#point-set-topology | Mathematics | Page 214

sleek thicket Mar 18, 2021, 5:20 AM

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nice!

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All forms on S^n are closed, so we just need to find a form with nontrivial integral

gritty widget Mar 18, 2021, 5:21 AM

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🧢

sleek thicket Mar 18, 2021, 5:21 AM

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You can take the volume form

shut moat Mar 18, 2021, 5:21 AM

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ty terra

sleek thicket Mar 18, 2021, 5:21 AM

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The volume form of a submanifold can be computed using a unit normal vector

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Although ig F isn't exactly the normal vector? Weird

shut moat Mar 18, 2021, 5:22 AM

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it's a solid angle

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so its integral over anything that encloses the sphere will be 4pi (if it doesn't do any funny stuff)

gritty widget Mar 18, 2021, 5:23 AM

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solid angle monkaS

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reminds me of that spivak exercise

sleek thicket Mar 18, 2021, 5:23 AM

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@dmashura

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Okay I remembered correctly

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But now I see

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Your F is just a nonzero function times the volume form

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errr

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I think I'm still confused but also need to do analysis sadcat

gritty widget Mar 18, 2021, 5:27 AM

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a lot of the stuff lee proves in ISM about riemannian manifolds are exercises in IRM chapter 2

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hmmm

sleek thicket Mar 18, 2021, 5:27 AM

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lmao hi gomez

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Wish me luck on my analysis final tomorrow

gritty widget Mar 18, 2021, 5:27 AM

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good luck

nimble jolt Mar 18, 2021, 5:27 AM

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https://tenor.com/view/cat-hi-hello-bye-gif-15388121

Tenor

gritty widget Mar 18, 2021, 5:27 AM

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remember triangle inequality

shut moat Mar 18, 2021, 5:28 AM

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you got this pandaHugg

sleek thicket Mar 18, 2021, 5:28 AM

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Legit the theorem I'm most worried about applying correctly on the test is the triangle inequality*

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*the Minkowski inequality for integrals

nimble jolt Mar 18, 2021, 5:28 AM

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https://tenor.com/view/kitty-hi-wave-waving-kitten-gif-17238300

Tenor

gritty widget Mar 18, 2021, 6:55 AM

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Screen_Shot_2021-03-18_at_5.54.51_pm.png

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how does may get the last les

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from the first

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why is $\pi_n(E,F)\cong\pi_n(B)$

gentle ospreyBOT Mar 18, 2021, 6:56 AM

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spinsicle

sleek thicket Mar 18, 2021, 7:44 AM

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I think this is essentially the lifting property of a fibration, although I'm not sure. A map I^n -> B with boundary mapped to the basepoint x0 of B can be lifted to a map I^n -> E, and the boundary is now going to be mapped into F = the fiber over x0

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I'm not comfortable with relative homotopy groups, so this might be wrong

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But I think "lifting property of a fibration" is the right idea

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Like we have a map (E, F) -> (B, x0), and the question is why this is bijective on homotopy groups. Surjective at πn should follow from being able to lift maps of cubes into B up to E and injective should follow from being able to lift homtopies between those maps

gritty widget Mar 18, 2021, 8:13 AM

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hmm I don’t really follow

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ya I guess my question is how does that map (E,F)\to (B,*) induce an isomorphism of homotopy groups

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fff concise is too concise

marsh forge Mar 18, 2021, 3:01 PM

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are you reading a talk of peters or something

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i didnt know concise had language like that

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Okay so I think this is the idea. If F is the fiber over the basepoint of B then maps (D^n,S^n-1)->(E,F) extend nicely to (B,x_0)

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Then the maps D^n/S^n-1->B are also maps D^n->B

sleek thicket Mar 18, 2021, 3:09 PM

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I mean you can just compose, can't you? The trick is going back the other way

marsh forge Mar 18, 2021, 3:09 PM

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you lift these

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and it turns out

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this is automatically a map of pairs for the lift to make sense

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And these processes should be inverse to eachother

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so sham had the correct idea

marsh forge Mar 18, 2021, 3:10 PM

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sleek thicket I mean you can just compose, can't you? The trick is going back the other way

my point is that if the pair was (E,Y) with Y \neq F it wouldnt work

sleek thicket Mar 18, 2021, 3:11 PM

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Ah sure

marsh forge Mar 18, 2021, 3:11 PM

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I have a final in a few hours but I think this should be correct @gritty widget

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also pls do tell me if that paragraph is actually in concise bc it looks like lecture notes

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and I think peter explains this better in real concise

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(these lifting properties for Serre fibrations might be more obvious if you think I^n instead of D^n)

gritty widget Mar 18, 2021, 3:52 PM

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Can any kings or queens help me with thesis topic? The things I’ve most liked have been learning about characteristic classes and various bundles. I really like these ideas of when conditions on the topology of the base space putting restrictions on the total space

marsh forge Mar 18, 2021, 3:53 PM

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~~serre spectral sequence~~

gritty widget Mar 18, 2021, 3:54 PM

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You are into homotopy theory? Can I ask you about how you got into that after your final/ at some later stage?

marsh forge Mar 18, 2021, 3:54 PM

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oh id rather talk about this than my final anyway

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honestly its like

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50% reus and 50% working w a specific professor

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if you are asking how I got into it in a literal sense

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I can give book recs

gritty widget Mar 18, 2021, 3:59 PM

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More so, what in homotopy theory interested you

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I’ve really enjoyed algebraic topology, but I think I preferred the cohomology side of things

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And it spooks me a little when people say that now always homotopy theory is really a seperate field

flint cove Mar 18, 2021, 4:01 PM

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marsh forge I can give book recs

Yes please

marsh forge Mar 18, 2021, 4:02 PM

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Oh so

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I'm not like

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a homotopy theorist homotopy theorist

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but I normally work in the (stable) homotopy category

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which is where modern alg top takes place

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I am a topologist moreso than homotopy theorist by contemporary definition probably

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My first rec is Hatcher

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once you are comfy w the material in there

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You should go through Concise

flint cove Mar 18, 2021, 4:03 PM

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Okay, then I'm successfully between point 1 and point 2

marsh forge Mar 18, 2021, 4:03 PM

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and pay special attention to things like (co)fibrations

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if you understood hatcher well concise should be kind of quick

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(perhaps a short detour in category theory at the level of the first half of Riehl would be helpful)

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But concise is a much more modern perspective

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than hatcher

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After that its a bit of a free for all

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dependning on what you want to learn

flint cove Mar 18, 2021, 4:06 PM

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Let's say I read 60% of hatcher, since I have more need for the cohomology stuff, but atm less for the homotopy stuff
My categorical foundations are decent I'd say
So probably I should go through chapter 4 of Hatcher and Read concise from beginnig to end 😄

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I do enjoy May's writing style very much tho

gritty widget Mar 18, 2021, 4:06 PM

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I’ve had two pretty good courses on algebraic topology

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And have seen a decent bit of his k-theory book

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I have this fomo

marsh forge Mar 18, 2021, 4:07 PM

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flint cove I do enjoy May's writing style very much tho

thats a first

gritty widget Mar 18, 2021, 4:07 PM

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I thought I wanted to work in differential topology doing topology of manifolds

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But I don’t really want to be doing PDEs the whole time

wide kayak Mar 18, 2021, 4:08 PM

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a lot of the interesting stuff about manifolds does seem to involve ODEs/PDEs

gritty widget Mar 18, 2021, 4:08 PM

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But I do want to be doing something a little more “hands on” working with spaces

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I don’t want to end up doing something super abstract homotopy theory

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Hmm maybe I should look into k-theory more

marsh forge Mar 18, 2021, 4:10 PM

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So maybe you'd be interested in the type of problems I personally care about

wide kayak Mar 18, 2021, 4:10 PM

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what about algebraic geometry? I think there's a lot of pretty concrete computational fun stuff there

marsh forge Mar 18, 2021, 4:10 PM

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where you are trying to like

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compute difficult invariants or other algebraic things

wide kayak Mar 18, 2021, 4:10 PM

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despite its reputation for being horribly abstruse

marsh forge Mar 18, 2021, 4:10 PM

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using a hodgepodge of topological and algebraic techniques

gritty widget Mar 18, 2021, 4:10 PM

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Yeah

marsh forge Mar 18, 2021, 4:11 PM

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But yeah in general if you like arguments that

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don't work up to homotopy

gritty widget Mar 18, 2021, 4:11 PM

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Wasn’t hugely gone algebraic geometry

marsh forge Mar 18, 2021, 4:11 PM

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or that don't even work up to homeomorphism

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alg top might not be your cup of tea

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Like many geometric structures are just out the window

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Maybe a problem worth looking into @gritty widget

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Is the Kervaire invariant one problem

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which starts as a simple question about the middle cohomology of homotopy spheres (manifolds htpy eq to a sphere)

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with differential structure

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and eventually becomes a homotopy theoretic problem

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after a few papers

gritty widget Mar 18, 2021, 4:13 PM

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I really enjoyed Milnors characteristic classes and the homotopy groups of sphere paper

marsh forge Mar 18, 2021, 4:13 PM

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Kervaire-Milnor have a paper titled

