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 Wolfgang Franz (1935) and Georges de Rham (1936).Analytic torsion (or Ray–Singer torsion) is an invariant of Riemannian manifolds defined by Daniel B. Ray and Isadore M. Singer (1971, 1973a, 1973b) as an analytic analogue of Reidemeister torsion. Jeff Cheeger (1977, 1979) and Werner Müller (1978) proved Ray and Singer's conjecture that Reidemeister torsion and analytic torsion are the same for compact Riemannian manifolds. 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 Wolfgang Franz (1935) and Georges de Rham (1936).Analytic torsion (or Ray–Singer torsion) is an invariant of Riemannian manifolds defined by Daniel B. Ray and Isadore M. Singer (1971, 1973a, 1973b) as an analytic analogue of Reidemeister torsion. Jeff Cheeger (1977, 1979) and Werner 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 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 (Milnor 1966). It has also given some important motivation to arithmetic topology; see (Mazur). For more recent work on torsion see the books (Turaev 2002) and (Nicolaescu 2002, 2003). 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 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 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 Let X {displaystyle X} be a finite connected CW-complex with fundamental group π := π 1 ( X ) {displaystyle pi :=pi _{1}(X)} and universal cover X ~ {displaystyle { ilde {X}}} , and let U {displaystyle U} be an orthogonal finite-dimensional π {displaystyle pi } -representation. Suppose that for all n. If we fix a cellular basis for C ∗ ( X ~ ) {displaystyle C_{*}({ ilde {X}})} and an orthogonal R {displaystyle mathbf {R} } -basis for U {displaystyle U} , then D ∗ := U ⊗ Z [ π ] C ∗ ( X ~ ) {displaystyle D_{*}:=Uotimes _{mathbf {Z} }C_{*}({ ilde {X}})} is a contractible finite based free R {displaystyle mathbf {R} } -chain complex. Let γ ∗ : D ∗ → D ∗ + 1 {displaystyle gamma _{*}:D_{*} o D_{*+1}} be any chain contraction of D*, i.e. d n + 1 ∘ γ n + γ n − 1 ∘ d n = i d D n {displaystyle d_{n+1}circ gamma _{n}+gamma _{n-1}circ d_{n}=id_{D_{n}}} for all n {displaystyle n} . We obtain an isomorphism ( d ∗ + γ ∗ ) odd : D odd → D even {displaystyle (d_{*}+gamma _{*})_{ ext{odd}}:D_{ ext{odd}} o D_{ ext{even}}} with D odd := ⊕ n o d d D n {displaystyle D_{ ext{odd}}:=oplus _{n,odd},D_{n}} , D even := ⊕ n even D n {displaystyle D_{ ext{even}}:=oplus _{n,{ ext{even}}},D_{n}} . We define the Reidemeister torsion where A is the matrix of ( d ∗ + γ ∗ ) odd {displaystyle (d_{*}+gamma _{*})_{ ext{odd}}} with respect to the given bases. The Reidemeister torsion ρ ( X ; U ) {displaystyle ho (X;U)} is independent of the choice of the cellular basis for C ∗ ( X ~ ) {displaystyle C_{*}({ ilde {X}})} , the orthogonal basis for U {displaystyle U} and the chain contraction γ ∗ {displaystyle gamma _{*}} .