/sci/ - Science & Math

[Anonymous No.11914020]

Hey everyone, I've been doing a study group with a few lads and we've decided to start studying algebraic topology (using Hatcher's book). We're all bored with nothing to do for the rest of summer, so anyone is welcome to join in and study. This week we plan to review some basic abstract algebra (group theory) so that everyone is on the same page when we start.

[Anonymous No.11914022]

oh yeah here's the discord link for the server we're using discord gg/ka3krwY

[Anonymous No.11914024]

currently in voice chat

[Anonymous No.11914120]

>>11914020
>sci working together
HAPPENING CONFIRMED

[Anonymous No.11914131]

Awful book. I got filtered hard by it.

[Anonymous No.11914134]

hatcher is a good book. Good idea reviewing before hand anon

[Anonymous No.11914140]

>>11914131
It will be better with frens to help you get comfy desu

[Anonymous No.11914147]

Topologists don't actually think in terms of proofs, that's a common misconception. Handwavy techniques are all that it it takes. For example, in topology to prove S^n and S^m are not homeomorphic it's enough to say that an embedding of it into R^k separates it into two components only for the appropriate k. You don't actually need to prove it because it's intuitively obvious. We've localized the topological nastiness from of the problem into an extremely particular and uninteresting situation. Such is the goal of topology

[Anonymous No.11914314]

[Anonymous No.11914340]

>>11914147
kek

[Anonymous No.11914375]

>>11914022
What are the people talking about? Is anyone welcome??

[Anonymous No.11914404]

What are the prerequisites for reading that book?

[Anonymous No.11914459]

>>11914375
Its a copypasta, but anyone can join
>>11914404
Basic familiarity with topology. We're reviewing abstract algebra this week so everyone is on the same page

[Anonymous No.11915098]

Hatcher is watered down for brainlets.

[Anonymous No.11915314]

Guys do you plan reading it cover to cover? Are you going to do all exercises?

[Anonymous No.11915587]

>>11914404
solid understanding of the basics of topology and group theory

[Anonymous No.11915598]

>>11915587
Any recs for group theory?

[Anonymous No.11915620]

>>11915314
Yes. We will do as many as we can.

>>11915598
Just know the basics: abelian/cyclic groups, homomorphisms, normal subgroups etc. Reference Artin or D&F and knowing something about free groups could help.

[Anonymous No.11915621]

[Anonymous No.11915627]

>discord
query me when there's an IRC channel

[Anonymous No.11915635]

>>11915627
Born in the 70's?

[Anonymous No.11915641]

>>11915627
>oh no the jews will steal the problem solutions I send to my math bros

[Anonymous No.11915650]

>>11914147
Low-dimensional topology doesn't even feel like math anymore. I enjoyed alg top when I took it, but I could never be a topologist solely due to not being philosophically comfortable with what passes as a proof there.

[Anonymous No.11915659]

>>11915650
give example of a proof which you find unsatisfactory (also you responded to a troll)

[Anonymous No.11915701]

>>11915659
Any proof in Saveliev where he avoids rigorous explanation by telling the reader to use their imagination.

[Anonymous No.11915735]

>>11915701
book is literally called "topology illustrated", does it actually claim to be a fully rigorous introduction?

[Anonymous No.11915746]

>>11915098
We'll probably do something more rigorous after Hatcher. There's also a complex analysis study group but they're mid book

[Anonymous No.11915763]

>>11915650
>>11915659
>>11915701
>>11915735
>>11915746
These really read like posts written by people who don't really enjoy topology much.
Here's the secret: no topologist gives a single shit about rigor and you fucking morons belong in algebra.

[Anonymous No.11915780]

>>11915763
>proposing something more rigorous after Hatcher = doesn't enjoy topology

[Anonymous No.11915837]

>>11914147
>>11915701
Brainlet who doesn’t understand the power of proof by visualization.
Also there are a good amount of topology proofs that boil down to
>write down the exact sequence
>simplify terms
Which is nothing but rigorous.

[Anonymous No.11915969]

>>11915627
not gonna happen, all the meetings happen in voice chat.

[Anonymous No.11915974]

>>11915650
But that's so much nicer than the alternative because you can actually get the idea behind the proof immediately. With most topology you literally can just visualise why things work.

