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Quantum principal bundles and their characteristic classes

Mićo Đurđević (1997)

Banach Center Publications

A general theory of characteristic classes of quantum principal bundles is presented, incorporating basic ideas of classical Weil theory into the conceptual framework of noncommutative differential geometry. A purely cohomological interpretation of the Weil homomorphism is given, together with a geometrical interpretation via quantum invariant polynomials. A natural spectral sequence is described. Some interesting quantum phenomena appearing in the formalism are discussed.

Stratifications of teardrops

Bruce Hughes (1999)

Fundamenta Mathematicae

Teardrops are generalizations of open mapping cylinders. We prove that the teardrop of a stratified approximate fibration X → Y × ℝ with X and Y homotopically stratified spaces is itself a homotopically stratified space (under mild hypothesis). This is applied to manifold stratified approximate fibrations between manifold stratified spaces in order to establish the realization part of a previously announced tubular neighborhood theory.

The Mumford conjecture

Geoffrey Powell (2004/2005)

Séminaire Bourbaki

The Mumford Conjecture asserts that the rational cohomology of the stable moduli space of Riemann surfaces is a polynomial algebra on the Mumford-Morita-Miller characteristic classes; this can be reformulated in terms of the classifying space B Γ ∞ derived from the mapping class groups. The conjecture admits a topological generalization, inspired by Tillmann’s theorem that B Γ ∞ admits an infinite loop space structure after applying Quillen’s plus construction. The text presents the proof by Madsen and...

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