point-set-topology

[unreal stratus]

That is good

[hidden crag]

this is what i had in mind as well

it's how may does it

[unreal stratus]

Oh nice

[hidden crag]

slightly different but same idea

[unreal stratus]

Yes iirc doesn't he like scale the polynomials themselves

Rather than like restrict to a big circle like here

[hidden crag]

yeah

[tiny ridge]

I have a cheeky proof

[hidden crag]

This is my sign to pick up my copy of may again for a bit after i'm done grading

i got a used version for like 7 euros i think, love it

[undone ridge]

that's the one I'm most familiar with

[hidden crag]

this dodges degree theory by plugging in the e^... stuff

lol

[undone ridge]

I'm taking a course in galois theory this semester tho and we'll probably prove it again in a different manner

[unreal stratus]

Oh this is funny because like

[undone ridge]

btw theorem 1.7 is that \pi_1(S^1) is isomorphic to \Z

[unreal stratus]

I did a past paper exam recently for my uni where the notation and even wording was near identical to this

Lol

[abstract saffron]

There's a good one using diff geo

Milnor proved it quite elegantly

[tiny ridge]

Gah, I think my proof requires some extra effort

I was trying to prove it by Atiyah Singer index theorem

[abstract saffron]

Bruh

[tiny ridge]

Let H be the space of all complex valued L^2 functions on the unit circle such that the Fourier coefficients a_n = 0 for all n < 0. For any polynomial p, consider the map T_p : H -> H, f(t) -> p(t) f(t). Then T_p is a Fredholm operator, and if the leading term is z^n, T_{p_t} is a homotopy of Fredholm operators where p_t = z^n + t(lower order terms in p). Index of a Fredholm operator is a homotopy invariant, Index of T_p = Index of T_{z^n} which can be checked by hand to be -n.

[unreal stratus]

Nice lmao

[tiny ridge]

But what next?…

Its not finished something has to be said. If p is nonzero, T_p should have zero index

Oh coz its invertible, as 1/p is holomorphic

T_{1/p}

[abstract saffron]

Not sure if anyone can read that

[gritty widget]

wtf is wrong with your pdf

[abstract saffron]

I had the pdf, the cap the screen and Paint-edited it

Imma try again

[undone ridge]

why is the formatting like that, is this an old paper/book? Looks like something from the 60s~70s or smth

[tiny ridge]

theres a funny copy of milnor like this online

i dont know whats wrong with it

[abstract saffron]

I gave up joining everything into one photo

[coral pivot]

this is extremely based ibsen.

[abstract saffron]

This is stupid

[abstract saffron]

It is from the 60s

[coral pivot]

which milnor book is this

[abstract saffron]

https://math.uchicago.edu/~may/REU2017/MilnorDiff.pdf
page 7

[coral pivot]

bc if its either char classes or morse theory, its latexed

[undone ridge]

well that explains it I guess

[tiny ridge]

thanks 🙇‍♂️

[abstract saffron]

Topology from the Differentiable Viewpoint

[coral pivot]

Oh I dont think that ones latexed :(

[abstract saffron]

It's not really well-known for some reasons, altho it's a hidden gem

[gritty widget]

"for some reason"

milnor was just that guy

[abstract saffron]

His proof is like, Homotopy theory but in disguise

[tiny ridge]

homotopos theory

[abstract saffron]

lmfao

I think this is cleanest proof using topology

Of course, if you point a gun at me, I'll pull the complex analysis way without a doubt, it's shorter and simpler

[thorny agate]

I'm having trouble coming up with a map that connects p and q

Say the map is f. My issue is that I know f(0) = p and so we should have f^-1(p) = [0, b) for some b < 1 which isn't an open set.

[gritty widget]

it's open in [0, 1]

[thorny agate]

oh

duh why am I thinking about R

thanks

[cosmic socket]

Given two line bundles on $\xi, \eta$ on a space $X$. These line bundles are can be thought of pull backs of of the universal bundle $\gamma^1$ via maps $f, g \in [X, \mathbb{C}P^\infty]$. I can consider the composition $X \rightarrow \mathbb{C}P^\infty \times \mathbb{C}P^\infty \rightarrow \mathbb{C}P^\infty$ (where the first map is $(f, g)$ and the second map is given by the multiplication on the H-space structure of $ \mathbb{C}P^\infty$). Now pulling back the universal bundle by this composition I should (I think) get the tensor product $\xi \otimes \eta$. Does anyone know where I can find a proof of this last part? Or a sketch of one if possible.

[gentle osprey]

bigradedSphere

[cosmic socket]

I think it is enough to prove it for $X = \mathbb{C}P^\infty$ and the first map to just be the diagonal.

[gentle osprey]

bigradedSphere

[cosmic socket]

Also unrelated question but does the phrase weakly-complex manifold

mean a finite CW complex that has weak homotopy type of a manifold

or a manifold with weak homotopy type of a complex manifold

oh wait maybe it just mean an almost complex manifold

that would make much more sense

[echo elk]

A question. From Wikipedia, boundary operators of a chain complex are also called differentials; this seems to imply that coboundary operators are called codifferentials. This doesn't make sense to me because exterior derivatives (i.e. “differentials”) are the operators of a cochain complex, so that'd make them the codifferentials (the naming is already a red flag). Then there is already the notion of a “codifferential” for a chain complex, so then that “codifferential” is really a differential. I must be misunderstanding something, does anyone have any clarifying insights?

[cosmic socket]

I think it’s just nomenclature

I’ve only seen it called differential whether it is cochain complex or not

That is to say coboundary operators are also called differentials

[echo elk]

Alrighty, I figured. Better to verify than to continue with the possibility of what seemed to be a fundamental misunderstanding. Thanks.

[fresh thunder]

Hi

[tiny ridge]

@cosmic socket This is nearly definitional depending on how you look at it. Given a line bundle L -> X, embed it fiberwise in the trivial vector bundle X x C^n+1 for some large n, then the “fiberwise Gauss map” is the map X -> CP^n. Tensor product of two such line bundles L1 o L2 embeds fiberwise inside C^n+1 o C^m+1 = C^(n+1)(m+1), and the fiberwise Gauss map lands inside CP^(n+1)(m+1)-1. This means that the tensor product L1 o L2 is induced from the Segre embedding CP^n x CP^m -> CP^(n+1)(m+1)-1, which is precisely just “tensor product of two lines”.

Taking a limit as n, m -> infty gives the H-space structure on CP^infty

[coarse night]

You can give a product on homology groups similar to cohomolgy for H spaces, is it any useful? I haven't seen much that's why asking

@tiny ridge

[tiny ridge]

@coarse night For an H-space, the homology becomes a Hopf algebra, in that it admits both a product and a coproduct (which is always the case, induced from the diagonal map X -> X x X and Kunneth). This is extremely useful, see Hatcher Section 3.C

[coarse night]

hmm okay

[tiny ridge]

Essentially, the high level of compatibility between the two allows one to compute the algebra from the coalgebra, or vice versa. I think a nice computation along these lines is H_*(\Omega S^n)

Which can incidentally be computed using Morse theory.

[coarse night]

i see

[urban zinc]

wowwww every group is the fundamental group of some 2-dim cw complex that's so COOL

[tiny ridge]

this is a good one

[unreal stratus]

Yeah and sorta intuitive construction which is noice

[hidden crag]

and generalizes

[unreal stratus]

ofc

Every group, whether abelian or not, is the pi2 of some 2-dim cw complex

/j

[tiny ridge]

@urban zinc What’s the fundamental group of S^1 x S^3?

[unreal stratus]

lol

[lean knot]

(X, t) is a second countable space, B is a basis for t. Is there always a countable subset C of B, such that C is also a basis for t?

[thorny agate]

Doesn't second countable mean that t has a countable basis

By definition

[unreal stratus]

This is a different question

[lean knot]

we don't know if B is countable

[thorny agate]

Oh NVM I see

I misread

[unreal stratus]

But anyway, so like

[thorny agate]

After some thinking I think the answer is yes but lemme try to type it out on my phone lol

[trail charm]

if X is second countable, then every open cover of X has a countable subcover; bases are open covers, but would a countable subcover of a basis be considered a basis?

