Analytic torsion – Wikipedia

Topological invariant of manifolds that can distinguish homotopy-equivalent 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).

Definition of analytic torsion[edit]

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 λ_j then the zeta function ζ_k is defined to be

${displaystyle zeta _{k}(s)=sum _{lambda _{j}>0}lambda _{j}^{-s}}$

${displaystyle Delta _{k}=exp(-zeta _{k}^{prime }(0))}$

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

${displaystyle T(M,E)=exp left(sum _{k}(-1)^{k}kzeta _{k}^{prime }(0)/2right)=prod _{k}Delta _{k}^{-(-1)^{k}k/2}.}$

Definition of Reidemeister torsion[edit]

Let

${displaystyle X}$

be a finite connected CW-complex with fundamental group

${displaystyle pi :=pi _{1}(X)}$

and universal cover

${displaystyle {tilde {X}}}$

, and let

${displaystyle U}$

be an orthogonal finite-dimensional

${displaystyle pi }$

-representation. Suppose that

${displaystyle H_{n}^{pi }(X;U):=H_{n}(Uotimes _{mathbf {Z} [pi ]}C_{*}({tilde {X}}))=0}$

for all n. If we fix a cellular basis for

${displaystyle C_{*}({tilde {X}})}$

and an orthogonal

${displaystyle mathbf {R} }$

-basis for

${displaystyle U}$

, then

${displaystyle D_{*}:=Uotimes _{mathbf {Z} [pi ]}C_{*}({tilde {X}})}$

is a contractible finite based free

${displaystyle mathbf {R} }$

-chain complex. Let

${displaystyle gamma _{*}:D_{*}to D_{*+1}}$

be any chain contraction of D_*, i.e.

${displaystyle d_{n+1}circ gamma _{n}+gamma _{n-1}circ d_{n}=id_{D_{n}}}$

for all

${displaystyle n}$

. We obtain an isomorphism

${displaystyle (d_{*}+gamma _{*})_{text{odd}}:D_{text{odd}}to D_{text{even}}}$

with

${displaystyle D_{text{odd}}:=oplus _{n,odd},D_{n}}$

,

${displaystyle D_{text{even}}:=oplus _{n,{text{even}}},D_{n}}$

. We define the Reidemeister torsion

${displaystyle rho (X;U):=|det(A)|^{-1}in mathbf {R} ^{>0}}$

${displaystyle (d_{*}+gamma _{*})_{text{odd}}}$

with respect to the given bases. The Reidemeister torsion

${displaystyle rho (X;U)}$

is independent of the choice of the cellular basis for

${displaystyle C_{*}({tilde {X}})}$

, the orthogonal basis for

${displaystyle U}$

and the chain contraction

${displaystyle gamma _{*}}$

.

Let

${displaystyle M}$

be a compact smooth manifold, and let

${displaystyle rho colon pi (M)rightarrow GL(E)}$

be a unimodular representation.

${displaystyle M}$

has a smooth triangulation. For any choice of a volume

${displaystyle mu in det H_{*}(M)}$

, we get an invariant

${displaystyle tau _{M}(rho :mu )in mathbf {R} ^{+}}$

. Then we call the positive real number

${displaystyle tau _{M}(rho :mu )}$

the Reidemeister torsion of the manifold

${displaystyle M}$

with respect to

${displaystyle rho }$

and

${displaystyle mu }$

.

A short history of Reidemeister torsion[edit]

Reidemeister torsion was first used to combinatorially classify 3-dimensional lens spaces in (Reidemeister 1935) by Reidemeister, and in higher-dimensional spaces by Franz. 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 E.J. Brody (1960) showed that this was in fact a classification up to homeomorphism.

