For any two graphs G and H Lovász has defined a cell complex Hom (G,H) having in mind the general program that the algebraic invariants of these complexes should provide obstructions to graph colorings. Here we announce the proof of a conjecture of Lovász concerning these complexes with G a cycle of odd length. More specifically, we show that If Hom (C2r+1, G) is k-connected, then χ(G) ≥ k + 4....

1. a) Compute the Stiefel-Whitney classes of the tangent bundle of RP . (Use the method from class for the tangent Chern classes of complex projectives spaces.) b) Conclude that if the tangent bundle is trivial, then n = 2 − 1 for some m. (In fact n must be 0, 1, 3, 7, but this is much harder to prove; one proof uses the Bott periodicity theorem.) c) Deduce (very easily!) a complete characteriz...

INTRODUCTION THE existence of a non-zero vector field on a differentiable manifold M yields geometric and algebraic information about M. For example, (I) A non-zero vector field exists on M if and only if the tangent bundle of M splits off a trivial bundle. (2) The kth Stiefel-Whitney class W,(M) of M is the primary obstruction to obtaining (n -k + I) linearly independent non-zero vector fields...

Proof. First consider the following example of the Conner-Floyd theorem. Let H,,,,,(C) denote a non-singular hypersurface of degree (1,l) in the product P,,,(C) x P.(C). b terms of homogeneous co-ordinates (wO, . . . , w,) and (z,,, . . . , .z”) with m 5 n this hypersurface can be defined as the locus w,,z, + wlzl + . . . + w,,,z, = 0. It can also be thought of as a P,_,(C)-bundle over P,(C).] ...

Two-dimensional (2D) Stiefel-Whitney (SW) insulator is a fragile topological state characterized by the second SW class in presence of space-time inversion symmetry. So far, SWIs have been proposed several electronic materials but seldom phononic systems. Here we recognize that large 2D buckled honeycomb crystals termed Xenes and their ligand-functionalized derivatives realize nontrivial topolo...

A generalization of classical theorems on the existence of sections of real, complex and quaternionic Stiefel manifolds over spheres is proved. We obtain a complete list of Lie group homomorphisms ρ : G → Gn, where Gn is one of the groups SO(n), SU(n), Sp(n) and G is one of the groups SO(k), SU(k), Sp(k), which reduce the structure group Gn in the fibre bundle Gn → Gn+1 → Gn+1/Gn.

We give a short elementary proof for a formula for the Hopf–Stiefel function that was found recently by Plagne via additive number theory. c © 2006 Elsevier Ltd. All rights reserved. The Hopf–Stiefel function ◦, introduced by Hopf [3] and Stiefel [8], plays an important role in additive number theory and bilinear algebra; see e.g. Plagne [6] for an overview and many references, for example [5] ...