On the homology spectral sequence for topological Hochschild homology

Marcel Bökstedt has computed the homotopy type of the topological Hochschild homology of using his definition of topological Hochschild homology for a functor with smash product. Here we show that easy conceptual proofs of his main technical result of are possible in the context of the homotopy theory of -algebras as introduced by Elmendorf, Kriz, Mandell and May. We give algebraic arguments based on naturality properties of the topological Hochschild homology spectral sequence. In the process we demonstrate the utility of the unstable ``lower' notation for the Dyer-Lashof algebra.

     

On K3 of Witt Vectors of Length Two Over Finite Fields

We prove that for W₂ the Witt vectors of length two over the finite field , we have in characteristic at least 5 and for (3,f) = 1. The result is proved by using the identity and calculating the right term with a group homology spectral sequence. Some information on the spectral sequence is achieved by using the action of the outer automorphism of SL on the homology groups and recent results on K-groups of local rings and the ring of dual numbers over finite fields.     

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.     

Über Konvergenz von Spektralfolgen der stabilen Homotopietheorie

We discuss the problem of convergence of spectral sequences that arise from a filtration of a spectrum in Boardman's stable homotopy category by applying a generalized homology, homotopy or cohomology theory. The criteria we get give e.g. the convergence of the Adams spectral sequence for a generalized homology theory in certain cases (using similar methods this equestion has been considered independently by J. F. Adams in his forthcoming Chicago lecture notes), and some results on the Adams cohomology spectral sequence including the well-known convergence properties in case of singular cohomology with Z_p-coefficients and complex cobordism.     

The Theory of Local Homology for Artinian Modules

In this paper, the theory of local homology for Artinian modules is developed, which in some aspect is similar to the theory of local cohomology. There are given some criteria for local homology modules to be good and dual of the ideal transform functor is studied.AMS Subject Classification (1991): 13E10 13D45     

The Brown-Peterson cohomology of the classifying spaces of the projective unitary groups and exceptional Lie groups

For a fixed prime , we compute the Brown-Peterson cohomologies of classifying spaces of and exceptional Lie groups by using the Adams spectral sequence. In particular, we see that and are even dimensionally generated.

     

The Hochschild cohomology of a closed manifold

Let M be a closed orientable manifold of dimension d and be the usual cochain algebra on M with coefficients in a field k. The Hochschild cohomology of M, is a graded commutative and associative algebra. The augmentation map induces a morphism of algebras . In this paper we produce a chain model for the morphism I. We show that the kernel of I is a nilpotent ideal and that the image of I is contained in the center of , which is in general quite small. The algebra is expected to be isomorphic to the loop homology constructed by Chas and Sullivan. Thus our results would be translated in terms of string homology.     

Equivariant homotopical homology with coefficients in a Mackey functor

Let M be a Mackey functor for a finite group G. In this paper, generalizing the Dold-Thom construction, we construct an ordinary equivariant homotopical homology theory with coefficients in M, whose values on the category of finite G-sets realize the bifunctor M, both covariantly and contravariantly. Furthermore, we extend the contravariant functor to define a transfer in the theory for G-equivariant covering maps. This transfer is given by a continuous homomorphism between topological abelian groups.We prove a formula for the composite of the transfer and the projection of a G-equivariant covering map and characterize those Mackey functors M for which that formula has an expression analogous to the classical one.     

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.