god the urge to shitpost
#point-set-topology
coarse night
Here's a module problem. M is fg R module and f: M → M is surjective. Show f is then an isomorphism
unreal stratus
Here's a module problem. M is fg R module and f: M → M is surjective. Show f is ...
Good question
solemn oar
I don't believe it's true. An fg module doesn't have a well defined basis in general, so it can have generating sets of varying sizes.
unreal stratus
I have done this question before lol
Maybe there is a missing hypothesis i need to check
coarse night
no the hypothesis is enough, just checked for sanity
unreal stratus
||you can turn M into a fg R[x] module by x.m = f(m). Since (x).M = M, there's some g in R[x] with g = 1 mod x such that g.M = 0. In other words, f is the root of a polynomial with constant coefficient 1. This clearly implies injectivity||
Is that the proof you gave too ryu?
I imagine there are similar things that work
coarse night
I saw it on Matsumura, it's the same yes
solemn oar
unreal stratus
I think my comm alg course was basically just a load of Matsumura and Atiyah-Macdonald questions lol
coarse night
Have you seen this one, Rⁿ → R^m is injective then n ≤m
unreal stratus
I'd need to think about it
Well
I know ways to do it i think
maybe we should go to #algebraic-geometry
lol
gritty widget
how can i proceed solving this problem?
gaunt linden
You should be able to find some accumulation points just by understanding what the set A is, and the definition of the term "accumulation point".
We can then see if we can find a proof that the ones you have found are all of them.
unreal stratus
interesting using "and" for 2
gritty widget
You should be able to find _some_ accumulation points just by understanding what...
i do understand the definition of accumulation point but i don't understand the set A and how should i use the definition do find the points
gaunt linden
Would you be able to find an accumulation point of B = { 1 + 1/m | m in N* }?
gritty widget
Would you be able to find an accumulation point of B = { 1 + 1/m | m in N* }?
one is an accumulation point
gaunt linden
Right. Now do you agree that B is a subset of A?
gritty widget
who is B?
gaunt linden
Would you be able to find an accumulation point of B = { 1 + 1/m | m in N* }?
I defined a set I call B here.
gritty widget
Would you be able to find an accumulation point of B = { 1 + 1/m | m in N* }?
oh oke
gritty widget
Right. Now do you agree that B is a subset of A?
i agree
gaunt linden
And since we have agreed that 1 is an accumulation point of B and B is a subset of A, does this tell you something about accumulation points of A?
gritty widget
1 is an accumulation point in A?
gritty widget
i can't
gaunt linden
Hmm, how did you conclude that 1 is an accumulation point of B?
gritty widget
Hmm, how _did_ you conclude that 1 is an accumulation point of B?
i did not
gaunt linden
But you suggested that number yourself and I agreed with it. Once you have the idea of trying that number, can you at least make an argument that it is in fact an accumulation point of B?
At this point I'm suspecting that at least part of your problem is that you're not actually imagining the sets visually. If you have trouble with this, I'd sugges actually using paper. Draw a number line and put dots on it for the points that are in B (or at least a representative selection of them).
gritty widget
gaunt linden
That's not a lot of points, and it's not quite clear the scale of that number line is. Can you try plotting more of them?
It does look like you have a dot at 1, though, and 1 is not an element of B.
gaunt linden
You're saying that the set of numbers that can be written as 1 + 1/m for some positive integer m, is the same as the set of numbers that are >1 but <= 2??
How do you write, for example, 1.8 in the form 1 + 1/m?
gritty widget
how?
gaunt linden
That's what I'm asking you. You're claiming that { 1 + 1/m | m in N* } = (1,2].
Do we agree that 1.8 is an element of (1,2]?
gaunt linden
If we agree about that, and you're right that { 1 + 1/m | m in N* } and (1,2] are the same set, then 1.8 must also be an element of { 1 + 1/m | m in N* }.
gritty widget
so should i draw a long segments with many point on it?
gaunt linden
Yes. Or perhaps, for better readability, many points slightly above your number line, such that the points won't collide visually with the number line itself and the ticks that tells us which places on the number line represent which numbers.
gritty widget
how many points?
gaunt linden
Enough that you have a feeling for how the rest of them go.
gaunt linden
,rccw
gentle ospreyBOT
gritty widget
sorry for ping
gaunt linden
The image you ought to be able to form in your head consists of the red dots in this one:
gritty widget
oke it is formed in my head
gaunt linden
And furthermore, which this image you should be able to see on it that 1 is an accumulation point.
gaunt linden
Good.
gritty widget
1/13 is an accumulation point
gaunt linden
Yes.
gritty widget
how can i proceed solving this problem?
that's the main problem
gaunt linden
By generalizing your experience with finding that 1 and 1/13 are accumulation points, do you now have some guesses for more of them?
gritty widget
i can take any value of n or m in N and find others
gaunt linden
That sounds like the right direction. Please describe the points you find in that way.
gaunt linden
I thought I was helping, but perhaps I'm not as good as being helpful as I hoped.
