Lecture notes

• A-infinity functors and higher torsion, v1 (2003), (ps), 22 pages, 5 figures. These are lecture notes from a series of lectures given in Goettingen on September 1-4, 2003 on the subject of higher Franz-Reidemeister torsion. Basically it explains contents of my book and the two papers ``Twisting cochains and higher torsion'' and ``Complex torsion and the Framing Principle.''
• Axioms for higher torsion, v1 (2003), (ps), 20 pages, 7 figures. This paper shows that nonequivariant higher torsion is characterized by two simple axioms: additivity and transfer. Any characteristic class of smooth bundles satisfying these conditions must be a linear combination of the higher Franz-Reidemeister torsion and the higher even Miller-Morita-Mumford class. These are notes from my last lecture at Goettingen on September 7, 2003 plus additional notes and proofs of the theorems. I didn't have the complete statement at the time of the lecture.

Recent papers

• Outer automorphisms and the Jacobian (joint work with John R. Klein and E. Bruce Williams, 2005), 19 pages, 2 figures. A graph of rank r has a canonical immersion into its Jacobian given by a variation of the Abel-Jacobi map. A neighborhood of the image is an immersed manifold. Therefore, by a theorem of Dwyer, Weiss and Williams, the map from the outer automorphism of the free group into the general linear group is trivial in rational homology in the stable range.
• Axioms for higher torsion invariants of smooth bundles (2005), 24 pages, 0 figures. This is a new version of the Axioms paper. A higher torsion invariant is defined to be one which satisfies additivity and transfer for linear sphere bundles. By restricting to bundles with closed fibers, it becomes clear that higher torsion invariants have even and odd parts which are nonzero for even and odd dimensional closed manifold fibers, respectively. Both components are unique up to a scalar multiple in every degree 4k>0. The even component must be proportional to the higher Miller-Morita-Mumford class. The odd component must be proportional to the odd part of higher Franz-Reidemeister torsion. The paper shows how higher torsion can be computed just from the axioms. It also shows how higher Franz-Reidemeister torsion can be computed from the ``framing principle'' and the computation verifies that it satisfies the axioms.

Other papers

• Twisting cochains and higher torsion, v2 (2003), (postscript), 18 pages, 1 figure. This short paper explains how an old construction of Ed Brown plays a central role in the theory of higher Franz-Reidemeister torsion and analytic torsion. This is Brown's idea of twisting (!) cochains and twisted tensor products. Basically, when two twisting cochains are A-infinity equivalent there is still a higher algebraic K-theory obstruction for deforming one into the other when they are based. When the simplices are infinitesimal the boundary map for the dual of the twisted tensor product is given by a flat superconnection. This paper, dedicated to Ed Brown, explains these ideas without much proof. Jim Stasheff pointed out to me that the correct terminology is ``twisting cochains'' so I changed the title in the second version.