Higher-Dimensional Reidemeister Torsion

• Higher-dimensional Reidemeister torsion is a generalization of classical torsion, defined using high-dimensional representations built from symmetric and tensor powers.
• It is constructed via twisted chain complexes of manifolds and relies on the acyclicity of local systems to yield well-defined topological invariants.
• Its asymptotic analysis connects geometric invariants like hyperbolic volume and Euler characteristics with analytic torsion and surgery formulas in manifold topology.

Higher-dimensional Reidemeister torsion is the generalization of the classical Reidemeister torsion invariant to contexts where the underlying representation of the fundamental group is high-dimensional, often constructed via compositions of base representations with irreducible symmetric-power or tensor representations. This invariant provides deep links between topology, representation theory, quantum invariants, and geometric/analytic structures, particularly in the study of 3-manifold topology, Seifert fibered spaces, and knot theory.

1. Construction of Higher-Dimensional Reidemeister Torsion

Let $M$ be a compact, oriented manifold (typically of dimension 3), and let $\rho:\pi_1(M)\to \mathrm{SL}_2(\mathbb{C})$ be an irreducible representation. For each $n\geq1$, one considers the $n$-dimensional irreducible complex representation of $\mathrm{SL}_2(\mathbb{C})$, i.e., the $(n-1)$st symmetric power: $$\sigma_n:\mathrm{SL}_2(\mathbb{C}) \to \mathrm{SL}_n(\mathbb{C}),$$ which acts on homogeneous polynomials of degree $n-1$ in two variables. The composite

$$\rho_n := \sigma_n \circ \rho:\pi_1(M) \to \mathrm{SL}_n(\mathbb{C})$$

is then used to define a twisted local coefficient system. The higher-dimensional Reidemeister torsion

$\tau_n(M, \rho_n) = \mathrm{Tor}(C_*(M; V_n))$

is constructed from the twisted chain complex $$C_*(M; V_n) = V_n \otimes_{\mathbb{C}[\pi_1(M)]} C_*(\widetilde{M}; \mathbb{C})$$, provided $H_*(M; V_n) = 0$ (acyclicity). The torsion is defined as the alternating product of determinants of boundary maps with respect to chosen bases, yielding an element in $\mathbb{C}^*$, up to certain sign ambiguities for odd-dimensional manifolds (Yamaguchi, 2012, Porti, 2015).

2. Criteria for Acyclic Representations and Character Varieties

The computation and well-definedness of higher-dimensional torsion depends on the acyclicity of the induced local system. For torus knot exteriors $E_K = S^3 \setminus N(K)$ with $K = T(p,q)$ and

$\rho: \pi_1(E_K) \to \mathrm{SL}_2(\mathbb{C}),$

acyclicity for all even-dimensional lifts $\rho_{2N}$ is achieved precisely when the central element maps to $-I$: $$\rho(x^p) = \rho(y^q) = -I, \quad a \equiv b \equiv 1 \pmod{2},$$ where $x^p = y^q$ is central and $R_{a,b}$ parametrizes the component of the character variety determined by traces of peripheral elements (Yamaguchi, 2012).

For hyperbolic 3-manifolds, it is established that for $n$-dimensional symmetric power representations (with suitable choices of spin structure for even $n$ in the cusped case) the associated cohomology groups vanish, ensuring the torsion is well defined for all $n$ (Menal-Ferrer et al., 2011, Porti, 2015).

On the level of the character variety $X(M, \mathrm{SL}_2(\mathbb{C}))$, higher Reidemeister torsion may be viewed as a (typically rational) function on the space of representations. In more advanced settings, e.g., for the adjoint representation or for $\mathrm{PSL}_{n+1}(\mathbb{C})$-character varieties, the torsion function encodes geometric data, with regularity properties and singularities where twisted cohomology jumps dimension (Porti, 2015, Porti et al., 2021).

3. Asymptotic Behavior and Relations to Geometric Invariants

A central result in higher-dimensional torsion theory concerns the asymptotic growth of

$\frac{\log|\tau_n(M, \rho_n)|}{n^2}$

as $n \to \infty$. For finite-volume hyperbolic $3$-manifolds $M$, results of Müller (closed case), Menal-Ferrer--Porti (cusped case), and others show that

$$\lim_{n \to \infty} \frac{\log|\tau_n(M)|}{n^2} = -\frac{\mathrm{Vol}(M)}{4\pi},$$

identifying the leading-order growth coefficient with the hyperbolic volume of $M$ (Menal-Ferrer et al., 2011, Porti, 2015).

In contrast, for manifolds such as torus knot exteriors and more generally Seifert fibered spaces, which have zero hyperbolic volume (they are non-hyperbolic), the analogous normalized torsion exhibits vanishing quadratic growth: $$\lim_{N \to \infty} \frac{\log|\tau_{2N}(E_K, \rho_{2N})|}{(2N)^2} = 0,$$ with linear (in $2N$) growth determined by data related to the base orbifold Euler characteristic for Seifert fibered spaces (Yamaguchi, 2012, Yamaguchi, 2012, 2002.01156).

