 
Summary: The Intersection RTorsion for Finite Cone
Xianzhe Dai
Xiaoling Huang
1 Introduction
Torsion invariants were originally introduced in the 3dimensional setting by K. Reidemeister [23] in
1935 who used them to give a homeomorphism classification of 3dimensional lens spaces. The Rei
demeister torsions (Rtorsions for short) are defined using linear algebra and combinational topology.
The salient feature of Rtorsions 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,
Rtorsion is a secondary invariant with respect to the Euler characteristic. For geometric operators
such as the GaussBonnet 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., Rtorsion, appears. The Rtorsions were generalized to
arbitrary dimensions by W. Franz [13] and later studied by many authors (Cf. [19]).
Analytic torsion (or RaySinger torsion), which is a certain combinations of determinants of
Hodge Laplacians on kforms, is an invariant of Riemannian manifolds defined by Ray and Singer
[22] as an analytic analog of Rtorsions. Based on the evidence presented by Ray and Singer, Cheeger
[4] and M¨uller [20] proved the RaySinger 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
