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Manifolds and modular forms. Transl. by Peter S. Landweber. (English) Zbl 0767.57014
Aspects of Mathematics. E 20. Wiesbaden: Friedr. Vieweg. xi, 211 p. (1992).
In the first author’s famous book ‘Neue topologische Methoden in der algebraischen Geometrie’ (1956; Zbl 0070.163) a genus was defined as a homomorphism from the oriented cobordism ring to the ring of integers, and methods were developed for obtaining and calculating such genera. Extending this to the case when the target ring is an algebraic function field, a genus can be defined using doubly periodic meromorphic functions. Evaluating it on a manifold gives an elliptic function; allowing the underlying elliptic curve also to vary gives a modular function. The notion extends to a genus of level $$N$$, which is defined on complex manifolds, and yields modular forms for the appropriate congruence subgroup. Of particular interest are the evaluations of such forms at the cusps. This gives a rich theory with numerous formulae arising from the topology on one side and the theories of elliptic and modular functions on the other. The development was originally motivated by ideas of Witten, applying classical formulae for genera and equivariant indices to the loop space on a manifold. Applications to topology include a study of multiplicativity of genera for fibre bundles, divisibility theorems and a rigidity theorem for circle actions on certain complex manifolds.
The book is an elaborated version of notes on lectures given by the first author in 1987/88 together with 4 appendices providing background and completing the theory in some respects. Although each section is clearly written, this piecemeal approach is somewhat unsatisfactory as the main part of the book relies heavily on Appendix 1, and ignores developments later than early 1988; there is also much repetition between sections. Nevertheless the book is very readable and provides a good introduction to this fascinating theory.

##### MSC:
 57R20 Characteristic classes and numbers in differential topology 57R77 Complex cobordism ($$\mathrm{U}$$- and $$\mathrm{SU}$$-cobordism) 57S15 Compact Lie groups of differentiable transformations 11F11 Holomorphic modular forms of integral weight 33E05 Elliptic functions and integrals