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Groups of Homotopy Spheres

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which is the origin of this longstanding problem

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that was recently almost fully solved by Hill-Hopkins-Ravenel

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using super high level homotopy-style algtop

gritty widget Mar 18, 2021, 4:14 PM

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But it kind of scared me that people were like this is now just a problem in stable homotopy kevaire invariant

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I do really like the idea of learning about surgery

marsh forge Mar 18, 2021, 4:15 PM

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in the words of my mentor in undergrad

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"Surgery theory is beautiful mathematics that has recently fell out of fashion among most topologists"

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So my advice would be to learn it if you like it because there is probably lots to do in it

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but to caution that its not super common in modern research

sinful pecan Mar 18, 2021, 4:15 PM

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low dimensional topology has surgery

marsh forge Mar 18, 2021, 4:16 PM

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Is surgery still common in contemporary work

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its possible my mentor was misinformed

sinful pecan Mar 18, 2021, 4:16 PM

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i remember an old prof who was still doing stuff with dehn surgery, so i think it's still happening

gritty widget Mar 18, 2021, 4:16 PM

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This also spooks me

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It seems the stuff I like is viewed as old tat

marsh forge Mar 18, 2021, 4:16 PM

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Idk about that

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just a little less hype

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but lots of great math is less hype

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If you want to look into stuff like this

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I know Andrew Putman works on problems you might like

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Benson Farb as well

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and their gang of collaborators

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are all very famous and work in more concrete contexts

gritty widget Mar 18, 2021, 4:17 PM

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I really appreciate all this

marsh forge Mar 18, 2021, 4:18 PM

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Benson in particular has a sort of distaste for homotopy theory memes

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whereas the read I get on Putman is they just don't fully click with him

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(He has often self-deprecatingly joked about it being above his head but given how smart he is I sort of doubt this is the case)

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np @gritty widget happy to help. I know @honest narwhal might be more well versed in this stuff bc his interests aligned more w benson and andy

gritty widget Mar 18, 2021, 4:21 PM

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Good luck with this final

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Another idea I did like the sound of was this moduli space of riemannian metrics

marsh forge Mar 18, 2021, 4:22 PM

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The idea of moduli spaces is sort of ubiquitous nowadays

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topologists sometimes use different names

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like classifying spaces

honest narwhal Mar 18, 2021, 4:24 PM

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Ooh we talking topology?

marsh forge Mar 18, 2021, 4:26 PM

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He was asking about modern research that isn't of the homotopy-esque flavor

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and that might employ things like surgery

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I know very little about this so I thought you might know something of use or have papers to rec

honest narwhal Mar 18, 2021, 4:33 PM

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@gritty widget how much do you think you might like, on the one hand geometric group theory, and on the other hand "algtop \cap alg geo" type stuff?

elder yew Mar 18, 2021, 4:54 PM

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marsh forge but to caution that its not super common in modern research

It comes and it goes through periods of popularity

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Sometimes it's the hot hot thing

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And then a decade or two of nothing

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Then bamm everyone's in it again

marsh forge Mar 18, 2021, 4:55 PM

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i'd imagine it has to do with how recently it was employed to solve something intersting

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I mean

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the slogan for perelmann's work

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was "Ricci Flow w surgery"

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so who knows maybe its in for a renaissance

gritty widget Mar 18, 2021, 5:20 PM

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honest narwhal <@456226577798135808> how much do you think you might like, on the one hand geom...

I did not know anything about geometric group theory, I did not like my undergrad course on groups focusing on finite groups, I did lie my grad course on Lie groups

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I like the idea of looking at the algebraic topology of the spaces you look at in algebraic geometry

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I did not love the tools being used in algebraic geometry, but I did find it very cool how easy it was to include non smooth spaces in the theory

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One thing that did seem cool that I’ve heard in passing is looking at how the topology changes around singularities

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So this example of the family of spaces going from two paraboloids, to two touching cones, to one hyperaboloid

gritty widget Mar 18, 2021, 6:16 PM

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also thanks very much to everyone to all of you

honest narwhal Mar 18, 2021, 6:16 PM

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So as someone who knows nothing about both

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They could be cool things to try out

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On the AG\cap AT side you have stuff like Hodge theory

gritty widget Mar 18, 2021, 6:16 PM

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this has been really helpful

honest narwhal Mar 18, 2021, 6:16 PM

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Which like okay kinda involves PDE but in a nifty way

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And yeah there's a prof at my school who works on topology of singular spaces

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And it seems like good stuff. I'm actually taking his class officially but I got very behind this semester so once I catch up on other stuff I'll just go hard on it lol

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But yeah that's I think one way to sorta do algebraic topology but not "straight algebra"

gritty widget Mar 18, 2021, 6:18 PM

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I did think about looking at hodge theory stuff, I was looking through my old notes from my undergrad and phone a set of notes on hodge theory i had printed after my electromag teacher said soemthing about it

honest narwhal Mar 18, 2021, 6:18 PM

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Geometric group theory is more... I mean "studying groups geometrically" is the slogan. Some vague things include

gritty widget Mar 18, 2021, 6:18 PM

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i don't know what i was thinking trying to read them then

honest narwhal Mar 18, 2021, 6:18 PM

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So if you give me a group with generators and relations you can form what's called the Cayley graph

gritty widget Mar 18, 2021, 6:18 PM

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i can barely read those notes now

honest narwhal Mar 18, 2021, 6:19 PM

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Now you can put a metric on these graphs and now you have a space on which the group acts

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So one thing is studying these for infinite groups

gritty widget Mar 18, 2021, 6:20 PM

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i was just about to ask

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it seems that this means you are only working with finite groups

honest narwhal Mar 18, 2021, 6:20 PM

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Oh yeah people study these a lot for infinite groups. Here's the one thing vaguely resembling a result I know

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So there's a notion of a "hyperbolic metric space"

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Basically means that you can find some delta > 0 such that every geodesic triangle is contained in a delta neighborhood of the union of its edges

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This includes, obviously finite groups

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But also fundamental groups of hyperbolic manifolds

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Now it turns out that hyperbolic groups don't contain copies of Z^2

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In particular this holds for pi_1(Sigma_g) for g > 2

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Which is pretty cool. This was actually a problem in my algebraic topology final which I solved using just basic covering space stuff

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But yeah I found it kinda cute that this had a GGT interpretation

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@marsh forge you might like this too

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But yeah lime soup the other thing I'm vaguely aware of GGT people caring about, and what I think you might actually like more from a super super vague impression

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Mapping class groups

marsh forge Mar 18, 2021, 6:25 PM

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I will read later

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i am pretending to study for physics

honest narwhal Mar 18, 2021, 6:26 PM

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If you give me a manifold (possibly with adjectives), you take diffeomorphisms up to isotopy

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I have less of a coherent elevator pitch about this stuff since what I know of is very scattered

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But there's a lot of super cool math going on there, and it links to topology, algebraic geometry (short version is that these groups are connected with moduli of algebraic curves), apparently even group cohomology, etc

obtuse meteor Mar 18, 2021, 6:42 PM

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geometric group theory is kinda cool imo

gritty widget Mar 18, 2021, 6:43 PM

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I mean also

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I suppose i don't have to do something that is very hot topic

marsh forge Mar 18, 2021, 6:51 PM

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someone post that Borcherds quote

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im lazy

viral atlas Mar 18, 2021, 6:57 PM

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marsh forge someone post that Borcherds quote

gritty widget Mar 18, 2021, 7:02 PM

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is there a real nice quote by him

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yes, ted just posted one

marsh forge Mar 18, 2021, 7:08 PM

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it was something like

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study what you like rather than what is fashionable

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as you will likely make more progress in the former

gritty widget Mar 18, 2021, 7:10 PM

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gritty widget yes, ted just posted one

Perhaps I should have said a nice real quote

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Damn non commutativity of adjectives

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What physics are you doing @marsh forge

marsh forge Mar 18, 2021, 7:21 PM

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magnetism

sleek thicket Mar 18, 2021, 7:52 PM

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@obtuse meteor how was your midterm?

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Alternatively, good luck on your midterm!

gritty widget Mar 18, 2021, 8:04 PM

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let M be a manifold

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let J be an almost complex structure on M

shut moat Mar 18, 2021, 8:15 PM

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J is a complex structure except on a set of measure zero tinktonk

gritty widget Mar 18, 2021, 8:15 PM

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realshit

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measure zero sets on manifolds hmmm

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what's the measure on M tinktonk

sleek thicket Mar 18, 2021, 8:19 PM

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sully

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you do not need a measure

gritty widget Mar 18, 2021, 8:19 PM

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(i know)

sleek thicket Mar 18, 2021, 8:19 PM

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Okay lol I was surprised

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I need to stop taking you seriously

gritty widget Mar 18, 2021, 8:19 PM

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you define measure zero for subsets of manifolds using charts and the usual notion in eucliean space iirc

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i havent used it in a while

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but its important for e.g. sard's theorem

sleek thicket Mar 18, 2021, 8:20 PM

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Pretty much, the key is that smooth => preserves measure zero

gritty widget Mar 18, 2021, 8:20 PM

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locally lipschitz hmmm

sleek thicket Mar 18, 2021, 8:20 PM

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And that a manifold is second countable

gritty widget Mar 18, 2021, 8:20 PM

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ah yeah that'd probably be important

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countable union of zero sets hmmm

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.

obtuse meteor Mar 18, 2021, 8:26 PM

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sleek thicket <@131041620131840000> how was your midterm?

it's in 30 minutes!

gritty widget Mar 18, 2021, 8:26 PM

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good luck

obtuse meteor Mar 18, 2021, 8:26 PM

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at a math talk rn about rational bitangents ^^

sleek thicket Mar 18, 2021, 8:27 PM

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Nice!

