point-set-topology

[obtuse meteor]

:O

I don't know but it sounds neat

[cedar pebble]

yea it's cool as hell!

thanks for coming to my Ted talk

idk I'm very excited because I hated this class all semester and then something finally clicked and I can compute examples and it's just so cool

[obtuse meteor]

nice!

[marsh forge]

i was zoned out until you said compute ext groups

[obtuse meteor]

lol

so I have discovered that it is OP to take all coordinate patches to be connected when proving things about orienting manifolds

so then your determinants all have a specified sign

[sleek thicket]

I almost always take mine to be R^n

Or I take them to be a ball inside a larger R^n

[obtuse meteor]

yeah I should start doing this

[marsh forge]

@cedar pebble Do you know of a source describing how if it all the UCT respects multiplication? I always see the theorem itself described only on groups but then people often remark that by UCT they get not only the groups but ring structure too

is this just laziness

am I missing something

[cedar pebble]

Oh hmm I do not know off the top of my head

[marsh forge]

My guess is its like

the groups are by UCT and the multiplication follows from vague inspection

but if theres actually a strong result

it seems useful to know (and write down lmao)

[frigid river]

Can't I just argue that I am unifying countable unions, and therefore this must be a countable union again?

I mean, a set is of first category, if it is a union of nowhere dense subsets. So if I unite such sets again, I still have a union of nowhere dense subsets.

[sleek thicket]

yup

[frigid river]

Bruh

Okay, thanks.

[sleek thicket]

Yeah haha this seems like a silly question

[marsh forge]

im gonna teach a grad class with ridiculously hard psets

until the final

where every question will be so trivial

that everyone doubts themselves

[frigid river]

What even is this xD The previous question was so easy and now this. My attention span isn't even long enough.

Soon, let me try understanding the question first.

Would it be important to understand what the metric p actually means, or could I just blindly try proving the triangle inequality?

[honest narwhal]

@frigid river try to understand it

[frigid river]

Not possible xD Well, I know that p(f,g) can never be bigger than 1, but that's about it.

[frigid river]

I still do not know how to prove the triangle inequality -.- How would I do this? $$sup(min(1,d(f(s),h(s))) \leq sup(min(1,d(f(s),g(s)))+ sup(min(1,d(g(s),h(s))).$$ Any hints?

[gentle osprey]

veryhappyperson

[frigid river]

Even the bot thinks the question is too long.

[gritty widget]

Maybe step 1: how do you prove sup blah1 <= blah2?

Also, you can try to understand this metric by considering the case f, g : R -> R

[frigid river]

Show that the minimum of blah2 is bigger or equal to blah1?

[gritty widget]

Maybe I should be more explicit, eg how would you prove sup_x f(x) <= A where A is a constant?

(Well, what's one way to prove it?)

[frigid river]

Determine the maxima of f(x)?

[gritty widget]

I guess so

What I had in mind was to show f(x) <= A for all x

[frigid river]

Okay, I get that. Yet I don't really see any way how I could apply that.

[gritty widget]

But your problem is in exactly that form

Another possible approach: try thinking about this metric p and get an intuition for it. Is there a slightly more simplified form for it? (Well, maybe it is subjective whether it is more simplified or not)

[frigid river]

Wait, am I dumb or was I dumb. If I do sup(stuff equal or smaller than 1) don't I just get 1?

[gritty widget]

Erm, you get something that is 1 or potentially smaller than 1 right?

[frigid river]

from the minimum, I think.

Oh wait, there doesn't have to be 1 in it at all, right?

[gritty widget]

Oh?

[shut moat]

yeah it doesn't have to be there, I think

[gritty widget]

I don't see that

[frigid river]

$$sup(min(1,d(f(s),h(s)))$$

[gentle osprey]

veryhappyperson

[shut moat]

given a metric d, you can always define a metric d'(x,y) = min{1, d(x,y)} called the standard bounded metric, which can be shown to induce the same topology

[gritty widget]

Yeah, this is what I meant as simplified above (assuming I am thinking the same thing as you)

[shut moat]

ah sorry I didn't read the scrollback

[gritty widget]

Ah no, I didn't say too much, since I didn't want to give it away (just left it at "this can be simplified" )

[shut moat]

right ok

[frigid river]

I am still lost. xD

[gritty widget]

Maybe I am being too terse about hints and actually being unhelpful 🙂

[shut moat]

lmk if I'm giving away too much 8da, but supposing you knew that min{1, d(f(s), g(s))} is a metric with the triangle inequality, are you comfortable with proving the triangle inequality for sup_{s in X} d(f(s), g(s))? (where d is some rando metric)

[frigid river]

No. I have never actually proven that such things are metrics. Only simpler things, and then with a given metric, like d(x,y)=|x-y|, not any metric.

[shut moat]

well a summary of my hint is that you can break your problem down into two steps:
$$ $$

1. show that $\sup_{s\in X}d(f(s), g(s))$ is a metric
   $$ $$
2. show that if $d$ is a metric, $d'(x,y) = \min\qty{1, d(x, y)}$ is a metric.

[gentle osprey]

Buncho Spheres

[shut moat]

and while the supremum metric might look intimidating, the proof is quite ok

because you already know that d is a metric

sry gtg back go working on the bio homework i've been procrastinating on

but hopefully this helped somewhat

[frigid river]

Yes, it helped, I think. Thank you.

[gritty widget]

Hi guys, I keep seeing online that the set Q subset R is not locally connected. But it never says why.
Why?

no open subset of Q in R is connected so it can't be locally connected

[shut moat]

wait huh?

do you mean like the empty set

in which case reeeee

[gritty widget]

this is why mathematicians are bad

i should just become a physicist

[shut moat]

based

[gritty widget]

@shut moat you might like chapters 5 and 6

[shut moat]

:o ty

[gritty widget]

(this pdf was obtained legally.)

[shut moat]

my collection grows

[gritty widget]

obtained legally, but distributed legally?

[shut moat]

i was thinking the same thing lol

not that i'm complaining

[sleek thicket]

Tterra criminal arc???

[gritty widget]

be gay do crimes

[honest narwhal]

With applications to physics what a nerd

[sleek thicket]

Be ultra do product.

