Hirzebruch, F. Manifolds and modular forms. (Mannigfaltigkeiten und Modulformen.) (German) Zbl 0752.57013 Jahresbericht der DMV, Jubiläumstag., 100 Jahre DMV, Bremen/Dtschl. 1990, 20-38 (1992). [For the entire collection see Zbl 0743.00050.]This is a report on the theory of elliptic genera which were studied by the author, Landweber, Ochanine, Stong, Witten and others during the last years. The lecture starts with the definition and properties of the signature and the \(\hat A\)-genus and the related twisted genera. Then the equivariant signature theorem is stated for the case of the circle group \(S^ 1\) acting on a closed oriented smooth manifold \(X\) of dimension \(4k\). Let \(LX\) denote the loop space of \(X\) consisting of all smooth mappings of \(S^ 1\) to \(X\). \(LX\) admits a natural \(S^ 1\) action with fixed point set \(X\). In this case the author discusses the equivariant signature as it was proposed by Witten. This equivariant signature is a formal power series and the coefficients are twisted signatures which may be expressed by Pontryagin numbers of \(X\). The elliptic genus of \(X\) is defined by this power series. The author studies its properties and states some results on the divisibility of the twisted signatures by powers of 2. If \(X\) admits a smooth \(S^ 1\) action, the equivariant elliptic genus is defined. Results of the author and P. Slodowy [Geom. Dedicata 35, 309-343 (1990; Zbl 0712.57010)] and R. Bott and C. Taubes [J. Am. Math. Soc. 2, 137-186 (1989; Zbl 0667.57009)] are discussed. Reviewer: K.H.Mayer (Dortmund) Cited in 1 ReviewCited in 8 Documents MSC: 57R20 Characteristic classes and numbers in differential topology 57S15 Compact Lie groups of differentiable transformations 11F11 Holomorphic modular forms of integral weight 58D15 Manifolds of mappings 57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes Keywords:modular forms; elliptic genera; signature; \(\hat A\)-genus; twisted genera; equivariant signature theorem; circle group; loop space; Pontryagin numbers; divisibility of the twisted signatures; smooth \(S^ 1\) action; equivariant elliptic genus Citations:Zbl 0712.57010; Zbl 0667.57009; Zbl 0743.00050 PDFBibTeX XMLCite \textit{F. Hirzebruch}, in: Jahresbericht der Deutschen Mathematiker-Vereinigung. Jubiläumstagung: 100 Jahre DMV, Bremen, Deutschland, 17.-22. September 1990. Hauptvorträge. Stuttgart etc.: B.G. Teubner. 20--38 (1992; Zbl 0752.57013)