(not an exercise-like question, genuine question)

[coarse night]

that would not necessarily be a basis

[trail charm]

kk that's what i was thinking

[coarse night]

check if you take the subcover (n, n+2) for n ∈ ℕ then it forms a subcover of the std basis but is not itself a basis

check why

[coarse night]

There's a another one that says any finite group can be realised as a fund group of a manifold

[tiny ridge]

shh

[coarse night]

and even stronger one saying ...

ok were you planning to give it as an exercise?

[tiny ridge]

my question is the first step to a proof of this

[coarse night]

oh you are doing the 4manifold one

[tiny ridge]

the question that everyone eat reacted to

[coarse night]

[gentle osprey]

Spamakin🎷

[coarse night]

help me on this one, section 2.2 problem 9d of hatcher

[thorny agate]

@lean knot

I think this works?

[quiet thorn]

latexing on phone

[thorny agate]

It was so bad but I was on couch and didn't want to get laptop

[tiny ridge]

king shit

[thorny agate]

Hey, no compilation errors

Ignore the fact that I used i for many things, uhhh read from context

[tiny ridge]

let me crack it open

[coarse night]

feels like those covering space action but not sure about the solution

[tiny ridge]

oh this was posted here a couple days ago

cellular homology

run it

[lean knot]

Maybe, I'll check it out latter

[tiny ridge]

and yes those identifications are done by the covering space action

[coarse night]

yeah thought about cellular, what I'm getting is Z →Z² → Z with all differentials 0 but not sure if that's correct

[tiny ridge]

correct

[coarse night]

okay that is correct

cool

9b without küneth

[tiny ridge]

mayer vietoris

its two torii glued along a pair of meridians

[coarse night]

did that with cellular

[tiny ridge]

oh, neat

[coarse night]

btw do you know the explicit π_1 of Hawaiian earring?

is it describable?

[tiny ridge]

its a messed up quotient of the inverse limit of free groups theres a site online that describes awful topological spaces which has the answer

[coarse night]

nah I'm good

[tiny ridge]

probably a sub not a quotient

there can be infinite length words but as long as a letter does not infinitely repeat youre good

[coarse night]

hmm makes sense actually otherwise image of [0,1] won't be compact

[tiny ridge]

https://wildtopology.com/2013/11/23/the-hawaiian-earring/

[tiny ridge]

exactly

[quiet thorn]

how does one get this?

i really cannot picture this quotient space in my head

[coarse night]

try giving it a CW structure

rather try to see what happens to the CW structure it already has

[tiny ridge]

heres a hint, if you take S^1 x [0, 1] and quotient the S^1 x {1} end by Z/nZ, thats the mapping cylinder of S^1 -> S^1, z-> z^n

@coarse night how was commalg

[urban zinc]

uhh Z?

ayyy

[coarse night]

not great

[tiny ridge]

yeah. what about (S^1 x S^3) # (S^1 x S^3)? (# = connect sum)

[tiny ridge]

neither is commalg so its ok

[coarse night]

she asked us to compute all galois subext of x⁸-2

[tiny ridge]

isnt that D_8 i forget

thats a lot of subgroups

[coarse night]

no quasi diheadral

didn't do it, nobody did

[tiny ridge]

oh right

right right right this is the modular group of order 16

dummit foote moment

[coarse night]

yeah

it was an assignment prob as well

I didn't do it back then as well

[unreal stratus]

lol

Yeah x^n - 2 just seems annoying for large enough n lol

[coarse night]

it is

[tiny ridge]

i think i did it for reasons unknown a long time ago but i dont remember why

[coarse night]

I will never compute those

[tiny ridge]

probably in my algebraic number theory course

[coarse night]

you don't just have to compute the group but the normal subgroups as well as their fixed fields

[tiny ridge]

pfeh

[cosmic socket]

I don’t understand how you know the gauss map will agree with the segre embedding.

[tiny ridge]

just definitions, the line fiber of L1 o L2 sitting over a point in X is tensor product of the line fibers of L1 and L2

the Segre embedding just sends a pair of lines to the tensor product of the lines

[cosmic socket]

Ok I think I see

That’s

Pretty cool

[tiny ridge]

in the spirit of italian and russian mathematicians

no classifying space, no mixed motives, no deligne mumford stacks

only projective geometry

[cosmic socket]

Yea I guess now how do you know the segre embedding agrees with the H space structure induced by the functoriality of classifying space

[tiny ridge]

[cosmic socket]

So are you a hater of higher algebra

[cosmic socket]

Actually this shouldn’t be too hard I think you can show the h space structure on CP^infty is unique

[tiny ridge]

probably eckmann hilton

im not going to check compatibility

[tiny ridge]

not necessarily, i think of an A_oo algebra in the spirit of Sullivan, just the loopspace

and also Peter May

[cosmic socket]

I don’t know much higher algebra but Vezzosi (in the sense of Toën and Vezzosi) is Italian. So you must like derived algebraic geometry right

[tiny ridge]

by italians i mean the old italian algebraic geometers

Segre, Veronese, Enriques, Cremona, Castelnuovo, …

[cosmic socket]

I’m curious what area of math you work on?

[lean knot]

X is complete metric space, O is an open subset of X, x is in O, a complete metric group G acts continuously on X. Are there an open neighborhood of 1_G, U, and an open neighborhood of x, P, s.t. U⋅P ⊆ O and for any y ∉ O, U⋅y ∩ P = ∅?

[tiny ridge]

broadly, geometric topology. so far my work has been more algebraic than geometric, but i am planning to change this :p

but topology, all in all

[urban zinc]

Z*Z? no idea

[tiny ridge]

That is correct!

So the connect sum of n copies of S^1 x S^3 has fundamental group the free group on n generators, yes?

[rough cedar]

[urban zinc]

ayyyy

[urban zinc]

That makes sense

The S^3 here is redundant right?

Like the connected sum of n copies of S^1 also has fundamental group the free group on n generators?

[tiny ridge]

Connect sum of two circles is a circle. Oops.

[urban zinc]

Oh wait

I confused wedge and #

I see now lol oops

[tiny ridge]

Wedge is absolutely right. The S^3 allows you to treat the connect sum as a thickened wedge, Thickened by a sphere of sufficiently high dimension (eg S^2 will also work) so that it doesn’t affect the fundamental group.

[urban zinc]

ahhh I get it

[tiny ridge]

Now, I want to think like you’re thinking. #^n (S^1 x S^3) is like a thickened wedge of n circles, fundamental group is F_n. Suppose I have a group with n generators, and some finitely many relations.

I want to do the Cayley complex stuff but with the thick version

[urban zinc]

👀

[tiny ridge]

Suppose you have some relation, r = 1, where r is a word on the generators. This is homotopy class of some loop in #^n(S^1 x S^3)

Pick a loop from this homotopy class. This can be an absolutely terrible, space filling loop. But homotopy it so that its a nice, smooth or even piecewiselinear loop.

It may self intersect at a bunch of points, but I claim you can remove all the self intersections and make it an embedded loop (Jordan loop) by a homotopy also. Why?

[thorny agate]

Not sure how to do this

I was trying contradiction since I know that X \ U is closed and so compact but that got me nowhere

Can I maybe do cases?

case 1: one of the C_alpha is empty, so then this is immediate

otherwise case 2: they are all non-empty so in particular finite intersections are non empty and we have the finite intersection property, so C is non empty

idk what that could get me though

[thorny agate]

nvm got it

very annoying

[nimble portal]

What is an inclusion map?

Can someone also remind me what an embedding/embedding map is?

[novel acorn]

injective continuous map

if you're working in diff geo continuity is probably replaced by smoothness

[novel acorn]

it's a map that includes a subspace of some larger space into that larger spaces

so like

take the unit ball and R^2

[nimble portal]

Nah Lee says assume continuity at most

[novel acorn]

then there's an inclusion map of S^1 -> R^2

[nimble portal]

Hes

Yes

I understand

But like

How does the function act on the subspace?

