The third homotopy group as a \(\pi _1\)-module
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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 \(\le 3\).
KeywordsThird homotopy group as module over the fundamental group Whitehead’s Certain Exact Sequence Quadratic modules
Mathematics Subject Classification55Q05
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