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, ${\pi }_{1}X$, 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 ${\pi }_{1}X$. 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 $\le 3$.

17 pages