Is it just the identity map

From the subspace to the larger space

LOL

[novel acorn]

yes

[nimble portal]

Icic

And an embedding?

Inclusion map sounds like what I’d think of as “embedding” intuitively

but I think it could be more general than that

[unreal stratus]

An embedding is a homeomorphism onto its image yeah

So like you can factor an embedding as a homeomorphism followed by an inclusion anyway so yeah

[nimble portal]

Ayaya

[urban zinc]

Okay I'm stuck here, I have no idea

[tiny ridge]

Suppose you have a very nice smooth loop in R^3, which is like a figure 8 contained in the xy plane

How do you homotope this 8 to be not self intersecting?

(Hint: This would not be possible within R^2)

[urban zinc]

could you move the one strand up in the z direction and the other strand down

[tiny ridge]

yup

in fact just need to move one of the strands slightly up

[urban zinc]

ahhh

how do you prove you can always do that

like what if there are a bunch of self intersections really close together

[tiny ridge]

If there are finitely many such intersections, and each of them look like an X, you can always zoom into one of them, and move one of the strands of X up in the extra dimension

And since you can do it in R^3, certainly you can do it in R^4 as well!

The loop we started off with was piecewise linear (we ensured this after a homotopy, but I didn’t tell you why. The reason is the simplicial approximation theorem, but let us blackbox this), so convince yourself that all the intersections necessarily look like such.

[coarse night]

that makes wonder which groups cannot be realised as fund groups of a 3manifold. Is there an obstruction measuring this?

(assuming finitely presented)

[tiny ridge]

These things are all understood after geometrization

Finite 3-manifold groups are discrete subgroups of O(4)

[coarse night]

I can kinda see how go from discrete subgroup to 3-mani

maybe not

just quotient right?

[tiny ridge]

That may have fixed points, not all such groups work I think.

But, say, its a discrete subgroup of SU(2). Then quotient works.

[coarse night]

finite discrete?

[tiny ridge]

Discrete subgroup of a compact group is necessarily finite

I’m saying if you pick a finite subgroup of O(4), that may not act freely on S^3.

[coarse night]

lol right

even central

hmm I see

[tiny ridge]

Uh, no. Central would mean abelian. In fact center of O(4) is Z/2

O(4) has tons of nonabelian discrete subgroups

[coarse night]

okay I misremembered something

[tiny ridge]

For infinite groups, I know Z^2 is not a 3-manifold group

Any Z^n with n =/= 1,3

[coarse night]

T²×R? although that's not compact

[tiny ridge]

Compact 3-manifolds, certainly

[coarse night]

how hard is it to prove it for Z²?

[gritty widget]

1 and 3...

[tiny ridge]

Uhm

Suppose M is a 3-manifold with fundamental group Z^2. Pass to the universal cover, thats simply connected. So \pi_2 = H_2 by Hurewicz. I claim that this has to be zero

If its not then there’s a non-nullhomotopic sphere. Push this sphere down to M, its still non-nullhomotopic (else youd be able to lift the nullhomotopy upstairs)

Now by some standard rigamarole this means that the sphere is separating ie if you cut along it the manifold splits into two, ie M is a connect sum of two manifolds M1 # M2 where neither M1 nor M2 is a sphere (theres some casework but lets ignore it)

That would means the fundamental group is a nontrivial free product by van Kampen which Z^2 is not

So \pi_2 = H_2 = 0

The universal cover is noncompact, so \pi_3 = H_3 = 0 by noncompact Poincare duality

All the higher homologies vanish by cellular homology because youre in dimension 3 hence so do the higher homotopies

This means that the universal cover has all homotopy groups zero so its contractible by Whitehead

So M is a K(Z^2, 1) space, so its homotopy equivalent to a torus T^2

But M is a compact 3-manifold! H_3(M; Z/2) = Z/2!

Contradiction

[coarse night]

oh wow that's a really nice argument

[tiny ridge]

This is where the hard stuff is hidden

[coarse night]

yeah I realised

[tiny ridge]

You need the Papakyriakapoulos sphere theorem

[gritty widget]

did you spell that from memory

[coarse night]

thought about asking but I guess I need more machinary

[tiny ridge]

yes lol

[novel acorn]

Ibsen is a crazy russian don't forget
Who knows the vastness of knowledge that is in his head

[coarse night]

lol lbsen russian

[tiny ridge]

morally, morally

[novel acorn]

He always calls himself a russian style mathematician so that's what I know him by lmao

[gritty widget]

someone was touched by that arnold rant

[novel acorn]

I am intrigued by that notion

(I didn't know someone ranted abt arnold lmao)

[gritty widget]

https://www.math.fsu.edu/~wxm/Arnold.htm

[novel acorn]

arnold was russian what

I didn't know that I thought he was french for some reason???

[tiny ridge]

wtf lmao

[gritty widget]

rolling in his grave

holy shit

[novel acorn]

His name is spelled like Arnol'd and I thought damn that looks french

[gritty widget]

the ' is supposed to be the soft sign in russian

the tiny little b thing i don't have it on my keyboard

[coarse night]

you have any reference for papyakfjwokpolos sphere?

[coral pivot]

my favourite frenchie

g'elfand

[tiny ridge]

https://pi.math.cornell.edu/~hatcher/3M/3M.pdf

[coarse night]

lol okay

[novel acorn]

Ibsen are you by any chance a hatcher fan

[tiny ridge]

chapter 3

[tiny ridge]

Sherlock Holmes here has my number

[novel acorn]

that's a controversial stance on this server

you are a brave one

[tiny ridge]

i think its good for geometric topologists

not everyone is a geometric topologist, which is fine

[novel acorn]

Also I have to say that this is a long proof wow

[coarse night]

60 is manageable lol

[cosmic socket]

I don’t know what I am but I like hatcher

Good book

[novel acorn]

the proof is the last 7-8 pages if I read it clearly

[coarse night]

I think he's setting up the basis

[tiny ridge]

Papa’s theorem is an old one but a hard one. There were no correct proofs of the theorem until 50 years of its announcement by Dehn (along with a wrong proof)

[coarse night]

I'll have a look after 6th

[novel acorn]

Thankfully it's not as long as the proof of the main theorem of Iwasawa theory that's like a solid 20 pages lol

Gonna learn Iwasawa theory over the summer can't wait

[tiny ridge]

dont read too much about 3-manifolds though

dead field kek

im just repeating what my advisor told me when i wanted to learn this stuff

3-manifold topology has become geometric group theory now

and i couldnt be bothered to learn more about gRoNps

[fading vale]

thanks to HF its also become combinatorics!

[tiny ridge]

HF is not 3-manifold topology

its (4-1) manifold topology

subtle difference

[fading vale]

So true

i would like to one day be able to feel on a moral level the n - m dimensional topology slogan

[tiny ridge]

theres 2+1 which is Thurston style hyperbolic geometry (and ggt grrrrr), and theres 4-1 which is using 4d invariants on nullcobordisms to spit out info about 3

[coral pivot]

and geometric group theory is von nuemann algebras

:))

[tiny ridge]

nuclear take

[coral pivot]

biggest open problem in GGT? there are many answer

all of them are wrong

the correct answer is

[novel acorn]

Mentally challenged zealots of "abstract mathematics"

[coral pivot]

is vN(Fn) iso to vN(Fm)

[novel acorn]

this rant is nuts

[tiny ridge]

what is vN(G)

[coral pivot]

the von nuemann algebra of G, for a discrete G it is simple enough to explain

[tiny ridge]

thats sufficient

[coral pivot]

take the representation G-> B(L^2(G)) the left regular representation

[novel acorn]

It seems to me that a reasonable explanation was given by I.G. Petrovskii, who taught me in 1966: genuine mathematicians do not gang up, but the weak need gangs in order to survive. They can unite on various grounds (it could be super-abstractness, anti-Semitism or "applied and industrial" problems)

[coral pivot]

this extends to a *-homomorphism L^1(G) -> B(L^2(G)) (L^1 with convolution product), look at the image and take the strong operator closure

(strong operator topology is the topology of pointwise convergence, i.e A_lambda ->A in SOT iff A_lambda x -> Ax for all x\in L^2(G))

[tiny ridge]

the image is vN?

