Higher complex torsion and the framing principle

Abstract

This paper contains a long summary of the basic properties of higher FR torsion. An attempt is made to simplify the constructions from my book Higher Franz-Reidemeister Torsion (IP/AMS Studies in Advanced Math 31). Some new basic theorems are also proved such as the Framing Principle in full generality. This is used to compute the higher torsion for bundles with closed even dimensional fibers. We construct a higher complex torsion for bundles with almost complex fibers. This is shown to generalize the real even dimensional higher FR torsion. We also show that the higher complex torsion is a multiple of generalized Miller-Morita-Mumford classes.