On the universal coefficients formula for shape homology

In this paper it is investigated whether various shape homology theories satisfy the Universal Coefficients Formula (UCF). It is proved that pro-homology and strong homology satisfy UCF in the class FAB of finitely generated abelian groups, while they do not satisfy UCF in the class AB of all abelian groups. Two new shape homology theories (called UCF-balanced) are constructed. It is proved that balanced pro-homology satisfies UCF in the class AB, while balanced strong homology satisfies UCF only in the class FAB.     

Abelian groups with a vanishing homology group

An abelian group A is called absolutely abelian, if in every central extension N ? G ? A the group G is also abelian. The abelian group A is absolutely abelian precisely when the Schur multiplicator H₂A vanished. These groups, and more generally groups with H_nA = 0 for some n, are characterized by elementary internal properties. (Here H₁A denotes the integral homology of A.) The cases of even n and odd n behave strikingly different. There are 2^?ο different isomorphism types of abelian groups A with reduced torsion subgroup satisfying H_2nA = 0. The major tools are direct limit arguments and the Lyndon-Hochschild-Serre (L-H-S) spectral sequence, but the treatment of absolutely abelian groups does not use spectral sequences. All differentials d^r for r ≥ 2 in the L-H-S spectral sequence of a pure abelian extension vanish. Included is a proof of the folklore theorem, that homology of groups commutes with direct limits also in the group variable, and a discussion of the L-H-S spectral sequence for direct limits.     

Localization of André-Quillen-Goodwillie towers, and the periodic homology of infinite loopspaces

Let K(n) be the nth Morava K-theory at a prime p, and let T(n) be the telescope of a v_n-self map of a finite complex of type n. In this paper we study the K(n)_*-homology of Ω^∞X, the 0th space of a spectrum X, and many related matters.We give a sampling of our results.Let PX be the free commutative S-algebra generated by X: it is weakly equivalent to the wedge of all the extended powers of X. We construct a natural map

s_n(X):L_T(n)P(X)→L_T(n)Σ^∞(Ω^∞X)₊     

Multiplicity results for problems involving the Hardy-Sobolev operator via Morse theory

We establish some multiplicity results for a class of boundary value problems involving the Hardy-Sobolev operator using Morse theory.     

Nontrivial solutions for resonant cooperative elliptic systems via computations of the critical groups

In this paper, we consider a class of resonant cooperative elliptic systems. Based on some new results concerning the computations of the critical groups and the Morse theory, we establish some new results about the existence and multiplicity of solutions under new classes of conditions. It turns out that our main results sharply improve some known results in the literature.     

Nontrivial solutions of Kirchhoff-type problems via the Yang index

We obtain nontrivial solutions of a class of nonlocal quasilinear elliptic boundary value problems using the Yang index and critical groups.     

A remark on K-theory and S-categories

It is now well known that the K-theory of a Waldhausen category depends on more than just its (triangulated) homotopy category (Invent. Math. 150 (2002) 111). The purpose of this note is to show that the K-theory spectrum of a (good) Waldhausen category is completely determined by its Dwyer-Kan simplicial localization, without any additional structure. As the simplicial localization is a refined version of the homotopy category which also determines the triangulated structure, our result is a possible answer to the general question: “To which extent K-theory is not an invariant of triangulated derived categories? ”     

Structures de dérivabilité

We introduce a very general framework in which Quillen's theorems of existence, composition and adjunction for derived functors can be proved. We thus generalize and unify previous results by Dwyer, Hirschhorn, Kan and Smith, obtained in their formalism of “homotopical categories,” and by Radulescu-Banu in the context of Cisinski's “derivable categories.”     

Three models for the homotopy theory of homotopy theories

Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the “homotopy theory” of the model category. There is a model category structure on the category of simplicial categories, so taking its simplicial localization yields a “homotopy theory of homotopy theories”. In this paper we show that there are two different categories of diagrams of simplicial sets, each equipped with an appropriate definition of weak equivalence, such that the resulting homotopy theories are each equivalent to the homotopy theory arising from the model category structure on simplicial categories. Thus, any of these three categories with the respective weak equivalences could be considered a model for the homotopy theory of homotopy theories. One of them in particular, Rezk’s complete Segal space model category structure on the category of simplicial spaces, is much more convenient from the perspective of making calculations and therefore obtaining information about a given homotopy theory.     

Field Theoretic Cogalois Theory via Abstract Cogalois Theory

Abstract Cogalois Theory for arbitrary profinite groups was initiated by T. Albu and ?.A. Basarab [An Abstract Cogalois Theory for profinite groups, J. Pure. Appl. Algebra 200 (2005) 227-250]. The aim of this paper is twofold: firstly, to present the abstract group theoretic versions of various types of Kummer field extensions, and secondly, to show how some basic results of the (Field Theoretic) Cogalois Theory can be very easily deduced from their abstract versions.     

CoHochschild homology of chain coalgebras

Generalizing the work of Doi and of Idrissi, we define a coHochschild homology theory for chain coalgebras over any commutative ring and prove its naturality with respect to morphisms of chain coalgebras up to strong homotopy. As a consequence we obtain that if the comultiplication of a chain coalgebra C is itself a morphism of chain coalgebras up to strong homotopy, then the coHochschild complex admits a natural comultiplicative structure. In particular, if K is a reduced simplicial set and C_∗K is its normalized chain complex, then is naturally a homotopy-coassociative chain coalgebra. We provide a simple, explicit formula for the comultiplication on when K is a simplicial suspension.The coHochschild complex construction is topologically relevant. Given two simplicial maps g,h:K→L, where K and L are reduced, the homology of the coHochschild complex of C_∗L with coefficients in C_∗K is isomorphic to the homology of the homotopy coincidence space of the geometric realizations of g and h, and this isomorphism respects comultiplicative structure. In particular, there is an isomorphism, respecting comultiplicative structure, from the homology of to H_∗L|K|, the homology of the free loops on the geometric realization of K.     

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.     

Crossed modules as homotopy normal maps

In this note we consider crossed modules of groups (N→G, G→Aut(N)), as a homotopy version of the inclusion N⊂G of a normal subgroup. Our main observation is a characterization of the underlying map N→G of a crossed module in terms of a simplicial group structure on the associated bar construction. This approach allows for “natural” generalizations to other monoidal categories, in particular we consider briefly what we call “normal maps” between simplicial groups.     

On Galois cohomology of semisimple groups over local and global fields of positive characteristic

We treat a case that was omitted from consideration in our article [2] in Math Zeit, 2007.     

The second bounded cohomology of hypo-Abelian groups

Fujiwara [K. Fujiwara, The second bounded cohomology of a group with infinitely many ends, math.GR/9505208] conjectured that the second bounded cohomology of a group is zero or infinite-dimensional as a vector space over R. However, it is known that there are some linear groups for which the second bounded cohomology is not zero but finite-dimensional. In this paper, by using the transfinitely extended derived series, we prove that Fujiwara's conjecture is true for the hypo-Abelian groups, that is, groups with no non-trivial perfect subgroups.