Analytic torsion

In mathematics, Reidemeister torsion (or R-torsion, or Reidemeister–Franz torsion) is a topological invariant of manifolds introduced by Kurt Reidemeister for 3-manifolds and generalized to higher dimensions by and . Analytic torsion (or Ray–Singer torsion) is an invariant of Riemannian manifolds defined by as an analytic analogue of Reidemeister torsion. and proved Ray and Singer's conjecture that Reidemeister torsion and analytic torsion are the same for compact Riemannian manifolds. Reidemeister torsion was the first invariant in algebraic topology that could distinguish between closed manifolds which are homotopy equivalent but not homeomorphic, and can thus be seen as the birth of geometric topology as a distinct field. It can be used to classify lens spaces. Reidemeister torsion is closely related to Whitehead torsion; see . It has also given some important motivation to arithmetic topology; see . For more recent work on torsion see the books and . If M is a Riemannian manifold and E a vector bundle over M, then there is a Laplacian operator acting on the k-forms with values in E. If the eigenvalues on k-forms are λj then the zeta function ζk is defined to be for s large, and this is extended to all complex s by analytic continuation. The zeta regularized determinant of the Laplacian acting on k-forms is which is formally the product of the positive eigenvalues of the laplacian acting on k-forms. The analytic torsion T(M,E) is defined to be Let be a finite connected CW-complex with fundamental group and universal cover , and let be an orthogonal finite-dimensional -representation. Suppose that for all n. If we fix a cellular basis for and an orthogonal -basis for , then is a contractible finite based free -chain complex. Let be any chain contraction of D*, i.e. for all . We obtain an isomorphism with , . We define the Reidemeister torsion where A is the matrix of with respect to the given bases. The Reidemeister torsion is independent of the choice of the cellular basis for , the orthogonal basis for and the chain contraction .