We prove that $\Omega S^n_{(2)}, S^n\{2^r\}$, and $\Omega^2 S^n_{(2)}$,are minimal atomic spaces for appropriate values of $n$. We do this by using secondary and tertiary cohomology operations to prove that, above the Hurewicz dimension, no elements in the mod 2 homology of the cited spaces are in the image of the Hurewicz homomorphism. In the case of ­$\Omega^2 S^n$, we construct and exploit an appropriate filtration to facilitate the use of higher order cohomology operations. An appendix consisting of an examination of the coefficients in Adams’ factorization is included. 1.

We define support varieties in an axiomatic setting using the prime spectrum of a lattice of ideals. A key observation is the functoriality of the spectrum and that this functor admits an adjoint. We assign to each ideal its support and can classify ideals in terms of their support. Applications arise from studying abelian or triangulated tensor categories. Specific examples from algebraic geometry and modular representation theory are discussed, illustrating the power of this approach which is inspired by recent work of Balmer. Contents

This paper contains some basic results on 2-groupoids, with special emphasis on computing derived mapping 2-groupoids between 2-groupoids and proving their invariance under strictification. Some of the results proven here are presumably folklore (but do not appear in the literature to the author’s knowledge) and some of the results seem to be new. The main technical tool used throughout the paper is the Quillen model structure on the category of 2-groupoids introduced by Moerdijk and Svensson.

The model structure on the category of chain functors $\mathfrak{Ch}$, developed in [4], has the main features of a simplicial model category structure, taking into account the lack of arbitrary (co-)limits in $\mathfrak{Ch}$. After an appropriate tensor and cotensor structure in $\mathfrak{Ch}$ is established ($\S 1, \S 3$), Quillen’s axiom SM7 is verified in $\S 5$ and $\S 6$. Moreover, it turns out that in the definition of a simplicial model structure, the category of simplicial sets can be replaced by the category of simplicial spectra endowing $\mathfrak{Ch}$ with the structure of an approximate simplicial stable model structure (= approximate ss-model structure) ($\S 7$). In $\S 8$ the model structure on $\mathfrak{Ch}$ is shown to be proper.

In this work, we explain the relations among braided regular crossed modules, simplicial groups, 2-crossed modules, quadratic modules and crossed squares, and the role of hypercrossed complex pairings in these structures.

Suppose given a Frobenius category $\mathcal{E}$, i.e. an exact category with a big enough subcategory $\mathcal{B}$ of bijectives. Let $\mathcal{E} := \mathcal{E/B}$ denote its classical stable category. For example, we may take $\mathcal{E}$, to be the category of complexes $C(\mathcal{A})$ with entries in an additive category $\mathcal{A}$, in which case $\underline{\mathcal{E}}$ is the homotopy category of complexes $K(\mathcal{A})$. Suppose given a finite poset $D$ that satisfies the combinatorial condition of being ind-flat. Then, given a diagram of shape $D$ with values in $\underline{\mathcal{E}}$ (i.e. stably commutative), there exists a diagram consisting of pure monomorphisms with values in $\mathcal{E}$ (i.e. commutative) that is isomorphic, as a diagram with values in $\underline{\mathcal{E}}$, to the given diagram.

We study the nested collection of left ideals of $\mathcal{A}$, the mod 2 Steenrod algebra, $L(k) : = \mathcal{A}\{Sq^{2^0}, Sq^{2^1}, Sq^{2^2}, ... , Sq^{2^k}$. We determine the smallest k such that $Sq^n \in l (k)$. We discuss an application which improves upon the results of F. R. Cohen and the first author in their paper comparing the loop of the degree 2 map on a sphere and the H-space squaring map on the loop of a sphere.

We compute the characteristic polynomials of the posets of hypertrees. We show that the generating series of the polynomials can be expressed using cyclic hypertrees. We also propose a conjecture on the action of symmetric groups on the Whitney homology of these posets. In addition, we show that Vallette’s poset of pointed partitions is homotopy equivalent to Pitman’s poset of forests. The implicit common theme of these topics is the combinatorics of the PreLie operad.

