The Homotopy Groups of the Inverse Limit of a Tower of Fibrations

We carefully present an elementary proof of the well known theorem that each homotopy group (or, in degree zero, pointed set) of the inverse limit of a tower of fibrations maps naturally onto the inverse limit of the homotopy groups (or, in degree zero, pointed sets) of the spaces in the tower, with kernel naturally isomorphic to lim of the tower of homotopy groups of one dimension higher. The main theorem here is Theorem 2.1. This is due to Gray [G], Quillen [Q, Proposition 3.8], Vogt [V], Cohen [C1,C2], and Bousfield and Kan [BK, Theorem 3.1 in Chapter IX, section 3], some of which only consider the case of simply connected spaces or spaces with abelian fundamental groups. The proof given here is at least morally the one in [C2]. 1. lim of a tower of not necessarily abelian groups This definition is as in [V] and [BK, Chapter IX, section 2].