[coral pivot]

the strong operator closure of it

[tiny ridge]

ah ok

got it

bizarre

[coral pivot]

yeah its surprising how much ggt and von nuemann interacts

basically von nuemann algebras are a good way to study certain groups and the groups ggt care about happen to fall into that

so one way to study these groups is the fourier algebra, which is the predual of vN(G), which is known to be rigid. But we want to know when is vN rigid, and the problem i stated is p much one of the biggest open problems in von nuemann algebras

we know for every amenable group for example, the have the same von nuemann algebra

[tiny ridge]

this feels reminiscent of one of many ways to construct boundaries of groups in ggt but everything is too operatorized so im having trouble orienting myself

[coral pivot]

ends? yeah idk how they connect I need to learn more ggt oof

[tiny ridge]

something like ends yeah

[tiny ridge]

curious

they’re also all one ended

[coral pivot]

oh interesting

[tiny ridge]

ehh maybe im lying

morally morally

[coral pivot]

all abelian groups are amenable btw

[tiny ridge]

yeah

end is a flippy notion, theres this thing called a poisson boundary

a countable group is amenable iff there is a symmetric random walk law with trivial poisson boundary

[coral pivot]

(oh btw not all amenable i mispoke, its all amenable that have each non trivial conjugacy class infinite, these are called ICC groups and von nuemann algebraist care bc this means vN(G) has trivial center, i.e is a type ii_1 factor!)

oh interesting

[tiny ridge]

maybe i can define one of the boundaries to illustrate my point regarding this similarity

let me recall from past life

@coral pivot ok

Say G is a countable group, pick a set S of generators and P be a probability measure on S. Run a random walk on G using P, by picking generators at random using P and concatenating one after the other, which generates a walk on the Cayley graph of G

[coral pivot]

right

[tiny ridge]

For any two points x, y in G compute K(x, y), the expected number of time the random walk starting at x returns to y

[coral pivot]

hmm is there a reason this would not just be infinite

unless you meant how many steps to y

rather than how many times x returns to y oof

[tiny ridge]

it would be if your graph is recurrent for example

but in Z^3 already it is finite

the random walk is transient

[coral pivot]

oh interesting

[tiny ridge]

it just escapes to infinity

[coral pivot]

i see

[tiny ridge]

Choose some basepoint o, then look at the map G -> C^0(G), y -> K(-, y)/K(o, y)

put the pointwise convergence topology on C^0(G) and take the closure of the image

cl(G) \ G is called the Martin boundary

[coral pivot]

interesting, yeah i see some similarities

[tiny ridge]

this is true for the Martin boundary as well (Poisson boundary is just Martin boundary with a natural measure I think)

for hyperbolic groups the Martin boundary becomes the Gromov boundary, which is closer to the notion of ends (eg for F_n it is exactly the space of ends, the Cantor set)

do you know what a hyperbolic group is

[coral pivot]

yeah gromov hyperbolicity

stuff like delta slim etc

[tiny ridge]

yup that

right

[coral pivot]

https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-137/issue-2/Hyperfinite-von-Neumann-algebras-and-Poisson-boundaries-of-time-dependent/pjm/1102650384.pdf

there seems to be literature on this connection

(by Connes himself even)

[tiny ridge]

oh lol

[coral pivot]

https://arxiv.org/pdf/2009.11787.pdf#:~:text=The Poisson boundary is a,map as introduced by Izumi.

haha these are people I know in real life who wrote this

[tiny ridge]

damn

people have discovered everything

no new ideas are possible anymore

[coral pivot]

lol

[tiny ridge]

ill tell you what a gromov boundary is tho

[coral pivot]

yeah sure

[tiny ridge]

let G be a gromov hyperbolic group, o be a basepoint. look at all geodesic rays in the Cayley graph of G starting at o

call two geodesics equivalent if uh

if theyre at bounded distance from each other

[coral pivot]

hmm

I thought geodesics diverged exponentially in hyperbolic groups

oof

[tiny ridge]

definitely interestingly distinct geodesics do that

thats why were identifying the unimportant ones

[coral pivot]

i see

[tiny ridge]

heres a dumb example

take F_2

look at the ray aaaaaaaaaaaaaaaa

um

ok yeah then look at the ray which is the same thing but lags behind in time 1

[coral pivot]

oh yeah i see

[tiny ridge]

yeah so the gromov boundary is all rays modulo equivalent rays

[coral pivot]

i see

[tiny ridge]

the topology is a bit more convoluted but basically imagine two points p, q in the gromov boundary and two geodesics joining o to these points at infinity

if these stay close together for a long time, you say p and q are close

[coral pivot]

I see, so something like how long they were close is the metric ig?

[tiny ridge]

thatd make nearby points have a very large distance so you have to do e^{-how long they were close}

[coral pivot]

right i see

oh yeah duh i did the opposite lol

[tiny ridge]

no yeah thats the most natural thing, i get it flipped also

its like the p-adic metric

[coral pivot]

right

[tiny ridge]

delta slimness basically guarantees this is a meaningful notion because the triangle joining o, p, q will be delta slim (two vertices of this triangle are at infinity but you get my drift)

which means the geodesics do tend to stay kind of close, just the measure of closeness is the metric

[coral pivot]

right that makes sense visually

[tiny ridge]

and yeah bobs your uncle

the Gromov boundary of F_2 is a Cantor set, of a surface fundamental group is a circle, …

its invariant under quasi isometries etc etc

[coral pivot]

i see

what about for F_n

[tiny ridge]

Cantor set

[coral pivot]

i see, so this is probably not something that can distinguish the von nuemann algs oof

[tiny ridge]

probably not

lol its open if a hyperbolic group with Gromov boundary S^2 is a 3-manifold group or not

false for all other dimensions

nuts

i mean no what it means is its probably false and no one has come up with a nutcase enough example

[coral pivot]

interesting

[tiny ridge]

gronps

[coral pivot]

maybe the solution is to look at a correct von Nuemann algebra 👁️

probably not von nuemann doesnt capture stuff like smooth structure well oof

[tiny ridge]

maybe! the only thing worse than a gronp is a C* algebra

so you expect the latter to solve a problem about the former

[coral pivot]

lol i see

(btw @bitter smelt you might find the above convo interesting, still trying to turn you over to the dark side lol!)

[tiny ridge]

whats the bright side

aka what does Migilonomy do

[coral pivot]

ggt

[tiny ridge]

[bitter smelt]

Yeah, I used to do a lot of ggt! But sadly not to many people here are into it so I've gradually shifted over to alg/diff top. I might try to hit up Chifan though, I should read more about von Neumann algebra

[fading vale]

i thought mig also did symplectic and contact stuff

[bitter smelt]

Sniped :)

Yeah very recently I've been into contact stuff, knot theory and whatknot

[tiny ridge]

oh fantastic

[bitter smelt]

Floer homologies etc. I plan to hammer down on something specific this semester though. My net is too wide 😅

[tiny ridge]

ive been thinking about symplectic/contact a lot since last year

[fading vale]

what sort of stuff r u into ibsen

[tiny ridge]

everyone here does ggt

[bitter smelt]

It's a trendy topic with lots of low hanging fruit I feel

Contact/symplectic that is

Well maybe less on the symplectic side. Geometry is frightening.

[tiny ridge]

my research dances around h principles, which is a way to attack geometric problems using softer tools from topology

[tiny ridge]

yeah lol looking for more problems on the contact side

id like to say that yeah

[fading vale]

oh this is cool stuff

[bitter smelt]

Oh wow you really were the person to ask about the h principle resources then huh!!

[tiny ridge]

yeah lol

we should make a discord cohort of symplectic people at this point

[bitter smelt]

It's not a bad idea, could also just start with a thread here to gauge interest

[fading vale]

i think just threads or posting here is better cause the offshoot discords tend to get very inactive

[tiny ridge]

yeah agreed

[bitter smelt]

Anyway I am going to go be social at this conference, thanks for the rec @coral pivot, I'll take a look later

[fading vale]

god i gotta learn more complex AG

[coral pivot]

oh nice gl

also oof didnt realize you moved away from ggt 😭

[tiny ridge]

bye migilonomy

[fading vale]

bye bye mig have fun at your conference

[coral pivot]

hey moth, been a while

what are you upto these days

[fading vale]

hi john!

