Iwasawa Invariants: Theory & Applications

• Iwasawa invariants are numerical measures quantifying the asymptotic growth of algebraic and topological structures in infinite cyclic covers.
• They are derived from the study of Alexander modules using p-adic techniques, mirroring classical class number formulas in number theory.
• These invariants provide explicit criteria linking factorization properties of Alexander polynomials with the growth of torsion in the homology of branched cyclic covers.

Iwasawa invariants are numerical invariants that appear in both number theory and low-dimensional topology, originating in the paper of the growth of algebraic and topological invariants in infinite towers of covers. Defined classically for the p-primary part of ideal class groups in cyclotomic ℤₚ-extensions of number fields, these invariants have rigorous analogues in the context of 3-manifolds and link complements, where they measure the asymptotic growth of torsion in the first homology groups of cyclic covers, extending the analogy between prime ideals and knots or links.

1. Definition of Iwasawa Invariants in the Topological Setting

For a link $L = K_1 \cup K_2 \cup \cdots \cup K_r$ in $S^3$ with complement $X$, the abelianization of the fundamental group $G_L = \pi_1(X)$ is $\mathbb{Z}^r$. Given a surjective homomorphism $$\Omega_z: H_1(X; \mathbb{Z}) \rightarrow \mathbb{Z}$$ induced by a vector $z = (z_1, \dots, z_r)$ with $\gcd(z_1, \ldots, z_r) = 1$, one studies the cyclic cover $X_z$ corresponding to $\ker(\Omega_z)$.

The multivariable Alexander module $A_L$ is a finitely generated module over the ring $$\mathcal{A} = \mathbb{Z}[t_1^{\pm1}, \dots, t_r^{\pm1}]$$, and the Alexander polynomial $\Delta_L(t_1, \dots, t_r)$ captures subtle topological information about the link complement. Specializing along $z$, set

$$A_{L,z}(t) = \begin{cases} (t-1)A_L(t^{z_1}, \dots, t^{z_r}), & r \geq 2, \ A_L(t), & r = 1. \end{cases}$$

For a fixed prime $p$, complete the ring at $p$ and the augmentation ideal, yielding $\mathcal{A}_1 = \mathbb{Z}_p[[T]]$ with $t = 1+T$. By the $p$-adic Weierstrass preparation theorem, $A_{L,z}(1+T)$ admits a unique factorization

$$A_{L,z}(1+T) = p^{\nu_{L,z}} \cdot P_{L,z}(T) \cdot U(T)$$

where:

• $p^{\nu_{L,z}}$ is the exact power of $p$ dividing $A_{L,z}(1+T)$,
• $P_{L,z}(T)$ is a distinguished (monic, constant coefficient divisible by $p$) polynomial of degree $\lambda_{L,z}$,
• $U(T)$ is a unit in $\mathbb{Z}_p[[T]]$.

The two primary Iwasawa invariants for $(L, z, p)$ are then:

• $\lambda_{L,z} = \deg P_{L,z}(T)$,
• $\mu_{L,z} = \nu_{L,z}$,

and the exponent $\nu_{L,z}$ serves as an additive correction (the so-called "constant term" invariant).

2. Analogy with Number Theory and Iwasawa's Class Number Formula

The link invariants $\lambda_{L,z}$ and $\mu_{L,z}$ are explicitly modeled on the classical Iwasawa invariants $\lambda_k$, $\mu_k$ (from cyclotomic $\mathbb{Z}_p$-extensions $k_\infty / k$), where the structure of the Iwasawa module (the Galois group of the maximal unramified abelian pro-$p$ extension) determines the growth of the $p$-part of the class number in analogy with the following table:

Topological	Arithmetic
Link complement $X$	Maximal unramified pro-$p$ extension of $k$
Alexander module $A_L$	Iwasawa module $Y$
Alexander poly. $\Delta_L$	Iwasawa polynomial $P_k(T)$
Specialization to $X_z$	Specialization to $\mathbb{Z}_p$-extension $k_\infty/k$
Invariants $(\lambda_{L,z}, \mu_{L,z}, \nu_{L,z})$	Invariants $(\lambda_k, \mu_k, \nu_k)$

In both settings, these invariants describe the extent and rate of growth for arithmetic or topological objects in towers.