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Just remember

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If you're ever stuck, reduce to algebra

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And actually do that if you're not stuck too

marsh forge Mar 18, 2021, 8:35 PM

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coho... what if we kissed... while you took your midterm and i take my midterm... both in 30 minutes... hahaha jk.... unless?

gritty widget Mar 18, 2021, 8:38 PM

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max dont u have a gf

marsh forge Mar 18, 2021, 8:39 PM

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what r u a cop

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(jk i love my gf i would not actually kiss coho do not worry)

gritty widget Mar 18, 2021, 8:40 PM

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ew no

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i am not a cop sully

rotund thicket Mar 18, 2021, 8:47 PM

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i reckon this discord server has a few narcs

proud ridge Mar 18, 2021, 8:50 PM

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Math Narcs smh

obtuse meteor Mar 18, 2021, 10:12 PM

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marsh forge coho... what if we kissed... while you took your midterm and i take my midterm.....

what are you fucking straight

obtuse meteor Mar 18, 2021, 10:13 PM

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sleek thicket If you're ever stuck, reduce to algebra

I did this, not via your advice, but I was stuck trying to draw something and just couldn't lol

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ohhhhhhhhh I am dummy now I have the picture in my head

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that is a much clearer argument than the group theory damnit

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not going to comment bc there are people taking the exam later

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but

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meh

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it's okay I don't think a proof was necessarily expected on that part

obtuse meteor Mar 18, 2021, 10:16 PM

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obtuse meteor that is a much clearer argument than the group theory damnit

this is always true in covering space theory except when the group theory is very trivial

cobalt seal Mar 18, 2021, 10:26 PM

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Hey guys, hoping this is the right spot for this, let me know if not. I'm working with tensors at the moment, and following some notes. At one point, they have the following lines:
a_i (T^i-T'^(alpha) x^i_(alpha) = 0
delta^k_i (T^i-T'^(alpha) x^i_(alpha) = 0
So in other words they've just done something to both sides such that a_i --> delta^k_i
I'm sure I'm missing something clear here, but I'm wondering how they got from the covariant component a_i to a kronecker delta^k_i

gritty widget Mar 18, 2021, 10:27 PM

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can you latex that

cobalt seal Mar 18, 2021, 10:27 PM

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ooh yep, sorry one sec

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$a_i (T^i - T'^\alpha x^i_\alpha ) = 0 \
\delta^k_i (T^i - T'^\alpha x^i_\alpha ) = 0$

gentle ospreyBOT Mar 18, 2021, 10:31 PM

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Spiff Munt

cobalt seal Mar 18, 2021, 10:31 PM

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Man, I need to work on my latex skills

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But yeah, I'm wondering how they get from the covariant $a_i$ in the first line to the kronecker delta $\delta^k_i$ in the second

gentle ospreyBOT Mar 18, 2021, 10:32 PM

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Spiff Munt

marsh forge Mar 18, 2021, 11:54 PM

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obtuse meteor what are you fucking straight

bi

obtuse meteor Mar 19, 2021, 12:07 AM

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Damnit I’ve been foiled

pulsar lynx Mar 19, 2021, 12:11 AM

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I foiled myself

rotund thicket Mar 19, 2021, 12:20 AM

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For every quotient map we can define an equivalence relation that turns it into a map from the space to the quotient space?

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Like the definition of a quotient map is via surjective, but you can rephrase it in terms of equivalence relations?

ivory dragon Mar 19, 2021, 12:27 AM

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yes

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this is often called the fundamental theorem of quotient maps or something similar

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munkres 22.2

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er, 22.3 actually

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corollary of 22.2

gentle ospreyBOT Mar 19, 2021, 1:18 AM

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slimvesus

sleek thicket Mar 19, 2021, 2:50 AM

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absolutely not, gfy

gritty widget Mar 19, 2021, 2:51 AM

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on the cotangent bundle of any smooth manifold there is a tautological 1-form, which differentiates to a symplectic form. is there an analogous construction on tangent bundles?

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tinktonk

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pls no "just pick a riemannian metric lol"

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by "analogous" i don't have anything specific in mind

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something along the lines of "is there anything nice that we get on a tangent bundle that we can define without picking coordinates / that isn't coordinate-dependent"

tough imp Mar 19, 2021, 2:57 AM

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yeah

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You could ask it its name without picking coordinates

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See if it likes tea

gritty widget Mar 19, 2021, 2:57 AM

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tea M

obtuse meteor Mar 19, 2021, 2:58 AM

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chm chose violence today

gritty widget Mar 19, 2021, 2:59 AM

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seems like maybe euler vector fields are related thonk

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vector field on TM

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geodesic field vibes

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monkaS

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ok so you get a flow on TM given by Fl(t, X) = e^tX and this differentiates to what's called the "euler vector field" apparently

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1-form on T*M and vector field on TM catshrug seems analogous enough

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https://math.stackexchange.com/questions/2847100/euler-field-is-invariant-under-the-cotangent-lift

Mathematics Stack Exchange

Euler field is invariant under the cotangent lift

Let $Q$ be a smooth manifold, $\alpha\in\Omega^1(T^*Q)$ the tautological $1$-form and $F:T^*Q\to T^Q$ a symplectomorphism such that $F^\alpha=\alpha$ ($F$ could be the cotangent lift of some

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tinktonk

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monkaS you can prove the darboux theorem with this http://personal.psu.edu/mjf5726/DarbouxEuler.pdf

tight agate Mar 19, 2021, 3:12 AM

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gritty widget on the cotangent bundle of any smooth manifold there is a tautological 1-form, w...

just pick a riemannian metric lmao

gritty widget Mar 19, 2021, 3:12 AM

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hypersully

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i am going to look into this tomorrow

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i finished almost all of my other homework flonshed

tight agate Mar 19, 2021, 3:13 AM

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I need to write 2 more essays for my writing class sadcat

gritty widget Mar 19, 2021, 4:20 AM

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is there an analogue of the exterior derivative of 1-forms for tangent vectors monkaS

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something that eats vector fields and gives me

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uh oh

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i don't know

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2-vector fields thonkzoom

shut moat Mar 19, 2021, 4:22 AM

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musical isomorphism--> exterior derivative--> musical isomorphism back? thonkeyes

gritty widget Mar 19, 2021, 4:22 AM

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i dont wanna make any choices of bundle isomorphisms (or in general i'd just like it to not rely on any extrinsic choices, e.g. riemannian metric)

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if possible

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vector field is a (1,0)-tensor field

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(2,0)-tensor field monkagiga

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what's a reasonable way to turn something that eats one 1-form into something that eats two 1-forms

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i have DG brain rot

fading vale Mar 19, 2021, 5:32 AM

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i think i mostly follow this argument up until the underlined part

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i dont see why independence of x in W implies continuity of the transition map?

honest narwhal Mar 19, 2021, 6:21 AM

#

Let's see

honest narwhal Mar 19, 2021, 6:44 AM

#

Or okay this time let's actually see

#

@fading vale so W is an open path connected neighborhood of x, seems to basically say that in the first coordinate you're constant on path components?

fading vale Mar 19, 2021, 6:56 AM

#

mhm

honest narwhal Mar 19, 2021, 6:56 AM

#

Pi(B) is fundamental groupoid?

#

What's TRA_B?

fading vale Mar 19, 2021, 6:57 AM

#

functors from PI(B) to SET

honest narwhal Mar 19, 2021, 7:00 AM

#

So okay we're taking the disjoint union of Phi(b), and okay looking up tom Dieck

#

transport-simple means any two paths sharing endpoints are homotopic

sinful pecan Mar 19, 2021, 7:00 AM

#

looks to me that the first underlined sentence is just stating well-definedness, and continuity follows from SET

wide kayak Mar 19, 2021, 7:49 AM

#

could anybody recommend a nice introduction to clifford algebras?

