Motivic bivariant characteristic classes and related topics

Journal of Singularities
volume 5 (2012), 124-152
Proceedings of the International Conference on Singularity Theory and Applications, Hefei, China, July 25-31, 2011

Received 31 January 2012. Received in revised form 18 April 2012.

DOI: 10.5427/jsing.2012.5j

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Abstract:

We have recently constructed a bivariant analogue of the motivic Hirzebruch classes. A key idea is the construction of a suitable universal bivariant theory in the algebraic-geometric (or compact complex analytic) context, together with a corresponding "bivariant blow-up relation" generalizing Bittner's presentation of the Grothendieck group of varieties. Before we already introduced a corresponding universal "oriented" bivariant theory as an intermediate step on the way to a bivariant analogue of Levine-Morel's algebraic cobordism. Switching to the differential topological context of smooth manifolds, we similarly get a new geometric bivariant bordism theory based on the notion of a "fiberwise bordism". In this paper we make a survey on these theories.

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