ar X iv : m at h / 04 09 06 8 v 2 [ m at h . G N ] 5 J ul 2 00 5 THE COMBINATORICS OF τ - COVERS

ar X iv : m at h / 04 09 06 8 v 3 [ m at h . G N ] 1 7 M ay 2 00 7 THE COMBINATORICS OF τ - COVERS

We solve four out of the six open problems concerning critical cardi-nalities of topological diagonalization properties involving τ-covers, show that the remaining two cardinals are equal, and give a consistency result concerning this remaining cardinal. Consequently, 21 open problems concerning potential implications between these properties are settled. We also give structural results based o...

ar X iv : m at h / 06 07 13 0 v 1 [ m at h . A G ] 5 J ul 2 00 6 TWISTED LOOP GROUPS AND THEIR AFFINE FLAG VARIETIES

ar X iv : m at h / 06 09 68 5 v 2 [ m at h . G T ] 2 1 D ec 2 00 6 EQUIVARIANT COVERS FOR HYPERBOLIC GROUPS

We prove an equivariant version of the fact that word-hyperbolic groups have finite asymptotic dimension. This is important in connection with our forthcoming proof of the Farrell-Jones conjecture for K∗(RG) for every word-hyperbolic group G and every coefficient ring R.

ar X iv : m at h / 04 06 24 5 v 2 [ m at h . N T ] 1 4 M ar 2 00 5 1 Graphical patterns in quadratic residues

Plots of quadratic residues display some visual features that are analyzed mathematically in this paper. The graphical patterns of quadratic residues modulo m depend only on the residues of m modulo the lowest positive integers.

ar X iv : m at h / 04 09 02 9 v 1 [ m at h . A G ] 2 S ep 2 00 4 ACM BUNDLES ON GENERAL HYPERSURFACES IN P 5 OF LOW DEGREE

In this paper we show that on a general hypersurface of degree r = 3, 4, 5, 6 in P 5 a rank 2 vector bundle E splits if and only if h 1 E(n) = h 2 E(n) = 0 for all n ∈ Z. Similar results for r = 1, 2 were obtained in [15], [16] and [1].