A spectral sequence for string cohomology

Let X be a 1-connected space with free-loop space ΛX. We introduce two spectral sequences converging towards H^*(ΛX;Z/p) and H^*((ΛX)_hT;Z/p). The E₂-terms are certain non-Abelian-derived functors applied to H^*(X;Z/p). When H^*(X;Z/p) is a polynomial algebra, the spectral sequences collapse for more or less trivial reasons. If X is a sphere it is a surprising fact that the spectral sequences collapse for p=2.     

Euler homology

We geometrically construct a homology theory that generalizes the Euler characteristic mod 2 to objects in the unoriented cobordism ring of a topological space X. This homology theory Eh _* has coefficients in every nonnegative dimension. There exists a natural transformation that for X = pt assigns to each smooth manifold its Euler characteristic mod 2. The homology theory is constructed using cobordism of stratifolds, which are singular objects defined below. An isomorphism of graded -modules is shown for any CW-complex X. For discrete groups G, we also define an equivariant version of the homology theory Eh _*, generalizing the equivariant Euler characteristic.     

Some algebraic properties of cyclic homology groups

In [10], see also [8], a cyclic homology theory HC _* ^– was introduced. The purpose of this paper is to study algebraically the properties of this version of cyclic homology. First we study its relation to Connes cyclic cohomology theory HC ^* and to the usual cyclic homology theory HC _* studied by Loday and Quillen in [15]. We explain the precise sense in which HC _* ^– is dual to HC ^*. Next we study products and describe a general method for constructing product operations in cyclic homology and cohomology theories. Finally we examine the relation between the theory HC _* ^– and algebraic K-theory.     

Anderson Duality in K-Theory and Im(J)-Theory

We study Anderson duality for connective K-theory l and Im(J)-theory A, the latter being the p-localization of Quillen's algebraic K-theory of a suitably chosen finite field. The interplay between Anderson duality and Postnikov decomposition results in a construction of two families of natural transformations from ordinary cohomology theory to l^* and A^*-theory, which are related to the Chern character. These operations are then used to study l^*(l) and A^*(A), the cohomology operations of connective K-theory and Im(J)-theory.     

A homotopy theoretic realization of string topology

Let M be a closed, oriented manifold of dimension d. Let LM be the space of smooth loops in M. In [2] Chas and Sullivan defined a product on the homology H _* (LM) of degree -d. They then investigated other structure that this product induces, including a Batalin -Vilkovisky structure, and a Lie algebra structure on the S¹ equivariant homology H _* ^{S 1} (LM). These algebraic structures, as well as others, came under the general heading of the ”string topology” of M. In this paper we will describe a realization of the Chas-Sullivan loop product in terms of a ring spectrum structure on the Thom spectrum of a certain virtual bundle over the loop space. We also show that an operad action on the homology of the loop space discovered by Voronov has a homotopy theoretic realization on the level of Thom spectra. This is the ” cactus operad” defined in [6] which is equivalent to operad of framed disks in . This operad action realizes the Chas - Sullivan BV structure on H _* (LM). We then describe a cosimplicial model of this ring spectrum, and by applying the singular cochain functor to this cosimplicial spectrum we show that this ring structure can be interpreted as the cup product in the Hochschild cohomology, HH ^* (C ^* (M); C ^* (M)). Received: 31 July 2001 / Revised version: 11 September 2001 Published online: 5 September 2002     

REMARKS ON UNIVERSAL COEFFICIENT THEOREMS FOR GENERALIZED HOMOLOGY THEORIES

Abstract

New proofs of universal coefficient theorems for generalized homology theories (cf. ∮ 2, ∮ 3) including L. G. Brown's result, relating Brown-Douglas-Fillmore's Ext (X) with complex K-theory are presented. They are all based on a theorem asserting the existence of a chain functor for a generalized homology theory (cf. ∮ 1), which was originally designed for the construction of strong homology theories on strong shape categories.     

Cochain model for thickenings and its application to rational LS-category

A thickening of a finite CW-complex X is by definition a compact manifold M of the same simple homotopy type as X. We give a model for the cochain complex of the boundary of that manifold, C ^*(δM), as a module over the cochain algebra C ^*(X). We also show how to construct an algebraic model of the rational homotopy type of δC ^*(M) from a model of X. Using this rational model, we prove a new formula for the rational Lusternik–Schnirelmann category of X. Received: 24 September 1999     

Bordisme de S1-fibrés vectoriels munis¶de structures supplémentaires

Bordism of S ¹-vector bundles with additional structures We give isomorphisms between equivariant bordism groups of certain S ¹-vector bundles and bordism groups of suitable “classifying” spaces determined by certain caracterestic classes. In the spinorial case, we detect the even or odd type of the S ¹-action and give a relationship with elleptic homology. Furthermore, we define a new type of $S^1$-actions, depending on the actions and the given slice type. This new type differs, in certain cases, from the classical odd or even type of S ¹-actions on spinorial manifolds. Received: 7 July 2000 / Revised version: 10 February 2001     

Generalized homology theories and chain complexes

Summary To each generalized homology theory h_* defined on a category of topological spacesK (definition 1.2)a chain functorC _*:K ch (=category of chain complexes) (cf. definition 2.1) is established, which is related to h _* (definition 2.4,theorem 8.1).In subsequent papers this result is used for the construction of a strong homology theory (i.e. an analogue of the Steenrod-Sitnikov homology theory for general topological spaces) cf. [4].To G.S. ogovili on the occasion of his 75th birthday     

Domain decomposition and splitting methods for Mortar mixed finite element approximations to parabolic equations

Summary We introduce in this article a new domain decomposition algorithm for parabolic problems that combines Mortar Mixed Finite Element methods for the space discretization with operator splitting schemes for the time discretization. The main advantage of this method is to be fully parallel. The algorithm is proven to be unconditionally stable and a convergence result in (Δt/h ^1/2) is presented.     

