Santana Afton

It seems less an honest question and more of a frustrated rant

I’d love to see this question closed:

This question is a duplicate of this one.

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@Behacad I know that “simply asking” isn’t the exciting answer you’re looking for, but it’s the correct one. Even now, two individuals in a relationship who could biological procreate will not necessarily do so. Disagreements about having kids is certainly a reason couples can break up, and having that conversation simply requires talking about it. I don’t see any difference with having it normalized in this society to talk about your biology just as you would your intention to have kids.

Glad to be of some help. Good night!

Oh, please do. Seriously. I usually have a group or two of undergraduates that I do reading courses with each semester because I like working through stuff like this.

I think I have my website linked on my math.se account.

I really enjoy this sort of thing, so if you'd like to reach out again, you can just email me.

@y9QQ Okay, I'm going to head off to bed now. I hope our discussions were helpful!

Yup! Exactly.

Any point in the same orbit will go to the same point in Y/G

Only if G acts trivially on Y!

Actually, if you're finding that your group theory chops are not up to snuff, you might want to consider taking a look at Algebra: Chapter 0 by Aluffi. It goes through group theory through the lens of category theory, which is the viewpoint a lot of your sources are taking.

The "Galois cover" that link mentions is exactly the normal cover/regular cover you know, and the "Galois group" of the cover is just the deck transformation group. The author is just trying to be heavy handed with the relationship between covering space theory and number theory.

Right, okay. Yeah, it turns out that number theory and algebraic topology and lots of other areas of math display the same behavior in different areas

Mm, where are you seeing it?

Plus I've been there -- self studying algebraic topology is a bit of a nightmare.

Plus this sort of stuff is what I love, so I'm having fun finding all the holes in my understanding here

Oh, of course not! For some reason my sleep schedule has been all sorts of weird lately, so this isn't quite getting late for me.

Note the example Putman gives for the trivial homomorphism in the 1-1 correspondence and the trivial G-covering.

This particular section about G-covers seems to be fairly notation light.

You might want to find a copy of Algebraic Topology: A First Course by W. Fulton if you can, and flip to page 160

Hm. So I think my thought above about the isomorphism bit is a little wrong. I'm looking at Fulton's book (which Putman references as the source for the proof you're looking at) to get a sense of what a G-cover actually is, and he defines what the isomorphism is.

You might appreciate these notes: math.jhu.edu/~jmb/note/gsetcov.pdf

That's how I'm understanding it, anyway.

This "modding out" is the author just saying they don't want to consider that sort of situation.

It's basically the same action, but technically different because the action (the map Y\times G \to Y) is a different function.

But, I could define a "different" action by first doing some automorphism of G before I let my elements of G act on Y.

This really means that there is an action of G on Y that when you look at the quotient of Y by the orbits of G, the natural quotient map gives a cover of Z

So, to give an example of what this means, say you have a G-cover Y of a space Z.

They're modding out by isomorphism of G-isomorphisms

Unfortunately, they aren't really giving all the specifics ...

Given a G-cover, this is exactly how they build a map from $\pi_1(X,x)$ to $G$

So, take a look at the proof of the bijection in this van_Beek paper

Yeah!

Same thing.

Right, which is exactly what the deck group \mathbb{Z} is doing to the helix.

Right? Covering space theory feels like a lot of magic to me

That's the perfect example.

Yeah, exactly!

I think you mentioned you were studying from that book

The reference for this discussion in Hatcher is around page 70, I believe.

Right! Because p and q are in the fiber above the basepoint, which will have however many points as the sheetedness of the cover

This gives a well defined map from \pi_1 of Z to the symmetric group of the fiber above the basepoint. If my head is on straight, I believe that the image of that map is the deck transformation group.

You're absolutely correct, the paths are only loops when the cover is degree 1 -- i.e. the trivial cover.

Ah, right. So the monodromy action is defined as taking a based loop downstairs, in Z, and then lifting that loop up into Y. When you do that lifting, you get paths that run between the points in the fiber above the basepoint. Now, pick a particular point, p, in that fiber. Then there is a unique lift of your based loop downstairs to a path upstairs that starts at p. If q is the endpoint, then the monodromy action is the action taking p to q.

Oh, sorry -- can you repeat your question then?

Yeah, they share endpoints.