The third homotopy group as a \(\pi _1\)-module

  • Hans-Joachim Baues
  • Beatrice BleileEmail author
Original Paper


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 \(\le 3\).


Third homotopy group as module over the fundamental group  Whitehead’s Certain Exact Sequence Quadratic modules 

Mathematics Subject Classification



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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Max Planck Institut für MathematikBonnGermany
  2. 2.School of Science and TechnologyUniversity of New EnglandArmidaleAustralia

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