Celebratio Mathematica

P. F. Baum, P. M. Hajac, R. Matthes, and W. Szymański: “The \( K \)-theory of Heegaard-type quantum 3-spheres,” \( K \)-Theory 35 : 1–2 (2005), pp. 159–186. Dedicated to the memory of Olaf Richter. An erratum to this was published in K-Theory 37:1–2 (2006). MR 2240219 Zbl 1111.46051 article

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P. F. Baum, P. M. Hajac, R. Matthes, and W. Szymański: “Erratum: ‘The \( K \)-theory of Heegaard-type quantum 3-spheres’,” \( K \)-Theory 37 : 1–2 (2006), pp. 211. Erratum to an article published in K-Theory 35:1–2 (2005). MR 2274673 Zbl 1210.46054 article

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Piotr Mieczysław Hajac	Related
Wojciech Szymański	Related

P. Baum, P. M. Hajac, R. Matthes, and W. Szymanski: Non-commutative geometry approach to principal and associated bundles. Preprint, 2007. ArXiv math/0701033v2 techreport

We recast basic topological concepts underlying differential geometry using the language and tools of noncommutative geometry. This way we characterize principal (free and proper) actions by a density condition in (multiplier) \( C^* \)-algebras. We introduce the concept of piecewise triviality to adapt the standard notion of local triviality to fibre products of \( C^* \)-algebras. In the context of principal actions, we study in detail an example of a non-proper free action with continuous translation map, and examples of compact principal bundles which are piecewise trivial but not locally trivial, and neither piecewise trivial nor locally trivial, respectively. We show that the module of continuous sections of a vector bundle associated to a compact principal bundle is a cotensor product of the algebra of functions defined on the total space (that are continuous along the base and polynomial along the fibres) with the vector space of the representation. On the algebraic side, we review the formalism of connections for the universal differential algebras. In the differential geometry framework, we consider smooth connections on principal bundles as equivariant splittings of the cotangent bundle, as 1-form-valued derivations of the algebra of smooth functions on the structure group, and as axiomatically given covariant differentiations of functions defined on the total space. Finally, we use the Dirac monopole connection to compute the pairing of the line bundles associated to the Hopf fibration with the cyclic cocycle of integration over \( S^2 \).

Rainer Matthes	Related
Piotr Mieczysław Hajac	Related
Wojciech Szymański	Related

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