#

what they have to do with physics / geometry

gritty widget Mar 19, 2021, 10:38 AM

#

it's easy to describe the set, but how do you actually picture it? since the only irreducible polynomials in ℝ[x] are of order 1 and 2, it's easy to imagine Spec ℝ[x] as kind of the orbit space of the action of Gal(ℂ/ℝ) on ℂ
but in ℚ[x], you can have irreducible polynomials of any degree at all, so how do you visualise Spec ℚ[x]

uncut surge Mar 19, 2021, 11:23 AM

#

https://pbelmans.ncag.info/blog/atlas/ I think visualizing this in any conventional way is gonna be really weird, but here's some motivating images that do the same for integer polynomials

Pieter Belmans

#

I guess if you wanted to stretch the analogy with the galois group of $\mathbb{C}/\mathbb{R}$, I guess maybe this space would be like the orbit space of the algebraic closure of $\bar{\mathbb{Q}}$ under the galois group of $\bar{\mathbb{Q}}/\mathbb{Q}$

gentle ospreyBOT Mar 19, 2021, 11:24 AM

#

Lartomato

uncut surge Mar 19, 2021, 11:25 AM

#

Which sounds really cool and fancy but probably doesn't get more intuitive than that

gritty widget Mar 19, 2021, 11:27 AM

#

i see

uncut surge Mar 19, 2021, 11:28 AM

#

(but my engagement with algebraic geometry and these kinda things never went further than chapter 3 of the vakil lecture notes (the ones you're looking at right now) so maybe there's some big boy AG-people who feel more comfy saying something about this)

gritty widget Mar 19, 2021, 12:12 PM

#

Does anyone (perhaps @marsh forge) have an recommendations for the more “abstract” of homotopy theory. I think I’ve actually just made myself irrationally afraid of the abstract side. I’ve worked through Hatcher fully and most of the topics in concise

fading vale Mar 19, 2021, 1:49 PM

#

sinful pecan looks to me that the first underlined sentence is just stating well-definedness,...

how does continuity follow from SET here :?

uncut surge Mar 19, 2021, 2:16 PM

#

@gritty widget As I understand it, there's not really any "natural operations" in differential geometry other than the identity map and the de Rham differential, if you're not willing to make an further choices

#

Section 34 of "Natural Operations in Differential Geometry" by Kolar, Michor, Slovak

#

"natural" is a bit cheeky here of course but basically if you're not willing to make choices like a Riemannian metric you gonna be sad

#

(i've not worked through it but the theorem sounds smart and cool enough for me)

gritty widget Mar 19, 2021, 2:47 PM

#

thanks

#

Also just a general thanks very much to everyone, I think I have wanted to do a thesis in algebraic topology for a while. But the first "advanced maths" i did was a summer reading course in differential geometry. Since them I've had this fomo about not doing a thesis in differential geometry.

#

In hindsight i think i did just like the topology that appears in differential geometry

gritty widget Mar 19, 2021, 2:56 PM

#

uncut surge Section 34 of "Natural Operations in Differential Geometry" by Kolar, Michor, Sl...

i completely forgot i had this book in my folder, thank you for reminding me of it

#

and ty for the section reference, i guess i have some reading to do catThink

obtuse meteor Mar 19, 2021, 2:57 PM

#

question I asked my algtop teacher that I think some of yall might be interested in

#

So the whole deal with homotopy groups is that the homotopy classes of maps S^n -> X carry a group structure which we can write down.

In a sense we're using S^n as a "test space" to examine X via maps into X. Are there any nice spaces which have this same property, that homotopy classes of maps Y -> X always carry nice a group structure. And if so, why are we more interested in the higher homotopy groups than in these other cases. Aka: Why aren't we always considering RP^2-loops in our space.

#

This seems like a @marsh forge question :)

#

Also I aced my algtop midterm and I'm very proud of myself ^^

cedar pebble Mar 19, 2021, 3:14 PM

#

obtuse meteor Also I aced my algtop midterm and I'm very proud of myself ^^

Congrats!!

obtuse meteor Mar 19, 2021, 3:14 PM

#

Thanks ^^

#

apparently my question has to do with H-cogroups ^^

cedar pebble Mar 19, 2021, 3:15 PM

#

Mhm!

#

In general suspensions are h cogroups

obtuse meteor Mar 19, 2021, 3:15 PM

#

inceresting

cedar pebble Mar 19, 2021, 3:16 PM

#

Spheres are the most natural example: they are suspensions of the two point space S^0

obtuse meteor Mar 19, 2021, 3:16 PM

#

mhm

cedar pebble Mar 19, 2021, 3:20 PM

#

@obtuse meteor https://mathoverflow.net/questions/299662/suspensions-are-h-cogroup-objects

MathOverflow

Suspensions are H-cogroup objects

Do you know any reference where you have a formal justification for the following statement that appears in nLab?

https://ncatlab.org/nlab/show/suspensions+are+H-cogroup+objects

"Let $\mathcal{C}...

#

lmfao Denis

#

does anyone know how to justify the statement that the pinch map exhibits the cogroup structure?

#

Denis: Ok, let me try to give you a proof of something that is a lot stronger than what you asked for, but which hopefully is a bit more natural. Recall that an associative monoid in an ∞-category...

#

MO moment

obtuse meteor Mar 19, 2021, 3:27 PM

#

bruh moment

#

@cedar pebble so here's a good answer

#

the only connected compact manifolds with an H-cogroup structure are the spheres

#

ripped from

#

https://math.stackexchange.com/questions/2106346/generalization-of-homotopy-groups-can-we-replace-sn-by-other-spaces/2106754#2106754

Mathematics Stack Exchange

Generalization of homotopy groups: can we replace $S^n$ by other sp...

For a given topological space $X$, the fundamental group $\pi_1(X)$ is defined by homotopy equivalence classes of continuous map $f : S^1 \to X$. Similarly, the $n^{\text{th}}$ homotopy group $\pi_...

cedar pebble Mar 19, 2021, 3:27 PM

#

ah nice

#

yea there we go

#

currently arguing with [redacted] that localizing at p versus localizing away from p are different things

#

he says they are the same and said I'm bad at CA

#

kek

obtuse meteor Mar 19, 2021, 3:28 PM

#

This is apparently like a really nontrivial result tho apparently

#

https://estrogen.fun/i/y5w6.png

cedar pebble Mar 19, 2021, 3:28 PM

#

oh god

obtuse meteor Mar 19, 2021, 3:29 PM

#

"by the generalized Poincare conjecture"

cedar pebble Mar 19, 2021, 3:29 PM

#

proof is easy except for n=4

#

kek

obtuse meteor Mar 19, 2021, 3:29 PM

#

yeah this is like

#

on god

cedar pebble Mar 19, 2021, 3:29 PM

#

well

#

okay no this is not easy for n≥5

#

that was also a fields medal proof

obtuse meteor Mar 19, 2021, 3:29 PM

#

brother

cedar pebble Mar 19, 2021, 3:30 PM

#

but you can at least do all of it at once for n≥5

#

smol_nozoomi

#

I've tried 3333 times but I still cannot show the Conway knot is not slice

obtuse meteor Mar 19, 2021, 3:30 PM

#

lmao Ultra

#

wasn't that proved recently nG?

cedar pebble Mar 19, 2021, 3:30 PM

#

yes

#

smol_nozoomi

#

a lot of people tried though

#

had meeting with Daniel

#

figured out what was wrong with my computation

#

now I have a way forward I just have to compute uhhh

#

a few hundred integrals

#

might have to be a separate paper

#

which means another paper smol_nozoomi

#

but it's fairly involved and is probably beyond the scope of the current paper

#

monkey

#

idk maybe there's a slick way to do it

obtuse meteor Mar 19, 2021, 3:39 PM

#

cedar pebble a few hundred integrals

christ

#

nG you're being brought along with Daniel to be his personal integral machine

#

nozoomi

cedar pebble Mar 19, 2021, 3:51 PM

#

I mean there's likely a better way to do this

#

using a lot of symmetry

#

but yea it still kinda sucks

#

scraeming

marsh forge Mar 19, 2021, 3:52 PM

#

obtuse meteor So the whole deal with homotopy groups is that the homotopy classes of maps S^n ...

if Y=\Sigma X for some X then [Y,Z] has a group structure induced in the same way

cedar pebble Mar 19, 2021, 3:52 PM

#

actually yea there's a load of symmetry I can use

#

that cuts it down a lot

#

smol_nozoomi

obtuse meteor Mar 19, 2021, 3:53 PM

#

marsh forge if Y=\Sigma X for some X then [Y,Z] has a group structure induced in the same wa...

yeah found this out ^^

cedar pebble Mar 19, 2021, 3:53 PM

#

OH okay yea this is manageable

#

sick

#

smol_nozoomi

marsh forge Mar 19, 2021, 3:53 PM

#

Alternatively you can play the same game with [X,\Omega Y]

#

but this is just adjointness stuff

obtuse meteor Mar 19, 2021, 4:04 PM

#

did you see the like

#

big theorem I talked about with nG max?

marsh forge Mar 19, 2021, 4:07 PM

#

what big theorem

#

oh that sphere is unique

#

I think that the dependence here on poincare is overstated

#

of course if there existed homotopy spheres not homeo to a sphere then you would have manifolds exhibiting the same homotopy properties of S^n

