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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 28th 2014
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexStarted [[arithmetic topology]].

   Started arithmetic topology.

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 28th 2014
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI added a reference to Morishita's book which makes the cute point that Gauss dealt with both sides of the analogy, i.e., his integral in what might be called abelian gauge field theory for the linking number of two embedded circles, and his work in what's now abelian class field theory on quadratic reciprocity. Hence the hope for further connections between non-abelian versions of each, non-abelian gauge field theory and Langlands.

   I added a reference to Morishita’s book which makes the cute point that Gauss dealt with both sides of the analogy, i.e., his integral in what might be called abelian gauge field theory for the linking number of two embedded circles, and his work in what’s now abelian class field theory on quadratic reciprocity. Hence the hope for further connections between non-abelian versions of each, non-abelian gauge field theory and Langlands.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 3rd 2014
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAdded some detail about the MKR dictionary.

   Added some detail about the MKR dictionary.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeSep 3rd 2014
   • (edited Sep 3rd 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks! So I suppose one point is that we are to think of a global field as being like a bundle of surfaces over a circle... From that point of view though, why are primes like knots? Maybe like this: instead of thinking of a prime as being a point, now it is an $S^1$-family of points, hence a section of the bundle of surfaces over the circle, hence a map from the circle to the total space 3-manifold. Hm, does that make sense?

   Thanks!

   So I suppose one point is that we are to think of a global field as being like a bundle of surfaces over a circle…

   From that point of view though, why are primes like knots? Maybe like this: instead of thinking of a prime as being a point, now it is an $S^1$-family of points, hence a section of the bundle of surfaces over the circle, hence a map from the circle to the total space 3-manifold.

   Hm, does that make sense?

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 3rd 2014
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexPeople most often point to $Spec(\mathbb{F}_p)$ being one-dimensional. Minhyong Kim does that, but has a more subtle [account](http://minhyongkim.files.wordpress.com/2013/05/baez13-12.pdf) of why this is embedded in a 3-dimensional space in terms of the normal bundle being 2-dimensional.

   People most often point to $Spec(\mathbb{F}_p)$ being one-dimensional. Minhyong Kim does that, but has a more subtle account of why this is embedded in a 3-dimensional space in terms of the normal bundle being 2-dimensional.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeSep 3rd 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, but how does that give that primes are knots? I see that it gives that primes are _some_ knots, namely the suitable sections of the bundle over the "circle" $Spec(\mathbb{F}_p)$. But somehow all knots are supposed to correspond to primes? I must still not quite have the right picture here...

   Yes, but how does that give that primes are knots? I see that it gives that primes are some knots, namely the suitable sections of the bundle over the “circle” $Spec(\mathbb{F}_p)$. But somehow all knots are supposed to correspond to primes?

   I must still not quite have the right picture here…

7. • CommentRowNumber7.
   • CommentAuthorZhen Lin
   • CommentTimeSep 3rd 2014
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexI really cannot believe that "people most often point to $Spec(\mathbb{F}_p)$ being one-dimensional". At the very least it is a misleading statement: the orthodox view is that it is zero-dimensional, like the spectrum of any other field.

   I really cannot believe that “people most often point to $Spec(\mathbb{F}_p)$ being one-dimensional”. At the very least it is a misleading statement: the orthodox view is that it is zero-dimensional, like the spectrum of any other field.

8. • CommentRowNumber8.
   • CommentAuthorZhen Lin
   • CommentTimeSep 3rd 2014
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItex@Urs I don't see where you're getting the idea that all global fields are supposed to be like 3-manifolds fibred over a circle. [Morishita](http://arxiv.org/abs/0904.3399v1) [p. 6] says quite clearly that "an algebraic curve $C$ over a finite field $\mathbb{F}_q$ [...] is seen homotopically as a surface bundle over a "circle" $Spec(\mathbb{F}_q)$ [...] Since a number ring has no constant field, there is no analogy between the structures $\pi_1 (Spec (\mathscr{O}_k))$ and $\pi_1 (C)$". (Besides, the Hopf fibration has $S^3$ fibred over $S^2$ with fibre $S^1$, not the other way around!)

