Higher Torsion Invariants in Differential Topology and Algebraic K-Theory

Abstract

The classical Franz–Reidemeister torsion and its cousins, the Whitehead torsion and Ray–Singer analytic torsion, are topological invariants of manifolds with local coefficient systems (or flat vector bundles) that can distinguish homotopy equivalent spaces that are not homeomorphic. The purpose of this Arbeitsgemeinschaft was to learn about several natural generalisations of these classical invariants to families of manifolds.

Regard a family $p:E\to B$ of compact manifolds $M$, equipped with a flat vector bundle $F\to M$. Then the fibrewise cohomology groups $H^{\bullet }(E/B;F)$ form flat vector bundles over the base $B$. The starting point for our investigations are analogues of the Atiyah–Singer family index theorem that relate $F$ to $H^{\bullet }(E/B;F)$. To a flat vector bundle $F\to M$, one associates Kamber–Tondeur characteristic classes $c_{\bullet }(F)$ in $H^{\text{odd}}(M;\mathbb{R})$, which vanish if $F$ carries a parallel metric. By Bismut–Lott [1], one has

where $e(TM)$ is the Euler class of the vertical tangent bundle, and the right hand side is the Becker–Gottlieb transfer in de Rham cohomology. If one specifies some additional geometric data, then all classes above are naturally represented by specific differential forms. On the level of differential forms, the equation above only holds up a correction term $d\mathcal{T}$. Here $\mathcal{T}$ is the higher analytic torsion, which depends naturally on the fibration and the geometric data. If both $H^{\bullet }(E/B;F)$ and $F$ admit parallel metrics, then $\mathcal{T}$ gives rise to a secondary characteristic class $\mathcal{T}(E/B;F)\in H^{\text{even},\ge 2}(B;\mathbb{R})$.

Dwyer–Weiss–Williams [2] construct Reidemeister torsion for a smooth fiber bundle $p:E\to B$ as a byproduct of a family index theory. If $p$ is any fiber bundle with fibers compact topological manifolds and base a CW complex, then the family index theory states that $\chi (p)$, the A-theory Euler characteristic of $p$ is determined by the A-theory Euler class of ${\tau }_{fib}(p),$ the tangent bundle along the fiber. Here A-theory is algebraic K-theory of spaces in the sense of Waldhausen. More precisely, by applying fiberwise Poincare duality, and then an assembly map to the A-theory Euler class, one gets the A-theory Euler characteristic. If $p$ is a smooth bundle, then one gets a stronger smooth index theorem where the A-theory Euler class is replaced by the Becker–Euler class, which lives in the (twisted) stable cohomotopy of $E$. When $B$ is a point this result is equivalent to the classical Poincare–Hopf theorem.

The third approach is due to Igusa–Klein [3], and is somewhat different in nature. Here, one regards a generalised fibrewise Morse function on $M\to B$. Together with a flat vector bundle $F\to M$, this gives rise to a classifying map from $B$ to a Whitehead space, and the higher Franz–Reidemeister torsion is the pullback of a universal class on the Whitehead space.

There are conjectural relations between all three definitions of higher torsion. In a special case, Igusa has characterized higher Franz–Reidemeister torsion axiomatically; checking these axioms for either of the other higher torsions would prove equality. For some bundles, equality of higher Franz–Reidemeister torsion and higher analytic torsion can be shown analytically using the Witten deformation. Finally, one expects that higher Franz–Reidemeister torsion can be recovered from Dwyer–Weiss–Williams torsion.

It turns out that higher torsion invariants are somewhat finer than classical FR torsion, since they detect higher homotopy classes of the diffeomorphism group of high-dimensional manifolds that vanish under the forgetful map to the homeomorphism group. In particular, these invariants distinguish differentiable structures on a given topological fibre bundle $M\to B$, where one may even fix differentiable structures on $M$, $B$ and the typical fibre. There are also applications of higher torsions to problems in graph theory and moduli spaces of compact surfaces. Some of these were sketched throughout this Arbeitsgemeinschaft.

The talks were grouped as follows.

1. The first talk gave a short introduction to classical torsion invariants.
2. In talks 2–7, we discussed the Dwyer–Weiss–Williams homotopy theoretical approach.
3. Parametrized Morse theory, Kamber–Tondeur classes and Igusa–Klein torsion were discussed in talks 8–16, and some applications were given.
4. Finally, based on talks 10 and 11, we introduced analytic torsion in the talks 17–19.

The meeting took place from April 2nd till April 8th 2006 and was organized by Sebastian Goette (Regensburg), Kiyoshi Igusa (Brandeis) and Bruce Williams (Notre Dame). It was attended by 43 participants, coming mainly from Europa and the USA.