Serre homotopy theory in subcategories of simplicial groups

Abstract

If S⊆ Z is a multiplicative system and C is the class of the S-torsion abelian groups, we study Serre mod C homotopy theory in the subcategories of simplicial groups whose objects have trivial Moore complex in dimensions less than r and greater than n for 0≤r≤n. This is carried by giving a closed model structure in these categories and then studying the associated homotopy theory. When n→∞ we obtain the Serre homotopy theory for r-reduced simplicial groups studied by Quillen in Ann. Math. 90 (1969) 205–295. If S= Z−{0} and r=1 we have the corresponding rational homotopy theory. The case n=r+1 allows to consider Serre homotopy theory in categories of cat-groups or crossed modules of groups.