Perhaps take smaller steps than "describe all of them"?
Name just one accumulation point other than 1 and 1/13 you can find.
gritty widget
d=(1/14+1/m,m in N*) the accumulator point is 1/14
gaunt linden
Okay, now we have 1, and 1/13 and 1/14. Then another?
gaunt linden
Yes, but there a many sets where the points "get closer to 0", and many of those sets are not the particular set we're looking at here.
Is 1/7 one of the accumulation points we know how to find yet? Is 3/7? Is pi?
gritty widget
yes but for a subset not A
gaunt linden
I'm speaking in particular about the numbers that we now know are accumulation points of A.
Which numbers are those?
We have identified 1 and 1/13 and 1/14 and 1/15, but listing them one by one is obviously going to take too long, so we need a way to explain which points they are which isn't based on writing each of them down separately.
gritty widget
should i create a set that describe them?
obtuse meteor
Please don’t use this term — very bannable
gaunt linden
Well, they are a set. I think what you mean might be "should I write down a symbolic description of that set?" That might be a way forward if you cannot get yourself to put it in words.
rough cedar
I missed it 
coarse night
become a mod to see deleted messages
gritty widget
d=(1/n;n in N*)
gaunt linden
Yes!
gritty widget
is that it?
gaunt linden
That's what I've been trying to get you to say for the last hour or so.
You could also have said in words: "they are 1 divided by each of the natural numbers". Or "they are the reciprocals of the natural numbers".
Not quite sure what went wrong here. I suspect I failed to explain what I meant by "describe".
Anyway, {1/n | n in N*} are some of the accumulation points. There's a single one missing.
But even if I explain what the missing point is, I'm not sure I have a good explanation of why that's the only missing accumulation point.
gritty widget
which is?
is it always when i have two variable i should make one of them fix then calculate for the accumulation point?
gaunt linden
No, that is a specific reasoning particular to this problem.
The best general advice I can give at this point is practice imagining how the sets look, so you're won't be restricted to staring at symbols.
Symbolic arguments are good for convincing yourself and others that you're right, but often not very good for coming up with a statement you can be right about.
coarse night
maybe we can try to show B is closed, B is the original set union the calculated acc points? That will be enough to show cl(A)=B is indeed the
doesn't look like an easy thing to show tbh
thin scarab
why does this require compactness?
isnt the intersection of countably many nonempty sets necessarily nonempty?
or is that only for finitely many
i guess it wouldnt be true for countably many open sets like (0, 1/n)
still, why is closed not enough?
unreal stratus
which book is this hm
thin scarab
lee topological manifolds
unreal stratus
Ah okay
thin scarab
wanted to read smooth manifolds but my point set topology was lacking
unreal stratus
As another example btw, consider the subsets of Q given by like
{ x : |x^2 - 2| <= 1/n }
I give this example because it is in a sense dual to diligent's answer
plain raven
Nice.
thin scarab
the case of the missing sup
unreal stratus
lol
thin scarab
i get it
unreal stratus
Noic
half sable
Let $\mathbb{Q}_p$ be the p-adic numbers and $\mathbb{Z}_p$ be the p-adic integers, is the quotient $\mathbb{Q}_p/\mathbb{Z}_p$ compact? I would think not, right?
gentle ospreyBOT
CelesteCrow
rotund ivy
How should I construct a real function (must be discontinuous) such that its graph ${(x,f(x)):x\in\mathbb{R}}$ has non-zero Lebesgue measure?
gentle ospreyBOT
CollinGao-Original
rotund ivy
Let $\mathbb{Q}_p$ be the p-adic numbers and $\mathbb{Z}_p$ be the p-adic intege...
how do you define the quotient in this case?
well if you consider it as those cosets, then the space is equivalent to "set of finite sums of a_n*p^-n with n being positive integers"
with the original distance function, the space is discrete and countable, which isn't compact
echo oyster
Can someone explain this please, i don't understand how they generate same topology
bitter smelt
The sketch there is that each open set about a point in the "circle" topology is contained in an open set of the "rectangle" topology and vice versa
stark fog
Indeed, because for each point x in an element of the rectangle topology, there is an element of the circle topology containing that point as its subset and vice versa, the topologies are contained in each other, which means they are the same.
echo oyster
Understood, thank you both of you!
stark fog
yw
pseudo coral
im asked to describe a triangulation of the torus
coarse night
"=" ?
unreal stratus
eh = is common notation in presentations right
coarse night
Surface presentation I'm familiar with just writes it as <abc, c'a'b'>
idk what's = supposed go mean here
pseudo coral
so how do i DESCRIBE a triangulation of the torus
plain raven
2d complexes are group presentations of the fundamental group 😄
coarse night
but that doesn't represent the π1(T²) to begin with
unreal stratus
2d complexes are group presentations of the fundamental group 😄
True
coarse night
oh actually that does
steel glen
idk what's = supposed go mean here
don’t group presentations implicitly have = e next to their relations (or whatever the identity is)?
plain raven
A deformation retract of a space onto a point like 5 would be a homotopy h : Q x I -> Q such that
h(1,x) = 5
h(0,x)= x
and h is continuous.