On calcule les polynômes caractéristiques des posets des hyperarbres. On montre que la série génératrice de ces polynômes fait intervenir les hyperarbres cycliques. On donne aussi uneconjecture pour l’action du groupe symétrique sur l’homologie de Whitney de ces posets. Par ailleurs, on montre que le poset des partitions pointées de Vallette est équivalent homotopiquement au poset des forêts d’arbres enracinés de Pitman. Le thème commun implicite à tous ces objets est la combinatoirede l’opérade PreLie.

A di-embedding of the $n$-cube $I^n$ into $\mathbb{r}^n$ is a map $I ^n\to \mathbb{R}^n$ which is a dihomeomorphism onto its image.We show that such a map is, up to a permutation of coordinates, an $n$-fold product of 1-dimensional orientation preserving embeddings $I^1 \to \mathbb{R}$.

We want to investigate ‘spaces’ where paths have a ‘weight’, or ‘cost’, expressing length, duration, price, energy, etc. The weight function is not assumed to be invariant up to pathreversion. Thus, ‘weighted algebraic topology’ can be developed as an enriched version of directed algebraic topology, where illegal paths are penalised with an infinite cost, and the legal ones are measured. Its algebraic counterpart will be ‘weighted algebraic structures’, equipped with a sort of directed seminorm.

In the fundamental weighted category of a generalised metric space, introduced here, each homotopy class of paths has a weight (or seminorm), which is subadditive with respect to composition. We also study a more general setting, spaces with weighted paths, which has finer quotients and strong links with noncommutative geometry. Weighted homology of weighted cubical sets has already be developed in a previous work, with similar results.

Over an associative ring we consider a class $\mathbb{X}$ of left modules which is closed under set-indexed coproducts and direct summands. We investigate when the triangulated homotopy category $\mathsf{K}(\mathbb{X})$ is compactly generated, and give a number of examples.

The classical Stallings-Stammbach five-term exact sequence associated with a short exact sequence of groups is shown to exist in a general categorical context. This makes it possible to establish a new result relating low-dimensional homology and central series, which extends a well known result from the theory of groups.

We prove that if $M$ is a CW-complex, then the homotopy type of the skeletal filtration of $M$ does not depend on the cell decomposition of $M$ up to wedge products with $n$-disks Dn, when the latter are given their natural CW-decomposition with unique cells of order $0, (n - 1$) and $n$, a result resembling J.H.C. Whitehead’s work on simple homotopy types. From the higher homotopy van Kampen Theorem (due to R. Brown and P.J. Higgins) follows an algebraic analogue for the fundamental crossed complex II$(M)$ of the skeletal filtration of $M$, which thus depends only on the homotopy type of $M$ (as a space) up to free product with crossed complexes of the type $\mathcal{D}^n \doteq \Pi (D^n), n \in \mathbb{N}$.This expands an old result (due to J.H.C. Whitehead) asserting that the homotopy type of $\Pi(M)$ depends only on the homotopy type of M. We use these results to define a homotopy invariant $I_{\mathcal{A}}$ of CW-complexes for each finite crossed complex $\mathcal{A}$. We interpret it in terms of the weak homotopy type of the function space $TOP ((M, *), (|\mathcal{A}|, *))$ where $|\mathcal{A}|$ is the classifying space of the crossed complex $\mathcal{A}$.

We determine the 2-adic $K$-localizations for a large class of $H$-spaces and related spaces. As in the odd primary case, these localizations are expressed as fibers of maps between specified infinite loop spaces, allowing us to approach the 2-primary $v_1$-periodic homotopy groups of our spaces. The present $v_1$-periodic results have been applied very successfully to simply-connected compact Lie groups by Davis, using knowledge of the complex, real, and quaternionic representations of the groups. We also functorially determine the united 2-adic $K$-cohomology algebras (including the 2-adic $KO$-cohomology algebras) for all simply-connected compact Lie groups in terms of their representation theories, and we show the existence of spaces realizing a wide class of united 2-adic $K$-cohomology algebras with specified operations.