The third homotopy group as a π₁-module

Abstract

It is well-known how to compute the structure of the second homotopy group of a space, X, as a module over the fundamental group π₁X, using the homology of the universal cover and the Hurewicz isomorphism. We describe a new method to compute the third homotopy group, π₃X as a module over π₁X. Moreover, we determine π₃X as an extension of π₁X-modules derived from Whitehead's Certain Exact Sequence. Our method is based on the theory of quadratic modules. Explicit computations are carried out for pseudo-projective 3-spaces X=S¹Ue²Ue³ consisting of exactly one cell in each dimension ≤ 3.