On the algebra of cubes

Abstract

This is the first of two papers whose main purpose is to prove a generalisation of the Seifert-Van Kampen theorem on the fundamental group of a union of spaces. This generalisation (Theorem C of [8]) will give information in all dimensions and will include as special cases not only the above theorem (without the usual assumptions of path-connectedness) but also the Brouwer degree theorem (rc,S” = h); the relative Hurewicz theorem; Whitehead’s theorem that rrY and earlier work [5] of the authors on the case of dimension 2. The Seifert-Van Kampen theorem describes the fundamental group of a space X with base-point as, under certain circumstances, the colimit of the fundamental groups of subspaces whose interiors cover X. To generalise this to all dimensions we replace the space X by a filtered space