Let $M$ denote a two-dimensional Moore space (so $H_2(M; \Z) = 0$), 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 question we address here is to characterize 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 theorem of the title is deduced from the equivalence between crossed complexes and cubical $ω$-groupoids with connections proved by the authors in 1981. In fact we prove the equivalence of five categories defined internally to an additive category with kernels.
The aim is the theorems of the title and the corollary that the tensor product of two free crossed resolutions of groups or groupoids is also a free crossed resolution of the product group or groupoid. The route to this corollary is through the equivalence of the category of crossed complexes with that of cubical omega-groupoids with connections where the initial definition of the tensor product lies. It is also in the latter category that we are able to apply techniques of dense subcategories to identify the tensor product of covering morphisms as a covering morphism.
Connection matrices are one of the central tools in Conley's approach to the study of dynamical systems, as they provide information on the existence of connecting orbits in Morse decompositions. They may be considered a generalisation of the boundary operator in the Morse complex in Morse theory. Their computability has recently been addressed by Harker, Mischaikow, and Spendlove in the context of lattice filtered chain complexes. In the current paper, we extend the recently introduced Conley theory for combinatorial vector and multivector fields on Lefschetz complexes by transferring the con…
We define a ring R of geometric objects G generated by finite abstract simplicial complexes. To every G belongs Hodge Laplacian H as the square of the Dirac operator determining its cohomology and a unimodular connection matrix L). The sum of the matrix entries of the inverse of L is the Euler characteristic. The spectra of H as well as inductive dimension add under multiplication while the spectra of L multiply. The nullity of the Hodge of H are the Betti numbers which can now be signed. The map assigning to G its Poincare polynomial is a ring homomorphism from R the polynomials. Especially t…
We define the Coxeter cochain complex of a Coxeter group (G,S) with coefficients in a Z[G]-module A. This is closely related to the complex of simplicial cochains on the abstract simplicial complex I(S) of the commuting subsets of S. We give some representative computations of Coxeter cohomology and explain the connection between the Coxeter cohomology for groups of type A, the (singular) homology of certain configuration spaces, and the (Tor) homology of certain local Artin rings.
The first example of a non-residually finite group in the classes of finitely presented small-cancelation groups, automatic groups, and CAT(0) groups was constructed by Wise as the fundamental group of a complete square complex (CSC for short) with twelve squares. At the same time, Janzen and Wise proved that CSCs with at most three squares, five or seven squares have residually finite fundamental group. The smallest open cases were CSCs with four squares and directed complete VH complexes with six squares. We prove that the CSC with four squares studied by Janzen and Wise has a non-residually…
We prove a conjecture of Bahri, Bendersky, Cohen and Gitler: if K is a shifted simplicial complex on n vertices, X_1,..., X_n are spaces and CX_i is the cone on X_i, then the polyhedral product determined by K and the pairs (CX_i,X_i) is homotopy equivalent to a wedge of suspensions of smashes of the X_i's. This generalises earlier work of the two authors in the special case where each X_i is a loop space. Connections are made to toric topology, combinatorics, and classical homotopy theory.
The goal of this article is to make explicit a structured complex whose homology computes the cohomology of the p-profinite completion of the n-fold loop space of a sphere of dimension d=n-m<n. This complex is defined purely algebraically, in terms of characteristic structures of E_n-operads. Our construction involves: the free complete algebra in one variable associated to any E_n-operad; and an element in this free complete algebra, which is associated to a morphism from the operad of L-infinity algebras to an operadic suspension of our E_n-operad. We deduce our main theorem from: a connecti…
A simplicial complex is a set equipped with a down-closed family of distinguished finite subsets. This structure, usually viewed as codifying a triangulated space, is used here directly, to describe "spaces" whose geometric realisation can be misleading. An intrinsic homotopy theory, not based on such realisation but agreeing with it, is introduced. The applications developed here are aimed at image analysis in metric spaces and have connections with digital topology and mathematical morphology. A metric space X has a structure of simplicial complex at each (positive) resolution e; the resulti…
Group interactions occur frequently in social settings, yet their properties beyond pairwise relationships in network models remain unexplored. In this work, we study homophily, the nearly ubiquitous phenomena wherein similar individuals are more likely than random to form connections with one another, and define it on simplicial complexes, a generalization of network models that goes beyond dyadic interactions. While some group homophily definitions have been proposed in the literature, we provide theoretical and empirical evidence that prior definitions mostly inherit properties of homophily…
We express the hairy graph complexes computing the rational homotopy groups of long embeddings (modulo immersion) of R^m in R^n as "decorated" graph complexes associated to certain representations of the outer automorphism groups of free groups. This interpretation gives rise to a natural spectral sequence, which allows us to shed some light on the structure of the hairy graph cohomology. We also explain briefly the connection to the deformation theory of the little discs operads and some conclusions that this brings.
The notion of $\times$-homotopy from \cite{DocHom} is investigated in the context of the category of pointed graphs. The main result is a long exact sequence that relates the higher homotopy groups of the space $\Hom_*(G,H)$ with the homotopy groups of $\Hom_*(G,H^I)$. Here $\Hom_*(G,H)$ is a space which parametrizes pointed graph maps from $G$ to $H$ (a pointed version of the usual $\Hom$ complex), and $H^I$ is the graph of based paths in $H$. As a corollary it is shown that $π_i \big(\Hom_*(G,H) \big) \cong [G,Ω^i H]_{\times}$, where $ΩH$ is the graph of based closed paths in $H$ and $[G,K]_…
We introduce new methods for understanding the topology of $\Hom$ complexes (spaces of homomorphisms between two graphs), mostly in the context of group actions on graphs and posets. We view $\Hom(T,-)$ and $\Hom(-,G)$ as functors from graphs to posets, and introduce a functor $(-)^1$ from posets to graphs obtained by taking atoms as vertices. Our main structural results establish useful interpretations of the equivariant homotopy type of $\Hom$ complexes in terms of spaces of equivariant poset maps and $Γ$-twisted products of spaces. When $P = F(X)$ is the face poset of a simplicial complex $…
