Effective generalized Seifert-Van Kampen: how to calculate $\Omega X$

Abstract

Suppose $X$ is a 1-connected simplicial set with finitely many nondegenerate simplices. We give an effective algorithm to calculate a simplicial set with the $n$-type of the loop space $\Omega X$. Iterating gives an algorithm to calculate the $\pi_i(X)$, different from the algorithms already known due to E. Brown and Kan-Curtis. The method is an effective version of the generalized Seifert-Van Kampen theorem of alg-geom/9704006. This can be viewed as a Van Kampen statement concerning the loop space $\Omega X$ with its delooping structure. We use Segal's delooping machinery but at the end we speculate on extensions to other delooping machinery.