Absolute torsion and eta-invariant |
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Authors: | M Farber |
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Institution: | (1) Department of Mathematics, Tel Aviv University, Tel Aviv, 69978, Israel (e-mail: Farber@math.tau.AC.IL), IL |
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Abstract: | In a recent joint work with V. Turaev 6], we defined a new concept of combinatorial torsion which we called absolute torsion. Compared with the classical Reidemeister torsion, it has the advantage of having a well-determined sign. Also, the absolute
torsion is defined for arbitrary orientable flat vector bundles, and not only for unimodular ones, as is classical Reidemeister
torsion. In this paper I show that the sign behavior of the absolute torsion, under a continuous deformation of the flat bundle,
is determined by the eta-invariant and the Pontrjagin classes. This result has a twofold significance. Firstly, it justifies
the definition of the absolute torsion by establishing a relation to the well-known geometric invariants of manifolds. Viewed
differently, the result of this paper allows to express (partially) the eta-invariant, which is defined using analytic tools,
in terms of the absolute torsion, having a purely topological definition. The result may find applications in studying the
spectral flow by methods of combinatorial topology.
Received January 11, 1999; in final form August 16, 1999 |
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Keywords: | Mathematics Subject Classification (1991): Primary 57Q10 Secondary 53C99 |
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