goodwillie's calculus of homotopy functors. | Moduli of suspension spectra | 12,548 |
Stasheff's $A(\infty)$-algebra $(M,\{m_i:\otimes^iM\to M, i=1,2,3,...\})$ in fact is a DG-algebra $(M,m_1,m_2)$ with not necessarily associative product $m_2$ but this nonassociativity is measured by higher homotopies $m_{i>2}$. Nevertheless such structure arises in the strictly associative situation too, namely in the homology algebra $H(C)$ of a DG-algebra $C$ with free $H_i(C)$-s, particularly in the cohomology algebra $H^*(X,\Lambda)$ of a topological space $X$. It is clear that the $A(\infty)$-algebra $(H^*(X,\Lambda),\{m_i\})$ carries more information than the cohomology algebra $H^*(B,\Lambda)$. Naturally arises a question when this structure is degenerate, that is when an $A(\infty)$-algebra $(M, \{m_i\})$ is isomorphic to one with higher operations $m_i, i\geq 3$ trivial? In this paper we introduce the obstructions for such degeneracy. Namely, operations $\{m_i\}$ we interpret as Hochschild twisting cochain $m=m_3+m_4+..., m_i\in C^n(M,M)$ satisfying $\delta m=m\smile_1m$ where $\smile_1$ is Gerstenhabers product in $C^*(M,M)$. Using the generalized product $f\smile_1(g_1,...,g_k)$ we define perturbations of Hochschild twisting cochains (i.e. of $A(\infty)$ structures) and in particular prove that if for a graded algebra $(M,\mu)$ all Hochschild cohomologies $Hoch^{n,2-n}(M,M)=0$ for $n\geq3$ then any $A(\infty)$-algebra structure $\{m_i\}$ on $M$ with $m_1=0, m_2=\mu $, is degenerate. | Structure of $A(\infty)$-algebra and Hochschild and Harrison cohomology | 12,549 |
Chas and Sullivan recently defined an intersection product on the homology $H_*(LM)$ of the space of smooth loops in a closed, oriented manifold $M$. In this paper we will use the homotopy theoretic realization of this product described by the first two authors to construct a second quadrant spectral sequence of algebras converging to the loop homology multiplicatively, when $M$ is simply connected. The $E_2$ term of this spectral sequence is $H^*(M;H_*(\Omega M))$ where the product is given by the cup product on the cohomology of the manifold $H^* (M)$ with coefficients in the Pontryagin ring structure on the homology of its based loop space $H_*(\Omega M)$. We then use this spectral sequence to compute the ring structures of $H_* (LS^n)$ and $H_* (L\bcp^n)$. | The loop homology algebra of spheres and projective spaces | 12,550 |
Let A be a finite abelian group. We set up an algebraic framework for studying A-equivariant complex-orientable cohomology theories in terms of a suitable kind of equivariant formal groups. We compute the equivariant cohomology of many spaces in these terms, including projective bundles (and associated Gysin maps), Thom spaces, and infinite Grassmannians. | Multicurves and equivariant cobordism | 12,551 |
This paper establishes an equivalence between existence of free involutions on $H{\Bbb C}P^3$ and existence of involutions on $S^6$ with fixed point set an imbedded $S^3$, then a family of counterexamples of the Smith conjecture for imbeddings of $S^3$ in $S^6$ are given by known result on $H{\Bbb C}P^3$. In addition, this paper also shows that every smooth homotopy complex projective 3-space admits no orientation preserving smooth free involution, which answers an open problem [Pe]. Moreover, the study of existence problem for smooth orientation preserving involutions on $H{\Bbb C}P^3$ is completed. | Smooth free involution of $H{\Bbb C}P^3$ and Smith conjecture for
imbeddings of $S^3$ in $S^6$ | 12,552 |
In this article, we give some conditions on the structure of an unstable module, which are satisfied whenever this module is the reduced cohomology of a space or a spectrum. First, we study the structure of the sub-modules of Sigma^sH^*(B(Z/2)^{oplus d};Z/2), i.e., the unstable modules whose nilpotent filtration has length 1. Next, we generalise this result to unstable modules whose nilpotent filtration has a finite length, and which verify an additional condition. The result says that under certain hypotheses, the reduced cohomology of a space or a spectrum does not have arbitrary large gaps in its structure. This result is obtained by applying Adams' theorem on the Hopf invariant and the classification of the injective unstable modules. This work was carried out under the direction of L. Schwartz. Resume Dans cet article, on donne des restrictions sur la structure d'un module instable, qui doivent etre verifiees pour que celui-ci soit la cohomologie reduite d'un espace ou d'un spectre. On commence par une etude sur la structure des sous-modules de Sigma^sH^*(B(Z/2)^{oplus d};Z/2), i.e., les modules instables dont la filtration nilpotente est de longueur 1. Ensuite, on generalise le resultat aux modules instables dont la filtration nilpotente est de longueur finie, et qui verifient une condition supplementaire. Le resultat dit que sous certaines hypotheses, la cohomologie reduite d'un espace ou d'un spectre ne contient pas de lacunes de longueur arbitrairement grande. Ce resultat est obtenu par application du celebre theoreme d'Adams sur l'invariant de Hopf et de la classification des modules instables injectifs. Ce travail est effectue sous la direction de L. Schwartz. | Sur la realisation des modules instables | 12,553 |
