Dan Burghelea, Stefan Haller
A Riemannian Invariant, Euler structures and some Topological Applications
Preprint series: ESI preprints

MSC:

57R20 Characteristic classes and numbers

58G26 Determinants and determinant bundles

Abstract: In this paper:
\begin{enumerate}
\item
We define and study a new numerical invariant $R(X,g,\omega)$ associated
with a closed Riemannian manifold $(M,g)$, a closed one form $\omega$ and a

vector field $X$ with isolated zeros. When $X=-\grad_gf$ with $f:M\to\R$
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.
\item
We define and study the sets of Euler structures and co-Euler structures
of a based pointed manifold $(M,x_0)$. When $\chi(M)=0$ 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.
\item
We apply these concepts to torsion related issues,
\cf~Theorems~\ref{T:marsik} and~\ref{T:tan}.
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.
\end{enumerate}