Triadic van Kampen and Hurewicz Theorems∗

In [BH5] it is shown how the Relative Hurewicz Theorem follows from a Generalised Van Kampen Theorem (GVKT) for the fundamental crossed complex of a filtered space, and in [BL3] it is shown how a new multirelative Hurewicz Theorem follows from a GVKT for the fundamental cat-group of an n-cube of spaces. The purpose of this paper is to advertise and explain some implications and special cases of these GVKTs, and also to show how they came to be found. 1 Colimits of relative homotopy groups and the Relative Hurewicz Theorem Although the GVKT is stated in [BH2,BH5] for crossed complexes (over groupoids), it is an important point that the main content of the final result ([BH5] Theorem C) can be summarised as a theorem on relative homotopy groups considered as modules or crossed modules over the fundamental group. Recall that if P is a group then a crossed P -module consists of a group M , an action of P on M on the left, say, written (m, p) 7→ pm, and a morphism of groups μ : M → P satisfying the axioms: CM1) μ(pm) = pmp−1; CM2) mnm−1 = μmn; for all m,n ∈ M,p ∈ P . For background in examples and applications of crossed modules, see [BHu] and [B6]. ∗This is a LTEXversion, with updated references and endnotes, of the paper: R. Brown, “Triadic Van Kampen theorems and Hurewicz theorems”, Algebraic Topology, Proc. Int. Conf. Evanston March 1988, Edited M.Mahowald and S.Priddy, Cont. Math. 96 (1989) 39-57.