A Riemannian Invariant, Euler Structures and Some Topological Applications
نویسندگان
چکیده
In this paper: (i) We define and study a new numerical invariant R(X, g, ω) associated with a closed Riemannian manifold (M, g), a closed one form ω and a vector field X with isolated zeros. When X = − gradg f with f :M → R a Morse function this invariant is implicit in the work of Bismut–Zhang. The invariant is defined by an integral which might be divergent and requires (geometric) regularization. (ii) We define and study the sets of Euler structures and coEuler structures of a based pointed manifold (M,x0). When χ(M) = 0 the concept of Euler structure was introduced by V. Turaev. The Euler resp. co-Euler structures permit to remove the geometric anomalies from Reidemeister torsion resp. Ray–Singer torsion. (iii) We apply these concepts to torsion related issues, cf. Theorems 3 and 4. In particular we show the existence of a meromorphic function associated to a pair (M, e∗), consisting of a smooth closed manifold and a co-Euler structure, defined on the variety of complex representations of the fundamental group of M whose real part is the Ray–Singer torsion (corrected). This function generalizes the Alexander polynomial for the complement of a knot.
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