Reidemeister Torsion

Reidemeister torsion in algebraic topology is a topological invariant initially used for 3-manifolds, and it has now been widely adapted to a variety of contexts. At the time of its discovery, the Reidemeister torsion was the first 3-manifold invariant able to distinguish between manifolds which are homotopy equivalent but not homeomorphic. Since then, the notion has been adapted to higher-dimensional manifolds, knots and links, dynamical systems, Witten's equations, and so on. In particular, it has a number of different definitions for various contexts.

For a commutative ring , let be a finite acyclic chain complex of based finitely generated free -modules of the form

(1)

The Reidemeister torsion of is the value defined by

(2)

where is the set of units of , is a chain contraction, is the boundary operator, and

(3)

is a map from to . In this context, Reidemeister torsion is sometimes referred to as the torsion of the chain complex (Nicolaescu 2002) and can be considered a generalization of the determinant of a matrix (Ranicki 1997).

Another common context for which to define Reidemeister torsion is in the case of CW-complexes. Begin with a compact metric space with finite CW-decomposition and consider the canonically induced chain complex of free Abelian groups,

(4)

Lifting to a CW-decomposition of the maximal Abelian covering map of yields an associated chain complex which has a basis. To use the acyclic determinant definition, assume that this chain complex becomes acyclic after extension to the total ring of fractions (or a specified coefficient field). In the nonacyclic case, additional homology basis data are required. In particular, defining

(5)

where denotes the group of permutations of a set , the torsion of the chain complex of free -modules with respect to the -orbit of -bases is called the Reidemeister torsion of . In this context, the Reidemeister torsion is a well-defined element of . In-depth details of this construction can be found in e.g., Nicolaescu (2002).

Reidemeister torsion is sometimes known as R-torsion or Reidemeister-Franz torsion. What's more, R-torsion is closely related to a number of other topological tools including Whitehead torsion. The Cheeger-Müller theorem identifies it with analytic torsion for a closed Riemannian manifold equipped with a flat vector bundle defined by a group representation into an orthogonal group and compatible homology bases (Cheeger 1977, Nicolaescu 2002).