   @Urs

   I don’t see where you’re getting the idea that all global fields are supposed to be like 3-manifolds fibred over a circle. Morishita [p. 6] says quite clearly that “an algebraic curve $C$ over a finite field $\mathbb{F}_q$ […] is seen homotopically as a surface bundle over a “circle” $Spec(\mathbb{F}_q)$ […] Since a number ring has no constant field, there is no analogy between the structures $\pi_1 (Spec (\mathscr{O}_k))$ and $\pi_1 (C)$”.

   (Besides, the Hopf fibration has $S^3$ fibred over $S^2$ with fibre $S^1$, not the other way around!)

9. • CommentRowNumber9.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 3rd 2014
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexRe 7, that was obviously meant in the context of people giving a rationale for a 3-dimensional $Spec(Z)$. E.g., Kim saying >$Spec(F_p)$ has dimension one because it’s a $K(\hat{Z}, 1)$. Try reading about the [Principle of Charity](http://en.wikipedia.org/wiki/Principle_of_charity).

   Re 7, that was obviously meant in the context of people giving a rationale for a 3-dimensional $Spec(Z)$. E.g., Kim saying

   $Spec(F_p)$ has dimension one because it’s a $K(\hat{Z}, 1)$.

   Try reading about the Principle of Charity.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2014
    • (edited Sep 3rd 2014)
    • PermaLink
    Author: Urs
    Format: MarkdownItexSo, indeed, Morishita writes it is > as a surface bundle over a “circle” and this is recently repeated [here](http://www.math.columbia.edu/~woit/wordpress/?p=7114&cpage=1#comment-214353) in a somewhat stronger form: > the geometric analog of a number field or function field in finite characteristic should not be a Riemann surface, but roughly a surface bundle over the circle. This explains the “categorification” (need for a function-sheaf dictionary, which is the weak part of the analogy) that takes place in passing from classical to geometric Langlands (Curiously, this is in a comment apparently meant to refute my statement that there is "educated guesswork" involved in formulating geometric Langlands -- while to me it reads rather like further strengthening this point. ) What I would enjoy is to understand the analogy -- or false analogy, as it may turn out -- well enough to have it guide one to a formal "inter-geometric" axiomatics that would allow one to turn the handwaving analogies into systematic theorems.

    So, indeed, Morishita writes it is

    as a surface bundle over a “circle”

    and this is recently repeated here in a somewhat stronger form:

    the geometric analog of a number field or function field in finite characteristic should not be a Riemann surface, but roughly a surface bundle over the circle. This explains the “categorification” (need for a function-sheaf dictionary, which is the weak part of the analogy) that takes place in passing from classical to geometric Langlands

    (Curiously, this is in a comment apparently meant to refute my statement that there is “educated guesswork” involved in formulating geometric Langlands – while to me it reads rather like further strengthening this point. )

    What I would enjoy is to understand the analogy – or false analogy, as it may turn out – well enough to have it guide one to a formal “inter-geometric” axiomatics that would allow one to turn the handwaving analogies into systematic theorems.

11. • CommentRowNumber11.
    • CommentAuthorZhen Lin
    • CommentTimeSep 4th 2014
    • PermaLink
    Author: Zhen Lin
    Format: MarkdownItexYes – I can see that curves over $\mathbb{F}_p$ are supposed to be like surface bundles over a circle. This is more or less clear once you accept that $Spec(\mathbb{F}_p)$ is supposed to be like a circle. But my point is that Morishita makes no such claim about number rings! (Recall that there are two kinds of global fields: number fields and function fields.) As far as I understand it, this "primes are knots" idea can be very quickly motivated like this: 1. $Spec(\mathbb{Z})$ has something that looks like 3-dimensional Poincaré duality in étale cohomology. (I know nothing about the details here.) 2. $Spec(\mathbb{Z}) \cup \{ \infty \}$ is supposed to have no non-trivial unramified (finite) coverings, hence is supposed to be (profinitely) simply connected. 3. The only simply-connected closed 3-manifold is $S^3$. 4. The (petit) étale topos of $Spec(\mathbb{F}_p)$ is the classifying topos for $\hat{\mathbb{Z}}$, so should be regarded as some kind of profinite $S^1$. 5. A knot is a (tame) map $S^1 \to S^3$.

