Homological Domination in Large Random Simplicial Complexes

In this paper we state the homological domination principle for random multi-parameter simplicial complexes, claiming that the Betti number in one specific dimension (which is explicitly determined by the probability multi-parameter) significantly dominates the Betti numbers in all other dimensions. We also state and discuss evidence for two interesting conjectures which would imply a stronger version of the homological domination principle, namely that generically homology of a random simplicial complex coincides with that of a wedges of k-dimensional spheres. These two conjectures imply that under an additional assumption (specified in the paper) a random simplicial complex collapses to a k-dimensional complex homotopy equivalent to a wedge of spheres of dimension k.Comment: 8 pages, 1 figur

Finite domination and Novikov homology over strongly Z-graded rings

Let L be a strongly Z-graded ring, and let C be a bounded chain complex of finitely generated L-modules. We give a homological characterisation of when C is homotopy equivalent, over L_0, to a bounded complex of finitely generated projective L_0-modules, generalising known results for twisted Laurent polynomial rings.Comment: 22 pages; v2: changed example in introduction, and corrected minor misprint

Simplicial volume with Fp\mathbb{F}_p-coefficients

For primes $p$, we investigate an $\mathbb{F}_p$-version of simplicial volume and compare these invariants with their siblings over other coefficient rings. We will also consider the associated gradient invariants, obtained by stabilisation along finite coverings. Throughout, we will discuss the relation between such simplicial volumes and Betti numbers.Comment: 21 pages; v2: small changes; to appear in Period. Math. Hun

On stability of non-domination under taking products

We show that non-domination results for targets that are not dominated by products are stable under Cartesian products.Comment: 6 page