We define an ``algebraic'' version of the Goodwillie tower, P_n^alg F(X), that depends only on the behavior of F on coproducts of X. When F is a functor to connected spaces or grouplike H-spaces, the functor P_n^alg F is the base of a fibration whose fiber is the simplicial space associated to a cotriple built from the (n+1) cross effect of the functor F. When the connectivity of X is large enough (for example, when F is the identity functor and X is connected), the algebraic Goodwillie tower agrees with the ordinary (topological) Goodwillie tower, so this theory gives a way of studying the Goodwillie approximation to a functor F in many interesting cases. | Algebraic Goodwillie calculus and a cotriple model for the remainder | 12,554 |
Modular forms appear in many facets of mathematics, and have played important roles in geometry, mathematical physics, number theory, representation theory, topology, and other areas. Around 1994, motivated by technical issues in homotopy theory, Mark Mahowald, Haynes Miller and I constructed a topological refinement of modular forms, which we call {\em topological modular forms}. At the Zurich ICM I sketched a program designed to relate topological modular forms to invariants of manifolds, homotopy groups of spheres, and ordinary modular forms. This program has recently been completed and new directions have emerged. In this talk I will describe this recent work and how it informs our understanding of both algebraic topology and modular forms. | Algebraic topology and modular forms | 12,555 |
We formulate and prove a chain rule for the derivative, in the sense of Goodwillie, of compositions of weak homotopy functors from simplicial sets to simplicial sets. The derivative spectrum dF(X) of such a functor F at a simplicial set X can be equipped with a right action by the loop group of its domain X, and a free left action by the loop group of its codomain Y = F(X). The derivative spectrum d(E o F)(X)$ of a composite of such functors is then stably equivalent to the balanced smash product of the derivatives dE(Y) and dF(X), with respect to the two actions of the loop group of Y. As an application we provide a non-manifold computation of the derivative of the functor F(X) = Q(Map(K, X)_+). | A chain rule in the calculus of homotopy functors | 12,556 |
Let T be a torus. We show that Koszul duality can be used to compute the equivariant cohomology of topological T-spaces as well as the cohomology of pull backs of the universal T-bundle. The new features are that no further assumptions about the spaces are made and that the coefficient ring may be arbitrary. This gives in particular a Cartan-type model for the equivariant cohomology of a T-space with arbitrary coefficients. Our method works for intersection homology as well. | Koszul duality and equivariant cohomology for tori | 12,557 |
We consider fully effective orientation-preserving smooth actions of a given finite group G on smooth, closed, oriented 3-manifolds M. We investigate the relations that necessarily hold between the numbers of fixed points of various non-cyclic subgroups. In Section 2, we show that all such relations are in fact equations mod 2, and we explain how the number of independent equations yields information concerning low-dimensional equivariant cobordism groups. Moreover, we restate a theorem of A. Szucs asserting that under the conditions imposed on a smooth action of G on M as above, the number of G-orbits of points x in M with non-cyclic stabilizer G_x is even, and we prove the result by using arguments of G. Moussong. In Sections 3 and 4, we determine all the equations for non-cyclic subgroups G of SO(3). | Fixed point data of finite groups acting on 3-manifolds | 12,558 |
For a fibration with the fiber $K(\pi,n)$-space, the algebraic model as a twisted tensor product of chains of the base with standard chains of $K(\pi,n)$-complex is given which preserves multiplicative structure as well. In terms of this model the action of the $n$-cohomology of the base with coefficients in $\pi$ on the homology of fibration is described. | An algebraic model of fibration with the fiber $K(π,n)$-space | 12,559 |
