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

  • Hans-Joachim Baues
  • Beatrice BleileEmail author
Original Paper
  • 95 Downloads

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 \(\le 3\).

Keywords

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

Mathematics Subject Classification

55Q05 

References

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    Whitehead, J.H.C.: Combinatorial homotopy II. Bull. AMS 55, 213–245 (1949)CrossRefzbMATHMathSciNetGoogle Scholar
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    Whitehead, J.H.C.: A certain exact sequence. Ann. Math. 52, 51–110 (1950)CrossRefzbMATHMathSciNetGoogle Scholar

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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