Reidemeister Torsion

In algebraic topology, the Reidemeister torsion is a notion originally introduced as a topological invariant of 3-manifolds which 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 R-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 map, and

(3)

is a map from to . In this context, Reidemeister torsion is sometimes referred to as the torsion of the 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 cover of yields an associated chain complex which has a basis. 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, and was proven by Cheeger and Müller to be identically equal to the analytic torsion in the case of compact Riemannian manifolds.