MIT Topology Seminar

This talk is about the axiomatic approach to higher Franz- Reidemeister (FR) torsion. We consider smooth manifold bundles M\to E\to B satisfying certain conditions (e.g., B simply connected is sufficient for all cases). For these bundles there are several real characteristic classes \tau (E)\in H^{4k}(B;R): higher FR torsion, analytic torsion classes, higher Dwyer-Weiss-Williams (DWW) classes and tautological (Miller- Morita-Mumford) classes. However, for fixed k and fixed parity of the dimension of M there is only one characteristic class (up to a scalar multiple) satisfying two axioms. Using this theorem, old theorems become clearer and there are some new results. So far, only the FR-torsion and tautological classes are known to satisfy the axioms. So they are proportional (old result). Sebastian Goette and I have some new results which imply equality with DWW torsion in some cases. Finally, I will attempt to extend axiomatic torsion to the equivariant case.

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