[Anonymous No.11915993]

>>11915974
>With most topology you literally can just visualise why things work.
I really wish this was true

[Anonymous No.11916837]

>>11915969
Does everybody yell "nigger" in the group chat?

[Anonymous No.11916972]

>>11916837
nope, its usually pretty chill. no politicals talk.

[Anonymous No.11917483]

>>11916972
Yeah, its pretty math focused but not in some faggot plebbitor way. It won't be r/math

[Anonymous No.11918987]

>>11914020
Bump

[Anonymous No.11919614]

[Anonymous No.11920337]

>>11914020
Bump

[Anonymous No.11920751]

>>11919614
Based and kacpiSELECT ALL THE IMAGES WITH CARS

[Anonymous No.11922792]

>>11914020
bumpity bump good idea

[Anonymous No.11922813]

>>11915837
>proof by visualization

[Anonymous No.11922836]

>>11922813
>proof by assigning as exercise

[Anonymous No.11922856]

>>11914020
pure math is for fkin retards what the fuk is ur problem lmaoooooo

[Anonymous No.11922857]

>>11922836
>the proof is obvious and intuitive

[Anonymous No.11922862]

>>11922856
>coomer engineer cope

[Anonymous No.11922867]

>>11922862
im a computer scientist

[Anonymous No.11922903]

>>11922857
>>11922862
>this book has been left as an exercise for the reader to write and publish

[Anonymous No.11922907]

>>11922903
>I leave it to you, dear humble reader, to expand this statement into an entirely brand new field of mathematics

[Anonymous No.11922965]

>>11922907
>Graduate Texts in Mathematics
>A Book Left as an Exercise and its Applications

[Anonymous No.11923430]

>>11914020
bamp eet

[Anonymous No.11924613]

>>11923430
Don't samefag

[Anonymous No.11924690]

>>11914147
I'm so fucking mad. I saw this 3 days ago when I was banned for whatever dumb reason. And good lord, it made me angry.
I posted that. Must have been MONTHS ago. And you saved it to post here. You fucking ingrate. You people ridiculed me for this post. Conveniently, you've left out the reply I made in which I stated the OBVIOUS FACT to ANYONE WITH A BASE LEVEL OF TOPOLOGY BACKGROUND that my proof of S^m not homeo to S^n is not necessarily a proof, but is a translation of the proof by computing sphere cohomology groups via the mayer vietoris sequence into an intuitive, geometric language. It's the same shit. Just because I didn't write down the fucking chain complexes doesn't make the essence of the proof any different.
The fact that you saved my post just to repost it here later proves definitively that you know I'm right and that you have no arguments against my points.

[Anonymous No.11924699]

>>11915763
I'm convinced that this is copied from another one of my posts in /mg/. Could you do me a favor and suck my fucking dick? Piece of ungrateful shit.

[Anonymous No.11924708]

>>11915746
While I'm here ranting about that guy who thinks it's funny that I understand the spirit and soul of topology better than he does, I highly, highly recommend Fuchs-Fomenko Homotopical Topology as your followup book. One of the best mathematics texts ever written.

[Anonymous No.11924835]

>>11924708
We'll be sure to check it out anon

[Anonymous No.11924951]

>>11924690
Lol

[Anonymous No.11924957]

>>11924690

[Anonymous No.11925101]

>>11924951
>>11924957
Based
>>11924690
Topoloshit cope. No wonder physicist are starting to use you like retards if you can handwave everything lol.

[Anonymous No.11925178]

>>11925101
Not an argument. It's not handwaving, it's providing a concise intuition for a proof off of which a correct proof may be formulated given the correct machinery.

[Anonymous No.11925188]

>>11925178
>its not handwaving
>its just handwaving

[Anonymous No.11925195]

>>11925101
>>11925188
proof sketches are superior where the actual rigorous techniques are well understood.
most topology is diagram chasing. the sketch just tells you which diagrams to chase

[Anonymous No.11925200]

>>11924690
You don’t know how to ban evade?

[Anonymous No.11925203]

>>11925188
>The braindead algebraist keeps responding to me
Imagine thinking I'm the one who's "coping." What would I even be coping with? The concision of my proofs?