    Yes – I can see that curves over $\mathbb{F}_p$ are supposed to be like surface bundles over a circle. This is more or less clear once you accept that $Spec(\mathbb{F}_p)$ is supposed to be like a circle. But my point is that Morishita makes no such claim about number rings! (Recall that there are two kinds of global fields: number fields and function fields.)

    As far as I understand it, this “primes are knots” idea can be very quickly motivated like this:

    1. $Spec(\mathbb{Z})$ has something that looks like 3-dimensional Poincaré duality in étale cohomology. (I know nothing about the details here.)
    2. $Spec(\mathbb{Z}) \cup \{ \infty \}$ is supposed to have no non-trivial unramified (finite) coverings, hence is supposed to be (profinitely) simply connected.
    3. The only simply-connected closed 3-manifold is $S^3$.
    4. The (petit) étale topos of $Spec(\mathbb{F}_p)$ is the classifying topos for $\hat{\mathbb{Z}}$, so should be regarded as some kind of profinite $S^1$.
    5. A knot is a (tame) map $S^1 \to S^3$.
12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexThe statement about number fields is the very first line of the [MKR dictionary](http://ncatlab.org/nlab/show/arithmetic+topology#VersionOfMKR), which I was replying to in #4. I suppose one issue is that there are two different analogies here. That MKR dictionary and the idea of having something like a 3-manifold fibered over a circle.

    The statement about number fields is the very first line of the MKR dictionary, which I was replying to in #4.

    I suppose one issue is that there are two different analogies here. That MKR dictionary and the idea of having something like a 3-manifold fibered over a circle.

13. • CommentRowNumber13.
    • CommentAuthorZhen Lin
    • CommentTimeSep 4th 2014
    • PermaLink
    Author: Zhen Lin
    Format: MarkdownItexI don't think the two analogies are so separate. A given 3-manifold might be fibred over $S^1$ or it might not. But here is a simpler objection to thinking about primes as sections of a bundle over $S^1$. Putting things back into the language of schemes, you are asking whether every (closed) point of a variety $X$ over $\mathbb{F}_p$ is necessarily a section of the unique morphism $X \to Spec(\mathbb{F}_p)$, and the answer is no in general: this only recovers the rational points of $X$. In general, a (closed) point of $X$ is a morphism $Spec(\mathbb{F}_{p^n}) \to X$; so back in this point of view where $Spec(\mathbb{F}_q)$ is a circle, what we have is not quite a section of a circle bundle but rather a diagram $$\begin{array}{ccc} S^1 & \to & M \\ & \searrow & \downarrow \\ & & S^1 \end{array}$$ where the diagonal arrow $S^1 \to S^1$ is the $n$-fold covering map.

    I don’t think the two analogies are so separate. A given 3-manifold might be fibred over $S^1$ or it might not.

    But here is a simpler objection to thinking about primes as sections of a bundle over $S^1$. Putting things back into the language of schemes, you are asking whether every (closed) point of a variety $X$ over $\mathbb{F}_p$ is necessarily a section of the unique morphism $X \to Spec(\mathbb{F}_p)$, and the answer is no in general: this only recovers the rational points of $X$. In general, a (closed) point of $X$ is a morphism $Spec(\mathbb{F}_{p^n}) \to X$; so back in this point of view where $Spec(\mathbb{F}_q)$ is a circle, what we have is not quite a section of a circle bundle but rather a diagram

    $$\begin{array}{ccc} S^1 & \to & M \\ & \searrow & \downarrow \\ & & S^1 \end{array}$$

    where the diagonal arrow $S^1 \to S^1$ is the $n$-fold covering map.

14. • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 4th 2014
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexI added quite a few things. I don't know if I got that pairing right in 'Explanations for the analogy'. I corrected the coefficient given in this [post](http://www.neverendingbooks.org/index.php/mazurs-dictionary.html). Does it look right? There seem to be plenty of related ideas out there, including a [paper](http://people.mpim-bonn.mpg.de/zagier/files/cntp-3-2/fulltext.pdf) discussing 'arithmetic topological quantum field theory' which reminds the authors, Gukov, Zagier and others, of arithmetic topology

    I added quite a few things. I don’t know if I got that pairing right in ’Explanations for the analogy’. I corrected the coefficient given in this post. Does it look right?

    There seem to be plenty of related ideas out there, including a paper discussing ’arithmetic topological quantum field theory’ which reminds the authors, Gukov, Zagier and others, of arithmetic topology

15. • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 4th 2014
    • (edited Sep 4th 2014)
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItex@Zhen but most 3-manifolds (the hyperbolic ones, which I gather are "most" of them), after Agol's theorem, are at least _virtually_ fibred over the circle.

    @Zhen but most 3-manifolds (the hyperbolic ones, which I gather are “most” of them), after Agol’s theorem, are at least virtually fibred over the circle.

16. • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeSep 5th 2014
    • (edited Sep 5th 2014)
    • PermaLink
    Author: Urs
    Format: MarkdownItexre #13: that sounds sensible! re #14: this is about the new section _[Explanations for the analogy](http://ncatlab.org/nlab/show/arithmetic%20topology#explanations_for_the_analogy)_, right? re #15: what's a good [link](http://ldtopology.wordpress.com/2012/03/12/or-agols-theorem/) (if not nLab page...) (in my dictionary, netiquette includes providing pertinent links :-)

    re #13: that sounds sensible!

    re #14: this is about the new section Explanations for the analogy, right?

    re #15: what’s a good link (if not nLab page…)

    (in my dictionary, netiquette includes providing pertinent links :-)

17. • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 5th 2014
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexRe #16 (re #14), yes. And I redid it according to Knots and Primes: An Introduction to Arithmetic Topology, Masanori Morishita. He talks there (p. 43) of "modified étale cohomology groups... which take the infinite primes into account". I haven't gone into that. Apparently the pairing is part of [Artin-Verdier duality](http://en.wikipedia.org/wiki/Artin%E2%80%93Verdier_duality). We have [[Verdier duality]]. Any relation?

    Re #16 (re #14), yes. And I redid it according to Knots and Primes: An Introduction to Arithmetic Topology, Masanori Morishita.

    He talks there (p. 43) of “modified étale cohomology groups… which take the infinite primes into account”. I haven’t gone into that.

    Apparently the pairing is part of Artin-Verdier duality. We have Verdier duality. Any relation?

18. • CommentRowNumber18.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 6th 2014
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexIn a rush, so <https://en.m.wikipedia.org/wiki/Virtually_fibered_conjecture>

    In a rush, so https://en.m.wikipedia.org/wiki/Virtually_fibered_conjecture

19. • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeSep 6th 2014
    • (edited Sep 6th 2014)
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks for the link. That does sound relevant, doesn't it. Have given it a minimum entry _[[virtually fibered conjecture]]_ and cross-linked a bit.

    Thanks for the link. That does sound relevant, doesn’t it. Have given it a minimum entry virtually fibered conjecture and cross-linked a bit.

20. • CommentRowNumber20.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 6th 2014
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexThere are ideas to connect the volumes of hyperbolic manifolds with the discriminant of a number field, as [here](http://people.math.umass.edu/~hajir/mypapers/30-mfo-fundamental-group-final.pdf). Ought to chase up that cusp/infinite place connection too.

    There are ideas to connect the volumes of hyperbolic manifolds with the discriminant of a number field, as here. Ought to chase up that cusp/infinite place connection too.

21. • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeSep 6th 2014
    • (edited Sep 7th 2014)
    • PermaLink
    Author: Urs
    Format: MarkdownItexSo the volumes of hyperbolic manifolds are supposed to be related to the imaginary part of [[analytically continued Chern-Simons theory]] (see the links there). This in turn should be given by the "arithmetic Chern-Simons theory" which is the goal of all that talk about arithmetic cohesion etc. elsewhere. (So, while I have nothing concrete to add for the moment, this just to highlight the bigger story here.)