But if 5 is not equal to x, then the path h(t, x) from 5 to x will pass through irrational numbers at some point.
It will not be a map which takes values in Q. Instead it will take values in R.
There is no continuous path in Q between 5 and x for x not equal to 5, because such a path would necessarily meet each irrational number between the two
tardy fjord
Similarly, deformation retracts induce isomorphisms in homology by homotopy invariance, and Q does not have the homology of a one point space.
plain raven
That is ...
grim knot
oh it's actually easy, thx a lot, I was thinking too much
grim knot
Similarly, deformation retracts induce isomorphisms in homology by homotopy inva...
I'm not this far in topology yet 😦
tardy fjord
My B, no worries
grim knot
That is ...
but you helpedd me a lot though, thank you
umbral panther
Similarly, deformation retracts induce isomorphisms in homology by homotopy inva...
Similarly, deformation retracts induce an isomorphism on the set of path components and the set of path components of Q is not the singleton
hidden crag
umbral panther
Disjoint union
It’s not exactly standard notation, but I can tell from the context
frank bolt
That makes sense. The two spaces are disjoint, but we glue them together by identifying certain points
Cheers
echo oyster
What is the motivation behind product topology, if we had the box topology already ?
patent quarry
functions on product spaces are continuous iff their coordinate functions are, tychonoff theorem, universal property,...
gritty widget
can someone explain to me how we've shown that it is neither open nor closed
nocturne basalt
can someone explain to me how we've shown that it is neither open nor closed
it contains some but not all limit points
gritty widget
can you please not be concise 🙂
nocturne basalt
can you please not be concise 🙂
are you familiar with why something of the form [a, b) is not open or closed in R
gritty widget
no
gritty widget
are you familiar with why something of the form [a, b) is not open or closed in ...
not open because we have a and not closed because we don't have b?
nocturne basalt
kinda, are you aware that a set is open iff every point has an open ball nbhd contained in it (assuming standard euclidean topology)
nocturne basalt
yes i've read the definition
ok, disregarding the sequence argument, do you see how neither the set or it's complement are open?
using this definition?
gritty widget
if y is 1 then there isn't any open ball that is a subset of E
gritty widget
and for the complement there isn't a an open ball for (0,0) that is a subset of E
i think
gritty widget
i could have only said that without the sequence ?
nocturne basalt
basically the sequence is a way to justify this argument
bc if u have a sequence, you get points arbitrarily close
like the sequence is an example for why the open balls don't exist
like the sequence to the origin tells us that for any epsilon we can find some point in E within epsilon from the origin
which means that there does not exist some epsilon such that an epsilon ball around the origin is completely in the complement of E
if that makes sense
@gritty widget
nocturne basalt
ok just checking in
is there any part you're still confused about?
like you are correct about how certain points in E and it's complement have no open balls completely contained in either E or it's complement
but the way we justify that is with the sequences
gritty widget
didn't understand the sequence part tbh
nocturne basalt
ok so how do we prove that there's no open ball at the origin completely contained in the complement of E
so assume for contradiction there exists at least one, with radius epsilon
i.e. any points within epsilon of the origin are in the complement of E
but we have a sequence of points in E going to the origin
so there do exist points within epsilon of the origin which are in E
nocturne basalt
ok so how do we prove that there's no open ball at the origin completely contain...
so we've proved this
this tells us that the complement of E is not open so E is not closed
it's similar for proving why E is not open
supple sable
What are we quotienting by/what relations are we adding to the group?
Like i understand why the group shrinks, since we are effectively filling in a hole with a face so the loops around the hole can be contracted but i don't quite get what we are quotienting exactly (for computation purposes)
uncut surge
I suppose you can read this as quotienting out the subgroup generated by the equivalence class of \psi?
unreal stratus
This is a bit sloppy I guess
Psi needn't even be a loop in the fundamental group
But the point is any loop (regardless of where it starts) defines a normal subgroup of pi1
high hill
huh it surely is isnt it?
unreal stratus
Oh wait I misread lol
high hill
base point is psi(1)
uncut surge
The equivalence class could be trivial but it's a fine loop aye
high hill
Like i understand why the group shrinks, since we are effectively filling in a h...