[Anonymous No.11925204]

>>11924690
Its one of the funniest things I've ever seen on this board. bless your heart anon

[Anonymous No.11925207]

>>11925200
Why would I ever willingly steal 3 relaxing days of freedom from myself like that?

[Anonymous No.11925246]

>>11925203
i was a different anon
but feel free to thank any of us for fixing your shit field

[Anonymous No.11925711]

[Anonymous No.11925865]

>>11925711
faggot piece of shit lier

[Anonymous No.11925883]

>>11924690
>For example, in topology to prove S^n and S^m are not homeomorphic it's enough to say that an embedding of it into R^k separates it into two components only for the appropriate k. You don't actually need to prove it because it's intuitively obvious.
So did you actually mean this or not?

[Anonymous No.11926028]

>>11924690
This really reads like a post written by someone who doesn't really enjoy topology much.
Here's the secret: no topologist gives a single shit about rigor and you fucking morons belong in algebra.

[Anonymous No.11926038]

>>11925883
Of course "enough" and "don't actually need" are hyperboles. What I mean is,
>For example, in topology to prove S^n and S^m are not homeomorphic the proof is in essence to say that an embedding of it into R^k separates R^k into two components only for the appropriate k. You can provide a formalization of this basic argument, but no one cares about that in topology, what they care about are the geometric consequences of that formal machinery.
My point being that if you still need to think about diagrams like >>11915650 in terms of some fucked up homotopy group bullshit, then you are still an undergrad who overvalues rigor.

[Anonymous No.11926240]

>>11926038
S^n separating R^k is highly non-trivial. if anyone takes it for granted, it's because it has been already proved and is now a part of "general knowledge", definitely not because it's intuitively obvious.

[Anonymous No.11926268]

>>11926240
Whether or not the proof's technicalities are messy, the idea is as simple as "draw a transverse line with nontrivial intersection, and segments of this line alternate between lying inside and outside of S^n." Sure, checking that these things are two different connected components is nasty. But the point of topology is that one relegates nasty things like that to elementary theorems and lemmas so that one may focus on the geometric picture.

[Anonymous No.11926298]

>>11926268
nobody just handwaves the technicalities away. they still need to be checked and topologists absolutely do that.

[Anonymous No.11926372]

[Anonymous No.11926404]

>>11926298
No, undergraduates and first year graduate students do that in an introductory topology sequence, and then topologists know these facts when they're actually working and use them.

[Anonymous No.11926439]

>>11926404
>my undergrad topology course was handwavy therefore topology is handwavy
?

[Anonymous No.11926506]

>>11926404
What book did you use? Some stupid Dover 20-page beta pamphlet?

[Anonymous No.11926596]

>>11926404
????
Would you like me to show you some of the rigorous solutions to my point set and algebraic topology courses? Even my most handwavy proofs (mostly on identifying compactified spaces with homeomorphisms) were rigorous aside from a single tedious detail like showing an obviously homeomorphism was homeomorphic.

Being terse is good

[Anonymous No.11927111]

>>11926268
>draw a transverse line with nontrivial intersection
what the fuck does that mean?

[Anonymous No.11927432]

>>11927111
draw a line that is transverse with a nonzero intersection

[Anonymous No.11927500]

>>11915620
Free groups are also important. Modules over rings will be useful later also.

[Anonymous No.11927830]

>>11927111
It means precisely what it says? Do you not know what transverse means?

[Anonymous No.11927915]

>>11927432
>>11927830
If it is transverse it already has a "nontrivial" intersection. He could have just said consider a fucking line going through an n-sphere. Can't even handwave properly lol.

[Anonymous No.11927924]

>>11927915
>If it is transverse it already has a "nontrivial" intersection
t. doesn't understand transversality

[Anonymous No.11927929]

>>11927924
Ok then, what is transversality?

[Anonymous No.11927934]

>>11926268
>calls others undergrads
>thinks drawing a picture in R^3 and claiming it applies to all R^n is valid reasoning

[Anonymous No.11927954]

>>11927929
f and g are transverse if at every point of intersection df and dg span whole tangent space of the ambient space. if they're disjoint, then they're automatically transverse. and this is actually *super* important, it's not like it's just some minor technical case.