J. H. C. Whitehead defined the “torsion” of a homotopy equivalence between finite complexes. This is a direct generalization of the Reidemeister, Franz, and de Rham concept; but is a more delicate invariant. Whitehead torsion provides a key tool for the study of combinatorial or differentiable manifolds with nontrivial fundamental group and is closely related to the concept of “simple homotopy type”, see (Milnor 1966)

In 1960 Milnor discovered the duality relation of torsion invariants of manifolds and show that the (twisted) Alexander polynomial of knots is the Reidemeister torsion of its knot complement in

${displaystyle S^{3}}$

. (Milnor 1962) For each q the Poincaré duality

${displaystyle P_{o}}$

induces

${displaystyle P_{o}colon operatorname {det} (H_{q}(M)){overset {sim }{,longrightarrow ,}}(operatorname {det} (H_{n-q}(M)))^{-1}}$

and then we obtain

${displaystyle Delta (t)=pm t^{n}Delta (1/t).}$

The representation of the fundamental group of knot complement plays a central role in them. It gives the relation between knot theory and torsion invariants.

Cheeger–Müller theorem[edit]

Let

${displaystyle (M,g)}$

be an orientable compact Riemann manifold of dimension n and

${displaystyle rho colon pi (M)rightarrow mathop {GL} (E)}$

a representation of the fundamental group of

${displaystyle M}$

on a real vector space of dimension N. Then we can define the de Rham complex

${displaystyle Lambda ^{0}{stackrel {d_{0}}{longrightarrow }}Lambda ^{1}{stackrel {d_{1}}{longrightarrow }}cdots {stackrel {d_{n-1}}{longrightarrow }}Lambda ^{n}}$

and the formal adjoint

${displaystyle d_{p}}$

and

${displaystyle delta _{p}}$

due to the flatness of

${displaystyle E_{q}}$

. As usual, we also obtain the Hodge Laplacian on p-forms

${displaystyle Delta _{p}=delta _{p+1}d_{p}+d_{p-1}delta _{p}.}$

Assuming that

${displaystyle partial M=0}$

, the Laplacian is then a symmetric positive semi-positive elliptic operator with pure point spectrum

${displaystyle 0leq lambda _{0}leq lambda _{1}leq cdots rightarrow infty .}$

As before, we can therefore define a zeta function associated with the Laplacian

${displaystyle Delta _{q}}$

on

${displaystyle Lambda ^{q}(E)}$

by

${displaystyle zeta _{q}(s;rho )=sum _{lambda _{j}>0}lambda _{j}^{-s}={frac {1}{Gamma (s)}}int _{0}^{infty }t^{s-1}{text{Tr}}(e^{-tDelta _{q}}-P_{q})dt, {text{Re}}(s)>{frac {n}{2}}}$

${displaystyle P}$

is the projection of

${displaystyle L^{2}Lambda (E)}$

onto the kernel space

${displaystyle {mathcal {H}}^{q}(E)}$

of the Laplacian

${displaystyle Delta _{q}}$

. It was moreover shown by (Seeley 1967) that

${displaystyle zeta _{q}(s;rho )}$

extends to a meromorphic function of

${displaystyle sin mathbf {C} }$

which is holomorphic at

${displaystyle s=0}$

.

As in the case of an orthogonal representation, we define the analytic torsion

${displaystyle T_{M}(rho ;E)}$

by

${displaystyle T_{M}(rho ;E)=exp {biggl (}{frac {1}{2}}sum _{q=0}^{n}(-l)^{q}q{frac {d}{ds}}zeta _{q}(s;rho ){biggl |}_{s=0}{biggr )}.}$

In 1971 D.B. Ray and I.M. Singer conjectured that

${displaystyle T_{M}(rho ;E)=tau _{M}(rho ;mu )}$

for any unitary representation

${displaystyle rho }$

. This Ray–Singer conjecture was eventually proved, independently, by Cheeger (1977, 1979) and Müller (1978). Both approaches focus on the logarithm of torsions and their traces. This is easier for odd-dimensional manifolds than in the even-dimensional case, which involves additional technical difficulties. This Cheeger–Müller theorem (that the two notions of torsion are equivalent), along with Atiyah–Patodi–Singer theorem, later provided the basis for Chern–Simons perturbation theory.

A proof of the Cheeger-Müller theorem for arbitrary representations was later given by J. M. Bismut and Weiping Zhang. Their proof uses the Witten deformation.

