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.     

Splitting off rational parts in homotopy types

It is known algebraically that any abelian group is a direct sum of a divisible group and a reduced group (see Theorem 21.3 of [L. Fuchs, Infinite Abelian Groups, vol. I, Academic Press, New York-London, 1970]). In this paper, conditions to split off rational parts in homotopy types from a given space are studied in terms of a variant of Hurewicz map, say and generalised Gottlieb groups. This yields decomposition theorems on rational homotopy types of Hopf spaces, T-spaces and Gottlieb spaces, which has been known in various situations, especially for spaces with finiteness conditions.     

The coarse shape groups

The (pointed) coarse shape category Sh^* (), having (pointed) topological spaces as objects and having the (pointed) shape category as a subcategory, was recently constructed. Its isomorphisms classify (pointed) topological spaces strictly coarser than the (pointed) shape type classification. In this paper we introduce a new algebraic coarse shape invariant which is an invariant of shape and homotopy, as well. For every pointed space (X,?) and for every k∈N₀, the coarse shape group , having the standard shape group for its subgroup, is defined. Furthermore, a functor is constructed. The coarse shape and shape groups already differ on the class of polyhedra. An explicit formula for computing coarse shape groups of polyhedra is given. The coarse shape groups give us more information than the shape groups. Generally, does not imply (e.g. for solenoids), but from pro-πk(X,?)=0 follows . Moreover, for pointed metric compacta (X,?), the n-shape connectedness is characterized by , for every k?n.     

Homology rings of homotopy associative H-spaces

Let X be a homotopy associative mod p H-space for p an odd prime. The homology H_*(X;Fp) is an associative ring, but not necessarily commutative. We study conditions when for elements of H_*(X;Fp). Under certain conditions imply for l=p−2 or p−1. These methods can be used to prove results about homology commutators that were previously obtained using the adjoint action [H. Hamanaka, S. Hara, A. Kono, Adjoint action of Lie groups on the loop spaces and cohomology of exceptional Lie groups, Transform. Group Theory (1996) 44-50, Korea Adv. Inst. Sci. Tech.; A. Kono, K. Kozima, The adjoint action of a Lie group on the space of loops, J. Math. Soc. Japan 45 (3) (1993) 495-509; A. Kono, J. Lin, O. Nishimura, Characterization of the mod 3 cohomology of E₇, Proc. Amer. Math. Soc. 131 (10) (2003) 3289-3295]. We also generalize results of Kane [R. Kane, Torsion in homotopy associative H-spaces, Illinois J. Math. 20 (1976) 476-485] to nonfinite mod p homotopy associative H-spaces.     

Homomorphisms of homotopy coalgebras

In this paper we consider the theory of higher order homotopy coalgebras as a collection of spaces between co-H-spaces and suspensions, which dualizes Stasheff's theory of Ak-spaces when these are defined through Ak-structures. Moreover we extend two Berstein-Hilton theorems which deal with the primitive homotopy type of a suspension and the class of a suspension map, respectively.     

Homotopy monomorphisms and H-splitting in loop space fibrations

Let be a fibration, the holonomy action of this fibration and the connecting map. It is shown that if the fibre F admits an H-structure ν such that ρ?ν○(1×∂) (principal fibrations of all kinds satisfy such a condition), then i is a monomorphism if and only if it is weak monomorphism, the latter is equivalent to that Ωp has a homotopy right inverse Γ. If in addition Γ is an H-map, then ΩE has the same H-type as ΩB×ΩF.     

The relationship between the diagonal and the bar constructions on a bisimplicial set

The aim of this paper is to prove that the homotopy type of any bisimplicial set X is modelled by the simplicial set , the bar construction on X. We stress the interest of this result by showing two relevant theorems which now become simple instances of it; namely, the Homotopy colimit theorem of Thomason, for diagrams of small categories, and the generalized Eilenberg-Zilber theorem of Dold-Puppe for bisimplicial Abelian groups. Among other applications, we give an algebraic model for the homotopy theory of (not necessarily path-connected) spaces whose homotopy groups vanish in degree 4 and higher.     

Iterated homotopy fixed points for the Lubin-Tate spectrum

When G is a profinite group and H and K are closed subgroups, with H normal in K, it is not known, in general, how to form the iterated homotopy fixed point spectrum (ZhH)^hK/H, where Z is a continuous G-spectrum and all group actions are to be continuous. However, we show that, if G=Gn, the extended Morava stabilizer group, and , where is Bousfield localization with respect to Morava K-theory, En is the Lubin-Tate spectrum, and X is any spectrum with trivial Gn-action, then the iterated homotopy fixed point spectrum can always be constructed. Also, we show that is just , extending a result of Devinatz and Hopkins.     

Toric complexes and Artin kernels

A simplicial complex L on n vertices determines a subcomplex TL of the n-torus, with fundamental group the right-angled Artin group GL. Given an epimorphism χ:GL→Z, let be the corresponding cover, with fundamental group the Artin kernel Nχ. We compute the cohomology jumping loci of the toric complex TL, as well as the homology groups of with coefficients in a field k, viewed as modules over the group algebra kZ. We give combinatorial conditions for to have trivial Z-action, allowing us to compute the truncated cohomology ring, . We also determine several Lie algebras associated to Artin kernels, under certain triviality assumptions on the monodromy Z-action, and establish the 1-formality of these (not necessarily finitely presentable) groups.     