I dont know about 'filling a hole with a face' (could be multiple holes, could be none)
unreal stratus
Like usually you allow for pi1(X,x) for any x, not necessarily psi(1)
high hill
I suppose you can read this as quotienting out the subgroup generated by the equ...
Yh. All the loops gotten by continuously deforming psi(S1)
and maintaining the basepoint
supple sable
ah fair i didnt make the connection it was a loop
but that makes sense since it maps from S1
storm dagger
I have had a hard time nailing down the details of a subspace topology. Given the upper-limit topology on R, (a, b], a < b, select intervals that are open in the subspace topology Y=[0,5). For example, given (2, 5). I believe this is open in the subspace topology, but doesn't it also need to be an open set in the upper-limit topology?
stark fog
I have had a hard time nailing down the details of a subspace topology. Given t...
not necessarily, so if Y is a subspace of X, the open sets of Y are of the form U cap Y, where U is open in X. But being open in the subspace topology doesn't imply it is open in X because for example
take the real line and [0, 1] with the subspace topology, [0, 1] is open in the subspace topology but is closed in the real line
storm dagger
Are you saying X=R, Y=[0,1], U=[0,1] ? I still have the same confusion, "where U is open in X" because [0,1] is closed in X.
prisma geyser
What are we quotienting by/what relations are we adding to the group?
Is this Lee’s top mfds?
I think it is, I recently started using the book :0
stark fog
Are you saying X=R, Y=[0,1], U=[0,1] ? I still have the same confusion, "where ...
no, Y = X cap Y and X is open
storm dagger
no, Y = X cap Y and X is open
Is that a typo since it's a recursive definition of Y?
stark fog
no
storm dagger
FML
next crystal
Is that a typo since it's a recursive definition of Y?
you can take X=R, Y=[0,1] w/ subspace topology
Then [0,1] is open in Y because [0,1] = [0,1] \cap R for example
or you can say [0,1] = [0,1] \cap (-1,2)
where U = R in the first example and (-1,2) in the second
thick oak
I am trying to show that given either $\times$ or $\smile$, the associated product $a\smile_\times b\mapsto\Delta^*(a\times b)$ and $a\times_\smile b\mapsto p_1^*a\smile p_2^*b$ are natural. However, I'm not sure how to go about this. Can anyone point me in the right direction?
gentle ospreyBOT
TheRedLotus
tidal lynx
I have had a hard time nailing down the details of a subspace topology. Given t...
Yes this is perfectly fine. In general the subspace topology only coincides with the ambient topology when Y itself is open in X
void gazelle
Hi, guys, why for a path f: I\to X, where X with discrete topology, f is constant?
unreal stratus
hint: connectedness
tidal lynx
what happens when f takes on two separate values f(x1)=y1 and f(x2)=y2
void gazelle
hint: connectedness
Thanks, i see
unreal stratus
Also like
If empty set is not counted as connected then X is connected iff every map from X into discrete {0,1} has exactly one ponit in the image
coarse night
If empty set is not counted as connected then X is connected iff every map from ...
technically it can be written as a union of 2 disjoint open sets
unreal stratus
True
thick oak
I'm still completely lost.
obtuse meteor
If empty set is not counted as connected then X is connected iff every map from ...
Just modify the defn. Has <= to one point in the image :)
Stupid but correct
unreal stratus
I mean to me empty set is not connected which is why i said it lol
obsidian socket
Are universal covers just initial objects in the category of covering spaces?
plain raven
Are universal covers just initial objects in the category of covering spaces?
They're not initial because they can have nontrivial automorphisms.
But if you take the basepoints into consideration, something like this might be true.
I don't remember the exact caveats you need.
gritty widget
i don't think that it is right , can someone tell me what i did wrong
deep ibex
i don't think that it is right , can someone tell me what i did wrong
It's quite hard to read the working and tell what you are doing.
Try looking at $\langle (1,1,\dots ,1),(|x_{1}|,|x_{2}|,\dots ,|x_{n}|)\rangle$
gentle ospreyBOT
384920
deep ibex
You're welcome!
void gazelle
Hi, guys, if we want to show \pi_1(X\times Y, (x,y))\cong \pi_1(X,x)\times \pi_1(Y,y), then we do not need X, Y are path connected. Is this true? because i did not see anywhere we use path-conneced assumption in the proof except the statement omits base point
coarse night
convince yourself it doesn't matter
void gazelle
Thanks
coarse night
lol
That's right you use it to omit the basepoint, otherwise you can just look at the parh component containing the point
unreal stratus
Think its a bit silly adding that hypothesis
feral copper
Isn't the empty set both path-connected and disonnected? 
coarse night
It’s connected
unreal stratus
These things are like edge cases where conventions differ right lol
unreal stratus
Isn't the empty set both path-connected *and* disonnected? <:trollge:10196573337...
