Invariance of the Number of Holes

  • Reinhold Remmert
Part of the Graduate Texts in Mathematics book series (GTM, volume 172)


Is it intuitively clear that biholomorphically (more generally, topologically) equivalent domains have the same number of holes? There is no direct proof of this invariance theorem. The property of “having the same number of holes” is defined by how G lies in ℂ and at first glance is not an invariant of G. In order to prove the invariance of the number of holes, we assign every domain in ℂ its (first) homology group. The rank of this group, called the Betti number of G, is a biholomorphic (even topological) invariant of the domain.


Homology Group Betti Number Homology Class Group Homomorphism Free Abelian Group 
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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Reinhold Remmert
    • 1
  1. 1.Mathematisches InstitutWestfälische Wilhelms—Universität MünsterMünsterGermany

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