[fading vale]

symplectic geo and related things

[coral pivot]

i see

[fading vale]

recovering emotionally from finishing hartshorne II

[coral pivot]

lol

[tiny ridge]

that sounds painful

[fading vale]

my joker moment was when one of the exercises in II.7 was unsolvable because he did the entire theory of weil divisors with like one extra assumption that didnt hold in the problem statement and i spent ages agonizing over it only to discover i wasted my time and it was fine

shoulda just read EGA man

[coral pivot]

I got talked into doing a project on langlands connection to QFT for a QM class, so thats how my days are going these days lol

[fading vale]

that sounds like fun

i want to learn about geometric quantization one day

[coral pivot]

yes except when you realize this is way above your paygrade 😭

https://arxiv.org/abs/hep-th/0512172

what we are reading

[fading vale]

ur starting grad school in the fall right

[coral pivot]

yep

[fading vale]

very nice

[tiny ridge]

doesnt witten have a 200 page paper where he “proved” langlands lol

[fading vale]

damn i remember when u were in hs

[coral pivot]

haha right, and I remember when u were a first yr in hs i think!

[fading vale]

I think so

or maybe i had just finished my first year

So long ago

[coral pivot]

wouldnt put it past him ibsen lol

yeah indeed

remember how i used to hate analysis in hs

now I am analyst lol

[fading vale]

character growth

surely if i take 3 analysis classes in a row starting next fall i will like it by the end 🙂

[coral pivot]

yes

I can operator pill you too then!

although I dont know anyone at your school who does operator algs i dont think

[tiny ridge]

i think i finally understand how to grow mushrooms

[cedar pebble]

the ramping up in difficulty between parts I, II, and III is insane

[tiny ridge]

These are the same pictures

Well, the right is only partially drawn

[unreal stratus]

Nice

#dynamical-systems

Jk

[tiny ridge]

lol

pretty much

[unreal stratus]

Brings me back to when I took physics ODEs in first year

Good times

[tiny ridge]

Phase portraits are great

[unreal stratus]

Yeah good fun too lol

Like nice doing lil things about each point and then it being clear how stuff glues together

Yummy

[tiny ridge]

Now I just need to do the above but with three rows of mushrooms

💀

(Each row of mushrooms blocks one third of the flow, I want to block everything)

[urban zinc]

how do you "extract from this a countable subcover"? I get the rest of the problem, but this one step is tripping me up

[nimble portal]

Something tells me you’ll want to cover M similar to the cover used in the proof for the fundamental group being countable

[sweet wing]

same

[sweet wing]

iirc there was some conjectured extension of langlands duality from some funny string duality??? idk man

[tiny ridge]

sounds believable lol

[sweet wing]

some equiv of categories of branes or some black magic im not qualified to understand

[tiny ridge]

mirror symmetry, is the term youre looking for, probably

or T-duality, who knows

[sweet wing]

what ever the fuck this is

[tiny ridge]

rofl

ok its S-duality

[sweet wing]

oh ya it is jus S duality memeing the middle one is supposedly stronger than langlands

im partially unconvinced by the physical arguments but ok physics goes brrr

[tiny ridge]

yeah idk where they get this stuff

the version of mirror symmetry i care about seems shockingly legit

[sweet wing]

tbh yea string theory actually producing good results wtf

iirc there are quite a few areas where its conjectures have been proven correct

would be nice to someday formalize the lolimaginephysics arguments tho

it's like back when physicists with their electromagnetism and newton laws and predicting motions while cauchy figuring how to define convergence

[tiny ridge]

its so much worse though because at least some guy would fiddle around with the magnets and prove it works

nowadays the explanation is typically that particles in CERN just vanish in one of the 11 complex dimensions

also its damn funny that originally mirror symmetry was conjectured for calabi yau manifolds but nobody on gods green earth knows if there are infinitely many of these

good thing Kontsevich came along and extended the conjecture for a much larger class

russians, once again

[sweet wing]

tbf i feel like there are 2 groups of string theorists these days

those who still firmly believe they have an accurate model of reality and publish copium and
those that just use it as a cute toy model for a hypothetical universe and study it to get these mathematical results

[tiny ridge]

lol yeah accurate.

[sweet wing]

i dont think it's bad to study what if we try to quantize n-dimensional particles, it is definitely a cute toy

[tiny ridge]

i dont mind 2

[sweet wing]

the first group is just massive copium and goes on the media to talk shit

[tiny ridge]

yes lol

[sweet wing]

then theres also those people in the second group who can publish insane "results"(conjectures) with minimal equations (see literally any of wittens papers it's purely physical arguments) and those that bash out all the math in hopes of something lulz

[tiny ridge]

i feel like without actual experiments they eventually run out of their stupid path integral bag of tricks

cuz none of it makes any sense

[sweet wing]

at some point i think witten is just trolling us

his intuit is so good he doesnt need to bash the math to know it is true and us noobs need to wack through all the math fucking polchinski

[sweet wing]

it's insane how many conjectures comes from jus considering path integrals tho

[tiny ridge]

his papers have barely any justification, he spams a shitload of words and quotes atiyah

[sweet wing]

lol fr

but somehow it holds

fucking black magic

[tiny ridge]

yeah

truly

[sweet wing]

you know thinking about it

the expression \int e^{t<thing>}
does appear a lot in math

and changing the <thing> gives you a lot

could say use <thing> = sqrt(eigenvalues of laplacian) and you get a lot of information out of nowhere

maybe physicists were onto something

[tiny ridge]

this is pretty much supersymmetry yeah?

[sweet wing]

it appears in normal stat mech too

[tiny ridge]

index of the dirac operator is integral that thing with t -> 0 (infrared limit), and the right hand side of APS for t -> oo (UV limit)

[sweet wing]

if your energy levels sre Ei, your partition function is sum_i e^Ei/T where T is temperature(normalized)

[tiny ridge]

yeah hannah pointed out this thing a while back

[sweet wing]

then for some fuck reason it gives you everything you want

path integrals are kinda continuous variants of it

[tiny ridge]

i mostly think of gaussians when i think of this, the statmech connection didnt click

[sweet wing]

maybe if all the stat mech people didnt suicide we would have a better understanding /j

[tiny ridge]

rofl

[fading vale]

HMS is so cool

[tiny ridge]

some people are working on lattice gauge theory in the hopes of making some sense of e^-iSt

i dont know if theyre successful

the results look hideous

[fading vale]

like it feels like an area i can very easily see myself ending up just in terms of how much all the stuff i think is neat interplays there

[sweet wing]

some (more legit) application of these path integrals appears when S is a finite sum

[tiny ridge]

i think if you get into it you should be careful about which version you want to get into

[sweet wing]

i heard witten did some black magic there i need to dig up my notes when i tried to study a bit of string theory LOL

[tiny ridge]

you know Lefschetz tried to make sense of path integrals

[fading vale]

wdym by version

[tiny ridge]

i was reading an old book of his where he formulates a mathematical version

[sweet wing]

oo

[tiny ridge]

what do you think of when you typically think of hms?