I think under the like "naive" definitions that is probably the case lol
Personally I quite like the idea of just saying it's (path) connected iff there's one (path) component and some do that but it depends
feral copper
I was just being a meanie troll 😉
unreal stratus
Lol
zealous acorn
(in which case the empty set is path-disconnected lool)
coarse night
empty set is connected even with the naive definition
unreal stratus
Many/some say it is disconnected lol
coarse night
unreal stratus
that is what mine has
coarse night
I want to meet them
unreal stratus
i am here lol
coarse night
what definition are they using
unreal stratus
Well you can give that one I mentioned above
I'm trying to think of good reasons for saying the empty set is disconnected though and can't remember any good ones lol
tbh probably doesn't matter
coarse night
but like it contradicts all other defintions
unreal stratus
wdym
feral copper
I'm not sure you can define a path-connected component without referring to an element in that component, and the thing is: any element of the emptyset has whatever property you need
unreal stratus
The point is there are 0 components, like this is the empty relation on the empty set
feral copper
Oh you could consider continuous functions from X to Z, but idk
coarse night
connected if there's no surjection onto {0,1} or even writing as uniton of 2 disjoint non empty open sets
unreal stratus
Well
I have also heard like "connected iff every continuous function into {0,1} is constant"
coarse night
more like should have ≤1 component
unreal stratus
I mean of course the equivalent definitions are all gonna say the empty set is connected if they are equivalent right lol
coarse night
the empty function is technically constant
feral copper
I have also heard like "connected iff every continuous function into {0,1} is co...
The range of a function from O/ to {0,1} is empty sooooo 😄
feral copper
Empty range = constant? But hey, in any case, you're right, who cares about the empty set 
hidden crag
empty set discussion 
unreal stratus
Oh yeah here's a good point lol
feral copper
Usually it should be kicked of any reasonable theorem/definition, so whatever 🙂
uncut surge
i guess you could define constant as "range = singleton" or as "forall x,y, f(x) = f(y)"
unreal stratus
I guess
uncut surge
latter seems more convenient
feral copper
i guess you could define constant as "range = singleton" or as "forall x,y, f(x)...
You should avoid using 'forall x [...]' because that's usually what fails with \empty
uncut surge
i think i'm fine excluding the empty space
feral copper
Oh because \empty x X = \empty?
uncut surge
that one has never done anything for me
unreal stratus
Oh because \empty x X = \empty?
yh
feral copper
Fair enough 🙂
coarse night
maybe add non empty product instead
unreal stratus
But I think decomposition into path connected components is only unique if you require empty set to be disconnected
etc
Meh
Doesn't rly matter
coarse night
uncut surge
yea i guess this is kinda like "should 1 be prime"
feral copper
A good thing to do is the following:
The empty set is neither connected nor disconnected, it's a third edge case that nobody ever needs to deal with
coarse night
they are mutually exclusive and exhaustive
feral copper
yea i guess this is kinda like "should 1 be prime"
Yeah well, the ring-theoretic definition of prime-ness excludes invertibles, for the reason that we want uniqueness in UFD's
coarse night
coarse night
Dedekind domains 
vapid imp
How do you define the union/intersection of a collection?
Open question
What I've seen:\
Given set $X$ and $\mathcal{C}\subseteq\mathcal{P}(X)$, we define $\bigcup\mathcal{C}={x\in X:\exists S\in\mathcal{C},x\in S}$ and $\bigcap\mathcal{C}={x\in X:\forall S\in\mathcal{C},x\in S}$
gentle ospreyBOT
Flip
vapid imp
Because then you can say that the union of an empty collection is the emptyset, whereas the intersection of the empty collection is X itself, so in the definition of a topology T of X, you don't need to specify that emptyset and X are both in T
vapid imp
It pertains to the definition of a topology so here I am
Thank you though
Anyway I heard this wasn't common today so now I'm sampling lol
steel glen
This may be a silly question, but can we say H_1(X/A) is contained in H_1(X), or is a subgroup even?
is there any relationship i can abuse to say that if H_1(X) is not free abelian, then neither can H_1(X/A)?
ok i think i can with the inclusion homomorphism
so i guess my original question is that it is indeed a subgroup
bitter smelt
This may be a silly question, but can we say H_1(X/A) is contained in H_1(X), or...
No. It is not true. Consider X/A = S^1 where X is the closed unit interval and A is the end points.
If X and A form a CW pair you can say certain nice things.
