nLab arithmetic topology

Contents

Idea

Arithmetic topology is a theory describing some surprising analogies between 3-dimensional topology and number theory (arithmetic), where knots embedded in a 3-manifold behave like prime ideals in a ring of algebraic integers. See also at Spec(Z) – As a 3d space containing knots. More broadly, the scope of arithmetic topology is now taken to include the intersection of arithmetic geometry, algebraic topology and low-dimensional topology (see GGW20).

Under the original analogy, the 3-sphere, $S^3$ corresponds to the ring of rational numbers $\mathbb{Q}$, or rather (the closure of) $spec(\mathcal{O}_{\mathbb{Q}})$ (i.e., $spec(\mathbb{Z})$), since the 3-sphere has no non-trivial (unbranched) covers while $\mathbb{Q}$ has no non-trivial unramified extensions. The linking number between two embedded knots in the 3-sphere then corresponds to the Legendre symbol between two primes in the ordinary integers.

The so-called M^2KR dictionary (Mazur-Morishita-Kapranov-Reznikov) relates terms from each side of the analogy (see sec 2.2 of Sikora).

The dictionary

Version of Mazur-Morishita-Kapranov-Reznikov (M^2KR)

1. Closed, orientable, connected 3-manifolds correspond to (the closure of) schemes $Spec \mathcal{O}_K$ for number fields $K$.
2. Links correspond to ideals in $\mathcal{O}_K$ and knots correspond to prime ideals (tame in both cases). Knots can be represented by immersions of $S^1$ into $M$, and prime ideals in $\mathcal{O}_K$ can be identified with closed immersions $Spec \mathbb{F} \to Spec \mathcal{O}_K$, where $\mathbb{F}$‘s are finite fields. Each link decomposes uniquely as a union of knots and each ideal decomposes uniquely as a product of primes.
3. An algebraic integer corresponds to an embedded surface (possibly with boundary), and the operation $a \to (a)$ corresponds to taking its boundary. Closed embedded surfaces correspond to units in $\mathcal{O}_K$. Ideals of the form $(a)$ represent the identity in $Cl(K)$, and the links of the form $\partial S$ represent the identity in $H_1(M,\mathbb{Z})$.
4. $Cl(K)$ corresponds to the torsion component of first integral homology. The free component of $H_1(M,\mathbb{Z})$ corresponds to the group of units in $\mathcal{O}_K$ after removing the torsion (roots of unity).
5. Finite extensions of number fields correspond to finite branched coverings.
6. $S^3$ is supposed to correspond to $\mathbb{Q}$. Notice that $S^3$ has no nontrivial unbranched covers, and similarly $\mathbb{Q}$ has no nontrivial unramified extensions.
7. A Galois extension $L/K$ with Galois group $G$ induces a morphism $Spec \mathcal{O}_L \to (Spec \mathcal{O}_L)/G = Spec \mathcal{O}_K$. Such maps correspond to the quotient maps $M \to M/G$ induced by orientation preserving actions of finite groups $G$ on 3-manifolds $M$. One can show that $M/G$ is always a 3-manifold and that the maps $M \to M/G$ are branched coverings.
8. Let $q = p^n$. Consider the cyclotomic extension $\mathbb{Q}(\zeta_q)$. It is ramified only at $p$. These correspond to cyclic branched covers of knots in $S^3$. The union of these as $q$ ranges over all powers of $p$ should correspond to the universal abelian cover of $S^3 \setminus K$. There is a natural action of $\mathbb{Z}$ on the first homology group of the infinite cyclic cover of the knot complement corresponding to the natural action of the $p$-adic integers on the $p$-torsion of $Cl(\mathbb{Q}(\zeta_{p^{\infty}}))$. This concerns the Alexander polynomial of the knot and Iwasawa theory. (Sikora, pp. 5-6, Koberda08, pp. 32-33)

Note: Regarding (4), some have argued that $Cl(K)$ should correspond to the full first integral homology group, (see, e.g., Goundaroulis & Kontogeorgis).

The correspondence between $\pi^{et}_1(\mathbb{Z} -\{p\})$ and $\pi_1(S^3 \setminus K)$ can be developed to relate the Legendre symbol for two primes to the linking number of two knots, and further to the Rédei symbol for three primes and Milner’s triple linking number. Thus we can find a ‘Borromean link’ of primes, such as $(13, 61, 397)$, where each pair is unlinked.