#

idk if this is particularly deep though

obtuse meteor Mar 19, 2021, 4:08 PM

#

maybe

marsh forge Mar 19, 2021, 4:08 PM

#

Like basically the proof is two parts

#

showing that you need to look homotopically like a sphere

#

and then trying to leave the homotopy world for a problem about homotopy

#

and it is very natural that this second part is much harder

obtuse meteor Mar 19, 2021, 4:12 PM

#

yeah that makes sense

cedar pebble Mar 19, 2021, 4:12 PM

#

yea I think this theorem is largely unrelated to issues in homotopy theory so to speak

#

still it's interesting

obtuse meteor Mar 19, 2021, 4:13 PM

#

I wonder if you could prove without poincare that the homotopy functors [S^n , -] are the unique H-cogroup functors coming from a connected compact manifold

#

this is a more homotopy type question

gritty widget Mar 19, 2021, 4:28 PM

#

You guys know how to calculate a curvature?

marsh forge Mar 19, 2021, 4:29 PM

#

@obtuse meteor You cannot prove it categorically (in hTop)

#

for the obvious reason that you cant even detect manifoldness

#

You might be able to do something in Top but it would probably have to be a model categorical approach of some kind?

obtuse meteor Mar 19, 2021, 4:30 PM

#

yeah of course

#

re: not being able to prove it categorically

#

IDK how a model categorical approach would look

#

I'm not big brained enough yet

marsh forge Mar 19, 2021, 4:33 PM

#

My point is more like, model category theory is probably the best way to simultaneously work in Top while respecting homotopy equivalences

#

I don't tend to think about these kinds of questions so idk

sinful pecan Mar 19, 2021, 4:43 PM

#

fading vale how does continuity follow from SET here :?

because the transition map just comes down to, for c and b fixed, \Phi(u_x v_x^{-1}) or something, and SET in Dieck has continuous maps as morphisms.

#

i should say, SET in alg top haha

fading vale Mar 19, 2021, 5:18 PM

#

wait what??

#

morphisms in SET are continuous functions??

#

arent they just functions

sinful pecan Mar 19, 2021, 5:18 PM

#

often in alg top people say functions when they mean continuous or even smooth functions

fading vale Mar 19, 2021, 5:18 PM

#

but if ur objects are in SET how is continuity even defined

#

u dont have a topology

sinful pecan Mar 19, 2021, 5:19 PM

#

oh tru tru

#

ummmm

fading vale Mar 19, 2021, 5:19 PM

#

pandaOhNo

obtuse meteor Mar 19, 2021, 8:33 PM

#

I am once again poasting questions about algebraic topology that are probably dumb but make sense to my small brain

#

https://twitter.com/alittledelta/status/1373008538260410372?s=20

Faye (@alittledelta)

Algtop questions two. Electric Boogaloo: Homology

So are there any nice sequences of spaces X_n so that for any space Y the chain groups C_n(Y) = free group on maps X_n -> Y form a chain complex functorially?

Of course X_n = simplices or X_n = cubes work. (1/?)

median glade Mar 19, 2021, 8:42 PM

#

empty space?

#

I need to write "examples in topology"

#

and every example is just "the empty space"

#

A locally compact hausdorff space?

#

A compact space?

#

A space where every point has a clopen and compact neighborhood?

#

8^)

obtuse meteor Mar 19, 2021, 8:45 PM

#

median glade empty space?

very cool thank you

#

nozoomi

#

my algtop professor said the words "operad" in office hours today so we're officially in gamer territory

fading vale Mar 19, 2021, 8:54 PM

#

obtuse meteor https://twitter.com/alittledelta/status/1373008538260410372?s=20

shamrock might know something about this

#

he talked about dold kan earlier and it seemed kinda similar

#

or maybe not idk thonkEyes this feels familiarish

#

i feel like CW complexes would work 🤔

obtuse meteor Mar 19, 2021, 9:05 PM

#

hmmm

#

in this case we'd want to expand a bit

#

like

#

we wouldn't get a sequence in that case

#

we'd get like a big filtery-looking thing

#

if we considered this for all CW complexes

tight agate Mar 19, 2021, 9:07 PM

#

https://ncatlab.org/nlab/show/test+category

#

might be relevant

obtuse meteor Mar 19, 2021, 9:08 PM

#

yeah this seems sorta relevant

#

the defined test categories on the page all seem very combinatorial tho

tight agate Mar 19, 2021, 9:09 PM

#

you sorta want them to be combinatorial tho

#

so that you can actually compute

obtuse meteor Mar 19, 2021, 9:09 PM

#

yeah but

#

who cares about computing

gritty widget Mar 19, 2021, 9:09 PM

#

sully

obtuse meteor Mar 19, 2021, 9:10 PM

#

unrelated

#

https://estrogen.fun/i/x8oa.png

#

(this person is a type theorist)

tight agate Mar 19, 2021, 9:10 PM

#

a huge part of AT is driven by computations

obtuse meteor Mar 19, 2021, 9:10 PM

#

(so, at least it's slightly more understandable...)

#

no I agree computations are good

#

I'm just meme

tight agate Mar 19, 2021, 9:10 PM

#

petthesully

obtuse meteor Mar 19, 2021, 9:11 PM

#

I do wonder though if there is this entire world of like

#

not computable things that are still weird and interesting

#

and maybe if you have some smooth structure on the test spaces

#

it's still computable

#

(maybe only in smooth cases)

tight agate Mar 19, 2021, 9:11 PM

#

obtuse meteor not computable things that are still weird and interesting

homotopy groups of spheres

obtuse meteor Mar 19, 2021, 9:11 PM

#

yes

#

so like

#

why not have the same weird things in homology

#

https://estrogen.fun/i/kw5d.png

tight agate Mar 19, 2021, 9:14 PM

#

cohomology/k-theory of classifying spaces would be the analogue of homotopy groups of spheres ig

obtuse meteor Mar 19, 2021, 9:14 PM

#

hyperhonk

tight agate Mar 19, 2021, 9:14 PM

#

people do study them extensively

#

and it is super cool

#

group cohomology is very interesting

fading vale Mar 19, 2021, 10:02 PM

#

obtuse meteor https://estrogen.fun/i/x8oa.png

Whats Openness

#

type theorists NEED to be stopped

sleek thicket Mar 19, 2021, 10:08 PM

#

fading vale he talked about dold kan earlier and it seemed kinda similar

Lmao

#

I swear

#

I did not see this

#

Before tweeting a reply

fading vale Mar 19, 2021, 10:08 PM

#

👀

#

also im like 3/5ths of the way done rereading ch 3 of tom dieck and proving everything myself

#

hmmm

#

ok well like

#

85-90% myself

obtuse meteor Mar 19, 2021, 10:12 PM

#

my algtop professor is now of the opinion that the like proofs of excision and similar things

#

are not important for us and are just gonna get sketched

fading vale Mar 19, 2021, 10:12 PM

#

are disgusting?

obtuse meteor Mar 19, 2021, 10:12 PM

#

bc we must learn to compute

fading vale Mar 19, 2021, 10:12 PM

#

barycentric division sucks anyway

#

its boring af

#

computing >>>>

obtuse meteor Mar 19, 2021, 10:13 PM

#

computing >>>>>>>>>

fading vale Mar 19, 2021, 10:13 PM

#

lol wait i wonder if tom dieck has some completely ridiculous proof of excision

#

woeisme nooooooo he does barycentric subdivision too

#

pain

sleek thicket Mar 19, 2021, 10:14 PM

#

That sucks Faye

#

Excision is a cool proof. I will stand by this take

fading vale Mar 19, 2021, 10:14 PM

#

shamrock im like 99% certain uve complained about barycentric subdivision and the proof of excision to me b4

sleek thicket Mar 19, 2021, 10:14 PM

#

I absolutely have

obtuse meteor Mar 19, 2021, 10:15 PM

#

we didn't even do proof that homotopies give you chain homotopies

#

we were just like

#

"so intutitively you subdivide Delta^n x I into simplices, but it's disgusting"

#

chad Jenny

fading vale Mar 19, 2021, 10:15 PM

#

#

cocatThink

sleek thicket Mar 19, 2021, 10:15 PM

#

cocatthimc

#

Excision is good and it's also very bad

obtuse meteor Mar 19, 2021, 10:16 PM

#

the entire proof

#

https://estrogen.fun/i/1yvt.png

fading vale Mar 19, 2021, 10:16 PM

#

make up ur mind smh

sleek thicket Mar 19, 2021, 10:16 PM

#

I am multifaceted

fading vale Mar 19, 2021, 10:16 PM

#

tbh faye that approach isnt even that bad

#

i think reading abt it independently is better than going thru the tedium

obtuse meteor Mar 19, 2021, 10:16 PM

#

no I agree lol

#

it's chad

sleek thicket Mar 19, 2021, 10:17 PM

#

Oh yeah Faye so what I'm thinking is like

#

For singular homology (and de rham coho)

