Calculation of Fundamental Groups

  • Richard H. Crowell
  • Ralph H. Fox
Part of the Graduate Texts in Mathematics book series (GTM, volume 57)


It was remarked in Chapter II that a rigorous calculation of the fundamental group of a space X is rarely just a straightforward application of the definition of π(X). At this point the collection of topological spaces whose fundamental groups the reader can be expected to know (as a result of the theory so far developed in this book) consists of spaces topologically equivalent to the circle or to a convex set. This is not a very wide range, and the purpose of this chapter is to do something about increasing it. The techniques we shall consider are aimed in two directions. The first is concerned with what we may call spaces of the same shape. Figures 14, 15, and 16 are examples of the sort of thing we have in mind. From an understanding of the fundamental group as formed from the set of classes of equivalent loops based at a point, it is geometrically apparent that the spaces shown in Figure 14 below have the same, or isomorphic, fundamental groups.


Topological Space Fundamental Group Homotopy Type Inclusion Mapping Continuous Family 
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  1. 1.
    Most of the above spaces under the same figure are distinguishable from one another because of the fact that the dimension of a topological space in the neighborhood of a point is a topological invariant, i.e., under a homeomorphism the local dimension for any point is the same as that of its image under the homeomorphism. See W. Hurewicz and H. Wallman, Dimension Theory, (Princeton University Press, Princeton, New Jersey, 1948).zbMATHGoogle Scholar
  2. 2.
    A generalization is the definition of a deformation of X in a containing space Z as a family of mappings hs: X → Z, 0 ≤ s ≤ I, satisfying h 0(p) = p for all p in X and the condition of simultaneous continuity. The more restricted definition of deformation given above is suitable for pur purposes.Google Scholar
  3. 3.
    However, the fundamental groups of R 3K and R 3W may not be isomorphic (the torus W may be “horned”); See J. W. Alexander, “An Example of a Simply Connected Surface Bounding a Region which is not Simply Connected,” Proceedings of the National Academy of Sciences, Vol. 10 (1924), pp. 8–10.CrossRefGoogle Scholar
  4. 4.
    E. Moise, “Affine Structures in 3-Manifolds, V. The Triangulation Theorem and Hauptvermutung,” Ann. of Math. Vol. 56 (1952), pp. 96–114.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    For the usual definition of homotopy type see P. J. Hilton, An Introduction to Homotopy Theory, Cambridge Tracts in Mathematics and Mathematical Physics, No. 43 (Cambridge University Press, Cambridge, 1953).zbMATHGoogle Scholar
  6. 5a.
    For proof that Hilton’s definition is the same as ours see R. H. Fox, “On Homotopy Type and Deformation Retracts,” Ann. of Math. Vol. 44 (1943), pp. 40–50.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 6.
    E. R. van Kampen, “On the Connection between the Fundamental Groups of Some Related Spaces,” American Journal of Mathematics, Vol. 55 (1933), pp. 261–267;Google Scholar
  8. 6a.
    P. Olum, “Nonabelian Cohomology and van Kampen’s Theorem,” Ann. of Math., Vol. 68 (1958), pp. 658–668.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© R. H. Crowell and C. Fox 1963

Authors and Affiliations

  • Richard H. Crowell
    • 1
  • Ralph H. Fox
    • 2
  1. 1.Department of MathematicsDartmouth CollegeHanoverUSA
  2. 2.Princeton UniversityPrincetonUSA

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