This paper begins with an exposition of the author's research on the category of BP_*BP-comodules, much of which is joint with Neil Strickland. The main result of that work is that the category of E(n)_*E(n)-comodules is equivalent to a localization of the category of BP_*BP-comodules (the localization is L_n, analogous to the topological L_n). The main new result in this paper is that, analogously, the stable homotopy category of E(n)_*E(n)-comodules is equivalent to a localization (the finite localization L_n^f this time, not L_n) of the stable homotopy category of BP_*BP-comodules. These stable homotopy categories were constructed in previous work of the author, and are supposed to model stable homotopy theory; it is like stable homotopy theory where there are no differentials in the Adams-Novikov spectral sequence. Our result embeds the Miller-Ravenel and Hovey-Sadofsky change of rings theorems as special cases of more general isomorphisms. | Chromatic phenomena in the algebra of BP_{*}BP-comodules | 12,560 |
In a previous paper, the authors showed that the category of E(n)_*E(n)-comodules is a localization of the category of BP_*BP-comodules. In this paper, we study the resulting localization functor L_n on the category of BP_*BP-comodules. It is an algebraic analogue of the usual topological localization L_n. It is left exact, so has right derived functors L_n^i. We show that these derived functors are closely related to the local cohomology groups of BP_*-modules studied by Greenlees and May; in fact, they coincide with Cech cohomology with respect to I_{n+1}. We also construct a spectral sequence of comodules analogous to the Greenlees-May spectral sequence (of modules) converging to BP_*(L_n X) whose E_2-term involves L_n^i(BP_*X). The proofs require getting a partial understanding of injective objects in the category of BP_*BP-comodules. | Local cohomology of BP_*BP-comodules | 12,561 |
We show that, if E is a Landweber exact ring spectrum, then the category of E_*E-comodules is equivalent to the localization of the category of BP_*BP-comodules with respect to the hereditary torsion theory of v_n-torsion comodules, where n is the height of E. In particular, the category of E(n)_*E(n)-comodules is equivalent to the category of (v_n^{-1}BP)_*(v_n^{-1}BP)-comodules. We also prove structure theorems for E_*E-comodules; we show every E_*E-comodule has a primitive, we classify the invariant radical ideals, and we prove a version of the Landweber filtration theorem. | Comodules and Landweber exact homology theories | 12,562 |
We consider partitions of a set with $r$ elements ordered by refinement. We consider the simplicial complex $\bar{K}(r)$ formed by chains of partitions which starts at the smallest element and ends at the largest element of the partition poset. A classical theorem asserts that $\bar{K}(r)$ is equivalent to a wedge of $r-1$-dimensional spheres. In addition, the poset of partitions is equipped with a natural action of the symmetric group in $r$ letters. Consequently, the associated homology modules are representations of the symmetric groups. One observes that the $r-1$th homology modules of $\bar{K}(r)$, where $r = 1,2,...$, are dual to the Lie representation of the symmetric groups. In this article, we would like to point out that this theorem occurs a by-product of the theory of \emph{Koszul operads}. For that purpose, we improve results of V. Ginzburg and M. Kapranov in several directions. More particularly, we extend the Koszul duality of operads to operads defined over a field of positive characteristic (or over a ring). In addition, we obtain more conceptual proofs of theorems of V. Ginzburg and M. Kapranov. | Koszul duality of operads and homology of partition posets | 12,563 |
In general the processes of taking a homotopy inverse limit of a diagram of spectra and smashing spectra with a fixed space do not commute. In this paper we investigate under what additional assumptions these two processes do commute. In fact we deal with an equivariant generalization which involves spectra and smash products over the orbit category of a discrete group. Such a situation naturally occurs if one studies the equivariant homology theory associated to topological cyclic homology. The main theorem of this paper will play a role in the generalization of the results obtained by Boekstedt, Hsiang and Madsen about the algebraic K-theory Novikov Conjecture to the assembly map for the family of virtually cyclic subgroups. | Commuting homotopy limits and smash products | 12,564 |
If A is a graded connected algebra then we define a new invariant, polydepth A, which is finite if $Ext_A^*(M,A) \neq 0$ for some A-module M of at most polynomial growth. Theorem 1: If f : X \to Y is a continuous map of finite category, and if the orbits of H_*(\Omega Y) acting in the homology of the homotopy fibre grow at most polynomially, then H_*(\Omega Y) has finite polydepth. Theorem 2: If L is a graded Lie algebra and polydepth UL is finite then either L is solvable and UL grows at most polynomially or else for some integer d and all r, $\sum_{i=k+1}^{k+d} {dim} L_i \geq k^r$, $k\geq$ some $k(r)$. | Graded Lie algebras with finite polydepth | 12,565 |