#

You prove homotopy invariance by constructing a chain homotopy

obtuse meteor Mar 19, 2021, 10:17 PM

#

yeah

sleek thicket Mar 19, 2021, 10:17 PM

#

It's not just that they abstractly are homotopic or whatever

obtuse meteor Mar 19, 2021, 10:17 PM

#

and hopefully this is functorial on the level of homotopies

sleek thicket Mar 19, 2021, 10:17 PM

#

So idk, feels 2 functorial

#

Right

obtuse meteor Mar 19, 2021, 10:17 PM

#

monkaS

sleek thicket Mar 19, 2021, 10:18 PM

#

Oh yeah I never proved that thing about 2 functors of vector bundles

#

Oh well

obtuse meteor Mar 19, 2021, 10:18 PM

#

my new favorite words

#

"it is probably functorial"

#

hmmm

sleek thicket Mar 19, 2021, 10:18 PM

#

Functorial is my favorite bullshit adjective

obtuse meteor Mar 19, 2021, 10:18 PM

#

are there any choices

sleek thicket Mar 19, 2021, 10:19 PM

#

It's so bad

obtuse meteor Mar 19, 2021, 10:19 PM

#

in doing this construction

#

of a chain homotopy from a homotopy

sleek thicket Mar 19, 2021, 10:19 PM

#

The subdivision one?

obtuse meteor Mar 19, 2021, 10:19 PM

#

ye

#

non-functoriality sometimes comes from choices

sleek thicket Mar 19, 2021, 10:19 PM

#

Yee

#

It all comes from a subdivision of Δ^n × I

#

Not anything about the space

#

So it feels like, canonical

obtuse meteor Mar 19, 2021, 10:20 PM

#

yeah

#

maybe you have to pick a subdivision of Delta^n x I

#

once

#

but once you do you're good right

sleek thicket Mar 19, 2021, 10:20 PM

#

so if you have a homotopy H : X×I -> Y

#

Let i0, i1 : X-> X×I include at the top and bottom

#

then you have these chain maps C(H°i0) and C(H°i1)

#

hmm

#

I'm trying to remember how this goes

#

Oh right

#

That's C(H) ° C(i0) and C(H) ° C(i1)

#

so what you really want

#

Is that i0, i1 are chain homotopic

#

For any X

#

Because then you can like, whisker or whatever it's called

#

So it comes down to why the maps C -> X×I induce chain homotopic maps on homology

sleek thicket Mar 19, 2021, 10:22 PM

#

obtuse meteor but once you do you're good right

Yeah, I think so

#

I'm just trying to like

#

Formally relate this all back to Δ^n×I

native raptor Mar 20, 2021, 12:15 AM

#

anyone down to help me check my work on Hatcher 2.2.28?

fading vale Mar 20, 2021, 12:27 AM

#

@native raptor so it says to use mayer-vietoris

#

what U and V are you thinking of?

native raptor Mar 20, 2021, 12:27 AM

#

#

sorry this thing is mislabeled

#

but i broke it up into the 2-torus and the mobius band

#

honestly it’s the algebra i feel least confident on

fading vale Mar 20, 2021, 12:29 AM

#

yeah exact sequence stuff can be wacky

#

i think you should put a delta complex on the space

#

you have it set up as squares so you can put one on X and Y respectively and just make sure they agree

native raptor Mar 20, 2021, 12:30 AM

#

okay yeah i'll try that

fading vale Mar 20, 2021, 12:32 AM

#

(this might be easier if you remember that a mobius band can also be obtained from a delta complex on a triangle)

jolly ice Mar 20, 2021, 12:53 AM

#

If anybody wants to work on point set topology rn pls dm me

#

Why the reaction lol

fading vale Mar 20, 2021, 12:54 AM

#

u can just post the question lol

#

JustAsk

jolly ice Mar 20, 2021, 12:55 AM

#

I want to go through a handout but don't want to do it alone

fading vale Mar 20, 2021, 12:55 AM

#

think

#

post the handout then

jolly ice Mar 20, 2021, 12:55 AM

#

Yea trying to find it

#

Metric and topological spaces.pdf

📎 Metric_and_topological_spaces.pdf

sleek thicket Mar 20, 2021, 12:57 AM

#

This is rather long

jolly ice Mar 20, 2021, 12:57 AM

#

Well sorry not the whole thing

#

Section 3

#

Topological spaces

gritty widget Mar 20, 2021, 12:58 AM

#

sleek thicket This is rather long

just do the first and last exericses smugsmug 89 pages done immediately

jolly ice Mar 20, 2021, 12:58 AM

#

:/

fading vale Mar 20, 2021, 12:58 AM

#

i feel like u should attempt these independently

#

and then come here if u need help after ruminating on a particular problem

jolly ice Mar 20, 2021, 1:00 AM

#

I guess so. I was just looking if anyone else wants to because studying with other people can be easier

#

I will be back promptly then if I have a question

jolly ice Mar 20, 2021, 1:42 AM

#

Is there are reason we define topological spaces with open subsets?

gritty widget Mar 20, 2021, 1:43 AM

#

you can also define a topology as a collection of sets containing empty, whole set, which is closed under arbitrary intersection and finite union

#

define it by closed sets catshrug

jolly ice Mar 20, 2021, 1:43 AM

#

Are those two definitions equivalent?

gritty widget Mar 20, 2021, 1:43 AM

#

try proving it

jolly ice Mar 20, 2021, 1:44 AM

#

Since the sets would be complements right?

#

Hmm let me try to prove it and I will be back

gritty widget Mar 20, 2021, 1:45 AM

#

if you define a topology this way (i.e. by its closed sets) then you take complements to get back to your open sets

#

and vice versa

jolly ice Mar 20, 2021, 1:46 AM

#

Yea that was my guess just trying to make it rigorous

#

Is there a reason why not a mix of both?

#

Just general subsets

fading vale Mar 20, 2021, 1:52 AM

#

the idea of "open" as a label is that we are distinguishing some subsets from others

#

so im not totally sure what u mean by "why not a mix of both"

jolly ice Mar 20, 2021, 1:54 AM

#

I meant why not define it with general subsets?

#

instead of putting the category of just open

fading vale Mar 20, 2021, 1:54 AM

#

"open" is an arbitrary label we assign to certain subsets of the set

jolly ice Mar 20, 2021, 1:54 AM

#

well its not arbitrary though?

fading vale Mar 20, 2021, 1:55 AM

#

fulfilling certain conditions (union of open sets must be open, as must finite intersections, and the entire space and empty set)

jolly ice Mar 20, 2021, 1:55 AM

#

isnt it the definition from real analysis

fading vale Mar 20, 2021, 1:55 AM

#

uh okay hold up what is your definition of a topology here

jolly ice Mar 20, 2021, 1:56 AM

#

oh I just got it

#

since its not induced by a metric we are referring to general open sets

fading vale Mar 20, 2021, 1:56 AM

#

yeah

#

and those general sets can be any collection of subsets of our space X as long as they fulfill our 3 rules

jolly ice Mar 20, 2021, 1:57 AM

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ahhh thank you!

fading vale Mar 20, 2021, 1:58 AM

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np

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you did most of the work yourself anyway :p

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kinda one of those cases where talking at someone abt it helps you clear it up for yourself lol

jolly ice Mar 20, 2021, 1:59 AM

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definitely haha

delicate hollow Mar 20, 2021, 2:05 AM

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ok

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i have a question which is:

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wait im scared to ask nvm

gritty widget Mar 20, 2021, 2:05 AM

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sully

delicate hollow Mar 20, 2021, 2:05 AM

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ill ask you

fading vale Mar 20, 2021, 2:06 AM

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JustAsk

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what is there to be scared of

delicate hollow Mar 20, 2021, 2:08 AM

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um

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its a homework question @fading vale

fading vale Mar 20, 2021, 2:09 AM

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thonk

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many hw questions are asked here

delicate hollow Mar 20, 2021, 2:09 AM

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f:X->Y is a function between two topological spaces
Graph(f)={(x,f(x)) in X x Y| x in X}
Prove if f is continuous and Y Hausdorff then Graph(f) is closed.
So my guess is that we WTS X x Y - {(x,f(x)) in X x Y| x in X} is open
me thinks X x Y - {(x,f(x)) in X x Y| x in X} reduces to X x (Y- f(X))
is that correct?

sleek thicket Mar 20, 2021, 2:10 AM

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This is a good question

#

./problem

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Unfortunately the "reduces to" thing is false

delicate hollow Mar 20, 2021, 2:11 AM

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equals*

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oh damn

sleek thicket Mar 20, 2021, 2:11 AM

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Think about a constant function R-> R

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What does the first set look like? The second?