(even better, a "good" pair)
steel glen
imm doing hatcher 2.1.26 so im not working with good pairs
the issue is im trying to show the first homology of the hawaiian earring is not free abelian
but apparently computing the homology of the hawaiian earring is pretty rough
plain raven
Because then you can say that the union of an empty collection is the emptyset, ...
I think you're right.
steel glen
ok it turns out to be a countability argument
nocturne basalt
ok it turns out to be a countability argument
yes it is
steel glen
i used abelianization of pi_1(H) so kind of
i was going the wrong way about it at first
nocturne basalt
yeah this one you get that one is countable and the other is uncountable
hallow moss
For something like this, am I supposed to be creating an explicit homeomorphism for two circles?
one with [0,1] and the other with [1,2] under the equivalence relation?
I'm not really sure how to get the explicit map with this stuff yet
nocturne basalt
what's the definition two circles with a single common point? S¹ v S¹?
hallow moss
I'm not sure, I have not seen it yet, this is just point set topology
I assume it's just a figure 8, idk
I tried something like this but, idk
abstract saffron
I'm not really sure how to get the explicit map with this stuff yet
see S1 as [0, 1]
hallow moss
yea I see that 1
abstract saffron
which, is basically [1, 2] translated
then you attach two circles at a common point, which is 1
nocturne basalt
yeah i guess embedding in R^2 works, but it would be simpler to write a map from two copies of [0, 1] w the proper identifications
abstract saffron
it can be anything in fact, you can define equivalent relation where you identify a number of your choice, x \in [0, 1] with another y in [1, 2]. The map is messy, however, but it's the same
That's why I generally avoid writing down explicit equations in topo 😄
(technically not, since [0, 1] \cup [1, 2] = {1}, but let's ignore that)
hallow moss
Hmm I think I'm still at the place at where I still need to find the explicit equations unfortunately 😦
the book is finding them 😦
nocturne basalt
yeah if you think of two circles with a point in common as (I_a \coprod I_b) / (0_a = 1_a = 0_b = 1_b) then you can take the map x_a -> x, x_b -> x + 1
hallow moss
For this one, am I supposed to see the lines in D^2 y=x and y=-x as the origin?
nocturne basalt
err the x and y here arent x and y axes
hallow moss
oh my god
nocturne basalt
yeah i think its a poor choice of name lol
abstract saffron
err the x and y here arent x and y axes
but not that far off 😄
There's a very nice way to see how that equivalence relation is basically folding the circle into a quarter of a circle
Then the explicit map construction is just technical
nocturne basalt
yeah the explicit map is annoying but i think you can just do something along the lines of projecting
abstract saffron
Wait, no, I'm fooling myself, it's not that intuitive, hmmm
hallow moss
sorry, but why would it be a quarter circle and not a hLD circle?
Wouldn't it just look like the upper half of the disk?
nocturne basalt
isn't it a semicircle
abstract saffron
ignore what I said, I was thinking of something else
nocturne basalt
two pairs of quadrants are identified
leaving two quadrants left
oh nvm the topology might be diff
abstract saffron
be careful of the boundary, you also have to re-attach them
if what I'm thinking is right, it'll look smth like a cone?
which is basically a disc
boy, the map is gonna be very painful to write
abstract saffron
when you don't do a Math degree 😄
hallow moss
so for this I can't just write like a homeomorphism between the upper half of the disk and entire disk since like the quadrants are flipped or something?
abstract saffron
when you don't do a Math degree 😄
no but srsly, you can draw stuff only when you can justify when it's right, and sense when it's wrong. I have played with topo far too long to sense when an intuition fails. And I'm not even a math student
abstract saffron
so for this I can't just write like a homeomorphism between the upper half of th...
.... I guess? not sure what you mean here
also it depends on the prof. Some won't tolerate that
nocturne basalt
yeah idt thats enough since its not just like the upper half of the disk
abstract saffron
so for this I can't just write like a homeomorphism between the upper half of th...
I can roughly describe what I imagine though: you cut the disc into 6 parts, Four quadrants, and two perpendicular lines at the cut, say x- and y-axis.
Identify two quadrants with the other two, an axis will become the common axis between the two quadrants. Identify the two halves of the other axis, you have two quadrants glued together at two edges
i.e. something looks like a cone
Yeah, good luck writing that down rigorously 
hallow moss
nocturne basalt
actually its not that bad if you think of it as the disc cut with one half flipped and then both halves identified
and then working in polar
abstract saffron
you don't flip the half circle in the last step, only identify the two segment
that's why I say it looks like a cone
actually, what is this? I didn't mean this pic 😄
nocturne basalt
this is one way to get it as a cone
but if you treat the cone as this quotient space it becomes easier to write down the homomorphism w the disc
abstract saffron
for me it looks something like this
regions of same colours are identified. AD ~ AC, so that's done, then AE ~ AB, so you have a cone out of the left half, glued together at AC and AE(AB)
civic verge
The best book for topology?
