We show that the
n-th power of the first Stiefel-Whitney class of the ℤ
2-action on the graph complex Hom(
C
2r+1,
K
n+2) is zero, confirming a conjecture by Babson and Kozlov. This yields a considerably simplified proof of their graph colouring
theorem, which is also known as the Lovsz conjecture.
This research was supported by the Deutsche Forschungsgemeinschaft within the European graduate program “ Combinatorics, Geometry,
and Computation” (No. GRK 588/2)