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It is well-known how to compute the structure of the second homotopy group of a space, $$X$$X, as a module over the fundamental group, $$\\pi _1X$$¿1X, using the homology of the universal cover and the Hurewicz isomorphism. We describe a new method to compute the third homotopy group, $$\\pi _3 X$$¿3X, as a module over $$\\pi _1 X$$¿1X. Moreover, we determine $$\\pi _3 X$$¿3X as an extension of $$\\pi _1 X$$¿1X-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 $$3$$3-spaces $$X = S^1 \\cup e^2 \\cup e^3$$X=S1¿e2¿e3 consisting of exactly one cell in each dimension $$\\le 3$$≤3.

The third homotopy group as a $$\pi _1$$ π 1 -module