We consider a generalization of the van Kampen-Flores Theorem and relate it to the long-standing g-conjecture for simplicial spheres.

We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopy-theoretic version of the classical identification of covering spaces with locally constant sheaves. We also prove a new version of the classical result that...

If L is a finite lattice, we show that there is a natural topological lattice structure on the geometric realization of its order complex ∆(L) (definition recalled below). Lattice-theoretically, the resulting object is a subdirect product of copies of L. We note properties of this construction and of some variants, and pose several questions. For M3 the 5-element nondistributive modular lattice...

We introduce equivariant twisted cohomology of a simplicial set equipped with simplicial action of a discrete group and prove that for suitable twisting function induced from a given equivariant local coefficients, the simplicial version of Bredon-Illman cohomology with local coefficients is isomorphic to equivariant twisted cohomology. The main aim of this paper is to prove a classification th...

This paper lays the foundations of an approach to applying Gromov’s ideas on quantitative topology to topological data analysis. We introduce the “contiguity complex”, a simplicial complex of maps between simplicial complexes defined in terms of the combinatorial notion of contiguity. We generalize the Simplicial Approximation Theorem to show that the contiguity complex approximates the homotop...

There are very strong parallels between the properties of Mal’tsev and Jónsson-Tarski algebras, for example in the good behaviour of centrality and in the factorization of direct products. Moreover, the two classes between them include the majority of algebras that actually arise “in nature”. As a contribution to the research programme building a unified theory capable of covering the two class...

We develop an iterated homology theory for simplicial complexes. This theory is a variation on one due to Kalai. For 1 a simplicial complex of dimension d − 1, and each r = 0, . . . , d , we define r th iterated homology groups of 1. When r = 0, this corresponds to ordinary homology. If 1 is a cone over 1′, then when r = 1, we get the homology of 1′. If a simplicial complex is (nonpure) shellab...