[thin jewel]

What happened to the other math server

There was like a really active one for graduate+ mathematics

[ivory dragon]

there are a few of those out there, you might have to be more specific

was its icon, by chance, a hopf fibration? if so, lots of drama went down which led to the server being deleted

[barren sleet]

yikes what kind of draam

drama

[sweet wing]

lets uh

not talk about it

or do you mean this

it is dead

[ivory dragon]

tl;dr people said things that they really didnt want associated with their identity, but also made their identities public

this was a bad mix

as it turns out

[barren sleet]

oh.

[ivory dragon]

and by "things they really didnt want associated with their identity", i mean like the kind of thing that is very very likely to affect your academic employability

i think that paints enough of a picture

[barren sleet]

that sounds like a very unfortunate situation

and pretty seruous

serious

[sweet wing]

unfortunate series of things happened best is tovjust never mention it

[strong heron]

Why is this being discussed in this channel though? This is subject-specific, no?

[tight agate]

hopf fibration

[barren sleet]

you should have seen #advanced-analysis like 2 hours ago

i asked a topology question and ppl were like "wrong channel" then we started talking about being gay and like... furries? and i guess that was fine

i dont actually care about the reaction to my top question, i just included that for contrast

[thin jewel]

Ah right

[thin jewel]

Yeah that one

[thin jewel]

Right

[sleek thicket]

.pin

[flint flicker]

Hello i am trying to compute the fundamental group of the pinched torus using van Kampen i am trying to apply the theorem on this am i going in the right direction?

and i identify the points at the end with each other

i got that the fundamental groupe is generated by two elements 1 and a but i do not know how to proceed

[sweet wing]

it may help to know that the pinched torus is homotopic to a figure 8

[flint flicker]

so the fundamental group of the pinched torus is zxz?

[sweet wing]

nop

[sweet wing]

how do you plan to use van kampen here

[flint flicker]

this is how i want to do it

[sweet wing]

alrite that works

so what pushout are you looking at

[strong heron]

Wait, how is this true? Isn't it homotopy equivalent to the wedge of a sphere and a circle? I believe the fundamental group is just the integers.

Well, is it a pinched torus or a punctured torus? Which one did you mean, @flint flicker?

[flint flicker]

I mean the pinched one

[strong heron]

Like you take a doughnut and just pinch it at a point, right?

In that case, your space is homotopic to the wedge of an S^2 and an S^1

[flint flicker]

So this is wrong?

[sleek thicket]

I think ariana must have been thinking of the punctured torus

[strong heron]

Yes

[strong heron]

If the interior is filled, then your diagram is correct.

[flint flicker]

Yes it is filled

[strong heron]

Now you can use van-Kampen, if you like. The boundary of your picture will just give one loop, so you're good.

[flint flicker]

So it is better if i work with s^2 and s^1 instead of what i did?

[strong heron]

I mean, that is a direct way to say the fundamental group is Z.

But if you want to write a presentation, you can use van-Kampen and see that one of the generators, of the initial torus that you started with, becomes trivial.

And you have only one generator left.

[flint flicker]

Okay thank you for help

[strong heron]

No problem

[sweet wing]

oop

probably mb

[marsh forge]

this belongs in #groups-rings-fields really

but does anyone know how to compute massey products

or have a good source on them

[gritty widget]

What exactly is a disc in a surface? is it just the image of a disk in a chart?

(if yes, it would intuitively make sense to me to call that a small disc, is there such a thing as a non-small disk?)

[marsh forge]

any locally euclidean neighborhood would do

when you are not working smoothly charts are kind of unhelpful

small is meaningless here

[gritty widget]

aren't charts just homeomorphisms between a neighbourhood of some point and a open subset of euclidean space?

[marsh forge]

Normally when I hear the term chart used it is under the assumption that the manifolds comes equipped with a system of charts

Which is not something that matters as much in the study of topological manifolds

where we just assume the space is compact locally euclidean

But yes the idea is to just take a disc on the surface of the manifold

But for example you could have a system of charts on your manifold, and like, maybe you take a disc where three of them intersect weirdly

that disc is still a valid choice even though its not compatible with that system of charts

[gritty widget]

i see

[shut moat]

https://en.wikipedia.org/wiki/Convergence_space

WHAT

[marsh forge]

?

[shut moat]

I just found out about these on twitter, just thought it was rly weird

[remote beacon]

I actually think that's really cool

[sleek thicket]

Approx do you follow me on twitter

[shut moat]

ya

I found it on that hausdorff space poll

[sleek thicket]

ah nice

Hmm I need a new counterintuitive poll

Let's go with the one I posed in #advanced-analysis

Namely "Let X be a countable set and S a collection of subsets of X. Suppose that for any two elements A, B in S, the intersection of A and B is finite. Is S necessarily countable?"

[marsh forge]

how is this

topology

or

geometry

😠

[sleek thicket]

😌

Sets are just trees max

trees are topology

[gritty widget]

shouldnt that be in #calculus

[sleek thicket]

It's not topology but I also wouldn't call it calculus

[gritty widget]

often talk these stuff in calc 1 tho

maybe in #discrete-math cuz its countable

Some thoughts: ||at least it wouldn't break inclusion exclusion for S to be uncountable I think, or that S can be modified to not break it, assuming inclusion exclusion works on infinite sets||

[sleek thicket]

Please discuss this problem in #calculus 8da, it is not appropriate here

(also I wasn't actually posing it rn, I asked it as a poll on Twitter)

[gritty widget]

Ah I see, still a fun problem to discuss 🙂

[rugged swan]

is this equivalent to the continuous hypothesis 🤓

#foundations

😔

🙏

[shut moat]

@sleek thicket I think this is false, a counterexample being ,texsp If you take a collection of finite subsets $X_i$ such that $$X_i \subset X_{i+1}$$ then obviously any two sets have finite intersection, and you can form this regardless of the size of $X$

[gentle osprey]

Buncho Spheres

[shut moat]

oh the tex spoiler thing didn't work :(

[sleek thicket]

Nope, this will be countable

There are countably many finite subsets of a countable set

[shut moat]

oh they don't have to be all of them

at least that's my impression from your phrasing

[sleek thicket]

Wait, what do you mean?