Disanalogies

1. The algebraic translation of the Poincaré Conjecture is false. $\mathbb{Q}$ is not the only number field with no unramified extensions. Nevertheless, $\hat{H}^i(Spec \mathcal{O}_K, \mathbb{Z}/n\mathbb{Z}) = \hat{H}^i(Spec \mathbb{Z}, \mathbb{Z}/n\mathbb{Z}) (i \in \mathbb{Z}, n \geq 2)$ if and only if $\mathcal{O}_K = \mathbb{Z}$.
2. Let $M_1 \to M$ be a covering of 3-manifolds. A knot $K$ in $M$ does not necessarily lift to a knot in $M_1$, while every prime ideal $p \triangleleft \mathcal{O}_K$ gives rise to an ideal $p \mathcal{O}_L$, where $L/K$ is a Galois number field extension. (Goundaroulis & Kontogeorgis).

Version of Deninger

Similar to M^2KR, but with the introduction of a 2-dimensional foliation on the 3-manifold and a flow such that finite primes $p$ correspond to periodic orbits of length $log N p$ and the infinite primes correspond to the fixed points of the flow (Deninger02). (See also the work of Baptiste Morin on the Weil-étale topos.)

Version of Reznikov

Reznikov has modified the dictionary (Reznikov 00, section 12) so as to associate a number field with what he calls a $3\frac{1}{2}$-manifold, that is a closed three-manifold $M$, bounding a four-manifold $N$, such that the map of fundamental groups $\pi_1(M) \to \pi_1(N)$ is surjective.

Explanations for the analogy

Barry Mazur observed that for an affine spectrum $X = Spec(D)$ of the ring of integers $D$ in a number field, the groups $H^n_{et}(X, \mathbb{G}_{m, X})$ vanish (up to 2-torsion) for $n \gt 3$, and is equal to $\mathbb{Q}/\mathbb{Z}$ for $n = 3$, where $\mathbb{G}_{m, X}$ is the étale sheaf on $X$ defined by associating to a connected finite étale covering $Spec(B) \to X$ the multiplicative group $\mathbb{G}_{m, X}(Y) = B^{\times}$.

Also, there is a non-degenerate pairing for any constructible abelian sheaf $M$,

$$H^r_{et}(X,M^{'}) \times Ext^{3-r}_X(M,\mathbb{G}_{m, X}) \to H^3_{et}(X,\mathbb{G}_{m, X})\simeq \mathbb{Q}/\mathbb{Z},$$

where $M^{'} = Hom(M, \mathbb{G}_{m, X})$. This resembles Poincaré duality for 3-manifolds.

Minhyong Kim argues that the normal bundle of an embedding of a circle corresponding to a prime in $Spec(\mathbb{Z})$ is 2-dimensional (Kim).

Baptiste Morin claims to provide a unified treatment via equivariant etale cohomology (Morin06).

Related concepts

• Alexander polynomial

• function field analogy

• The virtually fibered conjecture says that every closed, irreducible, atoroidal 3-manifold with infinite fundamental group has a finite cover which is a surface fiber bundle over the circle.

• arithmetic Chern-Simons theory

References

• Christopher Deninger, A note on arithmetic topology and dynamical systems, (arxiv:0204274)

• Dimoklis Goundaroulis, Aristides Kontogeorgis, On the Principal Ideal Theorem in Arithmetic Topology, (talk, paper)

• Minhyong Kim, note

• Thomas Koberda, Class Field Theory and the MKR Dictionary for Knots, (pdf)

• Baptiste Morin, Applications of an Equivariant Etale Cohomology to Arithmetic Topology, arxiv:0602064 and Utilisation d’une cohomologie étale équivariante en topologie arithmétique, Compositio Math. 144 (2008), no. 1, 32-60.

• Masanori Morishita, Analogies between Knots and Primes, 3-Manifolds and Number Rings, (arxiv:0904.3399)

• Alexander Reznikov, Embedded incompressible surfaces and homology of ramified coverings of three-manifolds, Selecta Math. 6(2000), 1–39

• Adam Sikora, Analogies between group actions on 3-manifolds and number fields, (arxiv)

• Toshitake Kohno, Masanori Morishita (eds.), Primes and Knots, Contemporary Mathematics, AMS 2006 (conm:416)

• Masanori Morishita, Knots and Primes: An Introduction to Arithmetic Topology, 2012, Springer, (web)

• Claudio Gómez-Gonzáles, Jesse Wolfson, Problems in Arithmetic Topology (arXiv:2012.15434)