  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 $\leq 3$.
