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Stream: theory: mathematics

Topic: topological rings and fields

Peter Arndt (Feb 12 2025 at 18:58):

I can't resist to advertise this result that I find really cool:

The following are equivalent for a topological field K:

1. There is a path from 0 to 1
2. K is contractible
3. K is path connected
4. K can be used as coefficients for a Gelfand duality. That is: For any compact Hausdorff space $X$, the map $X \to MaxIdeals(Cont(X,K))$ is a homeomorphism)

(From Caicedo, Mantilla-Soler: On a characterization of path connected topological fields)

Kevin Carlson (Feb 12 2025 at 19:00):

That's very pretty! It's nice to see how 2 <-> 4 appear to get the particularity of the real numbers entirely out of the Gelfand duality story.

John Baez (Feb 12 2025 at 19:48):

Does there exist a path-connected topological field that's not an algebra over the real numbers?

Jean-Baptiste Vienney (Feb 12 2025 at 19:56):

The paper says that there exist path-connected topological fields of positive characteristic. If $F$ is such a field, it can't be an algebra over the real numbers. Let $n=\mathrm{char}\,F$. If $F$ is an algebra over the real numbers, we get a ring homomorphism $\rho:\mathbb{Q} \rightarrow \mathbb{R} \rightarrow F$ and so $0_F=n.1_F=1_F+\dots+1_F=\rho(1)+\dots+\rho(1)=\rho(1+\dots+1)=\rho(n)$ is invertible with inverse $\rho(\frac{1}{n})$. So $0_F=\rho(\frac{1}{n})0_F=\rho(\frac{1}{n})\rho(n)=\rho(1)=1_F$ which implies $0_F=1_F$ and thus $F=\{0_F\}$ which is false by definition of a field.

Jean-Baptiste Vienney (Feb 12 2025 at 20:07):

Now the question is what are these path-connected topological fields of positive characteristic!!

Jean-Baptiste Vienney (Feb 12 2025 at 20:07):

We should take a look at the paper they cite to discover this.

John Baez (Feb 12 2025 at 20:08):

Yes, such fields sound pretty weird.

Peter Arndt (Feb 12 2025 at 20:42):

Apparently yes, there exist such fields with positive characteristic, constructed in Waterman/Bergman: Connected fields of arbitrary characteristic

Kevin Carlson (Feb 12 2025 at 22:41):

Roughly speaking, you just adjoin indeterminates $X_r$ for every $r\in (0,1)$ to whatever integral domain you like, then create a norm that makes $X_r$ close to $X_{r+\varepsilon},$ and manage to extend that to a whole topological ring structure and then to the field of fractions.

Mike Shulman (Feb 13 2025 at 00:36):

Could you just work in the category of topological rings, starting with a field $k$ of characteristic $p$ with the discrete topology, adjoin a family of indeterminates $x_i$ indexed by the space $[0,1]$ (which should make sense in the category of topological rings), quotient by the ideal $(x_0, x_1-1)$, and then take the topological-field of fractions? Is there some reason some of those operations don't make sense in the topological world?

Kevin Carlson (Feb 13 2025 at 02:28):

Ooh, interesting, so the universal property of this guy $k[x_{[0,1]}]$ is that topological ring maps into a $k$-algebra $R$ correspond with continuous paths $[0,1]\to R$? Interesting! Maps out of $[0,1]$ give an continuous functor on topological rings satisfying the solution set condition, so that's definitely fine, and topological rings should be topological over rings, creating colimits there, so you've made it to the correct integral domain...It's less obvious to me how to get general category theory to help construct the topological field of fractions, since topological fields are not complete or accessible, so I feel very short on tools here. Is there an adjunction lifting theorem that will apply to lift along the pair of solid functors forgetting the topologies on rings and fields, maybe?

Mike Shulman (Feb 13 2025 at 03:43):

Couldn't you take the ordinary field of fractions and equip it with the finest field topology making the inclusion continuous?

I guess the sticking point in any abstract-nonsense approach like this would be proving that the topology you get isn't indiscrete. Since, of course, any field admits the indiscrete topology, but that's not very interesting.

Kevin Carlson (Feb 13 2025 at 18:25):

Yeah, I think your construction exemplifies that you can in fact lift adjunctions across pairs of topological functors.

Mike Shulman (Feb 13 2025 at 18:53):

I wonder if the topology you get this way coincides with the Waterman-Bergman one.

John Baez (Feb 13 2025 at 20:14):

By the way: unless I missed it, we've never yet found a topological ring with a path component that is not simply connected, nor proven that one can't exist. The trick just explained - "sewing in a path of indeterminates" - might be a strategy for create such a topological ring. Can we form the free topological ring on a loop of indeterminates $x_t (t \in S^1)$, and will this loop be noncontractible?

Mike Shulman (Feb 13 2025 at 20:18):

We should certainly be able to form it, and I'd be pretty surprised if it turned out to be contractible.

Mike Shulman (Feb 13 2025 at 20:18):

But I don't know how to go about proving that it isn't.

John Baez (Feb 13 2025 at 20:27):

Yes, since "freedom's just another word for nothing left to lose", as Joplin observed, it seems unlikely that the free topological ring on a loop will be contractible! Maybe I'll pose this as a question on MathOverflow, to give the experts something to chew on.

Jean-Baptiste Vienney (Feb 13 2025 at 20:37):

Please if you ask this on MathOverflow, which seems like a good idea, say that I started this discussion on Zulip.