    So the volumes of hyperbolic manifolds are supposed to be related to the imaginary part of analytically continued Chern-Simons theory (see the links there). This in turn should be given by the “arithmetic Chern-Simons theory” which is the goal of all that talk about arithmetic cohesion etc. elsewhere.

    (So, while I have nothing concrete to add for the moment, this just to highlight the bigger story here.)

22. • CommentRowNumber22.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 6th 2014
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexThat's what's being described in this [MO question](http://mathoverflow.net/q/84522/447), volume being combined with Chern-Simon invariant. That [paper](http://people.mpim-bonn.mpg.de/zagier/files/cntp-3-2/fulltext.pdf) by Gukov, Zagier et al I mentioned elsewhere is all about such such things.

    That’s what’s being described in this MO question, volume being combined with Chern-Simon invariant.

    That paper by Gukov, Zagier et al I mentioned elsewhere is all about such such things.

23. • CommentRowNumber23.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 6th 2014
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexUrs, isn't it [[analytically continued Chern-Simons theory]] we want rather than holomorphic Chern-Simons theory, which works with an odd number of complex dimensions? But they're related somehow presumably.

    Urs, isn’t it analytically continued Chern-Simons theory we want rather than holomorphic Chern-Simons theory, which works with an odd number of complex dimensions? But they’re related somehow presumably.

24. • CommentRowNumber24.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 6th 2014
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexFor what it's worth, I added something to [[volume conjecture]]. Another immense morass awaits me if I go any further.

    For what it’s worth, I added something to volume conjecture. Another immense morass awaits me if I go any further.

25. • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeSep 6th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexYes, right, sorry, I meant analytically continued CS.

    Yes, right, sorry, I meant analytically continued CS.

26. • CommentRowNumber26.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 6th 2014
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexSo had to create [[colored Jones polynomial]]. Just a stub.

    So had to create colored Jones polynomial. Just a stub.

27. • CommentRowNumber27.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 7th 2014
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexI added links to the paper of Agol et al finalising the proof of the virtual fibering conjecture, and Agol's ICM talk this year.

    I added links to the paper of Agol et al finalising the proof of the virtual fibering conjecture, and Agol’s ICM talk this year.

28. • CommentRowNumber28.
    • CommentAuthorTim_Porter
    • CommentTimeJun 2nd 2018
    • PermaLink
    Author: Tim_Porter
    Format: MarkdownItexdeleted duplicate reference to Deninger's paper. <a href="https://ncatlab.org/nlab/revision/diff/arithmetic+topology/22">diff</a>, <a href="https://ncatlab.org/nlab/revision/arithmetic+topology/22">v22</a>, <a href="https://ncatlab.org/nlab/show/arithmetic+topology">current</a>

    deleted duplicate reference to Deninger’s paper.

    diff, v22, current

29. • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeJan 1st 2021
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to this, from yesterday: * Claudio Gómez-Gonzáles, [[Jesse Wolfson]], _Problems in Arithmetic Topology_ ([arXiv:2012.15434](https://arxiv.org/abs/2012.15434)) <a href="https://ncatlab.org/nlab/revision/diff/arithmetic+topology/26">diff</a>, <a href="https://ncatlab.org/nlab/revision/arithmetic+topology/26">v26</a>, <a href="https://ncatlab.org/nlab/show/arithmetic+topology">current</a>

    added pointer to this, from yesterday:

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

    diff, v26, current

30. • CommentRowNumber30.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 2nd 2021
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexI added > 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](#GGW20)) since, as seen in the contents of the article in #29, the meaning of arithmetic topology is broadening. <a href="https://ncatlab.org/nlab/revision/diff/arithmetic+topology/27">diff</a>, <a href="https://ncatlab.org/nlab/revision/arithmetic+topology/27">v27</a>, <a href="https://ncatlab.org/nlab/show/arithmetic+topology">current</a>

    I added

    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)

    since, as seen in the contents of the article in #29, the meaning of arithmetic topology is broadening.

    diff, v27, current