We prove that the bar construction of an $E_\infty$ algebra forms an $E_\infty$ algebra. To be more precise, we provide the bar construction of an algebra over the surjection operad with the structure of a Hopf algebra over the Barratt-Eccles operad. (The surjection operad and the Barratt-Eccles operad are classical $E_\infty$ operads.) | The bar construction of an algebra as an E-infinite Hopf algebra | 12,566 |
Stasheff showed that if a map between H-spaces is an H-map, then the suspension of the map is extendable to a map between cprojective planes of the H-spaces. Stahseff also proved the converse under the assumption that the multiplication of the target space of the map is homotopy associative. We show by giving an example that the assumption of homotopy associativity of the multiplication of the target space is necessary to show the converse. We also show an analogous fact for maps between higher homotopy associative H-spaces. | Retractions of H-spaces | 12,567 |
Let $F \hookrightarrow X \to B$ be a fibre bundle with structure group $G$, where $B$ is $(d{-}1)$-connected and of finite dimension, $d \geq 1$. We prove that the strong L-S category of $X$ is less than or equal to $m + \frac{\dim B}{d}$, if $F$ has a cone decomposition of length $m$ under a compatibility condition with the action of $G$ on $F$. This gives a consistent prospect to determine the L-S category of non-simply connected Lie groups. For example, we obtain $\cat{PU(n)} \leq 3(n{-}1)$ for all $n \geq 1$, which might be best possible, since we have $\cat{\mathrm{PU}(p^r)}=3(p^r{-}1)$ for any prime $p$ and $r \geq 1$. Similarly, we obtain the L-S category of $\mathrm{SO}(n)$ for $n \leq 9$ and $\mathrm{PO}(8)$. We remark that all the above Lie groups satisfy the Ganea conjecture on L-S category. | Lusternik-Schnirelmann categories of non-simply connected compact simple
Lie groups | 12,568 |
We calculate geometric and homotopical (or stable) bordism rings associated to semi-free $S^1$ actions on complex manifolds, giving explicit generators for the geometric theory. The classification of semi-free actions with isolated fixed points up to cobordism complements similar results from symplectic geometry. | Bordism of semi-free S^1 actions | 12,569 |
For any smooth free action of the unit circle S1 on a smooth manifold M, the Gysin sequence of M is a long exact sequence relating the DeRham Cohomology of M and the orbit space M/S1. If the action is not free then M/S1 is not a smooth manifold but a stratified pseudomanifold, the lenght of M/S1 depending on the number of orbit types; and there is a Gysin sequence relating their intersection cohomologies. The links of the fixed strata in M/S1 are cohomological complex projective spaces, so the conecting homomorphism of this sequences is the multiplication by the Euler class. In this article we extend the above results for any action of S1 on a stratified pseudomanifold X of lenght 1. We use the DeRham-like intersection cohomology defined by means of an unfolding. If the action preserves the local structure, then the orbit space X/S1 is again a stratified pseudomanifold of lenght 1 and has an unfolding. There is a long exact sequence relating the intersection cohomology of X and X/S1 with a third complex $\mathcal{G}$, the Gysin Term, whose cohomology depends on basic cohomological data of two flavours: global and local. Global data concerns the Euler class induced by the action; local information depends on the cohomology of the fixed strata with values on some presheaves. | Intersection Cohomology of S1-Actions on Pseudomanifolds | 12,570 |
We give in this paper an isomorphism theorem between derived functors over categories of modules.There is a nice class of categories that gives examples in which this theorem applies for a special construction. This leads us to a new interpretation of a theorem of Pirashvili and Richter about cyclic homology and Hochschild homology. | Changement de base pour les foncteurs Tor | 12,571 |
Let V(1) be the Smith-Toda complex at the prime 3. We prove that there exists a map v_2^9: \Sigma^{144}V(1) \to V(1) that is a K(2) equivalence. This map is used to construct various v_2-periodic infinite families in the 3-primary stable homotopy groups of spheres. | On the existence of the self map v_2^9 on the Smith-Toda complex V(1) at
the prime 3 | 12,572 |
Let G be a chordal graph, X(G) the complement of the associated complex arrangement and Gamma(G) the fundamental group of X(G). We show that Gamma(G) is a limit of colored braid groups over the poset of simplices of G. When G = G_T is the comparability graph associated with a rooted tree T, a case recently investigated by the first author, the result takes the following very simple form: Gamma(G_T) is a limit over T of colored braid groups. | Arrangements associated to chordal graphs and limits of colored braid
