Manifolds and Modular Forms

Abstract

Volumes of the German-Ianguage subseries 'Äspekte der Mathematik" are listed at the end of the book.

Some Remarks on Circle Action on Manifolds

“…For more details on the following paragraphs in this subsection, we recommend the readers the references [10] or Appendix III of [11]. Let (M 2n , J) be an almost-complex manifold with first Chern class c 1 ∈ H 2 (M ; Z) divisible by a positive integer d > 1.…”

Functional Equations and the Generalised Elliptic Genus

“…as multiplication by q k (see details, for example, in [21]). The Atiyah-Bott fixed point theorem says that a genus φ of an almost complex manifold X with a compatible circle action is equal to the sum over all connected components X s of the fixed point set of expressions…”

“…Normalising the last expression by −y −1 we arrive as in [21,23] to Definition 1.1. For a stable almost complex manifold M 2n the equivariant χ y -genus of the loop space LM 2n is defined up to a normalisation by…”

“…A particular case of this will be the Ochanine genus, when g(x) is odd (which corresponds to ν being a half-period), and the functional equation becomes that studied by Hirzebruch [21]. The connection with (3) is made using the Jacobi triple product formula…”

“…In topology the German and Russian schools applied functional equations powerfully to formal group laws and genera [8,11,12,13,16,20,21,22,23]. They have arisen in the study of integrable systems in several different ways.…”