Manifolds and Modular Forms

  • Friedrich Hirzebruch
  • Thomas Berger
  • Rainer Jung

Part of the Aspects of Mathematics book series (ASMA, volume 20)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Friedrich Hirzebruch, Thomas Berger, Rainer Jung
    Pages 1-21
  3. Friedrich Hirzebruch, Thomas Berger, Rainer Jung
    Pages 23-33
  4. Friedrich Hirzebruch, Thomas Berger, Rainer Jung
    Pages 35-40
  5. Friedrich Hirzebruch, Thomas Berger, Rainer Jung
    Pages 41-56
  6. Friedrich Hirzebruch, Thomas Berger, Rainer Jung
    Pages 57-72
  7. Friedrich Hirzebruch, Thomas Berger, Rainer Jung
    Pages 73-95
  8. Friedrich Hirzebruch, Thomas Berger, Rainer Jung
    Pages 97-112
  9. Friedrich Hirzebruch, Thomas Berger, Rainer Jung
    Pages 113-120
  10. Back Matter
    Pages 121-212

About this book

Introduction

During the winter term 1987/88 I gave a course at the University of Bonn under the title "Manifolds and Modular Forms". I wanted to develop the theory of "Elliptic Genera" and to learn it myself on this occasion. This theory due to Ochanine, Landweber, Stong and others was relatively new at the time. The word "genus" is meant in the sense of my book "Neue Topologische Methoden in der Algebraischen Geometrie" published in 1956: A genus is a homomorphism of the Thorn cobordism ring of oriented compact manifolds into the complex numbers. Fundamental examples are the signature and the A-genus. The A-genus equals the arithmetic genus of an algebraic manifold, provided the first Chern class of the manifold vanishes. According to Atiyah and Singer it is the index of the Dirac operator on a compact Riemannian manifold with spin structure. The elliptic genera depend on a parameter. For special values of the parameter one obtains the signature and the A-genus. Indeed, the universal elliptic genus can be regarded as a modular form with respect to the subgroup r (2) of the modular group; the two cusps 0 giving the signature and the A-genus. Witten and other physicists have given motivations for the elliptic genus by theoretical physics using the free loop space of a manifold.

Keywords

Signatur algebra manifold material

Authors and affiliations

  • Friedrich Hirzebruch
    • 1
  • Thomas Berger
    • 1
  • Rainer Jung
    • 1
  1. 1.Max-Planck-Institut für MathematikBonnGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-663-10726-2
  • Copyright Information Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH, Wiesbaden 1994
  • Publisher Name Vieweg+Teubner Verlag, Wiesbaden
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-528-16414-0
  • Online ISBN 978-3-663-10726-2
  • Series Print ISSN 0179-2156
  • About this book
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