Jean-Baptiste Vienney (Feb 13 2025 at 20:37):

Else, I will be a bit angry.

John Baez (Feb 13 2025 at 20:50):

Sure, I'll say that. But by the way: when asking people for favors, it works best not to say "Else, I will be a bit angry". Saying that actually reduced the chance of me citing you, since I like doing favors but respond negatively to threats. (When people say "otherwise I'll be angry", that gets interpreted as a threat even if it's not, because anger is always an implicit threat - and to keep people from threatening us, we often resist doing what they asked.)

Jean-Baptiste Vienney (Feb 13 2025 at 20:53):

That’s kind of a threat indeed because I’m not asking for a favor but just to cite your sources. But thank you. Now, I’m happy. :)

Jean-Baptiste Vienney (Feb 13 2025 at 20:54):

(I agree that I'm not an expert in diplomacy and communication though so thank you for the advice.)

John Baez (Feb 13 2025 at 20:58):

I guess I should cite you, Mike Shulman (who in this conversation pointed out the idea of constructing a free ring on a space), and maybe Kevin Carlson (who described how to get path-connected fields of characteristic $p$), Peter Arndt (who pointed out a paper that does that), and Waterman and Bergman (who actually wrote a paper doing that), and Caicedo and Mantilla-Soler (since I'm mentioning one of their results on path connected topological fields).

Jean-Baptiste Vienney (Feb 13 2025 at 21:00):

You can do that. There are papers with dozen or hundred of authors sometimes when a lot of people are involved.

John Baez (Feb 13 2025 at 21:01):

Normally I wouldn't cite all these people, since I'm just posting a question to a website, not writing a paper, and I feel the question is my own. But you're asking me to cite you, I feel that to be fair I have to cite them as well.

Now I want to work on that question.

Jean-Baptiste Vienney (Feb 13 2025 at 21:06):

You can explain what everyone did like that I guess. I think this is perfectly fine, polite and fair.

That’s your choice. As to me, I’ve already said what I’ve said!

Kevin Carlson (Feb 13 2025 at 21:10):

John Baez said:

as Joplin

following Kristofferson!

John Baez (Feb 13 2025 at 21:11):

Whoops, I forgot to cite Kristofferson! Do I have to cite him too in my MathOverflow question? :face_with_spiral_eyes:

Jean-Baptiste Vienney (Feb 13 2025 at 21:12):

Brr but some important facts to consider are that: 1) I’m the one who started the discussion 2) the other guys in the discussion are professors, postdocs or work at the Topos institute I think while I’m just a PhD student.

Jean-Baptiste Vienney (Feb 13 2025 at 21:16):

If I was a tenured prof, I don’t think I would care that much.

Jean-Baptiste Vienney (Feb 13 2025 at 21:24):

Another time, you cited a guy on Mathsodon who participated in a conversation on Zulip without citing me who started the conversation and asked the relevant question. I think it was unfair.

Jean-Baptiste Vienney (Feb 13 2025 at 21:24):

(It was about multiplicatively idempotent rigs.)

John Baez (Feb 13 2025 at 21:37):

In fact I predict that even if you get tenure you will be worrying that people don't cite you enough. And as you get older and older, you will become ever more resentful of how you haven't been cited enough.

Of course I can't really predict what you'll do. But this is a common pattern which I observe in many people, including many famous mathematicians but also myself. I try to fight it in myself, with mixed success.

I have a theory for why this happens. When mathematicians are young, they hope that eventually they will become famous, or at least widely respected. But as they get older, even if they are famous, they start realizing that they are going downhill, not up: they'll never become as respected as they hoped, and they'll eventually be a mere footnote in the history of mathematics. So they become resentful - unless they deal with this problem in their own minds, and transcend this pettiness.

Jean-Baptiste Vienney (Feb 13 2025 at 21:48):

Maybe you're right. For sure, I hope that eventually I will become famous and probably I will just be a footnote in the history of mathematics. But for now, I also know that the more people can read my name, the better it will be for me professionally, to be paid at the end of the month, stable etc... So right now, it is not only about ego. At least that's why I feel. But that's very hard for me to understand why I'm saying this or this, why I'm doing this or this, what are the real reasons, so maybe this is mostly about ego and I just try to find a justification. I don't know!

Kevin Carlson (Feb 13 2025 at 21:51):

This is all fundamentally tangential to the simple point that it’s inappropriately aggressive in a professional setting to tell someone they’ll make you angry if they don’t do something you want. This remains just as true even if you have a knock-down argument that it’s ethically correct for them to do that thing. That’s the takeaway here.

Jean-Baptiste Vienney (Feb 13 2025 at 21:57):

Ok, I will not say anymore "I will be a bit angry." if that's inappropriate. And I'm also doing this: I kindly ask people to cite my name if they talk about a math problem on the Internet, somewhere else that on this Zulip if I started the discussion where the question emerged. Especially if I don't just started the discussion but also asked at the same time a relevant question which is very close to what they are talking about on the other website.

John Baez (Feb 13 2025 at 21:58):

Here is my Mathoverflow question:

• Is there a non-simply-connected topological ring?

Note that I actually conjecture a construction of a topological ring that has nontrivial nth homotopy group at a certain basepoint (not $0$ or $1$).

Jean-Baptiste Vienney (Feb 13 2025 at 22:00):

Thank you, that's perfect!