[shut moat]

like you just want a collection of subsets such that the intersection of any two pairs is finite

right?

so take a finite collection of nested finite sets

[sleek thicket]

I want an uncountable such collection

Your collection is countable

[shut moat]

oh wait does countable include finite

ok im dumb then nvm lol

[sleek thicket]

Yeah haha sorry

Otherwise just take the empty set

[shut moat]

oh yeah even better lmao

[barren sleet]

I'm kind of a topology noob so I haven't seen a lot of official stuff about mobius bands but intuitively they feel like a product of S^1 and [0,1]. I know this product explicitly is not a mobius band, but is there any sense in which that is true

[sleek thicket]

Yes!!!!!!

You have discovered fiber bundles

[barren sleet]

what it feels like to me

[sweet wing]

fibre bundles time

[sleek thicket]

A fiber bundle is a space which is *locally* a product

[barren sleet]

is like group semidirect products, but for spaces. are those completely unrelated?

[barren sleet]

is that just a manifold? cause like R^m is a product

[sleek thicket]

Yeah, there's a connection here

[sweet wing]

manifolds is somewhat different

[sleek thicket]

With semidirect products

Semidirect products are a kind of group extension

[sweet wing]

n-manifolds means zoom in it will always look like R^n

[sleek thicket]

Have you seen this?

[barren sleet]

seen what

[sleek thicket]

Extensions of groups

[barren sleet]

yeah, so would that be a fibre bundle?

[sleek thicket]

Ah I see what you're thinking

I've given the definition unclearly

[barren sleet]

I mean there are 2 equivalent ways to introduce group semidirect products. one with normal subgroups, one with extending a group. is there more I should know

[sleek thicket]

A fiber bundle involves three spaces

• the model fiber F
• the total space E
• the base space B
  The fiber bundle (over B, with model fiber F) is a projection π : E -> B which locally looks like the projection U × F -> U

So for a manifold, there's no intrinsic map down to a base space like this

Fiber bundles have to be sitting over something else

Like how the mobius band sits over a circle

[sweet wing]

otherwise everything is a trivial fibre bundle over itself

[barren sleet]

well I didn't consider that a problem

but let me think

[sleek thicket]

It is true that every space is a fiber bundle over itself with fiber a point

[barren sleet]

what's U

[sleek thicket]

Or fiber bundle over a point with model fiber itself

[sleek thicket]

An open subset of the base B

[sweet wing]

so basically you need to somehow be able to find a open covering by open sets U_i

[barren sleet]

U is a neighborhood around a given point?

[sleek thicket]

Yes exactly

[barren sleet]

ok

[sleek thicket]

For each point there must be such a neighborhood

[barren sleet]

this seems very manifoldy I'm not quite seeing the difference

[sweet wing]

such that π^{-1}(U_i) is homeomorphic to U_i \times F

[sleek thicket]

They're both things defined in terms of local structure

[sweet wing]

your map π depends both on your choice of E and B

[sleek thicket]

One key difference is that a fiber bundle sits over a fixed base space

it locally looks like a product with that base space

[sweet wing]

manifolds you need a open cover such that every open set is homeomorphic to R^n

[barren sleet]

ok I see

[sweet wing]

theres a (significant) difference

[barren sleet]

let's go through with mobius band

[sleek thicket]

Nah, just the definition of a group extension. If you have nice topological groups G, E, H and a group extension 1 -> H -> E -> G -> 1 then the projection E -> G is a fiber bundle with model fiber H

[shut moat]

|| Sham what about dedekind cuts or sets of the form (-\infty, r)\cap Q||

[sweet wing]

fibre bundles can be very messed up spaces

[sleek thicket]

The "nice" here means "lie group", things can get ugly in infinite dimensions

[sleek thicket]

||Those will have infinite intersection||

[sleek thicket]

Sure

So the mobius band has a central circle to it

[barren sleet]

so the fibre is

[shut moat]

||shit I completely forgot about that condition||

[barren sleet]

I would like to attem0t

[marsh forge]

another fun fact is that a group extension yields a fibration (like a fiber bundle) on classifying spaces

[sleek thicket]

Oh yeah max I remember going to your talk

And then like

In my homological algebra class, my prof talked about a spectral sequence for group cohomology

And I was like "wtf this is the same thing max talked about"

[marsh forge]

oh yeah lyndon hoschikd serre

[sleek thicket]

Literally like two days later

[marsh forge]

i am an omniscient genius

[sleek thicket]

This is boring when E is a semidirect product though, since topologically those are just product spaces

[barren sleet]

your whole space E is the mobius band. is the projection π perhaps the "make the strip of paper less and less wide?"

[sweet wing]

essentially

[barren sleet]

ok

this definition seems strange to me

[marsh forge]

well think about it this way

there is a well defined circle

living in the very middle of the mob band

[barren sleet]

group analogies are welcome btw

that will help me

[sleek thicket]

That's pretty much right, yeah

[sweet wing]

you can also quickly verify the trivial fibre always exists, the map from a space to itself

[sleek thicket]

Retract onto the center circle

[marsh forge]

from any other point on the mob band

you just walk down

[shut moat]

i'm convinced the answer is no but I need to find a decent example

[barren sleet]

that's so cool

[marsh forge]

to the closest point on that circle

[sleek thicket]

😉

[barren sleet]

I didn't think there was actually a cool answer to my question

[sleek thicket]

I took a course on fiber bundles last quarter

[sweet wing]

it's a nice observation

[barren sleet]

so locally the map π looks like you are squeezing a closed cylinder

[sleek thicket]

So it's very much in my brain right now

[sweet wing]

nice enuf that theres a nice theory behind it waiting

[barren sleet]

this seems very similar to group semidirect products

but the thing is

for those

I have a way to articulate what I mean

toy have a homomorphism from H to aut(N)

that essentially builds up the group structure

[marsh forge]

you can think of a bundle as an extension of a space by the fiber

[barren sleet]

here I just see "the mobius band looks a whole lot like the closed cylinder, bit just trust me, they're different"

[marsh forge]

and this analogy often yields similar theorems

in particular the failure to be the “obvious” extension given by the product

[barren sleet]

so when π makes the mobius band into S^1, the fibre is S^1 or [0,1]?