[Anonymous No.11927967]

>>11927954
>muh vaccuous truth
Lmao if proper mathematicians usually don't care about that autistic shit, topoloshits would laugh at you.

[Anonymous No.11927984]

>>11927967
why are you pretending that you know what you're talking about when clearly you don't ? you think it's hard to tell ?

[Anonymous No.11927988]

>>11927984
The empty set cases is a pedantic wank made by autistic mathematicians so that they can define concepts without stating
>suppose [math]X[/math] is non empty
No one cares.

[Anonymous No.11927991]

>>11927988
why are you pretending that you know what you're talking about when clearly you don't ? you think it's hard to tell ?

[Anonymous No.11927992]

>>11927991
yes

[Anonymous No.11928000]

>>11927915
dude what the fuck are you talking about? nonintersecting manifolds are transverse. this is an extremely important fact.
>>11927991
is it not surprising that I, who made the original post months ago, am clearly the rational and knowledgeable one here?

[Anonymous No.11928003]

>>11927934
it's not a picture in R^3, what are you on about?
why do you people have no fucking clue what you're talking about?
you don't see why the same picture works in, say, R^4?

[Anonymous No.11928015]

>>11927992
do you know what the transversality theorem is ?

[Anonymous No.11928023]

>>11928000
>Transversality is a notion about how interesection
>Dude the case when this intersection is empty is acutally very important
No it is not, vaccuous truths are usefull to have elegant self contained definitions, but tell me ANY particular theorem or property that uses transversality that for the case when the intersection is empty is not retardedly trivial

[Anonymous No.11928027]

>>11928003
>it's not a picture in R^3, what are you on about?
you're right, technically it's a picture _in_ R^2, but it's depicting a situation in 3-dimensional space.
>you don't see why the same picture works in, say, R^4?
No, I don't, because I can't "see" R^4. Neither can you.
This kind of reasoning is not even handwaving, it's just bullshit. Can I draw a Hopf map and say "look at the picture can't you see it works in R^4 too"?

[Anonymous No.11928028]

>>11928023
>but tell me ANY particular theorem or property that uses transversality that for the case when the intersection is empty is not retardedly trivial
transversality theorem

[Anonymous No.11928043]

>>11928023
??? take two embedded one-manifolds in R^3, almost every translation of one of them doesn't intersect the other
this is nontrivial without a simple form of thom transversality theorem

[Anonymous No.11928054]

>>11928027
yes, i can obviously see how a hopf map works in R^4. i can't "see" it visually, but i have an intuition for how it works on a geometric level.
you can't be very good at geometry, can you?

[Anonymous No.11928058]

>>11928023
okay you want to know why is it important? because when you want to prove that a property P of smooth maps is generic (so for example any map is homotopic to a map satisfying P), 9 out of 10 you do it like this: express (not P) as lying in or intersecting some manifold or stratifold S with high enough codimension, and then use transversality theorem. this says that almost every map is transverse to S => almost every map is disjoint with S => almost every map satisfies P. this is extremely useful technique.

[Anonymous No.11928065]

>Claims I'm wrong about what topologists do
>Doesn't even understand what topologists do

[Anonymous No.11928074]

>>11928054
>yes, i can obviously see how a hopf map works in R^4
it's quite impressive that you can see something that doesn't exist

[Anonymous No.11928085]

>>11928074
What are you on about? We're talking about a map from S^3 to S^2, no? You can't understand this map intuitively from the corresponding fiber bundle it's the projection for?

[Anonymous No.11928101]

>>11928085
>We're talking about a map from S^3 to S^2, no?
Yes. Now since the picture you drew of the map from S^3 to S^2 clearly applies in all dimensions, please show me the analogous map from S^4 to S^3.