bitter smelt
pointset, algebraic, or differential?
civic verge
It is an introduction, so it would be the pointset one.
bitter smelt
Munkres is the standard
civic verge
Another book that complements it ?
bitter smelt
huh?
burnt blaze
I believe topology without tears is recommended a lot. It's what I'm planning on working through shortly.
as an aside, would I be yelled at for coming here with proofs for certain exercises from that book?
bitter smelt
you would be amazed...
civic verge
thanks
I will be keeping those two books in mind for my next semester looking at topology :'3
bitter smelt
people who complain that munkres isn't well motivated or interesting (as if point-set can be) tend to recommend Janich Topology and kelly's general topology
ive not read either so i cant speak to that
you'll note all of these books are old
now a book that can be some fun restroom reading or something is counterexamples in topology (part ii of the book)
abstract saffron
Another book that complements it ?
Spivac is not too intimidating
civic verge
Spivac is not too intimidating
abstract saffron
there's also Differential Topology: First steps by Wallace. Quite thin, but it goes up to Gauss-Bonnet theorem and classification of compact surfaces at the end.
Also a lot of intuition, drawings and examples, which I like
civic verge
Nope, I'm just looking for some general topology to learn, to be prepared.
abstract saffron
Nope, I'm just looking for some general topology to learn, to be prepared.
then I don't know about that anymore 😄 I usually refer to my old lecture notes for that one
hallow moss
Btw, what are these universal properties that keep showing up
I just showed the universal property of the quotient topology, I could have sworn I also showed something similarly named for the weak topology a chapter ago or something
abstract saffron
sounds like universal property from group theory, is that the one?
Idk why my prof used that name
Oops, nope, sorry, my bad
wait, this property has a name ??? 😄
hallow moss
no idea uhh I will get back to dying
abstract saffron
no idea uhh I will get back to dying
I'll join you
bitter smelt
A universal property is a category theoretic term which gives you a unique map in certain situations
you can read more here: https://en.wikipedia.org/wiki/Universal_property
hallow moss
Well I will maybe understand this when I have a reason to
someday
looks very useful though
I couldve skipped all that pointset mess in the proof
if i knew cat theory
bitter smelt
Just because you could doesn't mean you should. Cat theory is great, but when you're first learning a subject, sometimes it's best to work with proofs based solely in that subject, to better build intuition and give you more experience with it
novel acorn
Btw, what are these universal properties that keep showing up
maybe another way to see it is that a universal property of some object is some property of it that makes it unique amongst other such objects
this is a bit of a vague statement but take for example the universal property of an embedding
The condition sets it apart from just some general injective map
in most cases you'll see it tho is something like
Object X has a universal property such that for any other object Y that does sorta the same thing as X there exists a unique map X->Y or Y->X that makes some diagram commute
And this sets X apart
abstract saffron
I think the cleanest way would be considering a continuous map from X to {0, 1} equipped with discrete topo, then show that map is constant
grave solstice
Waita minute, does {U_alpha} cover X?
in previous discussions they did cover X, but idk why it is not mentioned in these exercises 
abstract saffron
lmao
abstract saffron
It does, because if U_alpha does not cover X, then God know what else X comprises of
you can at most talk about union of U_alpha here
grave solstice
tbh I dont see what doesn't work
take stupid case where there is just one U_alpha=Y, so that Y is a subset of X and Y has a topology. Now you want to define a topology on X by declaring a subset U of X to be open iff U cap Y is open in the topology of Y
- Empty set clearly open.
- Arbitrary unions open by distributive properties of union and intersection, ie, $Y\cap \bigcup U_n=\bigcup Y\cap U_n$.
- And finite intersection is like $Y\cap\bigcap_{n=1}^N U_n=\bigcap_{n=1}^N Y\cap U_n$
gentle ospreyBOT
Croqueta
abstract saffron
I see what you mean: X is connected wrt the induced topology by U_alpha. That'll also do. It's just that the statement mentioned it nowhere
void gazelle
Hi, guys, in this proof, how can i show that the map: H[g] \to \tilde{g}(1) is well-defined? Suppose gf^{-1}\in H, then why \tilde{g}\tilde{f^-1} = \tilde{gf^-1}?
unreal stratus
They've shown that each element of the coset has a lift ending at the same place
void gazelle
sorry, why if f,g is a loop in X at x_0, then their lift ending at the same place
Oh oh i see i see
Thanks
odd flame
im trying to fill in some of the details of this proof
for b -> c
is this the universal property of direct sum in action?
plain raven
The category of Abelian groups is somewhat special. The direct sum of two Abelian groups has two conceptually distinct universal properties that happen to coincide.
