Topology of bounded-degree graph complexes

The bounded-degree graph complexes were first introduced by Reiner and Roberts [J. Algebraic Combin. 11 (2000) 135–154]. They arise from the finite free resolution of quadratic Veronese rings and modules. We prove various results about the homotopy types of these complexes, and deduce corresponding characteristic-free results about the quadratic Veronese resolutions. In particular, we characterize the set of multidegrees which support at least one higher syzygy in this resolution. The answer turns out to be independent of the field characteristic.