@BalarkaSen $\gamma^2$ not being the identity reminds me of an old question I asked here: Can torsion in the fundamental group happen for something embedded in $\mathbb{R}^3$?

@LeakyNun I don't know this enough, so I wouldn't be able to say.

@TobiasKildetoft That's a good question. I remember thinking about this for unrelated reasons a few days ago, but can't remember how.

@BalarkaSen I asked it on main, and I think I was directed to an MO question about it

suppose $\varphi : X \to \Bbb R^3$ is the embedding. clearly this gives an injection $\pi_1(X) \to \pi_1(\Bbb R^3)$, but the latter is zero so the former is zero, so everything embedded in $\Bbb R^3$ has trivial fundamental group, hence no torsion /s

It's certainly false for any decent God fearing subset eg a simplicial subcomplex of R^3. Who cares about the rest imo

can you embed Lens space into R^4?

well surely you can embed RP^2 into R^4 so whatever

low dim top is weird

You certainly cannot embed RP^3 in R^4

Tangent bundle is not stably 1-trivial.

I also asked a related question on MSE a while back math.stackexchange.com/questions/3048957/…

@BalarkaSen delet this

an orientable 3-manifold is ______

Oh parallelizable. Fuck.

I think the answer is you can't embed RP^n into R^{n+1} for any n but I forget the proof, somehow it involves the Z/2 cup product.

I don't remember about lens spaces in R^4.

I only care about the fact that you can't embed $\Bbb P^n_\Bbb R$ into any $\Bbb A^m_\Bbb R$

ok

Hm, I can't remember the proof that RP^3 cannot be embedded in R^4. I am sure I worked this out before.

@Balarka @MikeMiller do you know a good source on the geometry of the hyperbolic plane?

what do you want to know

I need some basics like hyperbolic triangles, hyperbolic volume etc.

@MatheinBoulomenos @loch in what sense is $\operatorname{Spec}(A/fA)$ the scheme associated to $V(f) \subseteq \operatorname{Spec}(A)$?

or is it actually not

@LeakyNun for a closed subset of a scheme, there are multiple scheme structures

note that $V(f^2)=V(f)$, but the rings are different

then how do we translate "open of closed of P^n" into scheme?

Yeah I dunno there are so many books that should cover this it almost doesn't matter what you get

@TobiasKildetoft Oh I think I remember why I was thinking about this

Say $X$ is a nice subset of $\Bbb R^3$ (let's say compact locally contractible connected). Compactify and consider $S^3 \setminus X$.

This is a $K(G, 1)$ by the Papa sphere theorem. It has zero $\pi_2$ and zero $\pi_3$ and higher because it's an open 3-manifold

21/7 demonstration again...

@LeakyNun A standard tactic from the CPC. Of course, many people in HK are too smart for that.

Let's see how chaotic Hong Kong will become

Hong Kong is not China yet

Hm, so that just means $H_1(S^3 \setminus X) \cong H^1(X)$ by Alexander duality.

So I can only say $H^1(X)$ has no torsion

No

$G/[G, G] = H_1(S^3 \setminus X) = H^1(X)$. It may happen that $G$ has no torsion yet $G/[G, G]$ has. Meh

if X is a scheme, is $\Gamma(X,\mathcal O_X) = \operatorname{Hom}(\operatorname{Spec}(\Bbb Z),X)$?

@BalarkaSen I don't understand what you're trying to prove exactly

@LeakyNun no

I mean, that $\pi_1(X)$ has no torsion, @MikeMiller. Like Tobias asked.

is $\Bbb P^n_\Bbb C(\Bbb C) = \Gamma(\Bbb P^n_\Bbb C, \mathcal O_{\Bbb P^n_\Bbb C})$? ok obviously not

Well, maybe something weaker, like $H^1(X)$ doesn't have torsion.

Oh sure. But do like the quote you said earlier, and just restrict your hypothesis until it works.

@MatheinBoulomenos but $\Gamma(\Bbb P^n_\Bbb C, \mathcal O_{\Bbb P^n_\Bbb C}) = \Bbb C = \operatorname{Hom}(\Bbb P^n_\Bbb C, \Bbb A^1)$ right

If X is a simplicial subcomplex then it has a mapping cylinder neighborhood so you just need to prove it for open 3-manifolds in R^3.

@LeakyNun yeah

@LeakyNun $\Gamma(X,\mathcal{O}_X)=\mathrm{Hom}(X,\mathrm{Spec}(\Bbb Z[x]))$, though

yeah I was thinking about that

really?

@MikeMiller Ah alright then you do sphere theorem on that open manifold to prove it's a K(G, 1).

because it's obviously true for affine $X$

Then if pi_2 is nonzero there's an embedded sphere representing it and your open manifold splits as a connected sum of things which embed in R^3. Since the free product of torsion-free groups is torsion-free (is this true?) you reduce to the case pi_2 = 0. Then X is a K(G,1) by the usual argument and is equivalent to a 2-complex so G can't have torsion.

@LeakyNun use the adjunction of spec and global sections

S^3 \ S^0 is not a a K(G,1) so you need to be careful

did I just rediscover it lol

Yeah you should chuck out connected subsets of S^3

compact/connected

I'm not taking complements

I know, I'm just saying which things have good complements

Unrelated

Still reading your argument

global section is left adjoint to Spec... i.e. $\operatorname{Hom}(\Gamma (X,\mathcal O_X), A) = \operatorname{Hom}(X, \operatorname{Spec}(A))$? this feels wrong because something is contravariant @MatheinBoulomenos

I implicitly use the Alexander theorem that a 2-sphere bounds a ball on both sides

and for the thing we said to be true, the left hom should be reversed

@LeakyNun yes, the left hom goes the other direction

so $\operatorname{Hom}(A,\Gamma(X,\mathcal O_X)) = \operatorname{Hom}(X,\operatorname{Spec}(A))$?

right

what is the convention?

for locally ringed spaces, even

we're taking CRing^op as the category?