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These are sets in the plane

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wait, I may have fucked up and found an example where this works...

delicate hollow Mar 20, 2021, 2:12 AM

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R x R - R x {3}?

sleek thicket Mar 20, 2021, 2:12 AM

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Yes sorry, this is a bad example

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think about the graph of a parabola

delicate hollow Mar 20, 2021, 2:12 AM

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ok

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so R x R+

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or X and Y are both R i guess

sleek thicket Mar 20, 2021, 2:13 AM

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Yeah, that's what I had in mind

#

So what's the first set?

delicate hollow Mar 20, 2021, 2:13 AM

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R x R - R x (img(f)=R+) = R x R-

sleek thicket Mar 20, 2021, 2:13 AM

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That's the second set

delicate hollow Mar 20, 2021, 2:13 AM

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wait it isnt the first?

sleek thicket Mar 20, 2021, 2:13 AM

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X × (Y - f(X))

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Nope

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So the first set is the complement of the graph

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You just cut a parabola out of the plane

delicate hollow Mar 20, 2021, 2:14 AM

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yea

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wait a second

#

what

#

oh

#

wait

#

i guess that what it means to be a graph

sleek thicket Mar 20, 2021, 2:14 AM

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Yup

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{(x, f(x)) : x in X} is the graph of X

delicate hollow Mar 20, 2021, 2:15 AM

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I was thinking that graph was something weird like R x Img(f)

sleek thicket Mar 20, 2021, 2:15 AM

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Ah gotcha

delicate hollow Mar 20, 2021, 2:15 AM

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which i knew it wasnt

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i was trying to find shortcuts

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ty for that

sleek thicket Mar 20, 2021, 2:15 AM

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Shortcuts are how you do math :)

delicate hollow Mar 20, 2021, 2:16 AM

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so if I dont simplify it further and just try to show that X x Y - Graph(f) is open

#

would I just apply theorem stating what makes a set open in product topology?

#

can I go straight to that

#

and also see if I can apply hausdorff and continuous

#

or are there some intermediate steps I could be missing?

#

i.e. should I just follow my nose?

sleek thicket Mar 20, 2021, 2:16 AM

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Hmm

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So I have a proof in mind

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But it relies on a different result you may not have seen

delicate hollow Mar 20, 2021, 2:17 AM

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wait

sleek thicket Mar 20, 2021, 2:17 AM

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I would try following your nose

delicate hollow Mar 20, 2021, 2:17 AM

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what does it deal with?

sleek thicket Mar 20, 2021, 2:17 AM

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There's an equivalent condition to be hausdorff

delicate hollow Mar 20, 2021, 2:17 AM

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oh

sleek thicket Mar 20, 2021, 2:18 AM

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I would suggest trying to do this problem the way you were thinking of

delicate hollow Mar 20, 2021, 2:18 AM

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singleton sets closed iff?

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is that a condition

sleek thicket Mar 20, 2021, 2:18 AM

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Nope, singleton sets closed is weaker

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Example: the cofinite topology on an infinite set

delicate hollow Mar 20, 2021, 2:18 AM

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o

sleek thicket Mar 20, 2021, 2:19 AM

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There are no disjoint nonempty open sets at all, right?

delicate hollow Mar 20, 2021, 2:19 AM

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yea

sleek thicket Mar 20, 2021, 2:19 AM

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But points are finite

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So their complement is cofinite = open

delicate hollow Mar 20, 2021, 2:19 AM

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another guess b4 i start trying stuff out

sleek thicket Mar 20, 2021, 2:19 AM

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Mhm

delicate hollow Mar 20, 2021, 2:19 AM

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does it have to do with seperability?

sleek thicket Mar 20, 2021, 2:20 AM

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Do you mean separability in the sense of "hausdorff" being a separation axiom or are you talking about the property of a space having a countable dense subset?

delicate hollow Mar 20, 2021, 2:21 AM

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i meant like a seperation

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ive heard something about hausdroff being a seperation axiom

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but not rly sure what it means

#

cuz my line of thought

sleek thicket Mar 20, 2021, 2:21 AM

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It's sort of an informal term

delicate hollow Mar 20, 2021, 2:21 AM

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is if I think of it as a graph on R2

sleek thicket Mar 20, 2021, 2:21 AM

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You will need to use hausdorff at some point for sure

delicate hollow Mar 20, 2021, 2:21 AM

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removing the graph seperates the space into two sections

sleek thicket Mar 20, 2021, 2:22 AM

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In fact, having this for all spaces X and all maps f : X -> Y is equivalent to Y being hausrdoff

delicate hollow Mar 20, 2021, 2:22 AM

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oh wait

#

i think i visualized a proof for r2

#

but

#

it doesnt make sense

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because it isnt hausdroff

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or wait

#

it is nvm

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because like

#

if you remove parabola like you said

#

R2 - parabola

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is an open set

sleek thicket Mar 20, 2021, 2:23 AM

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Yup

delicate hollow Mar 20, 2021, 2:24 AM

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cus u can take open sets from everywhere as close to missing graph

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and union them

sleek thicket Mar 20, 2021, 2:24 AM

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That's exactly the intuition

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Imo

delicate hollow Mar 20, 2021, 2:24 AM

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to make same shape again

sleek thicket Mar 20, 2021, 2:24 AM

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so try to do that in general!

delicate hollow Mar 20, 2021, 2:24 AM

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uh

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oh

#

so like

sleek thicket Mar 20, 2021, 2:24 AM

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delicate hollow would I just apply theorem stating what makes a set open in product topology?

You already noticed you should try to go local

delicate hollow Mar 20, 2021, 2:24 AM

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so i was given that earlier I think

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U a subset of X x Y is open iff for every p in U, there is an A x B subset of U and A is open in X and B is open in Y

sleek thicket Mar 20, 2021, 2:25 AM

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Yup

sleek thicket Mar 20, 2021, 2:26 AM

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delicate hollow cus u can take open sets from everywhere as close to missing graph

And that's sort of what you were saying here

#

(if I understood right)

delicate hollow Mar 20, 2021, 2:26 AM

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I might be stuck now

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because I get Y is hausdroff

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but not rly sure what to do with X

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I take U = X x Y - Graph (f)

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so i just need to show that for every point in U, I can find an A open in X and a B open in Y where A x B is a subset of U

sleek thicket Mar 20, 2021, 2:28 AM

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Sure, so take a point p = (x, y) in U

delicate hollow Mar 20, 2021, 2:28 AM

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yea but what would my open A be?

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i think i have an easier time for B

sleek thicket Mar 20, 2021, 2:29 AM

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Let's start with B then

delicate hollow Mar 20, 2021, 2:29 AM

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ok

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since Y is hausdroff

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for any two points in Y there is are disjoint open neighborhoods around them

sleek thicket Mar 20, 2021, 2:29 AM

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Right

delicate hollow Mar 20, 2021, 2:30 AM

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so for y in Y, I can take two points being f(x),y

sleek thicket Mar 20, 2021, 2:30 AM

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Yup

delicate hollow Mar 20, 2021, 2:30 AM

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and there will be an open neighborhood around y ill call B for every point in U.

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for x I think i use continuous

sleek thicket Mar 20, 2021, 2:30 AM

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well hang on

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Sorry, continue

#

I was going to point something out but it's not super relevant rn

delicate hollow Mar 20, 2021, 2:31 AM

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I can take an open neighrbood around B not containing f(x) right?

sleek thicket Mar 20, 2021, 2:31 AM

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Yup

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Err

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B is open

delicate hollow Mar 20, 2021, 2:31 AM

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yea

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is there a problem with that choice?

sleek thicket Mar 20, 2021, 2:32 AM

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delicate hollow I can take an open neighrbood around B not containing f(x) right?

Well I'm not sure what you mean here

delicate hollow Mar 20, 2021, 2:32 AM

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like uh

sleek thicket Mar 20, 2021, 2:32 AM

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Isn't B such a neighborhood?

delicate hollow Mar 20, 2021, 2:32 AM

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ok this is what im thinking

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since Y hausdroff we can take two points f(x),y and they both have open disjoint neighborhoods O,B respectively. y is in U = X x Y - Graph(f) and f(x) in graph(f)

#

oh

sleek thicket Mar 20, 2021, 2:33 AM

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Sorry what do you mean by y is in U?

delicate hollow Mar 20, 2021, 2:34 AM

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oh ok

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y is a component of a point in p = (x,y)

sleek thicket Mar 20, 2021, 2:35 AM

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Yup

delicate hollow Mar 20, 2021, 2:35 AM

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so y is in B, and we can choose B to be a neighorhood around y that doesnt have f(x)

#

and in doing that the second component of p=(x,y) is done

#

now I need to find an open set A containing x where x is in the first component of X x Y - Graph(f)

sleek thicket Mar 20, 2021, 2:36 AM

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yup

delicate hollow Mar 20, 2021, 2:36 AM

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I think I would use continuous definition here

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f is continuous meaning U in Y is open implies f-1(U) is open in X

sleek thicket Mar 20, 2021, 2:38 AM

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Yup

delicate hollow Mar 20, 2021, 2:38 AM

#

can I take f-1(B) to be A?

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and then be on my way?

sleek thicket Mar 20, 2021, 2:38 AM

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You can try

#

So define A = f^-1(A)

#

What properties did we want A, B to satisfy

delicate hollow Mar 20, 2021, 2:38 AM

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A dont is not in graph f

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wow english

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I want A intersect graph f to be empty

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no

#

wait

#

something similar but not semantically correct

sleek thicket Mar 20, 2021, 2:40 AM

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delicate hollow U a subset of X x Y is open iff for every p in U, there is an A x B subset of U ...

