The Third Homotopy Group As a $$\pi _1$$ Π 1 -Module

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 $$\le 3$$ .