[fading vale]

an equivalence between the derived category of coherent sheaves and the derived fukaya category on some pair of calabi yau manifolds

[tiny ridge]

yeah this is awful

[fading vale]

[tiny ridge]

nobody knows if there are finitely many calabi yau manifolds

pointless fucking conjecture

[sweet wing]

you know whats the most ridiculous thing ive seen so far

[tiny ridge]

slightly better is the landau ginzberg version, with a superpotential.

that is a correspondence between the fukaya seidel category and coherent sheaves

[sweet wing]

"we have 3 generations of fermions and bosons, hence the euler characteristic of the compactified calabi yau is 6 or -6"

how the fuck

but ok im not gonna ask

[tiny ridge]

works for toric, toric degenerations eg fano, certain del pezzo surfaces, …

and more

i like the one with weinstein manifolds, formulated by Kontsevich

[fading vale]

i see

for context i know nothing about the details of any of this stuff i just think it seems conceptually very neat lmao

[tiny ridge]

lmao

[fading vale]

also hi ari been a while

[sweet wing]

you know whats the greatest thing

theres no reason to use calabi yau nothing i see suggests a complex structure even but physicist just see it and wow it fits the condition of ricci flat plop

[sweet wing]

henroooooo

[tiny ridge]

yeah exactly i think they should not bother with much of this highly symmetric junk

[sweet wing]

reading polchinski vol 2 is like a fever dream

[tiny ridge]

someone should just formulate string theory for almost Kahler guys. worldsheets are just pseudoholomorphic curves

[sweet wing]

polchinski vol 1 is passable

[tiny ridge]

lets go

[sweet wing]

honestly id bet it would work out considering how flexible it is

[tiny ridge]

lol my roommate struggled with this for a long time and eventually gave up

[fading vale]

i dont really care about the physical motivation ngl

[sweet wing]

[tiny ridge]

exactly my thoughts

@fading vale I’ll just tell you about the Weinstein version of HMS, its super natural. Take the cotangent bundle of any manifold T*Q, this has a symplectic form sum_i dq_i wedge dp_i where q is position (coordinates on Q) and p is momentum (coordinates on cotangent fibers).

[fading vale]

mhm

[tiny ridge]

Screw the Fukaya category, lets just think about its object. These are Lagrangians in T*Q. What are some natural examples? Graph of exact 1-forms df, treated as a section of T^starQ, yes?

[fading vale]

lol

discord markdown strikes again

But yeah

[tiny ridge]

In general a Lagrangian will just be a multigraph over Q, locally still of the form df

[fading vale]

Multigraph?

[tiny ridge]

Think of multi-valued functions, no reason a Lagrangian will map homeomorphically back to the zero section under the bundle projection

[fading vale]

Ah

[tiny ridge]

But locally it is nevertheless df

[fading vale]

Yes

[tiny ridge]

Extend the class a little bit and allow conical Lagrangians. What are these now? Well think T*R which is R^2 and consider union of the y-axis and the x-axis

The axes are Lagrangian subspaces

The union is a singular gadget but it deserves to be called a singular Lagrangian

[fading vale]

sure

[tiny ridge]

Conical means invariant under the scaling (q, p) -> (q, cp), c > 0

Intuitively, if you know some contact geometry, this means the Lagrangian is cone of some Legendrian object in the unit cotangent bundle (which is a contact manifold with contact form pdq)

Anyway so these are my objects. Make your Fukaya category using these by running the machine, lets call it Fuk(T^*M)

[fading vale]

are you saying take subsets of this form?

[tiny ridge]

yeah

take only conical singular Lagrangians

[fading vale]

oh i see

okay this makes sense

[tiny ridge]

On the other hand, consider the manifold M that it is a cotangent bundle of and consider constructible sheaves over M. This just means that you have a sheaf s.t. there is some stratification of M for which the sheaf restricts to constant sheaves on each stratum

Given such a sheaf, you can look at the support, closure of the set of points where the stalk is nonzero.

[fading vale]

right

[tiny ridge]

There is a steroids version of support, called microsupport. It is given by the set (x, p) of codirections such that you cannot “parallel transport” the the stalk F_x in the codirection given by p, ie sections do not propagate in the direction given by p

Intuitively this just means that if you draw a little ball B around x, the map F(B) -> F(B \cap {p < 0}) is not an isomorphism

But not quite, you should take derived functors and ask the same thing for cohomology ie H^i(B; F) -> H^i(B \cap {p < 0}; F) is not an isomorphism. its ok

Theorem (Kashiwara): the microsupport of a constructible sheaf is a singular Lagrangian

[fading vale]

[tiny ridge]

Beefed up version by Nadler-Zaslow: microsupport gives a dg equivalence DSh(M) -> DFuk(T*M)

This is mirror symmetry for the cotangent bundle

[fading vale]

[tiny ridge]

This has a bunch of “real life” impact. Not the theorem per se, but philosophically it says whatever symplectic geometry you can do in T*M with all your pseudoholomorphic crank can be done just by sheaf theory

This has been used to great effect by Kashiwara-Schapira-Guillermou. For example, the Gromov nonsqueezing theorem can be proved just by sheaf theory

Shende-Zaslow-Truemann-Ng-Rutherford has interpreted invariants like Legendrian contact homology from sheaf theory

These are pretty concrete phenomena. The Nadler-Zaslow theorem is not as important as these IMO

Its just the banner of the story

[fading vale]

wait this is extremely neat

[tiny ridge]

The conjectural extension to Weinstein manifolds (I am not sure if you know what these are so I will avoid details) is based on the fact that these are symplectic manifolds which conjecturally walks and quacks like the cotangent bundle. They can be thought of as cotangent bundles to some singular Lagrangians inside them

Conjecturally, at least

[fading vale]

do you think you could use this to recast some of the pseudoholomorphic stuff in T^*M in terms of microlocal analysis on M

disclaimer i know 0 microlocal analysis

but it would be neat

[tiny ridge]

a great deal can be and has been done but I am not sure the extent of this.

[fading vale]

[tiny ridge]

its a bit bizarre that the two sides are related at all and imo one should unpack the nadler zaslow proof to see whats exactly going on

a lot of mirror symmetry proofs are just bashing out the two categories on either side and show they’re equivalent

it doesnt explain much

[fading vale]

this is very pretty to me in particular

[tiny ridge]

its shocking yeah

[fading vale]

yeah this does sound maybe less ideal

like being able to actually translate the problems from one language to the other is sort of more of what i was vaguely hoping on a conceptual level for the relationship between the theories

[tiny ridge]

but the sheaf theorists write awfully. after grappling with this stuff for a couple of months im convinced you can do away with a lot of generality Kashiwara et al works in

[tiny ridge]

the conceptual is mostly lost in most proofs of mirror symmetry in test cases unfortunately

the idea of microsupport was conceptual breakthrough

and i feel like most people do not appreciate it enough because they read pages of six functor formalism

[fading vale]

i see

thank u for telling me about this this is neat stuff

on the list of things to read for sure

[tiny ridge]

@sweet wing https://link.springer.com/book/10.1007/978-1-4684-9367-2

this is the book

[nimble portal]

applications

Is homotopy theory algebraic topology?

[tiny ridge]

i suppose but maybe homotopy theorists disagree

[nimble portal]

I asked because of fundamental group stuff

[languid patrol]

Fundamental groups are definitely algebraic topology

[nimble portal]

Yes

But is homotopy theory in general alg top

Ig that’s a silly question

Lmao

Can be used in a lot more places than just for algebra

[languid patrol]

What are you thinking of when you say “homotopy theory?” I guess the answer is either obviously yes or yes depending on what you’re thinking of

[nimble portal]

LMAO

I’m not sure 😭

JWKEHDK

That’s all…I really…know…

[languid patrol]

Anyway homotopy theory has wide reaching impacts which have stretched to algebra, category theory, algebraic geometry, etc.

[nimble portal]

abt any cool use of paths at least

What else is in alg top? All I know is fundamental groups and homology

Is stuff about chains algebraic topology?

[tiny ridge]

yes

[nimble portal]

Would you call the study of Lee groups algebraic topology?

Idek what “the study of Lie groups” means

[tiny ridge]

not particularly

[nimble portal]

mfw I wrote Lee instead of Lie

[languid patrol]

Lee groups are when you study Lie groups but only from Lee

[tiny ridge]

algebraic topology could be useful studying Lie groups

[nimble portal]

My cat just bit my lip

She loves biting my nose and sometimes she catches my lip bc I try to kiss her but stay out of nose-getting-bitten range

[tiny ridge]

maybe your cat is a homotopy theorist who wouldn’t like to be called a topologist of any kind

clark barwick

[nimble portal]

Could be

[nimble portal]

What else?