Actually, this statement is a little off

#

Your sets A, B aren't related to p here

#

You could just take A, B to be empty always if this was the definition

delicate hollow Mar 20, 2021, 2:40 AM

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oh true

#

non-empty a and b?

#

oh wait

sleek thicket Mar 20, 2021, 2:41 AM

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not quite

delicate hollow Mar 20, 2021, 2:41 AM

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A and B need to have x and y

sleek thicket Mar 20, 2021, 2:41 AM

#

Yup

#

Is that true of your A, B?

delicate hollow Mar 20, 2021, 2:41 AM

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its def true for B

sleek thicket Mar 20, 2021, 2:41 AM

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Yup

delicate hollow Mar 20, 2021, 2:41 AM

#

but A = f-1(B)

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x in A

#

means x is in preimage

#

so it would be in the graph

#

ok

#

so

sleek thicket Mar 20, 2021, 2:43 AM

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right

delicate hollow Mar 20, 2021, 2:43 AM

#

I think I apply hausdroff

sleek thicket Mar 20, 2021, 2:43 AM

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So x is not in A

delicate hollow Mar 20, 2021, 2:43 AM

#

again

sleek thicket Mar 20, 2021, 2:43 AM

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Which isn't great

delicate hollow Mar 20, 2021, 2:43 AM

#

If I apply hausdroff

#

wait

#

f(x),y can have seperate open neighborhoods in Y

#

i wonder if the inverse image of the open neighorhood of f(x) is good then?

#

but that seems counter intuitive

#

because inverse img of whatever that is is def in graph me thinks

sleek thicket Mar 20, 2021, 2:46 AM

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Well the inverse image won't be in the graph

#

The inverse image is a subset of X, not X × Y

#

Maybe try working this out with the parabola example again?

#

Like draw separating sets in Y or whatever

delicate hollow Mar 20, 2021, 2:46 AM

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visualizing this is just funny though

#

like

#

if i think of {-1,0,1,2,...} -> {0,1,2,4,...}

#

but then a bunch of numbers inbetween those

#

and if I take a positive range from (1,4) in the second set

#

the preimage is what now?

#

never actually thought of it

#

shouldnt it be positive and negatives?

#

oh wait

sleek thicket Mar 20, 2021, 2:49 AM

#

I'm not sure what you mean, sorry

delicate hollow Mar 20, 2021, 2:49 AM

#

yea it was terribly explained my bad

#

but i have another idea ig

#

so if i have x^2

#

and I take a section of it

sleek thicket Mar 20, 2021, 2:49 AM

#

Like a cross section?

delicate hollow Mar 20, 2021, 2:49 AM

#

its preimage is gonna be the corresponding x axis?

#

yea

#

cross section

#

wait that isnt what I want

#

the cross section of the graph shouldnt be open

#

even though you said the inverse image wont be in the graph

#

is this always true?

#

like if you have 1/x as the function

#

isnt the inverse image in the graph?

#

or no

#

nvm

#

is it because the graph is a cartesian product

#

and the inverse image is just X

sleek thicket Mar 20, 2021, 2:52 AM

#

Exactly

delicate hollow Mar 20, 2021, 2:53 AM

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i cant visualize that well though

#

the way i usually think of inverse image is with sets and arrows pointing to elements

#

#

like this

sleek thicket Mar 20, 2021, 2:54 AM

#

so let's stick with the parabola

#

Here's a random point not on the graph: p = (0,1)

#

what are sets like B, O?

#

We want disjoint open sets in Y around 1 and f(0) = 0

delicate hollow Mar 20, 2021, 2:57 AM

#

mb

#

im back

#

so an example of an open set around 1 is (0.5,1.5)

#

and around f(0)= 0 is (-0.5,0.25) @sleek thicket

sleek thicket Mar 20, 2021, 2:59 AM

#

sure

#

So what's f^-1(O)

delicate hollow Mar 20, 2021, 2:59 AM

#

f-1(-.5,.25)

sleek thicket Mar 20, 2021, 2:59 AM

#

It's all points x in R such that x^2 in (-0.5,0.25), right ?

delicate hollow Mar 20, 2021, 2:59 AM

#

yea

sleek thicket Mar 20, 2021, 2:59 AM

#

Is there a more explicit way to write this?

delicate hollow Mar 20, 2021, 2:59 AM

#

we can write square root

#

so sqrt((-.5,.25))

#

so (0,.5)?

sleek thicket Mar 20, 2021, 3:01 AM

#

Not quite

#

that's only the positive bits

#

For example, 0^2 = 0 is in (-0.5,0.25)

delicate hollow Mar 20, 2021, 3:01 AM

#

oh thats true

#

what though

#

-.5^2 = .25

#

so is it [0,.5)?

#

wait lol

#

that isnt inverese

#

if we are only choosing x in R wouldnt it be [0,0.5)

#

because we cant include complexes

sleek thicket Mar 20, 2021, 3:03 AM

#

no

#

Negative numbers can square to something in (-0.5,0.25)

delicate hollow Mar 20, 2021, 3:04 AM

#

dont they square to positive numbers only?

sleek thicket Mar 20, 2021, 3:05 AM

#

Yes

#

(-0.3)^2 is between -0.5 and 0.25

#

right?

delicate hollow Mar 20, 2021, 3:05 AM

#

yes

#

oh damn

#

ok

#

i see lol

#

so inverse image of (-.5,.25) would be all numbers that when squared are in this range

sleek thicket Mar 20, 2021, 3:06 AM

#

Right

#

(-0.5,0.5)

#

So the hope is that we can take A to be this set

#

And you can draw a picture of this setup and see that it's disjoint from the graph

#

I think the example might have gotten us a little sidetracked though

delicate hollow Mar 20, 2021, 3:07 AM

#

and this isnt in the graph

#

the first component*

#

wait

#

i still dont see it

#

did stupid little sketch

sleek thicket Mar 20, 2021, 3:08 AM

#

We have this product (-0.5,0.5) × (0.5,1.5)

#

And the claim is this is disjoint from the parabola

delicate hollow Mar 20, 2021, 3:09 AM

#

it entirely is yes

sleek thicket Mar 20, 2021, 3:09 AM

#

Yeah

#

You agree with that?

delicate hollow Mar 20, 2021, 3:09 AM

#

yes

sleek thicket Mar 20, 2021, 3:09 AM

#

Well that's what we wanted out of A, B

delicate hollow Mar 20, 2021, 3:09 AM

#

ok

#

i guess this is part im confusd about

#

so when we take the x component of X x Y - Graph(f)

#

that X component can't be in the graph meaning it cant be in the domain or the preimage?

#

domain would be silly

sleek thicket Mar 20, 2021, 3:10 AM

#

I don't understand you here

#

Why are we taking the x component?

delicate hollow Mar 20, 2021, 3:11 AM

#

arent we trying to find an open set A around the x component of any point in X x Y - Graph(f)

sleek thicket Mar 20, 2021, 3:11 AM

#

Ah sure

#

So p = (x, y) isn't in the graph

#

And you're saying x can't be in the domain or preimage?

#

The domain or preimage of what?

delicate hollow Mar 20, 2021, 3:11 AM

#

of the function

sleek thicket Mar 20, 2021, 3:11 AM

#

We want to have that x is in the set A

delicate hollow Mar 20, 2021, 3:11 AM

#

but that sounds silly

sleek thicket Mar 20, 2021, 3:11 AM

#

Yeah, it is sorry

delicate hollow Mar 20, 2021, 3:12 AM

#

yea we want x is in the set A

#

and A is a subset of the first product in X x Y - Graph(f), correct?

sleek thicket Mar 20, 2021, 3:13 AM

#

It's a subset of X

delicate hollow Mar 20, 2021, 3:14 AM

#

yea

#

it cant be any subset though

#

because like you said the preimage of B doesnt work even though f-1(B) is a subset of X

sleek thicket Mar 20, 2021, 3:15 AM

#

Yeah

#

So let's return to our setup

delicate hollow Mar 20, 2021, 3:15 AM

#

and I agreed because that is in the graph

sleek thicket Mar 20, 2021, 3:15 AM

#

We have p = (x, y) not in the graph

delicate hollow Mar 20, 2021, 3:15 AM

#

correct

sleek thicket Mar 20, 2021, 3:15 AM

#

We have open sets O,B in Y

#

With f(x) in O and y in B

#

And O, B are disjoint

#

Set A = f^-1(O)

#

You're concerned that maybe x isn't in A

#

Yeah?

delicate hollow Mar 20, 2021, 3:16 AM

#

yes

sleek thicket Mar 20, 2021, 3:16 AM

#

Well, what would it mean to have x in f^-1(O)?

delicate hollow Mar 20, 2021, 3:17 AM

#

x is in the inverse image of an openset set containing the graph?

#

also known as the preimage?

sleek thicket Mar 20, 2021, 3:17 AM

#

Yeah

#

So what's the definition of the preimage?

delicate hollow Mar 20, 2021, 3:18 AM

#

if X -> Y then the preimage are all the Y that have mappings in X

#

wait

#

its the X that map to points in Y