groups | 12,573 |
We show that the homology over a field of the space of free maps from the n-sphere to the n-fold suspension of X depends only on the cohomology algebra of X and compute it explicitly. We compute also the homology of the closely related labelled configuration space on the n-sphere with labels in X and of its completion, that depends only on the homology of X. In many but not all cases the homology of the configuration space coincides with the homology of the mapping space. In particular we obtain the homology of the unordered configuration spaces on a sphere. | Configuration spaces on the sphere and higher loop spaces | 12,574 |
Building on a classical solution to the pentagon equation, constructed earlier by the author and E.V. Martyushev and related to the flat geometry invariant under the group SL(2), we construct an algebraic complex corresponding to a triangulation of a three-manifold. In case if this complex is acyclic (which is confirmed by examples), we use it for constructing a manifold invariant . | SL(2)-solution of the pentagon equation and invariants of
three-dimensional manifolds | 12,575 |
The difference between the quadratic L-groups L_*(A) and the symmetric L-groups L^*(A) of a ring with involution A is detected by generalized Arf invariants. The special case A=Z[x] gives a complete set of invariants for the Cappell UNil-groups UNil_*(Z;Z,Z) for the infinite dihedral group D_{\infty}=Z_2*Z_2, extending the results of Connolly and Ranicki (math.AT/0304016) and Connolly and Davis. | Generalized Arf invariants in algebraic L-theory | 12,576 |
We investigate Gamma-cohomology of some commutative cooperation algebras E_*E associated with certain periodic cohomology theories. For KU and E(1), the Adams summand at a prime p, and for KO we show that Gamma-cohomology vanishes above degree 1. As these cohomology groups are the obstruction groups in the obstruction theory developed by Alan Robinson we deduce that these spectra admit unique E infinity structures. As a consequence we obtain an E infinity structure for the connective Adams summand. For the Johnson-Wilson spectrum E(n) with n > 0 we establish the existence of a unique E infinity structure for its I_n-adic completion. | $Γ$-cohomology of rings of numerical polynomials and $E_\infty$
structures on K-theory | 12,577 |
For any stratified pseudomanifold $X$ and any suitable action of the unit circle $S^1$ on $X$ preserving the strata and the local topological structure, the orbit space $B=X/S^1$ is again a stratified pseudomanifold and the orbit map $\pi/X\to B$ is a stratified morphism. For each perversity in $X$ this arrow induces a long exact sequence relating the intersection cohomologies of $X$ and $B$; we call it the Gysin sequence of $X$ induced by the action. The third term of the Gysin sequence is called the Gysin term; its cohomology depends on basic global and local data. Global data concerns the intersection cohomology of $B$ and the Euler class induced by the action; while local data is related to the Euler class of the links of the fixed strata. The cohomology of the Gysin term is calculated trhough a residual constructible sheaf in $B$ with support in the fixed points set. In the last modification of this paper, we improved the definition of the Euler class in intersection cohomology, which is made by recursion and implies the definition of an Euler perversity. We also verify that the Euler class is functorial for a suitable family of equivariant morphisms (i.e., not every equivariant morphism preserves the Euler class). | The Gysin Sequence for $S^1$-actions on stratified pseudomanifolds | 12,578 |
We use obstruction theory based on the unstable Adams spectral sequence to construct self maps of finite quaternionic projective spaces. As a result, a conjecture of Feder and Gitler regarding the classification of self maps up to homology is proved in two new cases. | Self maps of HP^n via the unstable Adams spectral sequence | 12,579 |
The paper examines machines of the type of the $\Gamma$-spaces of Segal which describe homotopy structures on topological spaces. The main result of the paper shows that for any such machine one can find an algebraic theory characterizing the same structure on spaces as the original machine. | From $Γ$-spaces to algebraic theories | 12,580 |