[languid patrol]

bit weird if Clark barwick bit your lip

[tiny ridge]

@fading vale ok so hits blunt its not totally bizarre that the two sides are related from a physics POV

think of a constructible sheaf as some fucked up bundle with a flat connection on M

T*M is where classical mechanics happens, and what this is saying is we can formulate it in terms of sections of this strange bundle

so this is some kind of quantization

Dmitry Tamarkin has some good stuff on how all of this ties in with physics, but I have never really read any of it

the russians i tell you

always read the russians

[abstract saffron]

cohomology ring, bigraded, trigraded even

At least in low dimensional topo, there's this huge thing about quantum invariants

[abstract saffron]

And with them comes the real "algebraic" part of AT

Also, surprise surprise, we still have covering spaces

[sweet wing]

ohhhh

ill have a look!

[abstract saffron]

Also even "homology" itself is so huge there are a ton of them

I guess you can add persistent homology, which is not a homology theory, but a field in its own right.

[novel acorn]

spectral sequences, homotopy theory, cohomology etc.

[coral pivot]

lol character growth for us both then!

[sweet wing]

i refuse to accept

jkjk

[coral pivot]

lol

[white oxide]

Can someone give me an example of the former? A quotient map which has contractible fiber but no section?

[unreal stratus]

How about I -> S^1, viewing latter as identifying 0 and 1

[tiny ridge]

Two points isnt contractible I believe

[white oxide]

That was disconnected fiber

[unreal stratus]

Oh wait lmfao

I forgot one of the conditions

Sad times.

[tiny ridge]

Try quotienting an interval by a closed subinterval

[unreal stratus]

Yes that's better lol

[white oxide]

Yeah that works

[tiny ridge]

Take a nonmetallic hollow, flexible tube. Knot it, glue the ends together. Thread a wire around it so that it becomes a knotted solenoid. Once you pass some current through this, this gives a magnetic field which is confined to the interior of the tube. But the potential is a U(1) connection on the exterior of the tube ie on the knot complement; particles doing a loop de loop around the knot (so as to have positive linking number with the knot) experiences a change in phase by the Aharonov-Bohm effect.

Is there an interpretation of the Alexander polynomial using this?

The connection is just the flat connection corresponding to 1 in H^1(knot complement; Z) = Z

The Alexander module is the twisted first homology of the knot complement, twisted by this flat bundle, as a Z[t, t^-1] module where t acts by this monodromy. But I don’t know what this physically means.

[nimble portal]

What the fuck did I just read

[floral bear]

do homology sequences go forever?

like are there infinite homology groups to describe any space

bc that kinda seems weird, like how do you have a 4d chain on a 3d space

im using intuition from simplical homology

forgive me if this seems weird

[tiny ridge]

If the space is nice and n dimensional, then the homology groups vanish above the nth grade

The keyword is “CW complex”

But there are pathological examples beyond the realm of CW complexes that might challenge your notion of dimension, see https://www.jstor.org/stable/2034486

[urban zinc]

galois correspondence for covering spaces is so coolllll

[coarse night]

Can you show RP² cannot be a cover of something other than itself

@urban zinc

[nimble portal]

What is R^0?

Just a point?

[coarse night]

yes

[nimble portal]

Well issue is you could call an element of R^1 a point on the real line

R^0 being a point doesn't make sense to me

Empty set feels more proper

[coarse night]

take it as a definition

R⁰ is generated by the empty set

so is a point

as a vector space

[nimble portal]

I think I understand

LMAO

[urban zinc]

No I prob need to read more first haha

[white oxide]

For CW-complexes?

[coarse night]

it becomes a cw complex

[white oxide]

Wasn't there something with the euler characteristic for that then?

[coarse night]

yes

[tiny ridge]

too fancy

[coarse night]

(subgroup)

[tiny ridge]

too fancy

[coarse night]

lol

[hidden crag]

lift to universal cover S^2

[coarse night]

(too fancy )

[white oxide]

Uh... why?

[hidden crag]

I’ll leave this here as a hint without elaborating

[coarse night]

too fancy, ignore

original argument I had was EC

[cosmic socket]

No I don’t think that’s what it’s saying

It’s saying if you start with a ball of radius r (in d). Then there is a smaller ball of radius r’ (in d’) that is contained in your original ball

No what you wrote is wrong.

You wrote that there exists a ball contained in every ball. This is not true

[white oxide]

Well, you want to show that both metrics generate the same topology. If it does then the interior (relative to d') of any open set U in the topology induced by d must be U.

Right, so if you were given just a set you can ask the question what it's interior looks like, the answer of which depends on the topology.

Did you have something along the lines of:
A set U is open iff every point has a neighbourhood contained in U?

Right, that's actually what's needed. The interior would just be the union of all the open sets from the second half, that are contained in U.

[gentle osprey]

marc

[white oxide]

Or yeah sure, use the books formulation.

Well, take an open set U from the topology induced by d. Now look at it as a set, what does it mean for this set to be open in the topology induced by d' ?

Why?

[gentle osprey]

marc

[white oxide]

Utilize this and your initial problem statement to show that the set U, which is open in (M,d) is also open in (M,d')

[white oxide]

"The metric space" always means the pair (M,d), M is just a set. It could be amodule, topological space, or an projective variety but just knowing the set M won't tell you that.

Very often it is clear that one "thinks of" M as a metric space, in that case d is just implicit.

Yeah

As a prototypical question, can you reason why the topology induced by the taxicab metric and the euclidean metric on R^2 is the same?

One has open sets that are just of overlapping squares without borders and the other made out of balls without borders.

[white oxide]

It has to do with this

Well, is there an open ball (given by d') of some positive radius around x which is contained in U?

I mean, you can take a break and re-look at the problem. But that is indeed the argument

Abstract nonsense stuff just takes time to digest

[viral halo]

Doing an exercise on de Rham cohomology

[gentle osprey]

Syst3ms

Syst3ms

[viral halo]

But i'm stuck for some of higher cohomology groups

[gentle osprey]

Syst3ms

[viral halo]

By removing two points from S² separately, we get open sets homeomorphic to ℝ²×S⁴, i.e homotopy equivalent to S⁴, with intersection homotopy equivalent to S¹×S⁴. Using the Mayer-Vietoris sequence, this gives :

[gentle osprey]

Syst3ms

[viral halo]

but we don't know the dimension of the first and last ones, so we're stuck

Any ideas ?

[unreal stratus]

Is Kuenneth banned

[viral halo]

yes, but i'm curious anyway, what does it state ?

[thorny agate]

I'm having trouble proving the following statement:

Let $\{Y_{\alpha}\}$ be a collection of spaces and $Y = \times_{\alpha} Y_{\alpha}$.
Prove that the function $f\colon X \to Y$ is continuous if and only if each composition $\pi_{\alpha} \circ f$ is continuous.

My issue is that I'm not sure how to do this for arbitrary products. For finite products basically I'm using the fact that
\begin{align*}
  f^{-1}(U_1 \times U_2) &= f^{-1}((Y_1 \times U_2) \cap (U_1 \times Y_2)) \\
                         &= f^{-1}(\pi_1(U_1)) \cap f^{-1}(\pi_2(U_2))
\end{align*}
And I rely on the fact that it's a finite intersection.

[gentle osprey]

Spamakin🎷

[toxic coral]

Assume that each composition $\pi_{\alpha} \circ f$ is continuous.

To show that $f$ is continuous, you observe that for any open set $U \subseteq Y$, we have $f^{-1}(U) = \bigcap_{\alpha} (\pi_{\alpha} \circ f)^{-1}(\pi_{\alpha}(U))$.

Since each $\pi_{\alpha} \circ f$ is continuous, the sets $(\pi_{\alpha} \circ f)^{-1}(\pi_{\alpha}(U))$ are open in $X$ for all $\alpha$.

Therefore, their intersection $f^{-1}(U)$ is open in $X$, proving the continuity of $f$.

Now, assuming $f$ is continuous, we want to show that each composition $\pi_{\alpha} \circ f$ is continuous.

