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, \(\pi _1X\), 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 \(\le 3\).
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Baues, HJ., Bleile, B. The third homotopy group as a \(\pi _1\)-module. AAECC 26, 165–189 (2015). https://doi.org/10.1007/s00200-014-0240-5
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DOI: https://doi.org/10.1007/s00200-014-0240-5
Keywords
- Third homotopy group as module over the fundamental group
- Whitehead’s Certain Exact Sequence
- Quadratic modules