The Homotopy of Simplicial Algebras

Abstract

In [6] Walter Taylor investigated the relationship between the algebraic structure of a topological algebra A and the group structure of its fundamental group π1(A) and of the higher homotopy groups πn(A),n > 1. The main result is that a variety satisfies a group law λ in homotopy (that is, π1) if and only if every group in the idempotent reduct of obeys λ. (The relevant definitions are in [6] and also § 2 of this paper.) A similar result is stated for the higher homotopy groups. As Taylor points out in the introduction, the hard part of the theorem is constructing a topological algebra in whose fundamental group may fail to obey λ; indeed, in [6] this is only done in detail for the commutative law, and the proof is rather computational.