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).

arXiv comment27 pages, 4 figures