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Semisimpliziale Algebraische Topologie by Klaus Lamotke

Semisimpliziale Algebraische Topologie

by Klaus Lamotke
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Book cover type: Paperback
  • ISBN13: 9783662129890
  • Binding: Paperback
  • Subject: N/A
  • Publisher: Springer
  • Publisher Imprint: Springer
  • Publication Date:
  • Pages: 288
  • Original Price: EUR 56.07
  • Language: German
  • Edition: Softcover Repri
  • Item Weight: 422 grams
  • BISAC Subject(s): General

In diesem Buch werden einige Gebiete der algebraischen Topologie, die man heute grö tenteils zum klassischen Bestand rechnet, mit semi- simplizialen Methoden in einheitlicher Weise dargestellt. Der Begriff der semisimplizialen Menge ist dabei von grundlegender Bedeutung. Er wurde um 1950 von EILENBERG und ZILBER bei der Untersuchung der singulären Homologietheorie geprägt. Seine Nützlichkeit für die alge- braische Topologie, und zwar nicht nur für die Homologietheorie, erwies sich bald darauf durch die Arbeiten von DOLD, KAN, MACLANE, MOORE und POSTNIKOV. Durch sie wurde das vorliegende Buch angeregt. Die semisimpliziale Menge steht zwischen der Topologie und der Algebra. Einerseits ist ihre Struktur so "algebraisch", da man direkt Homologie-und Homotopiegruppen für sie definieren und allgemeine Zusammenhänge zwischen ihnen beweisen kann. Andererseits haben viele topologische Begriffe, wie z. B. die Faserung oder die Homotopie ein semisimpliziales Gegenstück. Der Zusammenhang zwischen der Topologie und der semisimplizialen Theorie beschränkt sich nicht auf diese Analogie: Es gibt einen Funktor S von der Kategorie der topo- logischen Räume in die Kategorie der semisimplizialen Mengen, der die topologischen Begriffe in die entsprechenden semisimplizialen über- führt. "Semisimpliziale algebraische Topologie" bedeutet am Beispiel der singulären Homologietheorie: Man ordnet dem Raum X seine semi- simpliziale Menge SX zu, definiert die Homologie von SX als singuläre Homologie des Raumes X und folgert die Eigenschaften der singulären Homologietheorie aus denen der Homologie semisimplizialer Mengen. In dieser Weise werden die Homotopietheorie, die Homologie-und Kohomologietheorie semisimplizial entwickelt.

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