[sweet wing]

S^1

[barren sleet]

ok so products are trivial fibre bundles

[sweet wing]

oh the fibre

fibre is [0.1]

base space is S1

[barren sleet]

S^1 is the underlying space

[0,1] is the fibre

[marsh forge]

yes

[sweet wing]

yesh

[barren sleet]

I'm in love

[sweet wing]

or you can sticking a [0,1] to every point of S^1

then glue them

[barren sleet]

yeah

but the issue I have

ia

is

[sleek thicket]

They're very very cool Ethan

[barren sleet]

I cannot tell you in this language

[marsh forge]

this is where i give my pitch that a ton of algebra actually embeds in spaces in interesting and cool ways

[barren sleet]

how exactly to form a mobius band out of S1 and [0,1]

[marsh forge]

take the trivial bundle

cut down the vertical strip

rotate

glue

[sweet wing]

you can also get a funny space from sticking a bunch of S1 to S2

[barren sleet]

like formally 😭😭

[sleek thicket]

Do you know the thing with the square

[marsh forge]

this is actually formal

[sleek thicket]

Polygonal presentation or whatever

[sweet wing]

formally it's what sham said

[barren sleet]

for mobius?

[sleek thicket]

yeah

[barren sleet]

the thing with the square

I think so

let me think

I saw it before

[sweet wing]

just check these

[marsh forge]

its a gluing procedure

[barren sleet]

you have a relation on the edges or something

[marsh forge]

yes

thats the glue

[barren sleet]

that basically tells you how its glued

[sweet wing]

here the fibre structure is immediate

[sleek thicket]

The red line is the central circle

Unless I fucked up

[barren sleet]

so is it that (x,0)~(1-x,1)?

[sleek thicket]

Yup

[barren sleet]

btw

I thought this was funny

same observation as ke

without seeing that I said that

ab8ve^^

above

[sweet wing]

wait wheres that lol

[sleek thicket]

hahaha

[barren sleet]

private

[sleek thicket]

Wy get in here

Let us tell you about the long exact sequence

Of a fibration

In homotopy groups

[drifting sundial]

hello

[sweet wing]

but yea fibre bundles/fibrations are a very nice tool in pokijg the fundamental group

[sleek thicket]

Hello

[sweet wing]

herro

[barren sleet]

yayayaya hello

[sleek thicket]

Suppose you have a fiber bundle F -> E -> B

[sweet wing]

do you like to see

puppe sequences

[sleek thicket]

Quick question do yall know what the homotopy groups of a space are

[barren sleet]

is this all for the mobius band

[barren sleet]

idk higher ones

[drifting sundial]

I know a bit

[sleek thicket]

Okay cool well here's the quick definition

[drifting sundial]

and smth about the fundamental group being the first order

[barren sleet]

I know fundamental group, is that like the first one?

omg

[sleek thicket]

Yip

[barren sleet]

lol

wy and i

[sleek thicket]

So the fundamental group of X

Is the set of homotopy classes of maps S^1 -> X preserving the basepoint

[barren sleet]

yes

[sleek thicket]

This might not be how you've seen it before

Okay cool

[barren sleet]

and a homotopy is what

a continuous map from [0,1]² to x

X

?

[sleek thicket]

Well in this case it would be S^1 × [0,1]

[barren sleet]

oh

yeah

[sleek thicket]

because we're looking at maps out of the circel, not the interval

[barren sleet]

yes

you can compose it

with a one way continous function

but ok

[sleek thicket]

sure, but there are functions from the interval that aren't of this form

[drifting sundial]

wait wdym by this

[sleek thicket]

Paths with different endpoints

[barren sleet]

yeah but there is an extra condition on the image of the boundary of the square I didn't say

[sweet wing]

a homotopy H from f:X->Y to g:X->Y is a map from H:XxI->Y

[drifting sundial]

are you referring to functions S^1 x [0,1] -> X

oh nvm I see

[barren sleet]

ah this is like a general version I see

[drifting sundial]

I forgot what's the term for that

ok ignore that

I was confusing homotopies and loops

[sweet wing]

you actually have a pretty general exact sequence that can include fibrations given by the puppe sequence

[barren sleet]

I don't really know exact sequences well

so I couldn't intuit that rn

[drifting sundial]

I know like the definition but that's basically it

[barren sleet]

same

[sweet wing]

in this case you have ...->π_n(F)->π_n(E)->π_n(B)->π_{n-1}(F)->...

[drifting sundial]

so does that relate to semidirect products?

[sleek thicket]

Sorry people were texting me

[sweet wing]

actually it's kinda more related to like

coverings

if you look at the last few terms

[barren sleet]

well idk covering spaces or exact sequences or homotopy groups

[sleek thicket]

It was about "fundamental group of a fiber bundle"

[sweet wing]

π_1(E)->π_1(B)->π_0(F)->π_0(E)

[drifting sundial]

me neither

[sweet wing]

if E is simply connected

[barren sleet]

is it the product of the fibre and the base space

[sweet wing]

we get π_1(E) is the trivial group

[sleek thicket]

nope

[sweet wing]

and π_0(E) is the singleton

[sleek thicket]

It's almost an extension of groups

[barren sleet]

ya

[sweet wing]

so in some sense the fibre tells you about the fundamental group in this case

[sleek thicket]

But this can fail if the base space has nontrivial second homotopy group

Or the fiber isn't path connected

[barren sleet]

oh

that's weird

I don't like that

ariana where did you learn all this

[sweet wing]

fibre being discrete gives you coverings

from pirating books

[marsh forge]

There is a map that shamrock wrote connecting the pi_n to the pi_n-1 called the connecting homomorphism. If all of the boundary homomorphisms are trivial you get an SES for each n, so in a sense these boundary homomorphisms measure the failure of a fibration to give you SES on homotopy groups

[barren sleet]

which books?