This dichotomy reflects a broad "volume-entropy" philosophy: hyperbolic manifolds contribute a negative volume-determined term to the torsion asymptotics, while Seifert fibered (and thus graph) manifolds yield zero (Yamaguchi, 2012, 2002.01156).

4. Surgery Formulas, Linear Growth, and SU(2)-Character Varieties

A detailed analysis is possible for Seifert fibered spaces via explicit surgery formulas. Applying a general Mayer–Vietoris "surgery formula" for asymptotics (Yamaguchi, 2012), for a Seifert fibered manifold $X$ with base orbifold of Euler characteristic $\chi$, the limits are: $$\lim_{N\to\infty} \frac{\log|T_{2N}(X,\rho)|}{(2N)^2} = 0, \ \lim_{N\to\infty} \frac{\log|T_{2N}(X,\rho)|}{2N} = -\chi \log 2,$$ with $T_{2N}(X,\rho)$ the higher-dimensional torsion, and $\chi = 2-2g-\sum_{j=1}^m (1-1/\alpha_j)$ for $X$ a Seifert space over a genus $g$ surface with exceptional fibers of orders $\alpha_j$. The maximum slope is realized exactly on the top-dimensional components of the irreducible $SU(2)$-character variety, and the assignment $$[\rho] \mapsto \lim_{N \to \infty} \frac{\log|T_{2N}(X,\rho)|}{2N}$$ is locally constant on each component (Yamaguchi, 2012).

5. Explicit Formulas and Formulaic Characterizations

For torus knot exteriors $E_K$, the explicit computation gives

$$\tau_{2N}(E_K, \rho_{2N}) = \frac{2^{2N} \prod_{k=1}^N 4^2 \sin^2\left(\frac{(2k-1)a\pi}{2p}\right) \sin^2\left(\frac{(2k-1)b\pi}{2q}\right)}{}$$

where $a,b$ parametrize the representation via the component $R_{a,b}$, and $p,q$ are the parameters of the torus knot $T(p,q)$. The bounding estimates

$$2^{-2N} \leq |\tau_{2N}| \leq 2^{-2N} \left[\sin\left(\frac{\pi}{2p}\right)\sin\left(\frac{\pi}{2q}\right)\right]^{-2N}$$

ensure the (normalized) torsion asymptotically vanishes in the quadratic normalization (Yamaguchi, 2012).

For hyperbolic cases, analytic techniques relate higher-dimensional torsion to Ruelle and Selberg zeta functions, with the absolute value of the Ruelle zeta function at zero equating to the torsion, and the leading identity-term in the functional equation controlling the asymptotics (2002.01156, Menal-Ferrer et al., 2011).

6. Structural and Multiplicative Properties

Reidemeister torsion in high dimensions exhibits precise multiplicativity under connected sum for certain classes of manifolds. For highly connected even-dimensional manifolds in a unique factorization monoid, the product formula holds exactly: $$\mathbb{T}_{RF}(W^{2n}) = \prod_{j=1}^{k} \mathbb{T}_{RF}(M_j)$$ with $W^{2n} = M_1 \# \cdots \# M_k$ a unique decomposition into indecomposables, generalizing and refining the classical Milnor multiplicativity theorem (Erdal, 3 Nov 2025).

7. Broader Contexts and Applications

Higher-dimensional torsion invariants extend classical torsion into the setting of twisted local systems, fiber bundles, and generalized cohomological frameworks. Twisted higher torsion invariants satisfy a suite of axioms (naturality, geometric additivity and transfer, additivity/induction in coefficients), with the "Igusa–Klein" (higher Franz–Reidemeister) torsion and twisted Miller–Morita–Mumford classes providing spanning examples. In the setting of smooth manifold bundles with finite holonomy cover, the entire space of twisted higher torsion invariants, under suitable restrictions, is two-dimensional or one-dimensional, completely determined by these characteristic classes (Ohrt, 2012).

Applications span the obstruction theory for fiberwise $h$-cobordisms, the study of diffeomorphism groups via classifying spaces, ties to analytic torsion (Bismut–Lott), and connections with spectral invariants, dynamical zeta functions, and quantum topology.

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References:

• (Yamaguchi, 2012) Y. Yamaguchi, "Higher even dimensional Reidemeister torsion for torus knot exteriors"
• (Porti, 2015) J. Porti, "Reidemeister torsion, hyperbolic three-manifolds, and character varieties"
• (Menal-Ferrer et al., 2011) P. Menal-Ferrer, J. Porti, "Higher dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds"
• (2002.01156) Y. Yamaguchi, "Dynamical zeta functions for geodesic flows and the higher-dimensional Reidemeister torsion for Fuchsian groups"
• (Yamaguchi, 2012) Y. Yamaguchi, "A surgery formula for the asymptotics of the higher dimensional Reidemeister torsion and Seifert fibered spaces"
• (Erdal, 3 Nov 2025) T. Schick, M. Wrochna, "On torsion of non-acyclic cellular chain complexes of even manifolds in a unique factorisation monoid"
• (Ohrt, 2012) I. Igusa, "Axioms for Higher Twisted Torsion Invariants of Smooth Bundles"
• (Porti et al., 2021) J. Porti, Y. Yoon, "The adjoint Reidemeister torsion for the connected sum of knots"