[Anonymous No.11928114]

>undergrads ITT thinking everything in topology can be understood geometrically
You guys are in for a rude awakening

[Anonymous No.11928176]

>>11928028
The proof for the case when trasnversality is satisfied because the intersection is empty is trivial what is your point?
>>11928043
>Almost every translation of one of them doesn't intersect the other
And? It makes the proof more elegant, but it doesn't change the fact that the proof is equally easy if you define transversality by considering no empty intersection, you just need an extra step that is trivial.
>>11928058
That you can always separte it in cases for empty and non empty intersection. And the case of empty intersection is trivial. Yes, it makes everything much more elegant but jesus christ it is not fundamental in the definition to prove waht you are saying.

[Anonymous No.11928318]

>>11928176
so what is your proposed definition of transversality ?

[Anonymous No.11928375]

>>11928318
Jesus christ, the point is that if you are trying to explain transversality with your hands you are going to think of examples when the intersección is non trivial and so it was just weird to say
>draw a transverse line with nontrivial intersection
for a visual proof. The point then was that really it doesn't matter if you define transversality like this >>11927954 AND also asking the intersection to be non empty. Which again for rigorous proofs it just makes things less elegant, but the point that you never actually need to separate the proof into two cases is the reason why it will always be trivial when the definition is satified vacuously.

[Anonymous No.11928401]

>>11928375
it's not about elegance. the point is that "non-empty transversality" and "disjointness" are not generic properties, but if you put them together you get something generic. this clearly indicates that the separation is artificial on a fundamental level.

[Anonymous No.11928446]

>>11928101
There is no such analogous map. What is your point? The hopf map is not the same thing as a transversality theorem. The hopf map is not special because it deals with compact hypersurfaces in R^n, it's special because it deals specifically with S^3.

[Anonymous No.11928453]

>>11928401
Not really? If you want to define transverality as a way to formalize the intuive notion of two objects intersecting trasnverse, the it is important to consider that they intersect. Then you can perfectly call the more general definition "extended transversality" or whatever and explain exactly what you say how extended transversality is a generic property. Yes, naming shit is arbitrary in the end, it just stiked me as odd that if you already are going to do a non rigorous explanation of something, i.e. handwave that you need to put in that detail.

[Anonymous No.11928454]

>>11928375
keep in mind that you can draw any transverse line, nontrivial intersection or disjoint, and you still get the alternating in and out property.

[Anonymous No.11928459]

>>11928453
>If you want to define transverality as a way to formalize the intuive notion of two objects intersecting trasnverse, the it is important to consider that they intersect.
no
stop
the intuition for transversal manifolds is "WHEN they intersect, they do so transversally" not "they intersect and they do so transversally"
you just have the proper intuiton wrong. both ideas are equally intuitive.

[Anonymous No.11928460]

>>11928453
>If you want to define transverality as a way to formalize the intuive notion of two objects intersecting trasnverse
transversality formalizes the intuitive notion of two objects being in a general position, not what you have written.

[Anonymous No.11928472]

>>11914020
Wildberger has some lectures if you're interested https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8

[Anonymous No.11928490]

>>11928446
>There is no such analogous map. What is your point?
That objects do not behave the same in all dimensions, and that it's completely invalid, retarded reasoning to draw/visualize a picture of a <=3D case and insist that it works for all dimensions because you can see the picture.

[Anonymous No.11928499]

>>11928459
>>11928460
Are you autistic? The word "transverse" in geometry has a meaning that you learn in primary school and it has to do with how a line crosses another fucking line. Yes I understand that you can say "now what WE mean by transversality is this in the context of differential topology" and there is a perfectly good reason for doing so, but you haven't explain in what fucking way a line not crossing the thing is transverse in the usual sense. I fucking hate the inflexibility and autism with terms people get when learning math. People use the term transverse like that in every day life when explaining geometry ffs.

[Anonymous No.11928554]

>>11928499
Transversality means something different than you think, get over it. It's not like it was me who invented the terminology.
>you haven't explain in what fucking way a line not crossing the thing is transverse in the usual sense
I did. It's the general position.

[Anonymous No.11928572]

>>11928499
>The word "transverse" in geometry has a meaning that you learn in primary school and it has to do with how a line crosses another fucking line.
No, it has to do with how a line crosses TWO other lines. Those two other lines are perfectly allowed to be parallel.