3. Growth Formula for Branched Cyclic Covers

Let $M_{z,p^n}$ denote the $p^n$-fold cyclic branched cover of $S^3$ along $L$ determined by $(z, p)$. The $p$-part of the first homology group grows according to the Iwasawa-type formula: $$v_p(|H_1(M_{z,p^n}; \mathbb{Z})|) = \lambda_{L,z} \cdot n + \mu_{L,z} p^n + \nu_{L,z},$$ for all sufficiently large $n$. This formula mirrors the classical Iwasawa formula for the growth of the $p$-part of class numbers in a $\mathbb{Z}_p$-extension and is obtained from algebraic properties of the Alexander module under specialization.

4. The Greenberg-Type Conjecture and Pseudonullity for Link Modules

Greenberg's conjecture in number theory predicts that the unramified Iwasawa module $Y_k$ is pseudonull over the multi-variable Iwasawa algebra $\mathbb{Z}_p[[T_1, \ldots, T_r]]$, i.e., its support has codimension at least 2. Translating this to the link setting, the authors define the analogue

$$Y_L = B_L / (\operatorname{Image}(M_L) \cap \ker(\partial_2)),$$

where $M_L$ is generated by meridional elements and $\partial_2$ arises from a Fox–calculus-based Crowell exact sequence. $Y_L$ is an $\mathcal{A}_r = \mathbb{Z}_p[[T_1, \ldots, T_r]]$-module.

The principal problem is whether $Y_L$ is pseudonull as an $\mathcal{A}_r$-module. The main results specify:

• If the multivariable Alexander polynomial $\Delta_L$ has no prime factor $f$ with $f(1,1,...,1)=\pm1$, then $Y_L$ is pseudonull.
• For $r=1$ (knot case), $Y_L$ is not pseudonull.

This establishes a topological counterpart to Greenberg's conjecture for links, with strong dependencies on the properties of the Alexander polynomial.

5. Algebraic and Homological Techniques in the Study of Iwasawa Invariants

Key methodologies developed or employed include:

• Fox calculus and Alexander–Fox invariants to connect combinatorial presentations of $G_L$ with $\mathcal{A}$-module structures and effective presentations of the Alexander module.
• Application of the $p$-adic Weierstrass preparation theorem to the completed Alexander polynomial, facilitating the precise isolation of the invariants $\lambda_{L,z}$ and $\mu_{L,z}$.
• Inverse limit and localization techniques to analyze the structure and support of Alexander modules and their analogues in infinite cyclic covers.
• Use of covering space theory and explicit computation of homology in towers, often relying on the eigenvalues of deck transformation actions.

6. Explicit Criteria, Consequences, and Applications

The structural analysis provides explicit criteria linking the vanishing or nonvanishing of Iwasawa invariants to factorization properties of the Alexander polynomial and to the value $A_L(1,1,\dots,1)$. In particular:

• The existence of cyclic covers of $L$ with arbitrary prescribed values of $(\lambda, \mu, \nu)$ can be engineered by detecting suitable specializations.
• The pseudonullity (or lack thereof) of the "unramified" link module $Y_L$ is controlled by the absence (or presence) of factors in $\Delta_L$ vanishing at $(1, \ldots, 1)$.
• The analogy not only provides computational tools for the paper of cyclic covers of links but also furnishes topological models for conjectures and phenomena originally formulated in arithmetic, such as smallness of unramified Iwasawa modules and the connection between torsion growth and topological data.

7. Summary of Key Formulas and Conceptual Dictionary

The crucial algebraic and topological statements and their formulas include:

Invariant/Formulation	Formula/Description
Specialization of Alexander poly.	$A_{L,z}(1+T) = p^{\nu_{L,z}} P_{L,z}(T) U(T)$
Iwasawa invariants	$\lambda_{L,z} = \deg P_{L,z}(T)$, $\mu_{L,z} = \nu_{L,z}$
Growth formula for cyclic covers	$$v_p(|H_1(M_{z,p^n}; \mathbb{Z})|) = \lambda_{L,z} n + \mu_{L,z} p^n + \nu_{L,z}$$
Greenberg-type pseudonullity (criterion)	Absence of factors $f$ in $\Delta_L$ with $f(1,...,1) = \pm 1$ implies $Y_L$ pseudonull
Unramified link module	$$Y_L = B_L / (\operatorname{Image}(M_L) \cap \ker(\partial_2))$$

These results establish a robust parallelism between Iwasawa theory in number theory and the homological theory of links in three-manifolds, providing deep insight into the structure and growth of homology in covering towers of links within $S^3$ (Kadokami et al., 2012).