The Third Homotopy Group as a pi_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, ${\pi }_{1}X$, 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 $\pi_1 X$\mathrm{.}Moreover,wedetermine$\pi_3 X$asanextensionof$\pi_1 X$-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 $\le 3$.