[Anonymous No.11928593]

>>11928554
I'm note denying the terminology exists you fucking retard, I'm just stating the obvious fact that the word transverse has more than one meaning. Draw three lines, the first one however you want, the second one paralel to the first one but disjoint from it and the third one not paralel to the first. If someone asks you to say which of the lines are transverse to the first one you are going to sound like a fucking retard if you also chose the paralel line.
>>11928572
Yes a transversal line is defined to be a line that crosses two other lines at two distincts point, never in your line have you heard that a line is "transverse" to another fucking line meaning it crosses it? The street that is transverse to another street?
https://www.merriam-webster.com/dictionary/transverse
Not saying there is any controversy with the definitio of "transversality" in differential topology, but do you understand that sometimes mathematicians use the same word even if they mean different things right? The autism part comes when they don't acknowledge this and correct people for trivialites.

[Anonymous No.11928635]

glad to see all these fine transverse gentlemen enjoying the thread. Feel free to hop in the discord if you'd like to join the study group!

[Anonymous No.11928677]

>>11928593
>never in your line have you heard that a line is "transverse" to another fucking line meaning it crosses it? The street that is transverse to another street?
Why are you giving examples of colloquial English usage? That's completely irrelevant. You posted an incorrect planar geometry definition, I responded to it. Please try to focus.
>If someone asks you to say which of the lines are transverse to the first one
Why do you write in one sentence that the definition of a transversal requires you to draw _three_ lines and then you proceed to immediately ask "which line is transverse to this line?". This is meaningless. The planar geometry definition of a transverse line _requires three lines_. Questions about "transverse lines" involving two lines in isolation are syntactic garbage from definition you've given.

[Anonymous No.11928680]

>>11928490
but where the fuck did i draw a picture of a <= 3d case

[Anonymous No.11928686]

>>11928635
We're not transverse apparently, since we're clearly disjoint. This is how math works now.

[Anonymous No.11928694]

>>11928635
And no, I'm not interested in fapping to furry porn with schizos on this website under the guise of a "study group."

[Anonymous No.11928744]

>>11928677
>Why are you giving examples of colloquial English usage?
Because this whole retarded dicussion came from >>11926268 saying draw a transverse line with nontrivial intersection. Using the term "draw" clearly shows that he is not wrtiting a rigorous argument and usuing colloquial english so yes you could also say "a line transverse to the sphere" and it fucking obvious what he means.
>Why do you write in one sentence that the definition of a transversal requires you to draw _three_ lines and then you proceed to immediately ask "which line is transverse to this line?". This is meaningless. The planar geometry definition of a transverse line _requires three lines_. Questions about "transverse lines" involving two lines in isolation are syntactic garbage from definition you've given.
Pure autism. Being purposely dense doesn't make you a good mathematician. Given a line in the euclidean plane what do you call an element of the set of all lines that aren't parallel to the given line? Maybe you call it the complement of the set of parallel lines, but anyone who has graduated primary will tell you it is a a line transverse to the original line.

[Anonymous No.11928754]

>>11928694
This study group has completed the first 3 chapters of munkres and there have been no problems.

[Anonymous No.11928756]

>>11928754
>munkres
meme book

[Anonymous No.11928765]

>>11928756
Okay, if you'd like to recommend what you see as the "big dick" Point Set topology book youre welcome to if it helps you cope with the fact the "schizos" were more productive than you were this entire summer.

[Anonymous No.11928829]

>>11928754
I don't need to study basic point set topology, thanks.

[Anonymous No.11928862]

>>11928744
>using terms as they've been defined is autism
I don't think mathematics is for you, buddy. Maybe think about a different major.

[Anonymous No.11929669]

>>11928744
dude, you're the one who called out another anon for misusing a concept from topology in a topology thread here >>11927915. you've made assumptions about the concept based on a fucking high school knowledge and argued about which cases are trivial, whether the definition is only for elegance, what does it formalize, how would one go about explaining it with their hands etc. while obviously not knowing anything about it. you've been explained several times why you're wrong, and now you're trying to save your ass with "in colloquial use it means something different so technically I'm not wrong and you're a bunch of autists". I think the autistic retard title goes to you today.

[Anonymous No.11929753]

>>11928756
Name a better book for point set topology

>>11928829
We went through it pretty fast. It was just a review really. We are about to start Hatcher.

[Anonymous No.11929813]

>>11914020
>algebraic topology
what are prereqs?

[Anonymous No.11929816]

>>11929813
addition and multiplication

[Anonymous No.11929818]

>>11929816
Damn, exactly the two things I'm bad at.

[Anonymous No.11929824]

>>11929813
Modern Algebra and Point Set Topology

Algebra: Group Theory (Including Free Groups); Module Theory

Topology: Topological Spaces, Continuity, Interior/Boundary/Limit Points, Product/Quotient Topology

I probably missed some stuff but that should give you an idea. You don't need to know all that either; you can learn as we go along and we are still doing some preliminary review.

[Anonymous No.11929836]

>>11929813
the main prereq is enough mathematical maturity to not get stumped by geometrical arguments, mistaking them for being nonrigorous. besides that, just basic point set topology and basic group theory (AT is heavy on commutative algebra but it can be picked up as you go).

[Anonymous No.11930189]

>>11929836
>mathematical maturity
How do you know when you're "mathematically mature"?

[Anonymous No.11930192]

>>11930189
You don't, lmao.
Mathematical maturity is a meme.

[Anonymous No.11930337]

Ok so can anyone explain to me how the proof using transversality actually works?

[Anonymous No.11930461]

>>11930192
>Mathematical maturity is a meme.
GUYS MATHEMATICAL MATURITY IS A MEME I CAN DO LANG ALGEBRA RIGHT NOW GUYS

[Anonymous No.11930884]

>>11930337
Anyone?

[Anonymous No.11930901]

>>11930884
stop caring about proofs u fuken retard lol just use the formula or a library what the fuk i learned math in liek 2 weeks

[Anonymous No.11930916]

>>11930901
I mean the anon is asserting it can be proven using transversality. I'm genuinely interested in how to do it.
The way I see it there must be two parts of the proof:
1. An embedding S^n -> R^(n+1) separates R^ (n+1) into two components.
2. An embedding S^k -> R^n has connected complement for k<n-1.
From the replies ITT it's not at all clear how to prove either of these parts.

[Anonymous No.11930964]

>>11930337
>>11930916
i did not state it can be proven using transversality. i merely said the first step is to take a transverse line through the manifold (which you can't do unless it's S^n -> R^(n + 1)). this transverse line picks out two connected components. that needs to be proved, but from there the proof is just this techinical check.

[Anonymous No.11930965]

>>11930461
Lang is a meme.

[Anonymous No.11930972]

>>11930964
>transverse line picks out two connected components
How would you prove this?
Btw, we can move this to
>>11930966

[Anonymous No.11931010]

>>11930189
When you can read a book on a given topic and say, "hey, doesn't it related to [other dumb shit]?"
When you can predict the techniques used in a proof or do the intermediates steps yourself.
When you can actually understand why the proof is trivial and you've got no problem doing it yourself.
When you can easily convert a geometric argument to one using explicit functions and constructions (and the reverse).

And most importantly, when you're not spooked by symbols and you can keep a clear head of a complicated proof, writing out the essential idea in a sentence and understanding what it shows about the structure you study.

[Anonymous No.11931035]

>>11930972
Please don't.

[Anonymous No.11931566]

>>11924690
kek, topolofag seething

[Anonymous No.11931571]

>>11930972
stay in your containment thread discord faggot

[Anonymous No.11931729]

>>11930972
you still have questions ?

[Anonymous No.11931733]

>>11931729
Yes. I want to know how the proof works or at least for someone to point to where I can find the proof myself.

[Anonymous No.11931760]

>>11931733
it's a special case of Jordan-Brouwer separation theorem: any smooth closed connected submanifold [math]M \subseteq \mathbb{R}^{n+1}[/math] of codimension 1 separates into two connected components. this is how the proof goes.

1. there are at most two components
the crucial fact is that every SMOOTH submanifold has a nice neighborhood called tubular neighborhood. in our case it's diffeomorphic to the product [math]M \times [-1,1][/math]. this is absolutely not true for non-smooth submanifolds. now that you know what does it look like around [math]M[/math], you easily prove that any point in [math]\mathbb{R}^{n+1} \setminus M[/math] can be joined to a point in [math]U \setminus M[/math]. it follows that number of components of [math]\mathbb{R}^{n+1} \setminus M[/math] is not larger than that of [math]U \setminus M = [-1,0) \cup (0,1] \times M[/math] which is two.