References[edit]

• Bismut, J. -M.; Zhang, W. (1994-03-01), “Milnor and Ray-Singer metrics on the equivariant determinant of a flat vector bundle”, Geometric & Functional Analysis GAFA, 4 (2): 136–212, doi:10.1007/BF01895837, ISSN 1420-8970, S2CID 121327250
• Brody, E. J. (1960), “The topological classification of the lens spaces”, Annals of Mathematics, 2, 71 (1): 163–184, doi:10.2307/1969884, JSTOR 1969884, MR 0116336
• Cheeger, Jeff (1977), “Analytic torsion and Reidemeister torsion”, Proceedings of the National Academy of Sciences of the United States of America, 74 (7): 2651–2654, Bibcode:1977PNAS…74.2651C, doi:10.1073/pnas.74.7.2651, MR 0451312, PMC 431228, PMID 16592411
• Cheeger, Jeff (1979), “Analytic torsion and the heat equation”, Annals of Mathematics, 2, 109 (2): 259–322, doi:10.2307/1971113, JSTOR 1971113, MR 0528965
• Franz, Wolfgang (1935), “Ueber die Torsion einer Ueberdeckung”, Journal für die reine und angewandte Mathematik, 173: 245–254
• Milnor, John (1962), “A duality theorem for Reidemeister torsion”, Annals of Mathematics, 76 (1): 137–138, doi:10.2307/1970268, JSTOR 1970268
• Milnor, John (1966), “Whitehead torsion”, Bulletin of the American Mathematical Society, 72 (3): 358–426, doi:10.1090/S0002-9904-1966-11484-2, MR 0196736
• Mishchenko, Aleksandr S. (2001) [1994], “Reidemeister torsion”, Encyclopedia of Mathematics, EMS Press
• Müller, Werner (1978), “Analytic torsion and R-torsion of Riemannian manifolds”, Advances in Mathematics, 28 (3): 233–305, doi:10.1016/0001-8708(78)90116-0, MR 0498252
• Nicolaescu, Liviu I. (2002), Notes on the Reidemeister torsion (PDF) Online book
• Nicolaescu, Liviu I. (2003), The Reidemeister torsion of 3-manifolds, de Gruyter Studies in Mathematics, vol. 30, Berlin: Walter de Gruyter & Co., pp. xiv+249, doi:10.1515/9783110198102, ISBN 3-11-017383-2, MR 1968575
• Ray, Daniel B.; Singer, Isadore M. (1973a), “Analytic torsion for complex manifolds.”, Annals of Mathematics, 2, 98 (1): 154–177, doi:10.2307/1970909, JSTOR 1970909, MR 0383463
• Ray, Daniel B.; Singer, Isadore M. (1973b), “Analytic torsion.”, Partial differential equations, Proc. Sympos. Pure Math., vol. XXIII, Providence, R.I.: Amer. Math. Soc., pp. 167–181, MR 0339293
• Ray, Daniel B.; Singer, Isadore M. (1971), “R-torsion and the Laplacian on Riemannian manifolds.”, Advances in Mathematics, 7 (2): 145–210, doi:10.1016/0001-8708(71)90045-4, MR 0295381
• Reidemeister, Kurt (1935), “Homotopieringe und Linsenräume”, Abh. Math. Sem. Univ. Hamburg, 11: 102–109, doi:10.1007/BF02940717, S2CID 124078064
• de Rham, Georges (1936), “Sur les nouveaux invariants topologiques de M. Reidemeister”, Recueil Mathématique (Matematicheskii Sbornik), Nouvelle Série, 1 (5): 737–742, Zbl 0016.04501
• Turaev, Vladimir (2002), Torsions of 3-dimensional manifolds, Progress in Mathematics, vol. 208, Basel: Birkhäuser Verlag, pp. x+196, doi:10.1007/978-3-0348-7999-6, ISBN 3-7643-6911-6, MR 1958479
• Mazur, Barry. “Remarks on the Alexander polynomial” (PDF).
• Seeley, R. T. (1967), “Complex powers of an elliptic operator”, in Calderón, Alberto P. (ed.), Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Proceedings of Symposia in Pure Mathematics, vol. 10, Providence, R.I.: Amer. Math. Soc., pp. 288–307, ISBN 978-0-8218-1410-9, MR 0237943