Resumen de Cohomology of colorings of cycles

Dmitry N. Kozlov

• We compute the cohomology groups of the spaces of colorings of cycles, i.e., of the prodsimplicial complexes ${\tt Hom}(C_m,K_n)$. We perform the computation first with ${\Bbb Z}_2$, 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 {\it torus front complexes}. As an application of our method, we demonstrate how to collapse each connected component of ${\tt Hom}(C_m,C_n)$ onto a~garland of cubes.