[sweet wing]

rotman is a nice intro

[marsh forge]

Sorry SES=Group extension here

sham you either are the fastest reader I know or have a lot of faith in me

[barren sleet]

is this at

[sweet wing]

then the usual is hatcher but too much geom intuit so im using may/tom dieck

[sweet wing]

yesh

[marsh forge]

Okay i am going to step in with my pedagogy hat

please for the love of god

do not learn topology

from may or tammo tom dieck

[drifting sundial]

I'm heavily leaning towards geom intuition

[sleek thicket]

I was told this by Ethan yesterday

I'm a fast reader

Also seconding max

Really hard

Don't do that to yourself

[marsh forge]

I am one of the biggest fans of the concise series in particular

i've learned from a lot of mays writings

[sweet wing]

like as a second pass should be ok right

[marsh forge]

but it is not a first intro

May yes

tammo maybe

[sweet wing]

if i know how to compute π_1 and H_n for nice stuff

[barren sleet]

ok then max, what is a good first AT book

[marsh forge]

tammo is just off the rails

Hatcher is good

not great

Tammo like

doesn't present things

[barren sleet]

is anything great?

[sleek thicket]

Hatcher (regrettably) is my take too

[marsh forge]

how alg top people actually think about them

[sweet wing]

i went with rotman it was like pretty chill

[drifting sundial]

me and ethan were actually planning to do some of this in the future

[sweet wing]

tom dieck first few chapters is more like

[marsh forge]

its okay everyone once i write my AT textbook

[sweet wing]

homotopy theory stuff

[marsh forge]

there will be another bad choice

tom dieck is just wacky

[drifting sundial]

the plan is like pugh -> top/difftop -> at

[sleek thicket]

> max AT book
> no proofs

[sweet wing]

he really gets close to introducing model categories

[marsh forge]

i don't get his approach or what hes trying to do

hey

[sweet wing]

but doesnt mention the word model category

[marsh forge]

heres the thing sham i actually

probably write more detailed and thorough proofs

than 99% of topology people

[sleek thicket]

This is true

And it's the reason I don't like topology

[marsh forge]

ive been slaving over a set of lecture notes recently because i am trying to fill in horrible missing details

[drifting sundial]

does point set get boring later on

[sleek thicket]

Trying to read geometric topology stuff this summer

[marsh forge]

like progress is so slow bc i stg the original writers don't know why the results they cite are true

and they provide no reference

[sleek thicket]

From knot theory

Was so bad

Just fucking write a proof

[drifting sundial]

what I've been learning so far (munkres ch2) seems quite interesting

[marsh forge]

Hatcher for what it is worth

actually writes

[sleek thicket]

Most people find point set boring wy

[marsh forge]

very good proofs

[sweet wing]

yes point set topology gets very dry later on

haha

[sleek thicket]

But you're fine if you like it

I wasn't talking about that

[drifting sundial]

I guess I havent seen enough of it

[barren sleet]

I'm watching a "beginners" AT series on youtube and it's not very rigorous, so I keep getting these curiosities that I don't know how to figure out

[marsh forge]

May writes good proofs if you already know the results or you really like treating a textbook as one big exercise

[sleek thicket]

I was talking about people not proving things

[marsh forge]

boarbarktree???

[sleek thicket]

You could just have different tastes

[marsh forge]

I wanna meet boarbarktree

[sleek thicket]

Oh they have me blocked on Twitter

Idk why

[barren sleet]

idk what boarbarktree is

[marsh forge]

really

lmfao

[sleek thicket]

Wait

The way I found out

Is you talked with them

A while ago

And I was like "wtf who is max talking with that has me blocked"

[marsh forge]

i will admit that like there are a few harmless people i blocked on twitter

because of personal stuff

like during grad app szn

[sleek thicket]

I have no idea why, we interacted in the past and it seemed perfectly fine. I have 0 feelings towards them

valid

[marsh forge]

i should unblock them

[shut moat]

point set is ok ngl

especially after like the first few chapters

like metrization theorems are kinda satisfying

[sweet wing]

in some sense yes

munkres was kinda written with that in mind i think

[marsh forge]

tbh

you can like

stop reading

[drifting sundial]

it feels almost like black magic at firat

[marsh forge]

after the metrization theorems

lol

[barren sleet]

ok so to go back to the question. if you want to formulate and define the mobius band in terms of it's base space and fibre (S^1 and [0,1]) how do you do that?

[sweet wing]

or even before lol

[sweet wing]

you cant rlly

[shut moat]

what about paracompactness

or the alg top stuff

oh wait, paracompactness is sorta in the metrization theorem section nvm

[sweet wing]

there are many spaces with base space S1 and fibre [0,1]

[sleek thicket]

Fiber bundles can't be constructed out of the base and fiber directly

The same way group extensions can't be

[barren sleet]

there are other algtop books. does. munkres suck for that?

[sleek thicket]

Aren't there 2 lol

[drifting sundial]

is it like the mobius strip with n twists

[sweet wing]

it feels kinda eh other than using it for metricizatioj

[sweet wing]

wait i need to unstone myself ah

[barren sleet]

well I meant not alone, like what information do you need to add

[marsh forge]

Okay this is where sham checks my details, but given a base space B, and a nice cover of B on which you are going to build the total space E, you can specify how the fibers should "transition" on overlaps of the cover and make it trivial everywhere else to build your unique fiber bundle

[shut moat]

my impression is that it's basic prerequisite knowledge for manifold memes

[sweet wing]

true

[barren sleet]

I didn't mean like S^1 and [0,1] uniquely define a fibre bundle

[marsh forge]

So like

[sleek thicket]

Yeah max you need an external topological group acting on the fiber too

To be really precise like

[marsh forge]

yeah thats what i meant by transition

but yes

[sleek thicket]

Otherwise the theory craps itself a little

[sweet wing]

theres 2 that comes to me

[marsh forge]

Anyway ethan the idea is like

you take your fiber

and you specify some symmetries of it with something called a (topological) group action

[sweet wing]

afaik it is homeoprphic to one with n%2 twists

[marsh forge]

and you take your base space

and turn it into a blueprint

and then that data can be shown to uniquely specify a bundle

[barren sleet]

are you serious

[sweet wing]

yes

[sleek thicket]

You can do this with paper

Right?

A two twisted mobius band

Like you can literally construct one of these by gluing

Physically

[barren sleet]

well als9

[sweet wing]

idt theres an isometry in R^2 from one to another

but a homeomorphism

[barren sleet]

physical moving around isn't exactly homeomorphism

[drifting sundial]

R^3?

