Higher elliptic genera

We show that elliptic classes introduced in our earlier paper for spaces with infinite fundamental groups yield Novikov's type higher elliptic genera which are invariants of K-equivalence. This include, as a special case, the birational invariance of higher Todd classes studied recently by J.Rosenberg and J.Block-S.Weinberger. We also prove the modular properties of these genera, show that they satisfy a McKay correspondence, and consider their twist by discrete torsion.Comment: Comments on lemma 3.1 deleted. Final versio

Zariski-van Kampen theorem for higher homotopy groups

This paper gives an extension of the classical Zariski-van Kampen theorem describing the fundamental groups of the complements of plane singular curves by generators and relations. It provides a procedure for computation of the first non-trivial higher homotopy groups of the complements of singular projective hypersurfaces in terms of the homotopy variation operators introduced here.Comment: 37 pages, LaTeX2e with amsmath, amsthm and amscd packages. To appear in J. Inst. Math. Jussieu (2003) with the first proof of Theorem 7.1 significantly developped and new references added. Due to copyright restrictions, this final version will only be available at Cambridge Journals Online (http://journals.cambridge.org) when published. Thus the content of the paper here is the same as that of version 1 of 3 March 200