L^2 torsion without the determinant class condition and extended L^2 cohomology

Details L^2 torsion without the determinant class condition and extended L^2 cohomology L^2 torsion without the determinant class condition and extended L^2 cohomology

2004-06-10

arxiv.org/abs/math/0406222v1

Commun. in Contemporary Math. vol. 7 no. 4 (2005) 421-462

Mathematics

Differential Geometry

39 pages

Scientific paper

10.1142/S0219199705001866

We associate determinant lines to objects of the extended abelian category built out of a von Neumann category with a trace. Using this we suggest constructions of the combinatorial and the analytic L^2 torsions which, unlike the work of the previous authors, requires no additional assumptions; in particular we do not impose the determinant class condition. The resulting torsions are elements of the determinant line of the extended L^2 cohomology. Under the determinant class assumption the L^2 torsions of this paper specialize to the invariants studied in our previous work. Applying a recent theorem of D. Burghelea, L. Friedlander and T. Kappeler we obtain a Cheeger - Muller type theorem stating the equality between the combinatorial and the analytic L^2 torsions.

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