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). And it had given one of important motivation to arithmetic topology. (Mazur)

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 λ_j then the zeta function ζ_i 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 $\pi := \pi_1(X)$ and universal cover ${\tilde X}$, and let $U$ be an orthogonal finite-dimensional $\pi$-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 $\mathbf{R}$-basis for $U$, then $D_* := U \otimes_{\mathbf{Z}[\pi]} C_*({\tilde X})$ is a contractible finite based free $\mathbf{R}$-chain complex. Let $\gamma_*: D_* \to D_{*+1}$ be any chain contraction of D_*, i.e. $d_{n+1} \circ \gamma_n + \gamma_{n-1} \circ d_n = id_{D_n}$ for all n. We obtain an isomorphism $(d_* + \gamma_*)_{odd}: D_{odd} \to D_{even}$ with $D_{odd} := \oplus_{n \, odd} \, D_n$, $D_{even} := \oplus_{n \, even} \, D_n$. We define the Reidemeister torsion

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

where A is the matrix of $(d_* + \gamma_*)_{odd}$ 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 $\gamma_*$.

Let $M$ be a compact smooth manifold, and let $\rho:\pi(M)\rightarrow GL(E)$ be a unimodular representation. $M$ has a smooth triangulation. For any choice of a volume $\mu\in\mathop{det}H_*(M)$, we get an invariant $\tau_M(\rho:\mu)\in\mathbf{R}^+$. Then we call the positive real number $\tau_M(\rho:\mu)$ the Reidemeister torsion of the manifold $M$ with respect to $\rho$ and $\mu$.

A short history of Reidemeister torsion

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 (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 Reidemister torsion of its knot complement in S³. (Milnor 1962) For each q the Poincaré duality $P_o$ induces

$P_o:\operatorname{det}(H_q(M))\stackrel{\sim}{\longrightarrow}(\operatorname{det}(H_{n-q}(M)))^{-1}$

and then we obtain

$\Delta(t)=\pm t^n\Delta(1/t).$

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

Let $(M,g)$ be an orientable compact Riemann manifold of dimension n and $\rho:\pi(M)\rightarrow\mathop{GL}(E)$ a representation of the fundamental group of $M$ on a real vector space of dimension N. Then we can define the De Rham complex

$\Lambda^0\stackrel{d_0}{\longrightarrow}\Lambda^1\stackrel{d_1}{\longrightarrow}\cdots\stackrel{d_{n-1}}{\longrightarrow}\Lambda^n$

and the formal adjoint $d_p$ and $\delta_p$ due to the flatness of $E_q$. And we also obtain the Laplacian on p-form as usual

$\Delta_p=\delta_p d_p+d_{p-1}\delta_{p-1}.$

We assume $\partial M=0$, then the Laplacian is a symmetric positive simipositive elliptic operator with pure point spectrum

$0\le\lambda_0\le\lambda_1\le\cdots\rightarrow\infty.$

As same as the above definition we can define the zeta function associated with the Laplacian $\Delta_q$ on $\Lambda^q(E)$ by

$\zeta_q(s;\rho)=\sum_{\lambda_j >0}\lambda_j^{-s}=\frac{1}{\Gamma(s)}\int^\infty_0 t^{s-1}\mathop{Tr}(e^{-t\Delta_q} - P_q)dt,\ \ \ \mathop{Re}(s)>\frac{n}{2}$

where $P$ is the projection of $L^2 \Lambda(E)$ onto the kernel space $\mathcal{H}^q(E)$ of the Laplacian $\Delta_q$.

In 1967 Seeley proved that $\zeta_q(s;\rho)$ extends to a meromorphic function of $s\in\mathbf{C}$ which is holomorphic at $s=0$. (Seeley 1967)

As in the case of an orthogonal representation, we define the analytic torsion $T_M(\rho;E)$ by

$T_M(\rho;E) = \exp\biggl(\frac{1}{2}\sum^n_{q=0}(-l)^qq\frac{d}{ds}\zeta_q(s;\rho)\biggl|_{s=0}\biggr).$

In 1971 D.B. Ray and I.M. Singer conjectured that $T_M(\rho;E)=\tau_M(\rho;\mu)$ for any unitary representation $\rho$.Ray and Singer (1971) Independently, J. Cheeger Cheeger (1977, 1979) and W. Muller Müller (1978) proved the Ray–Singer conjecture. Their idea is considering the logarithm of torsions and their traces. Firstly for odd-dimensional manifolds they had proved the equality of two torsions and then for even-dimensional, which have some technical difficulties.

In later years, along with Atiyah–Patodi–Singer theorem, the Cheeger–Müller theorem, i.e. the equivalence of two torsions, forms the basis of Chern–Simons perturbation theory.

References

• Online book