O ct 1 99 7 Effective generalized Seifert - Van Kampen : how to cal - culate

A central concept in algebraic topology since the 1970's has been that of delooping machine [4] [23] [29]. Such a " machine " corresponds to a notion of H-space, or space with a multiplication satisfying associativity, unity and inverse properties up to homotopy in an appropriate way, including higher order coherences as first investigated in [33]. A delooping machine is a specification of the extra homotopical structure carried by the loop space ΩX of a connected basepointed topological space X, exactly the structure allowing recovery of X by a " classifying space " construction. The first level of structure is that the component set π 0 (ΩX) has a structure of group π 1 (X, x). Classically the Seifert-Van Kampen theorem states that a pushout diagram of connected spaces gives rise to a pushout diagram of groups π 1. The loop space construction ΩX with its delooping structure being the higher-order " topologized " generalization of π 1 , an obvious question is whether a similar Seifert-Van Kampen statement holds for ΩX. The aim of this paper is to describe the operation underlying pushout of spaces with loop space structure, answering the above question by giving a Seifert-Van Kampen statement for delooping machinery. We work with Segal's machine [28] [36]. Our Seifert-Van Kampen statement is actually contained (in an n-truncated version) in [31]. In the present paper, we don't concentrate on the formal aspects of this, but rather on the aspect of effectivity. It turns out that the situation for higher homotopy is actually much better than for π 1 : one can effectively calculate the pushout of connected loop spaces (of course in the nonconnected case, i.e. when the component groups are nontrivial, one has the well-known effectivity problems for pushout of groups). Again, rather than concentrate on formal aspects we present this in an applied way as an algorithm to describe a finite cell complex representing the n-type of ΩX, for a given finite simply connected simplicial complex X. Iterating in a relatively obvious way gives an algorithm for calculating π i (X). At the end of the paper we briefly discuss the various formal aspects of the situation and possible generalizations to other delooping machines. The problem of giving an effective calculation of the π i (X) of a simply connected finite complex was first solved by E. Brown ([5], 1957)—and apparently also by A. Shapiro, unpublished. …