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Homology and Fixed Points

  • Andrzej Granas
  • James Dugundji
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

In this chapter we develop the algebraic and geometric notions needed to formulate and prove the main result, the Lefschetz-Hopf theorem for polyhedra. We further illustrate the use of homology by studying the special case of maps S n S n , showing that the Brouwer degree of a map not only completely characterizes its homotopy behavior, but also gives considerable information about special topological features that such a map may have. We come full circle with the beginning of the last chapter by deriving Borsuk’s antipodal theorem within this homological framework.

Keywords

Vector Field Exact Sequence Simplicial Complex Periodic Point Homology Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Andrzej Granas
    • 1
    • 2
  • James Dugundji
  1. 1.Département de Mathématiques et StatistiqueUniversité de MontréalMontréalCanada
  2. 2.Department of Mathematics and Computer ScienceUniversity of Warmia and MazuryOlsztynPoland

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