[sleek thicket]

https://www.quora.com/If-instead-of-one-half-twist-we-use-two-half-twists-when-making-a-mobius-strip-is-it-still-called-a-mobius-strip-even-though-it-is-two-sided-What-are-these-alternate-mobius-strips-called-if-they-are-not-called

[sweet wing]

oop yea

[sleek thicket]

You're right I'm dumb lol

[sleek thicket]

Why not?

[gritty widget]

If-instead-of-one-half-twist-we-use-two-half-twists-when-making-a-mobius-strip-is-it-still-called-a-mobius-strip-even-though-it-is-two-sided-What-are-these-alternate-mobius-strips-called-if-they-are-not-called

[barren sleet]

the left handed mobius band is homeomorphic to the right handed one but with physical intuition they are different

also knots

are all homeomorphic

[sweet wing]

it's kinda like how all knots are the same

[drifting sundial]

basically they're not ambiently homeomorphic

[sleek thicket]

Ah sure, sorry

[barren sleet]

hahaha I was gonna add "just not ambiently

[sleek thicket]

I thought you were saying physical movement won't always be a homeomorphism

[barren sleet]

well

[sleek thicket]

Not that not all homeomorphism arise as ambient ones

[barren sleet]

left and right mobius bands are also ambiently homeomorphix

[sleek thicket]

but yeah I was wrong on that count

[barren sleet]

cant you just reflect R^3 over a plane

[sleek thicket]

The important thing to keep in mind is that I should think before I speak

I'm not sure anymore about my original claim

[sweet wing]

knots are only interesting when you embed in like H^3 and study the complement

[sleek thicket]

That there's only two

But I only know two

[marsh forge]

sham

just google the k group

and interpret

[barren sleet]

I would be surprised to find out that the 2 twisted loop is ambiently homeomorphic to the cylinxer

[sleek thicket]

Yes max, I agree there's two line bundles

My concern is the structure group

We're allowing arbitrary automorphisms of [0,1]

[marsh forge]

I'm confused sorry whats the claim

[sleek thicket]

Not just like reflections across 1/2 or whatever

Classify all locally trivial fiber bundles over S^1 with model fiber [0,1]. The structure group can be all of Homeo([0,1])

if you restrict to like O(1) then what you said applies

[barren sleet]

wait

what's locally trivial

these chats fly by way too quick sormtimes

[marsh forge]

locallly looks like part of the base space times the fiber

[sleek thicket]

it means locally a product, it's just the definition we gave earlier

[barren sleet]

is there a locally nontrivial def?

[sleek thicket]

Sometimes the definition of a fiber bundle includes what's called a structure group

I'm trying to emphasize that I'm not including that data

[barren sleet]

oh ok

[sleek thicket]

Or allowing the largest possible structure group or w/e

Anyways max compute BAut([0,1]) right this second

Or else

[marsh forge]

[sleek thicket]

🔫

[sweet wing]

[sleek thicket]

wait unironically though does my concern make sense to you?

[marsh forge]

$\Omega^{-1}Aut(0,1)$

yes it does

[gentle osprey]

MaxJ

[sleek thicket]

hahaha

[marsh forge]

i think w a large structure group

the claim is false

[sleek thicket]

Yeah, it might be

[marsh forge]

i have no reason for this

but vibes

[sleek thicket]

So like

You can still take a trivialization

On two open sets

S^1 \ pt

Your thing is determined by the gluing data

I think it comes down to something like what is π0 Homeo([0,1])

[barren sleet]

this makes me very excited to learn AT

more excited than difftop by far

[marsh forge]

Good

[barren sleet]

idek anything cool I will find in difftop

[sleek thicket]

blocked

I think it might still be true

[barren sleet]

although I expect it to be cool

[marsh forge]

Admittedly bundles are in the intersection of the two given the important role of tangent and normal bundles in diff top

[barren sleet]

fine it's sexy again

[marsh forge]

Diff top is the subject I will always think "oh thats cool about" but will never seriously work on myself

maybe

who knows

[sleek thicket]

I should learn difftop someday

[marsh forge]

max and sham bott tu arc

[barren sleet]

shit I need to write down when ppl give me book reccomend actions

reccomendations

[marsh forge]

if ONLY

someone would LISTEN

[barren sleet]

what is a good difftop book

[marsh forge]

to my SUGGESTION

[sleek thicket]

lmfao

[marsh forge]

about #books-old

[sleek thicket]

angery max arc

have mods been ignoring you

[barren sleet]

lol max what is your book thing

[marsh forge]

no one seems to like my suggestion

i did the calendar thing after getting ignored

bc i realized

they couldnt stop me

this is a safe place from the mods

except fiona and faye

but they arent narcs

[barren sleet]

I looked at the calendar and I was blocked from seeing it

[sleek thicket]

also dami

[marsh forge]

wait forreal

[sleek thicket]

who is a narc

[barren sleet]

yeah

[marsh forge]

ugh ill fix that

thanks for telling me

why goes google assume i only ever interact with uchicago students

[sleek thicket]

also rip max I am sad that you're moving out of chicago. I think I'm probably going to stay in Seattle this summer bc I've been away for a year and all my friends are there

if you or someone I knew was definitrly going to be at uc I would go

[marsh forge]

try now

[sleek thicket]

But I don't want to be all alone in Chicago for the summer. And like, idk how many other people are going to be there irl

[marsh forge]

Yeah I mean

if peter told me right now

it would be seriously partially online

i'd renew my lease for a little

but it feels like something that could just get canceled

[barren sleet]

difftop book suggestions?

[marsh forge]

Bott Tu is a good follow up to hatcher I am told

[sleek thicket]

Yeah makes sense max

[barren sleet]

so you need AT to do difftop?

[sleek thicket]

And I guess me buying into it would make it less likely to be canceled

but eh I don't want to be isolated again

[barren sleet]

btw is difftop ever called DT

[sleek thicket]

After this hell year

[marsh forge]

Yeah not worth

Stay near friends

is my suggestion

Plus Im def gonna be in CA at this point

for grad

so me being in CA for the summer is nice

[barren sleet]

ca is 😌

not if you are poor though.....

[sleek thicket]

Oh nice!

