A Riemannian Invariant, Euler Structures and Some Topological Applications
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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.
KeywordsEuler structure co-Euler structure combinatorial torsion analytic torsion theorem of Bismut-Zhang Chern-Simons theory geometric regularization
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