In 1983, C. McGibbon and J. Neisendorfer have given a proof for one conjecture in J.-P. Serre's famous paper (1953). In 1985, another proof was given by J. Lannes and L. Schwartz. Since then, one considers a more general conjecture: if the reduced mod 2 cohomology of any 1-connected polyGEM is of finite type and is not trivial, then it contains at least one element of infinite height, i.e., non nilpotent. This conjecture has been verified in several special situations, more precisely, by Y. Felix, S. Halperin, J.-M. Lemaire and J.-C. Thomas in 1987, by J. Lannes and L. Schwartz in 1988, and by J. Grodal in 1996. In this note, we construct an example, for which this conjecture fails. | Un 3-polyGEM de cohomologie modulo 2 nilpotente | 12,581 |
The mod p cohomology of a space comes with an action of the Steenrod Algebra. L. Schwartz [A propos de la conjecture de non realisation due a N. Kuhn, Invent. Math. 134, No 1, (1998) 211--227] proved a conjecture due to N. Kuhn [On topologicaly realizing modules over the Steenrod algebra, Annals of Mathematics, 141 (1995) 321--347] stating that if the mod $p$ cohomology of a space is in a finite stage of the Krull filtration of the category of unstable modules over the Steenrod algebra then it is locally finite. Nevertheless his proof involves some finiteness hypotheses. We show how one can remove those finiteness hypotheses by using the homotopy theory of profinite spaces introduced by F. Morel [Ensembles profinis simpliciaux et interpretation geometrique du foncteur T, Bull. Soc. Math. France, 124 (1996) 347--373], thus obtaining a complete proof of the conjecture. For that purpose we build the Eilenberg-Moore spectral sequence and show its convergence in the profinite setting. | Espaces profinis et problemes de realisabilite | 12,582 |
It is shown that Segal's theorem on the spaces of rational maps from CP^1 to CP^n can be extended to the spaces of continuous rational maps from CP^m to CP^n for any m less than or equal to n. The tools are the Stone-Weierstrass Theorem and Vassiliev's machinery of simplicial resolutions. | Spaces of rational maps and the Stone-Weierstrass Theorem | 12,583 |
Let G be a topological group such that its homology H(G) with coefficients in a principal ideal domain R is an exterior algebra, generated in odd degrees. We show that the singular cochain functor carries the duality between G-spaces and spaces over BG to the Koszul duality between modules up to homotopy over H(G) and H^*(BG). This gives in particular a Cartan-type model for the equivariant cohomology of a G-space. As another corollary, we obtain a multiplicative quasi-isomorphism C^*(BG) -> H^*(BG). A key step in the proof is to show that a differential Hopf algebra is formal in the category of A-infinity algebras provided that it is free over R and its homology an exterior algebra. | Koszul duality and equivariant cohomology | 12,584 |
We survey various approaches to axiomatic stable homotopy theory, with examples including derived categories, categories of (possibly equivariant or localized) spectra, and stable categories of modular representations of finite groups. We focus mainly on representability theorems, localisation, Bousfield classes, and nilpotence. | Axiomatic stable homotopy - a survey | 12,585 |
In this paper, we show that Goodwillie calculus, as applied to functors from stable homotopy to itself, interacts in striking ways with chromatic aspects of the stable category. Localized at a fixed prime p, let T(n) be the telescope of a v_n self map of a finite S--module of type n. The Periodicity Theorem of Hopkins and Smith implies that the Bousfield localization functor associated to T(n) is independent of choices. Goodwillie's general theory says that to any homotopy functor F from S--modules to S--modules, there is an associated tower under F, {P_dF}, such that F --> P_dF is the universal arrow to a d--excisive functor. Our first theorem says that P_dF --> P_{d-1}F always admits a homotopy section after localization with respect to T(n) (and so also after localization with respect to Morava K--theory K(n)). Thus, after periodic localization, polynomial functors split as the product of their homogeneous factors. This theorem follows from our second theorem which is equivalent to the following: for any finite group G, the Tate spectrum t_G(T(n)) is weakly contractible. This strengthens and extends previous theorems of Greenlees--Sadofsky, Hovey--Sadofsky, and Mahowald--Shick. The Periodicity Theorem is used in an essential way in our proof. The connection between the two theorems is via a reformulation of a result of McCarthy on dual calculus. | Tate cohomology and periodic localization of polynomial functors | 12,586 |
For finite coverings we elucidate the interaction between transferred Chern classes and Chern classes of transferred bundles. This involves computing the ring structure for the complex oriented cohomology of various homotopy orbit spaces. In turn these results provide universal examples for computing the stable Euler classes (i.e. Tr^*(1)) and transferred Chern classes for p-fold covers. Applications to the classifying spaces of p-groups are given. | Transfer and complex oriented cohomology rings | 12,587 |