I'm not going to be in CA :(

which is nice for me

[marsh forge]

i will be making that sick just above the poverty line

[sleek thicket]

Where are you going?

I know you were waiting to talk about it

Idk if you feel okay doing so now

[marsh forge]

I am 95% going to UCSD. I haven't heard from UCLA and honestly given the rumors I'd only begrudgingly accept for career reasons maybe.

[sleek thicket]

Nice!

That's awesome dude

[marsh forge]

Yeah I'm excited it'll be a good time

[sleek thicket]

I know a grad student there but he's in CS

[marsh forge]

I liked all the people I met at the open house and the guy I'd be working with there is great

So Im well set

[sleek thicket]

Nice!!

What field do they work in?

Field of algebraic topology I mean lol

[marsh forge]

So his big thing is that he found a new way to compute homotopy groups of spheres that led to the biggest jump in our understanding of them ever

the ever part is a bit citation needed but im pretty sure its true

[sleek thicket]

Oh wow

you're literally doing the algebraic topology meme

[marsh forge]

So he studies like the intersection of AG with AT

Basically like

[sleek thicket]

"algebraic topology is when you study the homotopy groups of spheres"

[marsh forge]

it turns out that because motivic homotopy groups are more complicated

[sleek thicket]

Right

[marsh forge]

you get more information

and you can pretty easily

forget the motivic parts

it ends up normally being about like killing one well-understood generator

[sleek thicket]

Oh is this why you were like

Looking at A^1 \ { 0 } as the circle?

[marsh forge]

yeah exactly

[sleek thicket]

what do the higher spheres look like?

Like those above dim 2

Oh I guess you just take A^n \ {0}

And that's a model of an odd dim sphere

[marsh forge]

Motivic spheres are bigraded and are just motivic smash products of the two you know

[sleek thicket]

ah okay

[marsh forge]

fun fact

for some reason

smashing by the tate circle

is already invertible

so when you pass to motivic spectra

you only are trying to invert your standard sphere

[tight agate]

someone I know is going to SD too

you know him too

[marsh forge]

oh rly

I only recognized one person at the open house

and i dont think they are going to SD

[tight agate]

oh maybe you dont know him?

[sleek thicket]

harry gindi is transferring actually 😌

[marsh forge]

god

i would fucking

[tight agate]

he was at the chicago reu last year

[sleek thicket]

Wait did you hear from chmonkey

[tight agate]

and did topology

[sleek thicket]

he came to the stacks class chm took last quarter

at the start

Before being snitched on and then told to leave

[marsh forge]

really

thats so funny

[sleek thicket]

yeah haha

Oh and like, vakil was there too iirc

[marsh forge]

its like negi wants to ruin his own career

[gritty widget]

[tight agate]

I might be going

[sleek thicket]

Yes but you've been very clear about remaining anonymous

[gritty widget]

[tight agate]

[marsh forge]

i feel like i would be able to pick brofib out of the ~10 topology adjacent people that will be there

[sleek thicket]

I think I will too

[obtuse meteor]

lel

[marsh forge]

i probably wont be at peters talks this time around unless he does something im excited about

[obtuse meteor]

Max how do I get big brain at topology

pls halp

[tight agate]

should be pretty easy

[obtuse meteor]

I'm gonna try and go to Uchicago reu next year

and max is gonna be gone

🥲

[tight agate]

there's only one other person who matches whatever details I've given out so far

you should be able to very quickly figure out which one spends his time shitposting on discord

[sleek thicket]

Well Faye hopefully peter may is still alive at u chicago

[obtuse meteor]

Lol

[shut moat]

so we just went through a problem on retracts in topology. The problem asks you to conjecture if S^1 is a retract of R^2. So the stackexchange post I found argues that it implies that the fundamental groups of R^2 and S^1 are isomorphic, but isn't that only true of deformation retracts? And also, is there an argument for this that doesn't use algebraic topology?

[sleek thicket]

Yes to the first and "not that I know of" to the second

Approx do you want me to tell you the correct proof using the fundamental groups?

Or do you want to figure it out yourself?

[shut moat]

well I only have a vague qualitative understanding of fundamental groups from skimming when I was bored, so can you show it to me?

[sleek thicket]

I can do better than qualitative understandings

Abstract nonsense :)

[obtuse meteor]

,,,topology and groupoids

[sleek thicket]

I'm not groupoiding!

[obtuse meteor]

as long as you believe in functoriality, pi_1(R^2) = 0, and pi_1(S^1) = Z

this should follow from abstract nonsense

[sleek thicket]

yes, this is the argument I was going to give

but I figured I should like

Define "functor"

[obtuse meteor]

go for it. I was just making fun of yoou

as I like to do

[sleek thicket]

and explain why the equalities you wrote give a contradiction

Alright approx

so

Oh wait Faye I have a minor nitpick about your tweet

I wasn't sure if I should bring it up

[obtuse meteor]

which one?

[sleek thicket]

I think the pseudocircle only has the same weak homotopy type as the circle

I don't think there's a homotopy inverse

[obtuse meteor]

¯_(ツ)_/¯

weak homotopy type is all HoTT people care about anyway

[sleek thicket]

Alright approx

Fundamental groups

We want to rule out the existence of a retraction r : R^2 -> S^1

yeah?

[shut moat]

yeah

[sleek thicket]

So first things first

We want to look at the fundamental group

This is defined relative to a basepoint

yeah?

[shut moat]

right

[sleek thicket]

So pick some b in S^1

maybe we can call it b = (1,0), it doesn't matter

We need three pieces of input from algebraic topology

The first is that the space R^2 is simply connected, and so π1(R^2, b) = 0 for our basepoint b

have you seen this before?

[shut moat]

yeah, you can just use straight line homotopy or something right

[sleek thicket]

Right exactly

Convex sets are contractible

Second input

π1(S^1,b) = Z

You may have seen the "winding number" if you've done any complex analysis

[shut moat]

ironically I've only seen it in physics

[sleek thicket]

still good!

