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The Intersection R-Torsion for Finite Cone Xianzhe Dai
 

Summary: The Intersection R-Torsion for Finite Cone
Xianzhe Dai
Xiaoling Huang
1 Introduction
Torsion invariants were originally introduced in the 3-dimensional setting by K. Reidemeister [23] in
1935 who used them to give a homeomorphism classification of 3-dimensional lens spaces. The Rei-
demeister torsions (R-torsions for short) are defined using linear algebra and combinational topology.
The salient feature of R-torsions is that it is not a homotopy invariant but rather a simple homo-
topy invariant; hence a homeomorphism invariant as well. From the index theoretic point of view,
R-torsion is a secondary invariant with respect to the Euler characteristic. For geometric operators
such as the Gauss-Bonnet and Dolbeault operator, the index is the Euler characteristic of certain
cohomology groups. If these groups vanish, the Index Theorem has nothing to say, and secondary
geometric and topological invariant, i.e., R-torsion, appears. The R-torsions were generalized to
arbitrary dimensions by W. Franz [13] and later studied by many authors (Cf. [19]).
Analytic torsion (or Ray-Singer torsion), which is a certain combinations of determinants of
Hodge Laplacians on k-forms, is an invariant of Riemannian manifolds defined by Ray and Singer
[22] as an analytic analog of R-torsions. Based on the evidence presented by Ray and Singer, Cheeger
[4] and M¨uller [20] proved the Ray-Singer conjecture, i.e., the equality of analytic and Reidemeister
torsion, on closed manifolds using different techniques. Cheeger's proof uses surgery techniques to
reduce the problem to the case of a sphere, while M¨uller's proof examines the convergence of the

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics