Cohomology of colorings of cycles

Abstract

We compute the cohomology groups of the spaces of colorings of cycles, i.e., of the prodsimplicial complexes Hom(C_m,K_n). We perform the computation first with Z_2, and then with integer coefficients. The main technical tool is to use spectral sequences in conjunction with a detailed combinatorial analysis of a family of cubical complexes, which we call torus front complexes. As an application of our method, we demonstrate how to collapse each connected component of Hom(C_m,C_n) onto a garland of cubes.