Algebraic properties of classes of path ideals of posets

We consider path ideals associated to special classes of posets such as tree posets and cycles. We express their property of being sequentially Cohen–Macaulay in terms of the underlying poset. Moreover, monomial ideals, which arise in algebraic statistics from the Luce-decomposable model and the ascending model, can be viewed as path ideals of certain posets. We study invariants of these so-called Luce-decomposable monomial ideals and ascending ideals for diamond posets and products of chains. In particular, for these classes of posets, we explicitly compute their Krull dimension, their projective dimension, their Castelnuovo–Mumford regularity and their Betti numbers.     

The reconstruction conjecture and edge ideals

Given a simple graph G on n vertices, we prove that it is possible to reconstruct several algebraic properties of the edge ideal from the deck of G, that is, from the collection of subgraphs obtained by removing a vertex from G. These properties include the Krull dimension, the Hilbert function, and all the graded Betti numbers β_i,j where j<n. We also state many further questions that arise from our study.     

The t-singular homology of orbifolds

For an orbifold M we define a new homology group, called t-singular homology group t-Hq(M) by using singular simplicies intersecting ‘transversely’ with ΣM. The rightness of this homology group is ensured by the facts that the 1-dimensional homology group t-H₁(M) is isomorphic to the abelianization of the orbifold fundamental group π₁(M,x₀). If M is a manifold, t-Hq(M) coincides with the usual singular homology group. We prove that it is a ‘b-homotopy’ invariant of orbifolds and develop many algebraic tools for the calculations. Consequently we calculate the t-singular homology groups of several orbifolds.     

Algebraic models for equivariant homotopy theory over Abelian compact Lie groups

We describe some algebraic models for equivariant rational and p-adic homotopy theory over Abelian compact Lie groups. Received: 12 February 2001; in final form: 15 August 2001 / Published online: 28 February 2002     

Homotopy theory of associative rings

A kind of unstable homotopy theory on the category of associative rings (without unit) is developed. There are the notions of fibrations, homotopy (in the sense of Karoubi), path spaces, Puppe sequences, etc. One introduces the notion of a quasi-isomorphism (or weak equivalence) for rings and shows that—similar to spaces—the derived category obtained by inverting the quasi-isomorphisms is naturally left triangulated. Also, homology theories on rings are studied. These must be homotopy invariant in the algebraic sense, meet the Mayer-Vietoris property and plus some minor natural axioms. To any functor X from rings to pointed simplicial sets a homology theory is associated in a natural way. If X=GL and fibrations are the GL-fibrations, one recovers Karoubi-Villamayor's functors KVi, i>0. If X is Quillen's K-theory functor and fibrations are the surjective homomorphisms, one recovers the (non-negative) homotopy K-theory in the sense of Weibel. Technical tools we use are the homotopy information for the category of simplicial functors on rings and the Bousfield localization theory for model categories. The machinery developed in the paper also allows to give another definition for the triangulated category kk constructed by Cortiñas and Thom [G. Cortiñas, A. Thom, Bivariant algebraic K-theory, preprint, math.KT/0603531]. The latter category is an algebraic analog for triangulated structures on operator algebras used in Kasparov's KK-theory.     

Second order homological obstructions on real algebraic manifolds

Let Y be a compact nonsingular real algebraic set whose homology classes (over Z/2) are represented by Zariski closed subsets. It is well known that every smooth map from a compact smooth manifold to Y is unoriented bordant to a regular map. In this paper, we show how to construct smooth maps from compact nonsingular real algebraic sets to Y not homotopic to any regular map starting from a nonzero homology class of Y of positive degree. We use these maps to obtain obstructions to the existence of local algebraic tubular neighborhoods of algebraic submanifolds of Rn and to study some algebro-homological properties of rational real algebraic manifolds.     

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.     

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.     

A continuum with no prime shape factors

As the main result of this paper, we prove that there exists a continuum with non-trivial shape without any prime factor. This answers a question of K. Borsuk [K. Borsuk, Concerning the notion of the shape of compacta, in: Proc. Internat. Symposium on Topology and Its Applications, Herceg-Novi, 1968, pp. 98-104]. We also show that for each integer n?3 there exists a continuum X such that Sh(X,x)=Shn(X,x), but Sh(X,x)≠Shn−1(X,x). Therefore we obtain the negative answer to another question of K. Borsuk [K. Borsuk, Some remarks concerning the shape of pointed compacta, Fund. Math. 67 (1970) 221-240]: Does Sh(X,x)=Shn(X,x), for a compactum X and some integer n?3, implie that Sh(X,x)=Sh²(X,x)?     

