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\begin{document}

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\section*{arithmetic topology}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{knot_theory}{}\paragraph*{{Knot theory}}\label{knot_theory}

[[!include knot theory - contents]]

\hypertarget{arithmetic_geometry}{}\paragraph*{{Arithmetic geometry}}\label{arithmetic_geometry}

[[!include arithmetic geometry - contents]]

\hypertarget{contents}{}\subsection*{{Contents}}\label{contents}

\noindent\hyperlink{contents}{Contents}\dotfill \pageref*{contents} \linebreak
\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{the_dictionary}{The dictionary}\dotfill \pageref*{the_dictionary} \linebreak
\noindent\hyperlink{VersionOfM^2KR}{Version of Mazur-Morishita-Kapranov-Reznikov (M{\tt \symbol{94}}2KR)}\dotfill \pageref*{VersionOfM^2KR} \linebreak
\noindent\hyperlink{disanalogies}{Disanalogies}\dotfill \pageref*{disanalogies} \linebreak
\noindent\hyperlink{version_of_deninger}{Version of Deninger}\dotfill \pageref*{version_of_deninger} \linebreak
\noindent\hyperlink{ReznikovVariant}{Version of Reznikov}\dotfill \pageref*{ReznikovVariant} \linebreak
\noindent\hyperlink{explanations_for_the_analogy}{Explanations for the analogy}\dotfill \pageref*{explanations_for_the_analogy} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\textbf{Arithmetic topology} is a theory describing some surprising [[analogies]] between [[low-dimensional topology|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 \emph{\href{Spec%28Z%29#As3dSpaceContainingKnots}{Spec(Z) -- As a 3d space containing knots}}.

Under this 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 ([[branched cover|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 \emph{M{\tt \symbol{94}}2KR dictionary} (Mazur-Morishita-Kapranov-Reznikov) relates terms from each side of the analogy (see sec 2.2 of \hyperlink{Sikora}{Sikora}).

\hypertarget{the_dictionary}{}\subsection*{{The dictionary}}\label{the_dictionary}

\hypertarget{VersionOfM^2KR}{}\subsubsection*{{Version of Mazur-Morishita-Kapranov-Reznikov (M{\tt \symbol{94}}2KR)}}\label{VersionOfM^2KR}

\begin{enumerate}%
\item [[closed manifold|Closed]], [[orientation|orientable]], [[connected topological space|connected]] [[3-manifolds]] correspond to (the closure of) schemes $Spec \mathcal{O}_K$ for number fields $K$.
\item 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.
\item 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 $H_1(M,\mathbb{Z})$.
\item $Cl(K)$ 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).
\item Finite extensions of number fields correspond to finite branched coverings.
\item $S^3$ 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.
\item 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.
\item 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]]. (\hyperlink{Sikora}{Sikora, pp. 5-6}, \hyperlink{Koberda08}{Koberda08, pp. 32-33})

\end{enumerate}
Note: Regarding (4), some have argued that $Cl(K)$ should correspond to the full first integral homology group, (see, e.g., \hyperlink{Goundaroulis}{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 R\'e{}dei 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.

\hypertarget{disanalogies}{}\paragraph*{{Disanalogies}}\label{disanalogies}

\begin{enumerate}%
\item The algebraic translation of the Poincar\'e{} 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 $\mathcal{O}_K = \mathbb{Z}$.
\item 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. (\hyperlink{Goundaroulis}{Goundaroulis \& Kontogeorgis}).

\end{enumerate}
\hypertarget{version_of_deninger}{}\subsubsection*{{Version of Deninger}}\label{version_of_deninger}

Similar to M{\tt \symbol{94}}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 (\hyperlink{Deninger02}{Deninger02}). (See also the work of \href{http://www.math.uni-muenster.de/reine/u/baptiste.morin/}{Baptiste Morin} on the Weil-\'e{}tale topos.)

\hypertarget{ReznikovVariant}{}\subsubsection*{{Version of Reznikov}}\label{ReznikovVariant}

Reznikov has modified the dictionary (\hyperlink{Reznikov00}{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.

\hypertarget{explanations_for_the_analogy}{}\subsection*{{Explanations for the analogy}}\label{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 \'e{}tale sheaf on $X$ defined by associating to a connected finite \'e{}tale 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 $M$,

\begin{displaymath}
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},
\end{displaymath}
where $M^{'} = Hom(M, \mathbb{G}_{m, X})$. This resembles Poincar\'e{} 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 (\hyperlink{Kim}{Kim}).

[[Baptiste Morin]] claims to provide a unified treatment via equivariant etale cohomology (\hyperlink{Morin06}{Morin06}).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Alexander polynomial]]


\item [[function field analogy]]


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


\item [[arithmetic Chern-Simons theory]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Christopher Deninger]], \emph{A note on arithmetic topology and dynamical systems}, (\href{http://arxiv.org/abs/math/0204274}{arxiv:0204274})


\item Dimoklis Goundaroulis, Aristides Kontogeorgis, \emph{On the Principal Ideal Theorem in Arithmetic Topology}, (\href{http://users.uoa.gr/~kontogar/talks/GkountPSATHA.pdf}{talk}, \href{http://arxiv.org/abs/0705.3937}{paper})


\item Minhyong Kim, \href{http://minhyongkim.files.wordpress.com/2013/05/baez13-12.pdf}{note}


\item Thomas Koberda, \emph{Class Field Theory and the MKR Dictionary for Knots}, (\href{http://users.math.yale.edu/users/koberda/minorthesis.pdf}{pdf})


\item [[Baptiste Morin]], \emph{Applications of an Equivariant Etale Cohomology to Arithmetic Topology}, \href{http://arxiv.org/abs/math/0602064}{arxiv:0602064} and Utilisation d'une cohomologie étale équivariante en topologie arithmétique, Compositio Math. 144 (2008), no. 1, 32-60.


\item Masanori Morishita, \emph{Analogies between Knots and Primes, 3-Manifolds and Number Rings}, (\href{http://arxiv.org/abs/0904.3399}{arxiv:0904.3399})


\item [[Masanori Morishita]] 2012, \emph{Knots and Primes: An Introduction to Arithmetic Topology}, Springer


\item Alexander Reznikov, \emph{Embedded incompressible surfaces and homology of ramified coverings of three-manifolds}, Selecta Math. 6(2000), 1--39


\item Adam Sikora, \emph{Analogies between group actions on 3-manifolds and number fields}, (\href{http://arxiv.org/abs/math/0107210}{arxiv})


\item [[Toshitake Kohno]], [[Masanori Morishita]] (eds.), \emph{Primes and Knots}, Contemporary Mathematics, AMS 2006 (\href{http://www.ams.org/bookstore-getitem/item=CONM-416}{conm:416})


\item [[Masanori Morishita]], \emph{Knots and Primes: An Introduction to Arithmetic Topology}, 2012 (\href{https://books.google.co.uk/books?id=DOnkGOTnI78C&pg=PA156#v=onepage&q&f=false}{web})



\end{itemize}
[[!redirects MKR dictionary]] [[!redirects MKR analogy]]



\end{document}