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Journal of Topology Advance Access published online on October 25, 2007
Journal of Topology, doi:10.1112/jtopol/jtm011
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© 2007 London Mathematical Society

Axioms for higher torsion invariants of smooth bundles

Kiyoshi Igusa

Department of Mathematics, Brandeis University, P O Box 9110, Waltham, MA 02454-9110, USA igusa{at}brandeis.edu


  Abstract

This paper attempts to explain the relationship between various characteristic classes for smooth manifold bundles which are known as ‘higher torsion’ classes. We isolate two fundamental properties that these cohomology classes may or may not have: additivity and transfer. We show that higher Franz–Reidemeister torsion and higher Miller–Morita–Mumford classes satisfy these axioms. Conversely, any characteristic class of smooth bundles satisfying the two axioms must be a linear combination of these two examples.

We also show how any higher torsion invariant, that is, any characteristic class satisfying the two axioms, can be computed for a smooth bundle with a fiberwise Morse function with distinct critical values. Finally, we explain the statements of the conjectured formulas relating higher analytic torsion classes, higher Franz–Reidemeister torsion and Dwyer–Weiss–Williams smooth torsion.

Received February 22, 2007.

2000 Mathematics Subject Classification 55R40 (primary), 57R50, 19J10 (secondary)

I am in great debt to John R. Klein and E. Bruce Williams for their help in completing the final crucial steps in the proof of the main theorem. I also benefitted greatly from conversations with Sebastian Goette, Xiaonan Ma, Wojciech Dorabiala and Gordana Todorov. Also, I would like to thank Bernard Badzioch for his inspired presentation of this work during the Arbeitsgemeinshaft at Oberwolfach in April, 2006. Finally, I should not forget to thank the organizers Thomas Schick, Ulrich Bunke and Sebastian Goette of the conference on higher torsion in Göttingen in September 2003 at which the first version of these results were developed and announced. Research for this paper was supported by NSF grants DMS 02-04386 and DMS 03-09480.


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