We give a proof of the Jardine-Tillmann generalized group completion theorem. It is much in the spirit of the original homology fibration approach by McDuff and Segal, but follows a modern treatment of homotopy colimits, using as little simplicial technology as possible. We compare simplicial and topological definitions of homology fibrations. | Homology fibrations and "group-completion" revisited | 12,588 |
The first purpose of this paper is to examine the relationship between equivariant elliptic genera and orbifold elliptic genera. We apply the character theory of Hopkins et. al. to the Borel-equivariant genus associated to the sigma orientation of Ando-Hopkins-Strickland to define an orbifold genus for certain total quotient orbifolds and supersingular elliptic curves. We show that our orbifold genus is given by the same sort of formula as the orbifold ``two-variable'' genus of Dijkgraaf et al. In the case of a finite cyclic orbifold group, we use the characteristic series for the two-variable genus to define an analytic equivariant genus in Grojnowski's equivariant elliptic cohomology, and we show that this gives precisely the orbifold two-variable genus. The second purpose of this paper is to study the effect of varying the BU<6>-structure in the Borel-equivariant sigma orientation. We show that varying the BU<6>-structure by a class in the third cohomology of the orbifold group produces discrete torsion in the sense of Vafa. This result was first obtained by Sharpe, for a different orbifold genus and using different methods. | Discrete torsion for the supersingular orbifold sigma genus | 12,589 |
In this work, the continuously controlled assembly map in algebraic $K$-theory, as developed by Carlsson and Pedersen, is proved to be a split injection for groups $\Gamma$ that satisfy certain geometric conditions. The group $\Gamma$ is allowed to have torsion, generalizing a result of Carlsson and Pedersen. Combining this with a result of John Moody, $K_0(k\Gamma)$ is proved to be isomorphic to the colimit of $K_0(kH)$ over the finite subgroups $H$ of $\Gamma$, when $\Gamma$ is a virtually polycyclic group and $k$ is a field of characteristic zero. | Splitting With Continuous Control in Algebraic K-theory | 12,590 |
We apply a version of the Chas-Sullivan-Cohen-Jones product on the higher loop homology of a manifold in order to compute the homology of the spaces of continuous and holomorphic maps of the Riemann sphere into a complex projective space. This product makes sense on the homology of maps from a co-H space to a manifold, and comes from a ring spectrum. We also build a holomorphic version of the product for maps of the Riemann sphere into homogeneous spaces. In the continuous case we define a related module structure on the homology of maps from a mapping cone into a manifold, and then describe a spectral sequence that can compute it. As a consequence we deduce a periodicity and dichotomy theorem when the source is a compact Riemann surface and the target is a complex projective space. | Rational maps and string topology | 12,591 |
We give an explicit formula for the rational category of an elliptic space whose minimal model has a homogeneous-length differential. We also show that for such a space, there are no gaps in the sequence of integers realized as the rational Toomer invariant of some cohomology class. With an additional hypothesis, we show a result from which we deduce the relation dim(H^*(X;Q)) >= 2 cat_0(X). | The Rational Toomer Invariant and Certain Elliptic Spaces | 12,592 |
The notion of a cyclic map g: A -> X is a natural generalization of a Gottlieb element in pi_n(X). We investigate cyclic maps from a rational homotopy theory point of view. We show a number of results for rationalized cyclic maps which generalize well-known results on the rationalized Gottlieb groups. | Cyclic Maps in Rational Homotopy Theory | 12,593 |
Let f: X -> Y be a based map of simply connected spaces. The corresponding evaluation map w: map(X,Y;f) -> Y induces a homomorphism of homotopy groups whose image in pi_n(Y) is called the nth evaluation subgroup of f. The nth Gottlieb group of X occurs as the special case in which Y = X and f = 1_X. We identify the homomorphism induced on rational homotopy groups by this evaluation map, in terms of a map of complexes of derivations constructed using Sullivan minimal models. Our identification allows for the characterization of the rationalization of the nth evaluation subgroup of f. It also allows for the identification of several long exact sequences of rational homotopy groups, including the long exact sequence induced on rational homotopy groups by the evaluation fibration. As a consequence, we obtain an identification of the rationalization of the so-called G-sequence of the map f. This is a sequence--in general not exact--of groups and homomorphisms that includes the Gottlieb groups of X and the evaluation subgroups of f. We use these results to study the G-sequence in the context of rational homotopy theory. We give new examples of non-exact G-sequences and uncover a relationship between the homology of the rational G-sequence and negative derivations of rational cohomology. We also relate the splitting of the rational G-sequence of a fibre inclusion to a well-known conjecture in rational homotopy theory. | Rationalized Evaluation Subgroups of a Map and the Rationalized