Let $V_{\alpha} \subseteq Y_{\alpha}$ be open. Then $(\pi_{\alpha} \circ f)^{-1}(V_{\alpha}) = f^{-1}(\pi_{\alpha}^{-1}(V_{\alpha}))$.

Since $\pi_{\alpha}^{-1}(V_{\alpha})$ is open in $Y$, its preimage under $f$, which is $(\pi_{\alpha} \circ f)^{-1}(V_{\alpha})$, is open in $X$. Thus, $\pi_{\alpha} \circ f$ is continuous.

So, we conclude that $f$ is continuous if and only if each composition $\pi_{\alpha} \circ f$ is continuous.

[gentle osprey]

∯ | caltech

[toxic coral]

it might be wrong, but i tried

[white oxide]

This is pretty much the reason why the topology on arbitrary product spaces is not the box topology if I didn't mix something up.

The pre-image of an open set under the projection map is precisely an element of base of the topology of your space, so by saying the composition is cts it follows that the pre-image of any basis element is open in X.

But since any open set is the union of basis elements, the pre-image is a union of such pre-images, which is then again open

[thorny agate]

The issue is that arbitrary intersection of open sets isn't necessarily open

[thorny agate]

Ah ok

It's a basis argument, got it

[white oxide]

It's part of the definition of the product topology

Apparently even it's universal property, funny thing

[violet summit]

I'm not sure how the dot products described in (a) could be turned into a function, and also how to use that to show O(n) is closed

[untold lily]

they can be turned into a function like how the map f(a,b) = ab is a function from R^2 to R

as to how this helps to show O(n) is closed, use the hint that the columns form an orthonormal basis, try to deduce something about the dot product

i.e. something these maps must fulfill

[violet summit]

wouldn't those be maps from R^{2n} to R though instead of n^2?

[untold lily]

your input is an n by n matrix

the function f(a,b,c) = ab is a map from R^3 to R even though I only use two of the coordinates

[violet summit]

I see

So I would end up with n^2 different functions?

[untold lily]

yeah

[urban zinc]

Suppose M is locally Euclidean, Hausdorff, connected, and has a locally finite cover by precompact coordinate domains. How do we know that the cover has a countable subcover?

I'm really stuck on this step

[urban zinc]

@nimble portalhave you done this problem yet 🥺

[nimble portal]

No but let me think about it a bit 🤔

I have a feeling that “extracting a countable subcover” is related to the fact that by local finiteness, any nbhd of a point intscts finitely many sets in your cover

I forgor what a connected component is, let me look

oh never mind I know

[urban zinc]

I see why all the other steps of the problem work, but this step specifically of turning from a locally finite cover to a countable cover is idk

😔

[nimble portal]

Could you do like

[urban zinc]

The fact they're precompact is probably important too

[nimble portal]

Yeah I got no clue beyond this

Could you just sample points in your component so that nbhds about them cover the component

Then take the intersections of those nbhds with your cover

Wouldn’t that cover the component, then since it’s locally finite you’re guaranteed a countable # of intersections, and since they’re precompact they’ll be open?

[urban zinc]

Why do you only need to sample countably many points

[nimble portal]

uh

good question

maybe that comes from paracompactness too

since any nbhd intersects finitely many sets in our cover

then only finitely many sets contain our point??

yeah I’m not sure…

[urban zinc]

yeah I'm really stuck

[violet summit]

How does this sound? (sry for tag)

Let \({f_{i,j}}\colon{\RR^{n^2}}\to{\RR}\) be the dot product 
    of \(\vec{v}_i\cdot\vec{v}_j\) in the matrix \(A\). Define 
    \[
        f(A) := \begin{bmatrix}
            f_{1,1}(A) \\
            f_{1,2}(A) \\
            \vdots \\
            f_{n,n}(A)
        \end{bmatrix}.
    \]
    \(f_{i,j}(A)\) equals \(0\) when \(i\neq j\) and 
    \(1\) when \(i=j\) for all \(1\leq i,j\leq \)if and only if \(A\) is in \(O(n)\).

    I claim \(f\) is continuous. This reduces to showing \(f_{i,j}\) is 
    continuous for all \(i,j\). Let \(A\in \RR^{n^2}\). 
    Given \(\eps>0\), we can change the values in \(A\) by less than 
    \(\delta=\frac{\eps}{\text{max entry in A}\cdot 2n}\), and still 
    get a vector that was changed by less than \(\eps\).

    Since \(f\) is continuous, the preimage of the closed set of 
    the vector that is \(0\) when \(i\neq j\) and \(1\) when \(i=j\) is 
    closed, which is precisely \(O(n)\).

[gentle osprey]

pramana

[untold lily]

perfect

[sweet wing]

iirc yea you also jus use the fact the closure is compact and stuff to eventually get down to a countable cover

~~mildly related: https://arxiv.org/abs/0910.0885 ~~

oh yes i recall the proof now

the idea is similar to the fundamental group being countable (it was proved earlier iirc)

your goal now is to construct $M=\bigcup K_n$ where $K_n$ is compact

and since you are given a locally finite cover whose sets are precompact, youd probably want to construct $K_n$ as some form of closure of unions of those sets

[gentle osprey]

ari 亲

[sweet wing]

can try to go on from here

a construction of K0 and K1 (afterwards it may be obvious i hope?):

K0=||closure of any set inside it||

K1=||closure of the [union of all open sets in the cover intersecting K0]||

[quiet summit]

do you guys know any journal that is easy to read, regarding application on topology??

[solemn oar]

Journals tend to be quite hard to read by themselves, and when it comes to applications of math, you'd want some background in the area of application to get much out of it.
The first thing that pops into my head is persistent homology and topological data analysis. Rather than journals, I'd recommend you pick up a textbook on those.

[quiet summit]

alright tnx

[urban zinc]

ty :))) I will look at this shortly

[urban zinc]

hmmm okay I still don't get why you only need countably many of these Kn to cover M

[sweet wing]

so the idea is to show that for every y in M, y is in some Kn right

[urban zinc]

yeah

[sweet wing]

the idea is ||the exists a path from x to y, i.e. a map [0,1], 0->x, 1->y||

[urban zinc]

ohhh wait and that's because connected components are open and therefore path-connected?

[unreal stratus]

Are you path connected, locally path connected, semi-locally simply connected and in want of insurance? We've got you covered!

[urban zinc]

wait no I don't see why it's path-connected

oh wait it's because M is locally Euclidean so locally path connected as well

I understand nowwww

thank you!

[urban zinc]

every group with two generators is the symmetry group of some covering space of the figure eight
wtf

[solemn oar]

Fancy way of saying that pi1( S^1 v S^1 ) is free on two generators.

[urban zinc]

this galois correspondence is so cool

[hidden crag]

Did you think about Ryus exercise

[urban zinc]

I thought about it and haven't gotten anywhere lol I think I need to finish reading this section first

[coarse night]

I am not sure what way you were suggesting

The solution I came up with using universal cover is not elementary

What was your intended solution

[white oxide]

Well, the figure eight has two generators and there is a way to add any relations you like by just gluing in two-cells in the right way

[hidden crag]

||decktrafos on S^2 are iso to pi_1(X) (where p:RP^2->X is a covering) or alternatively you look at free group actions on S^2||

I can elaborate tmrw if you want, I’m out rn

[coarse night]

yeah that's what I had

that's unnecessary fancy no?

[hidden crag]

Is it

The second one is more elementary ig?

[coarse night]

Ig but euler char is the easiest imo

Though you need a bit of homology

[hidden crag]

Yeah I agree

[grizzled ibex]

what do we get by glueing the boundary of a moebius strip into a single point?

kinda lazy to take the pen

and it doesn't seem a obvious answer to me

[tiny ridge]

rp2

[bitter smelt]

Draw the fundamental polygon and

Ok well nvm then

[grizzled ibex]

makes sense

[bitter smelt]

[tiny ridge]

sorry lol

[grizzled ibex]

its horrible to visualize it

[bitter smelt]

It's not.

[tiny ridge]

nah

[grizzled ibex]

i mean

visualizing rp2

[bitter smelt]

If you "draw the fundamental polygon and"

Oh, sure