Localization and duality in topology and modular representation theory

We develop a duality theory for localizations in the context of ring spectra in algebraic topology. We apply this to prove a theorem in the modular representation theory of finite groups.Let G be a finite group and k be an algebraically closed field of characteristic p. If p is a homogeneous nonmaximal prime ideal in H^∗(G,k), then there is an idempotent module κ_p which picks out the layer of the stable module category corresponding to p, and which was used by Benson, Carlson and Rickard [D.J. Benson, J.F. Carlson, J. Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997) 59-80] in their development of varieties for infinitely generated kG-modules. Our main theorem states that the Tate cohomology is a shift of the injective hull of H^∗(G,k)/p as a graded H^∗(G,k)-module. Since κ_p can be constructed using a version of the stable Koszul complex, this can be viewed as a statement of localized Gorenstein duality in modular representation theory. Various consequences of this theorem are given, including the statement that the stable endomorphism ring of the module κ_p is the p-completion of cohomology , and the statement that κ_p is a pure injective kG-module.In the course of proving the theorem, we further develop the framework introduced by Dwyer, Greenlees and Iyengar [W.G. Dwyer, J.P.C. Greenlees, S. Iyengar, Duality in algebra and topology, Adv. Math. 200 (2006) 357-402] for translating between the unbounded derived categories and . We also construct a functor to the full stable module category, which extends the usual functor and which preserves Tate cohomology. The main theorem is formulated and proved in , and then translated to and finally to .The main theorem in can be viewed as stating that a version of Gorenstein duality holds after localizing at a prime ideal in H^∗(BG;k). This version of the theorem holds more generally for a compact Lie group satisfying a mild orientation condition. This duality lies behind the local cohomology spectral sequence of Greenlees and Lyubeznik for localizations of H^∗(BG;k).In a companion paper [D.J. Benson, Idempotent kG-modules with injective cohomology, J. Pure Appl. Algebra 212 (7) (2008) 1744-1746], a more recent and shorter proof of the main theorem is given. The more recent proof seems less natural, and does not say anything about localization of the Gorenstein condition for compact Lie groups.     

Evaluation maps in rational homotopy

In the rational category of nilpotent complexes, let E be an H-space acting on a space X. With mild hypotheses we show that the action on the base point factors through a map Γ_E:S_E→X, where S_E is a finite product of odd-dimensional spheres and Γ_E is a homotopy monomorphism. Among others, the following consequences are obtained: if and only if is essential and if and only if X satisfies a strong splitting condition.     

A connection between cellularization for groups and spaces via two-complexes

Let M denote a two-dimensional Moore space (so ), with fundamental group G. The M-cellular spaces are those one can build from M by using wedges, push-outs, and telescopes (and hence all pointed homotopy colimits). The issue we address here is the characterization of the class of M-cellular spaces by means of algebraic properties derived from the group G. We show that the cellular type of the fundamental group and homological information does not suffice, and one is forced to study a certain universal extension.     

A description of continued fraction expansions of quadratic surds represented by polynomials

The principal thrust of this investigation is to provide families of quadratic polynomials , where e_k²−f_k²C=n (for any given nonzero integer n) satisfying the property that for any , the period length of the simple continued fraction expansion of is constant for fixed k and lim_k→∞?_k=∞. This generalizes, and completes, numerous results in the literature, where the primary focus was upon |n|=1, including the work of this author, and coauthors, in Mollin (Far East J. Math. Sci. Special Vol. 1998, Part III, 257-293; Serdica Math. J. 27 (2001) 317) Mollin and Cheng (Math. Rep. Acad. Sci. Canada 24 (2002) 102; Internat Math J 2 (2002) 951) and Mollin et al. (JP J. Algebra Number Theory Appl. 2 (2002) 47).     

Hyperbolic spaces at large primes and a conjecture of Moore

A simply connected finite complex X is called elliptic if its rational homotopy Lie algebra is of finite dimension and hyperbolic otherwise. According to a conjecture of Moore, there exists an exponent for the p-torsion part of if and only if X is elliptic. In this note, it is shown that, provided the prime p is sufficiently large, a hyperbolic space with p-torsion free loop space homology has no exponent in the p-torsion of the homotopy groups. For a class of formal spaces, this result is obtained for every odd prime.     

Rationalized evaluation subgroups of a map II: Quillen models and adjoint maps

We identify the long exact sequence induced on rational homotopy groups by the evaluation map , and in particular the rationalization of the evaluation subgroups of f, in terms of derivations of Quillen models and adjoint maps. We consider a generalization of a question of Gottlieb within the context of rational homotopy theory. We also study the rationalization of the G-sequence of a map. In a separate result of independent interest, we give an explicit Quillen minimal model of a product A×X, in the case in which A is a rational co-H-space.     

Sheaf coalgebras and duality

In this paper we construct an adjoint pair of functors between the category of sheaves on a smooth manifold M and the category of coalgebras over the ring of smooth functions with compact support on M. We show that the sheaf coalgebra associated to a sheaf E on M determines the sheaf E uniquely up to an isomorphism, and that the adjoint functors restrict to an equivalence between sheaves on M and sheaf coalgebras over .     

Exposed sets in potential theory

Let U be a relatively compact open subset of a harmonic space, and H(U) be the function space of all continuous functions on which are harmonic on U. We give a complete characterization of the H(U)-exposed subsets of . This extends the results of [J. Lukeš, T. Mocek, M. Smr?ka, J. Spurný, Choquet like sets in function spaces, Bull. Sci. Math. 127 (2003) 397-437].