Abstract
First we discuss a numerical invariant associated with a Riemannian metric, a vector field with isolated zeros, and a closed one form which is defined by a geometrically regularized integral. This invariant, extends the Chern-Simons class from a pair of two Riemannian metrics to a pair of a Riemannian metric and a smooth triangulation. Next we discuss a generalization of Turaev’s Euler structures to manifolds with non-vanishing Euler characteristics and introduce the Poincarée dual concept of co-Euler structures. The duality is provided by a geometrically regularized integral and involves the invariant mentioned above. Euler structures have been introduced because they permit to remove the ambiguities in the definition of the Reidemeister torsion. Similarly, co-Euler structures can be used to eliminate the metric dependence of the Ray-Singer torsion. The Bismut-Zhang theorem can then be reformulated as a statement comparing two genuine topological invariants.
The second author is supported by the Fonds zur Fürderung der wissenschaftlichen Forschung (Austrian Science Fund), project number P14195-MAT.
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Burghelea, D., Haller, S. (2006). A Riemannian Invariant, Euler Structures and Some Topological Applications. In: Bojarski, B., Mishchenko, A.S., Troitsky, E.V., Weber, A. (eds) C*-algebras and Elliptic Theory. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7687-1_3
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