Chromatic fracture cubes

In this note, we construct a general form of the chromatic fracture cube, using a convenient characterization of the total homotopy fiber, and deduce a decomposition of the E(n)-local stable homotopy category

Positive graphs

We study ‘‘positive’’ graphs that have a nonnegative homomorphism number into every edge-weighted graph (where the edge weights may be negative). We conjecture that all positive graphs can be obtained by taking two copies of an arbitrary simple graph and gluing them together along an independent set of nodes. We prove the conjecture for various classes of graphs including all trees. We prove a number of properties of positive graphs, including the fact that they have a homomorphic image which has at least half the original number of nodes but in which every edge has an even number of pre-images. The results, combined with a computer program, imply that the conjecture is true for all but one graph up to 10 nodes. © 2015 Elsevier Ltd.All rights reserved

On commensurable hyperbolic Coxeter groups

For Coxeter groups acting non-cocompactly but with finite covolume on real hyperbolic space Hn, new methods are presented to distinguish them up to (wide) commensurability. We exploit these ideas and determine the commensurability classes of all hyperbolic Coxeter groups whose fundamental polyhedra are pyramids over a product of two simplices of positive dimensions