Manifolds and modular forms

Includes index

Includes bibliography (p. 199-204) and references

ch. 1 Background 1.Cobordism Theory 2.Characteristic classes 3.Pontrjagin classes of quaternionic projective spaces 4.Characteristic classes and invariants 5.Representations and vector bundles 6.Multiplicative sequences and genera 7.Calculation of φ(Pj(H)) 8.Complex genera ch. 2 Elliptic genera 1.The Weierstraβ function 2.Construction of elliptic genera 3.An excursion on the lemniscate 4.Geometric complement on the addition theorem ch. 3 A universal addition theorem for genera 1.Virtual submanifolds 2.A universal genus ch. 4 Multiplicativity in fibre bundles 1.The signature and the L-genus 2.Algebraic preliminaries 3.Topological preliminaries 4.The splitting principle 5.Integration over the fibre 6.Multiplicativity and strict multiplicativity ch. 5 The Atiyah-Singer Index Theorem 1.Elliptic operators and elliptic complexes Contents note continued: 2.The Index of an elliptic complex 3.The de Rham complex 4.The Dolbeault complex 5.The signature as an Index 6.The equivariant Index 7.The equivariant Xy-genus for S1-actions 8.The equivariant signature for S1-actions ch. 6 Twisted operators and genera 1.Motivation for elliptic genera after Ed Witten 2.The expansion at the cusp 0 3.The Witten genus 4.The Witten genus and the Lie group Es 5.Plumbing of manifolds ch. 7 Riemann-Roch and elliptic genera in the complex case 1.Elliptic genera of level N 2.The values at the cusps 3.The equivariant case and multiplicativity 4.The loop space and the expansion at a cusp 5.The differential equation 6.The modular curve ch. 8 A divisibility theorem for elliptic genera 1.The theorem of Ochanine 2.Proof of Ochanine's theorem

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.