You can assign an integer to each loop, intuitively the net number of times it wraps around the circle (counterclockwise is positive, clockwise is negative)

[shut moat]

right

[sleek thicket]

and it turns out
(1) this is the same for homotopic loops
(2) if it's the same for two loops, the maps are homotopic
(3) the winding number of f(z) = z^n is n
(4) if r is the loop which travels p and then q then w(r) = w(p) + w(q)

I'm being a little fast and loose with the basepoint, but it doesn't matter too much

The important thing is that the winding number gives an isomorphism of π1(S^1,b) with Z

Sound good?

[shut moat]

yeah

[sleek thicket]

So there's only last step we need

The fundamental group is a functor from the category of pointed topological spaces to the category of groups

[shut moat]

oh dear

[sleek thicket]

so this means that for any space X with a basepoint x0, we have a group π1(X, x0) and for any continuous map f : (X, x0) -> (Y, y0) we have a group homomorphism f_* : π1(X, x0) -> π1(Y, y0)

The pair stuff just means f is a continuous map X -> Y and f(x0) = y0

This group homomorphism is easy to describe. Given a loop in X based at x0, apply f to the loop to get a loop in Y based to y0

does that make sense so far?

[shut moat]

yeah, this is nice

[sleek thicket]

Yeah!

This is really important to the fundamental group

It doesn't just give a group for each space

It plays nicely with continuous maps too

But to be a functor we actually need two more conditions

id_* = id and (f°g)_* = f_* ° g_*

does it make sense what these say?

[shut moat]

what does the asterisk mean?

(also oh shit does this mean pullbacks are functors?)

[sleek thicket]

Asterisk is the "push forward a loop" map

[obtuse meteor]

taking tangent bundle is a functor on manifolds yes

and I think pushfoward is the functor acting on maps

(something similar can probably be said for pullback?)

[sleek thicket]

Misread, sorry

[obtuse meteor]

👍

[sleek thicket]

pullback is the cotangent bundle functor

[obtuse meteor]

ah yeah makes sense ^^

[sleek thicket]

Or the differential forms

It's actually much nicer for forms. The tangent bundle is a functor into spaces, but you can't take global sections

Like "vector fields on M" isn't a functor

There's no global pushfoward

[shut moat]

ah ok then yeah this makes sense

[sleek thicket]

But there is a global pullback, so you can talk about the "differential forms on M" functor

[shut moat]

my brain wrinkles every time I come here 🤯

[sleek thicket]

hahaha

okay so

[sleek thicket]

Do these make sense?

[shut moat]

yeah

[sleek thicket]

Do you see why they're true?

[shut moat]

I think so

the trivial loop is homotopic to the constant loop so its image is a point which is the identity element on the other side.

and (foh)*g =f(h(g)) = f*h(g)==f* h* g

[sleek thicket]

Yup

Sorry I had a thing

But I'm back

Wait no

Second explanation is right

[sleek thicket]

But this is wrong

We want to say id_* α = α for any loop α

Right?

The constant loop doesn't factor into it

(ping me when you get back)

[shut moat]

oh whoops, I misinterpreted it

ok yeah I agree

[sleek thicket]

Cool

But it's still clear why this is true, right?

[shut moat]

yeah

[sleek thicket]

I will note, this isn't exactly valid. π1 is homotopy classes of loops, so you need to like take a class of loops α and then take a representative g and then this works

But you get the idea

So why do I care that π1 is a functor?

Well we have this map r : R^2 -> S^1

And r(b) = b, so it gives a map r_* : π1(R^2,b) -> π1(S^1,b)

Make sense?

[shut moat]

yeah

and this is a homomorphism

[sleek thicket]

Right

Is this a problem?

[shut moat]

so it would imply the existance of a homomorphism {e} --> Z, which doesn't make sense

[sleek thicket]

why not?

[shut moat]

wait my reasoning for that was flawed

[sleek thicket]

Yup, there are certainly homomorphisms like that

[shut moat]

I assumed they had to be surjective smh

[sleek thicket]

I mean, there are continuous maps R^2 -> S^1 at least :P

Okay good figure out why I reacted

[shut moat]

there's definitely no surjective map between a one point set and Z

[sleek thicket]

Definitely

Why does r_* have to be surjective?

[shut moat]

wait it does?

ooh because a curve in S^1 as a subset of R^2 maps to that same curve in S^1

because it's a retract

[sleek thicket]

Oh that's a great way to see it!

I was thinking slightly more abstractly, but it's really the same thing

We have the inclusion map ι : S^1 -> R^2 and this satisfies ι(b) = b and r ° ι = id

Right?

[shut moat]

ah

[sleek thicket]

So $r_* \circ \iota_* = (r \circ \iota)* = id* = id$

[gentle osprey]

shamgry rock

[sleek thicket]

And a map with a right inverse is surjective

But what you said is also totally valid

[shut moat]

that's a fairly simple proof!

[sleek thicket]

Yeah!!!

Three takeaways:

The fundamental group is really powerful

algebraic topology is really powerful

Functorality is really powerful

[shut moat]

until it tries to distinguish between a circle and a non hausdorff space with 4 points

[sleek thicket]

hahaha

[shut moat]

but yeah my interest in learning more about category is increasing

seems really interesting

[sleek thicket]

That's because they're the same space

[gritty widget]

shamrock corrupts another

[shut moat]

speaking of alg top I should probably start learning about group theory so I know what's going on when we start doing fundamental groups in a few weeks

[gritty widget]

just pick it up as u go

[sleek thicket]

Tterra corrupts another

[shut moat]

i've considered that but I think I'd appreciate its power more if i have a better handle on group theory as a whole

[sleek thicket]

You don't actually need that much group theory to use it in algebraic topology. I think it's still worth learning though, and it will randomly be important sometimes

I guess, the parts of group theory that an intro group theory course focuses on are largely not the parts important in algebraic topology

[shut moat]

oh wtf

[gritty widget]

u don't need the shitty sylow stuff to do fundamental groups so that already rules out 90% of an intro group theory class

[sleek thicket]

I agree, except that sylow stuff rocks

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i didn't like it in intro GT

but my intro GT also sucked balls

[sleek thicket]

I think people pretty commonly dislike the sylow/finite group counting stuff

I have weird taste

[gritty widget]

Maybe I liked it from a puzzle perspective, but it doesn't seem very relevant to anything

[sleek thicket]

It's relevant to finite group theory