Some algebraic aspects of mesoprimary decomposition

Recent results of Kahle and Miller give a method of constructing primary decompositions of binomial ideals by first constructing “mesoprimary decompositions” determined by their underlying monoid congruences. Monoid congruences (and therefore, binomial ideals) can present many subtle behaviors that must be carefully accounted for in order to produce general results, and this makes the theory complicated. In this paper, we examine their results in the presence of a positive A-grading, where certain pathologies are avoided and the theory becomes more accessible. Our approach is algebraic: while key notions for mesoprimary decomposition are developed first from a combinatorial point of view, here we state definitions and results in algebraic terms, which are moreover significantly simplified due to our (slightly) restricted setting. In the case of toral components (which are well-behaved with respect to the A-grading), we are able to obtain further simplifications under additional assumptions. We also provide counterexamples to two open questions, identifying (i) a binomial ideal whose hull is not binomial, answering a question of Eisenbud and Sturmfels, and (ii) a binomial ideal I for which $I_{\mathrm{toral}}$ is not binomial, answering a question of Dickenstein, Miller and the first author.     

On the γ-filtration of oriented cohomology of complete spin-flags

We study the characteristic map of algebraic oriented cohomology of complete spin-flags and the ideal of invariants of formal group algebra. As an application, we provide an annihilator of the torsion part of the γ-filtration. Moreover, if the formal group law determined by the oriented cohomology theory is congruent to the additive formal group law modulo 2, then at degree 2 and 3, we show that the γ-filtration of complete spin-flags is torsion free.     

Derivation radical subspace arrangements

In this note we study modules of derivations on collections of linear subspaces in a finite dimensional vector space. The central aim is to generalize the notion of freeness from hyperplane arrangements to subspace arrangements. We call this generalization ‘derivation radical’. We classify all coordinate subspace arrangements that are derivation radical and show that certain subspace arrangements of the Braid arrangement are derivation radical. We conclude by proving that under an algebraic condition the subspace arrangement consisting of all codimension c intersections, where c is fixed, of a free hyperplane arrangement are derivation radical.     

Isomorphism Conjecture for homotopy K-theory and groups acting on trees

We discuss an analogon to the Farrell-Jones Conjecture for homotopy algebraic K-theory. In particular, we prove that if a group G acts on a tree and all isotropy groups satisfy this conjecture, then G satisfies this conjecture. This result can be used to get rational injectivity results for the assembly map in the Farrell-Jones Conjecture in algebraic K-theory.     

The realization space of a Π-algebra: a moduli problem in algebraic topology

A Π-algebra A is a graded group with all of the algebraic structure possessed by the homotopy groups of a pointed connected topological space. We study the moduli space R(A) of realizations of A, which is defined to be the disjoint union, indexed by weak equivalence classes of CW-complexes X with , of the classifying space of the monoid of self homotopy equivalences of X. Our approach amounts to a kind of homotopical deformation theory: we obtain a tower whose homotopy limit is R(A), in which the space at the bottom is BAut(A) and the successive fibres are determined by Π-algebra cohomology. (This cohomology is the analog for Π-algebras of the Hochschild cohomology of an associative ring or the André-Quillen cohomology of a commutative ring.) It seems clear that the deformation theory can be applied with little change to study other moduli problems in algebra and topology.     

The Balmer spectrum of rational equivariant cohomology theories

The category of rational G-equivariant cohomology theories for a compact Lie group G is the homotopy category of rational G-spectra and therefore tensor-triangulated. We show that its Balmer spectrum is the set of conjugacy classes of closed subgroups of G, with the topology corresponding to the topological poset of [7]. This is used to classify the collections of subgroups arising as the geometric isotropy of finite G-spectra. The ingredients for this classification are (i) the algebraic model of free spectra of the author and B. Shipley [14], (ii) the Localization Theorem of Borel–Hsiang–Quillen [21] and (iii) tom Dieck's calculation of the rational Burnside ring [4].     

Preserving asphericity of virtually nilpotent spaces under P-localization

For a set of prime numbers P, we study when the P-localization of an Eilenberg-Mac Lane space with virtually nilpotent fundamental group is again aspherical. While investigating this problem, we devote special attention to infra-nilmanifolds.We prove that the P-localization of an orientable infra-nilmanifold is aspherical if and only if its holonomy group is P-torsion. The same holds for non-orientable infra-nilmanifolds if 2 is in P. We also develop computational techniques to check preservation of asphericity. These are explicitly applied to show that the P-localization of a non-orientable infra-nilmanifold of dimension ?3 is always aspherical. We point out that this is no longer true from dimension 4 onwards.     

Cellular covers of groups

In this paper we discuss the concept of cellular cover for groups, especially nilpotent and finite groups. A cellular cover is a group homomorphism c:G→M such that composition with c induces an isomorphism of sets between and . An interesting example is when G is the universal central extension of the perfect group M. This concept originates in algebraic topology and homological algebra, where it is related to the study of localizations of spaces and other objects. As explained below, it is closely related to the concept of cellular approximation of any group by a given fixed group. We are particularly interested in properties of M that are inherited by G, and in some cases by properties of the kernel of the map c.