Axioms for higher torsion invariants of smooth bundles

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.