On galois descent for Hochschild and cyclic homology

LetG be a finite group acting by automorphisms on an algebraS over some commutative ringk. We show that if the action ofG restricted to the center ofS is Galois in the sense of [C-H-R], thenHH _*(S ^G)≊HH _* (S) ^G. An analogous result holds for cyclic homology, provided the order ofG is invertible ink. The author was supported in part by a grant from the NSF.     

Strong homology of inverse systems of spaces. II

Recently the authors have defined a coherent prohomotopy category of topological spaces CPHTop [5]. In the present paper, which is a sequel to Part I [6], the authors define a strong homology functor H^s:CPHTop→Ab. The results of this paper are essential for the construction of a Steenrod-Sitnikov homology theory for arbitrary spaces.     

On the reissner-naghdi elasticity relationships : PMM vol. 38, n 6, 1974, pp. 1063–1071

Shell theory equations are constructed by the method in [1] to the accuracy of quantities of the order of h_*^2+k, where and k = 2−4t for (h_* is the relative semithickness of the shell and t is the index of the state of stress variation). Without being within the framework of the Lovetype theory, the equations obtained are compared with the Reissner-Naghdi equations. [2, 3] in which the transverse shear is taken into account, and it is shown that from the asymptotic viewpoint these latter are inconsistent. It is also shown that if the shell resists shear weakly, then from the asymptotic viewpoint the Reissner-Naghdi theory is completely well founded.The three-dimensional equations of elasticity theory are reduced to two-dimensional equations in [1] by using an asymptotic method, i.e. all members of the same order relative to the small parameter h_* are taken into account at each stage of the calculations. It has been shown that without going outside the framework of the ordinary concepts of the Love-type theory of shells (in particular, without taking account of transverse shear), the shell theory equations can be constructed to the accuracy of quantities of the order of h^2−2t_*, but it is impossible to exceed this limit without a qualitative complication in the theory.     

An algebraic model of fibration with the fiberK(π,n)-space

For a fibration with the fiberK(,n)-space, the algebraic model as a twisted tensor product of chains of the base with standard chains ofK(,n)-complex is given which preserves multiplicative structure as well. In terms of this model the action of then-cohomology of the base with coefficients in on the homology of fibration is described.     

The bar spectral sequence converging toh *(SO(2n+1))

We study the bar spectral sequence converging toh _*(SO(2n+1)), whereh is an algebra theory overBP. The differentials are determined completely ifh=P(l) andn<2 ^l . These results will be used in a future paper on the MoravaK-theories ofSO(2n+1), with no restriction onn. As another application, we determineBP _*(Spin(7)) including much of its algebra structure.AMS Subject Classification: 57T10, 57T30, 55N22     

Periodic equivariant realK-theories have rational Tate theory

Summary We study a generalized equivariantK-theory introduced by M. Karoubi. We prove, that it is anRO (G, U)-graded cohomology-theory and that the associated Tate spectrum is rational whenG is finite. This implies that for finite groups, the Atiyah-Segal Real equivariantK-theories have rational Tate theory. This article was processed by the author using the L^AT_EX style filecljour1 from Springer-Verlag     

Mod 3 cohomology algebras of finite H-spaces

Let X be a 3 local, finite, simply connected H-space with associative homology ring . Some known examples are the Lie group , Harper's H-space X(3) and any odd dimensional sphere . We prove the cohomology algebra is isomorphic to the cohomology algebra of a finite product of and odd dimensional spheres. Received: 15 May 2001; in final form: 22 May 2001 / Published online: 28 February 2002     

Gamma homology,Lie representations and <Emphasis Type="Italic">E</Emphasis><Subscript>∞</Subscript> multiplications

We prove that the stable homotopy of any Γ-module F is the homology of a bicomplex Ξ(F), in which the (q−1)st row is the two-sided bar construction ℬ(Lie^* _q ,Σ _q ,F[q]). This gives a natural homotopical cotangent bicomplex for graded commutative algebras, in a form suitable for use in a new obstruction theory for classifying E _∞ ring structures on spectra. The E _∞ structure on certain Lubin-Tate spectra is a corollary. Oblatum 15-X-2001 & 14-X-2002?Published online: 24 February 2003     

Singuläre definition der äquivarianten Bredon homologie

Let G be a discrete group,o(G) the orbit category of G and M:o(G)a a covariant (contravariant) functor to abelian groups. We define a singular equivariant homology theory H_*(X;M) (resp. H^*(X;M)) which satisfies a dimension axiom, in the sense of Bredon (Lecture notes 34). It turns out, that all fundamental properties of these theories directly follow by naturality from the analogous theorems in the classical non equivariant case.