[Anonymous No.11931794]

>>11931733
>>11931760
2. there are at least two components
this uses for example the winding number. for a point [math]p \in \mathbb{R}^{n+1}\setminus M[/math], the integer [math]W(p,M)[/math] measures how many times does [math]M[/math] wind around [math]p[/math]. it's a generalization of that thing from complex analysis. you need two things:

A. [math]W(p,M)[/math] is locally constant in [math]p[/math]. It follows that points with distinct winding numbers lie in distinct components of [math]\mathbb{R}^{n+1} \setminus M[/math].

B. you can compute it as follows. draw a ray [math]R[/math] emanating from [math]p[/math] which is transverse to [math]M[/math]. [math]W(p,M)[/math] is the signed intersection number of [math]R[/math] with [math]M[/math].

by A. we just need to find two points with distinct winding numbers. [math]M[/math] is compact, so if you pick [math]p[/math] very far from it, [math]W(p,M) = 0[/math] because you can find an arc which doesn't intersect [math]M[/math] at all. now we need a point with [math]W(q,M) = 1[/math]. since [math]M[/math] is compact, there is a point with the [math]x_1[/math]-coordinate minimal (let's say). pick [math]q[/math] to be slightly to the right from that point and pick [math]R[/math] to be the negative [math]x_1[/math]-direction. [math]R[/math] now crosses [math]M[/math] exactly once, so [math]W(q,M) = \pm 1[/math].

[Anonymous No.11931970]

>>11931760
>>11931794
Excellent post, thank you for this.
I'm sure you'll get tons of stupid questions about whether or not steps in your proof sketch can be formalized from the idiot you're responding to.
Also, note importantly the fact that transversal might mean disjoint.

[Anonymous No.11933027]

bump

[Anonymous No.11933291]

[Anonymous No.11934804]

>>11933027
I think we have enough people in the Discord now, I'd prefer a limited number of potential shitposter

[Anonymous No.11935361]

Lmao what the fuck algebraic cucks, the simplest reason a k-sphere is not homeomeorphic to an n-sphere is because open sets of different dimensions cannot be homeomorphic. becuase that is inuitively obvious

[Anonymous No.11936441]

>>11935361
The simplest reason is that they have different homology.
>open sets of different dimensions cannot be homeomorphic. becuase that is inuitively obvious
That's true and intuitive, but it absolutely requires a proof (and the standard proof of this actually requires knowledge of homology of spheres)

[Anonymous No.11936561]

>>11915627
What about a Telegram?

[Anonymous No.11937086]

>>11936441
Shitposting aside you could prove it with calculus really, well I maybe some measure theory to be perfectly rigorous. Just embbed the ball of lower dimensiion into the higher dimension and prove that the lower dimensiónal one must have volume 0 and all open sets non empty sets have positive lebsegue measure. Yea is another way of counting "dimensions" and more convoluted, but more people learn about the lebesgue measure than the homology of spheres lol.

[Anonymous No.11937265]

>>11914131
It's the best intro to algebraic topology; you're just a brainlet.

[Anonymous No.11937591]

>>11937086
Invariance of domain/invariance of dimension is easy for smooth maps, that's just inverse function theorem. Continuous maps behave way worse, for example image of an open ball in a larger space can absolutely have non-zero measure. I'm not saying it's not possible to do it only with analysis, but I doubt it's simpler than the proof using homology (which is now standard). According to Wikipedia, even the first proof by Brouwer used algebraic topology.

[Anonymous No.11937598]

>>11935361
this is much less of a proof than my proof was
they're in bijection, why can't they be homeomorphic?

[Anonymous No.11937603]

>>11937086
>mixing measure theory and topology

[Anonymous No.11937635]

>>11937598
That's a theorem, invariance of domain

[Anonymous No.11937646]

>>11937635
well, yes, but it's not an easy theorem to prove

[Anonymous No.11938010]

>>11914020
is the invite still working?