G-Sequence | 12,594 |
We prove the toral rank conjecture of Halperin in some new cases. Our results apply to certain elliptic spaces that have a two-stage Sullivan minimal model, and are obtained by combining new lower bounds for the dimension of the cohomology and new upper bounds for the toral rank. The paper concludes with examples and suggestions for future work. | Free Torus Actions and Two-Stage Spaces | 12,595 |
We study the moduli space of instantons on a simply connected positive definite four manifold by analyzing the classifying map of the index bundle of a family of Dirac operators parametrized by the moduli space. As applications we compute the cohomology ring for the charge 2 moduli space in the rank stable limit. | Dirac operator coupled to instantons on positive definite 4 manifolds | 12,596 |
This is a survey paper, based on lectures given at the Workshop on "Structured ring spectra and their applications" which took place January 21-25, 2002, at the University of Glasgow. The term `Morita theory' is usually used for results concerning equivalences of various kinds of module categories. We focus on the covariant form of Morita theory, so the basic question is: When do two `rings' have `equivalent' module categories ? We discuss this question in different contexts and illustrate it by examples: (Classical) When are the module categories of two rings equivalent as categories ? (Derived) When are the derived categories of two rings equivalent as triangulated categories ? (Homotopical) When are the module categories of two ring spectra Quillen equivalent as model categories ? There is always a related question, which is in a sense more general: What characterizes the category of modules over a `ring' ? The answer is, mutatis mutandis, always the same: modules over a `ring' are characterized by the existence of a `small generator', which plays the role of the free module of rank one. The precise meaning of `small generator' depends on the context, be it an abelian category, a derived category or a stable model category. | Morita theory in abelian, derived and stable model categories | 12,597 |
Poincare duality lies at the heart of the homological theory of manifolds. In the presence of the action of a group it is well-known that Poincare duality fails in Bredon's ordinary, integer-graded equivariant homology. We give here a detailed account of one way around this problem, which is to extend equivariant ordinary homology to a theory graded on representations of fundamental groupoids. Versions of this theory have appeared previously for actions of finite groups, but this is the first account that works for all compact Lie groups. The first part of this work is a detailed discussion of RO(G)-graded ordinary homology and cohomology, collecting scattered results and filling in gaps in the literature. In particular, we give details on change of groups and products that do not seem to have appeared elsewhere. We also discuss the relationship between ordinary homology and cohomology when the group is compact Lie, in which case the two theories are not represented by the same spectrum. The remainder of the work discusses the extension to grading on representations of fundamental groupoids, concentrating on those aspects that are not simple generalizations of the RO(G)-graded case. These theories can be viewed as defined on parametrized spaces, and then the representing objects are parametrized spectra; we use heavily foundational work of May and Sigurdsson on parametrized spectra. We end with a discussion of Poincare duality for arbitrary smooth equivariant manifolds. | Equivariant ordinary homology and cohomology | 12,598 |
We give an explicit simplicial model for the Hopf map S^3 -> S^2. For this purpose, we construct a model of S^3 as a principal twisted cartesian product K x_{eta} S^2, where K is a simplicial model for S^1 acting by left multiplication on itself, S^2 is given the simplest simplicial model and the twisting map is eta:(S^2)_n -> (K)_{n-1}. We construct a Kan complex for the simplicial model K of S^1. The simplicial model for the Hopf map is then the projection K x_{eta} S^2 -> S^2. | A simplicial model for the Hopf map | 12,599 |
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