Analytic torsion

In mathematics, Reidemeister torsion (or R-torsion, or Reidemeister–Franz torsion) is a topological invariant of manifolds introduced by Kurt Reidemeister (Reidemeister (1935)) for 3-manifolds and generalized to higher dimensions by Franz (1935) and de Rham (1936). Analytic torsion (or Ray–Singer torsion) is an invariant of Riemannian manifolds defined by Ray and Singer (1971, 1973a, 1973b) as an analytic analogue of Reidemeister torsion. Cheeger (1977, 1979) and Müller (1978) 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 spaces 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 (Milnor 1966). For later work on torsion see the books (Turaev 2002), (Nicolaescu 2002, 2003).

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Definition of analytic torsion

If M is a Riemannian manifold and E a vector bundle over M, then there is a Laplacian operator acting on the i-forms with values in E. If the eigenvalues on i-forms are λj then the zeta function ζi is defined to be

\zeta_i(s) = \sum_{\lambda_j>0}\lambda_j^{-s}

for s large, and this is extended to all complex s by analytic continuation. The zeta regularized determinant of the Laplacian acting on i-forms is

\Delta_i=\exp(-\zeta^\prime_i(0))

which is formally the product of the positive eigenvalues of the laplacian acting on i-forms. The analytic torsion T(M,E) is defined to be

T(M,E) = \exp\left(\sum_i (-1)^ii \zeta^\prime_i(0)/2\right) = \prod_i\Delta_i^{-(-1)^ii/2}.

Definition of Reidemeister torsion

Let X be a finite connected CW-complex with fundamental group π := π1(X) and U an orthogonal finite-dimensional π-representation. Suppose that

H^\pi_n(X;U)�:= H_n(U \otimes_{\mathbf{Z}[\pi]} C_*({\tilde X})) = 0

for all n. If we fix a cellular basis for C_*({\tilde X}) and an orthogonal R-basis for U, then D_*�:= U \otimes_{\mathbf{Z}[\pi]} C_*({\tilde X}) is a contractible finite based free R-chain complex. Let \gamma_*: D_* \to D_{*%2B1} be any chain contraction of D*, i.e. d_{n%2B1} \circ \gamma_n %2B \gamma_{n-1} \circ d_n = id_{D_n} for all n. We obtain an isomorphism (d_* %2B \gamma_*)_{odd}: D_{odd} \to D_{even} with D_{odd}�:= \oplus_{n \, odd} \, D_n, D_{even}�:= \oplus_{n \, even} \, D_n. We define the Reidemeister torsion

\rho(X;U)�:= |\mathop{det}(A)| \in \mathbf{R}^{>0}

where A is the matrix of (d* + γ*)odd with respect to the given bases. The Reidemeister torsion \rho(X;U) is independent of the choice of the cellular basis for C_*({\tilde X}), the orthogonal basis for U and the chain contraction γ*.

Examples

Reidemeister torsion was first used to classify 3-dimensional lens spaces in (Reidemeister 1935). The classification includes examples of homotopy equivalent 3-dimensional manifolds which are not homeomorphic – at the time (1935) the classification was only up to PL homeomorphism, but later (Brody 1960) showed that this was in fact a classification up to homeomorphism.

References