The third homotopy group as a $\pi _1$-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, $\pi _1X$, using the homology of the universal cover and the Hurewicz isomorphism. We describe a new method to compute the third homotopy group, $\pi _3 X$, as a module over $\pi _1 X$. Moreover, we determine $\pi _3 X$ as an extension of $\pi _1 X$-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 $X = S^1 \cup e^2 \cup e^3$ consisting of exactly one cell in each dimension $\le 3$.