@MikeMiller OK so you reduce to the case pi_2 = 0 on each of those pieces. Got it. Nice.

there is no convention, I think you're confused about the definition of a morphism of schemes

global sections is contravariant for locally ringed spaces

so... what on earth is an adjunction

@BalarkaSen what is $\pi_1(\Bbb Q \times \Bbb Q \cup (\Bbb R \setminus \Bbb Q) \times (\Bbb R \setminus \Bbb Q))$?

Is the thing about free products true? Maybe by Grigorchuk?

@LeakyNun lol fuck you

@LeakyNun idfk man

lol why

I mean Kurosh, not Grigorchuk

Yes, something like that.

My brain is a mush

My brain has been Kuroshed

@Leaky $\mathrm{Spec}$ and global sections (of ringed spaces, not of sheaves on a fixed base!) are both contravariant. Adjunctions for contravariant functors are defined as you would expect

oh right I proved the adjunction in Lean lmfao

therefore it is true

@BalarkaSen Here is the question I asked back then math.stackexchange.com/questions/631177/…

@MatheinBoulomenos what can be said about the "forgetful functor" Sch=>Top?

@LeakyNun it commutes with coproducts

@MikeMiller In this case you don't need to use Kurosh, I guess. You break up the submanifold of R^3 into connected sum of submanifolds with pi_2 = 0 in R^3 and each of those are finite-dimensional models of K(G_i, 1). The big submanifold has pi_1 = G_1 * ... * G_n and K(G_1 * ... * G_n, 1) = bigvee K(G_i, 1) which is a finite-dimensional model, so G_1 * ... * G_n is torsion-free.

@BalarkaSen yes Kurosh is sufficient: if A * B has torsion, it has a finite subgroup, which is the free product of a free group, a subgroup of A, and a subgroup of B. This can only happen if the free product has only one nontrivial factor

@MatheinBoulomenos that is not good

we care about products not coproducts right

what is the coproduct even

disjoint union

oh...

stupid construction

@BalarkaSen Smart. You still need to know that B(H * G) = BH vee BG, but I guess this follows from clever universal cover trickery

@MatheinBoulomenos is there left adjoint to any of the functor in the chain Sch => LRS => RS => AS => S => Top?

locally ringed spaces, ringed spaces, abelian-grouped spaces (?), sheaves, (I forgot presheaves)

@MikeMiller No, that's a good point. B(H * G) = BH v BG is basically the same circle of ideas as Kurosh.

oh Shf => PShf is sheafification

@LeakyNun that chain doesn't even make sense, you need "sheaved" spaces, not sheaves

AS => S is free abelian group? does that work?

oh right, sheaved spaces

Lol what the fuck is Leaky even doing

"sheaved spaces"

JESUS

The forgetful functor Sch=>Top has really bad properties. you can have a product of two one-point schemes that is uncountable

what

in fact the product of two copies of $\mathrm{Spec}(\overline{\Bbb Q})$ is in bijection to $\mathrm{Gal}(\overline{\Bbb Q}/\Bbb Q)$

The forgetful functor from schemes to topological spaces essentially forgets everything we care about (so it is indeed very forgetful)

yeah

it's better to work with $S$-valued points than with the forgetful functor

basechange whoosh fwoosh

@LeakyNun there's no canonical way to do that for the reason mathei mentioned that you can have multiple scheme structures on a closed set - so usually you should be clear what scheme structure you're taking (most of the time its the reduced one)

@BalarkaSen It feels strictly easier to me

his name is mathemi now lmao

taking open subschemes is fine though

@MikeMiller Yeah, you can just draw BH v BG as a graph of spaces. I'm sulking because I should have seen Kurosh lol

@MatheinBoulomenos that makes sense... somehow

I know how to write down the universal cover, and by cutting it into pieces I understand and how they fit together I get either an inductive MV description or an MV specseq depending on how you want to think about it

field theoretically

but not topologically

the topologies agree though

what is S-valued point?

morphism S -> X

oh

but that doesn't have a topology?

yeah, but who cares

so you're basically saying, it's better to work with the functor to set

It has... the Grothendieck topology!!! how exciting

@LeakyNun A scheme is a functor to the category of sets

lol ^

a matrix is a linear map

not the functor to set. It's better to work with the Yoneda embedding, basically

sry im really sleepy and i shouldnt dunk on algebraic geometers

@BalarkaSen Yes you should

@MatheinBoulomenos that is the functor to set

how does X(S)=Hom(S,X) carry the Grothendieck topology?

it isn't even a category...?

p sure he's messing w/ you

:c

-_-

Question ------> Does a homeomorphism exist between the sphere and the twice pointed sphere?

What do you think?

what on earth is the twice pointed sphere?

yes because they have genus 1

but

Oh I didn't even noticed, my brain replaced it with punctured automatically

I don't know how to deal with the points that are "pinched"

What does twice pointed mean then?

t's a sphere with two cusps

on the opposite poles

picture forthcoming

that second one looks like a bloody raindrop

can someone make a better or more elegant fit of this?

desmos.com/calculator/nvaso2liey

I'm guessing they first wrote it as $\displaystyle \prod_{p \le x} \sum_{n \ge 0}\frac{1}{p^n}$

But how does that weird index follow?

I'm only really interested in showing that the LHS > log(x).

There's a technique on the wiki page on zeta function but the method requires Re(s) > 1. This one diverges. And when we get to the harmonic series we easily get that this is > log(x).

@BalarkaSen meme paper arxiv.org/pdf/1907.05004.pdf