Non-abelian cohomology and Seifert-Van Kampen theorem

Abstract

The first goal of the present paper it to present a simple and elementary proof of the standard Seifert-van Kampen theorem based on ideas of P. Olum. The key tool is the singular cohomology theory with non-abelian coefficients in dimensions 0 and 1. After this we apply non-abelian cohomology to prove Crowell-Fox version of Seifert-van Kampent theorem and its improvement die to Brown-Salleh (this proof shows that the Lebesgue dimension of the disc is irrelevant for this improvement). Finally, we apply non-abelian cohomology to prove some theorem of van Kampen, which are more general than the standard version (and, in particular, include the computation of the fundamental group of the circle).