It has the universal property of the product and the universal property of the coproduct.
The universal property of the product gives necessary and sufficient conditions to construct a homomorphism into the direct sum.
The universal property of the coproduct gives necessary and sufficient conditions to construct a homomorphism out of the direct sum.
odd flame
hmmmm
so it's both of those properties that make that an isomorphism then?
also doesnt the fact that it's a finite direct sum mean that we get the property of the product anyways? like independent of them being abelian
plain raven
also doesnt the fact that it's a finite direct sum mean that we get the property...
The term 'direct sum' in my experience is only used for Abelian groups. It would be inappropriate to refer to the Cartesian product of non-Abelian groups as a direct sum. The coproduct of not-necessarily-Abelian groups could perhaps be called a sum but that is a matter of taste and it is not to my taste at all.
plain raven
so it's both of those properties that make that an isomorphism then?
I won't answer that directly because it's a little vague for me.
If an object A has a universal property, then we have that:
For each object X, and given data (....) such that (...), there exists a unique map A -> X such that (...).
(or sometimes, there exists a unique map X -> A such that ...)
One strategy for using universal properties to prove that A and B are isomorphic is to use the existence part of the universal property to construct the map from A to B, and/or from B to A; then use uniqueness to prove that the composition A -> B -> A is equal to the identity, because this map and the identity both have the desired property
odd flame
i'll keep it in mind then thank you clerk
novel acorn
for b -> c
I have not really seen it done this way
Rather people usually prove (c) implies (b)
unreal stratus
Hm if A is a contractible closed subspace of a compact Hausdorff space X, is X homotopy equivalent to X/A?
More specifically, I wanted to show that if A is a closed subspace of CH X then X U CA is homotopy equivalent to (X U CA)/CA I think
It is proven in some books that this induces an isomorphism in K theory (but they use methods only working for C) and other places state but do not prove this is a homotopy equivalence, lol
unreal stratus
Contractible as a space in itself
I found a counterexample for the first bit online w the hawaiian earring
Hm
plain raven
Oof.
Can you assume that the inclusion is a cofibration?
Or can you assume similar strengthenings, i.e. that (X, A) is a good pair
umbral panther
The topologist sine curve should give an easier counter example
unreal stratus
Yeah inclusion being a cofibration would be enough
Hm
I'm just confused lol cause I was trying to change a proof from complex K theory to real K theory and it seemed to break down because GL_n(R) isn't connected
So this was how I was gonna patch it up lol
But thanks
wise ruin
In this definition, do we care if \tilde{f} is equivariant wrt some homomorphism G_1 -> G_2? If we do, is that condition implicit in the phrase "lift"? And if so, do we just need there to exist a suitable homomorphism or is a collection of homomorphisms implicitly contained in the map?
wise ruin
It just feels like this definition has too much leniency given that it doesn’t even depend on how G1 and G2 act on these charts and, further, is this definition of smooth somehow strong enough to guarantee that all diffeomorphisms between orbifolds are (Ruan-Chen) good maps, or are there diffeomorphisms between orbifolds that are not good maps?
coarse night
is it true that $\varinjlim \pi_n(X^{(i)})= \pi_n (X)$ where $X^{(0)} \subseteq X^{(1)} \subseteq \cdots $ is a CW filtration or even a filtration where evey compact set lies in one of the $X^i$'s?
gentle ospreyBOT
lunar yoke
is it true that $\varinjlim \pi_n(X^{(i)})= \pi_n (X)$ where $X^{(0)} \subseteq ...
Yes, if you have a sequence of closed embeddings of T1 spaces then the induced map is an iso
because for such a sequence, for any compact space K, a map K -> colim X_i factors through one of the X_i
coarse night
Why do I need T1
Is this a different filtration you are taking about than what I said?
lunar yoke
idk what you mean by filtration tbh, but what i said works for the chain of inclusions of CW-skeleta
The T1 is just some pointset stuff because arbitrary spaces can be weird. Don't know a counterexample off the top of my head though
if you're working in the nice category of compactly generated weak hausdorff spaces you also don't need the T1 assumption and just need the maps to be closed embeddings
novel acorn
What definition of continuity is being used here?
Maybe I'm just having a moment lol
lunar yoke
What definition of continuity is being used here?
Maybe I'm just having a moment...
preimage of nbhd of point is nbhd of point
novel acorn
preimage of nbhd of point is nbhd of point
yeah but I'm confused why they only check that the nbhd maps into the nbhd in the codomain?
They never check that the preimage of the evaluation map is actually open just that it contains a nbhd
lunar yoke
the preimage need not be open, just a nbhd of the chosen point (f,x). He shows that the preimage contains the nbhd W(K,U) x K of (f,x) and is thus a nbhd of (f,x) itself
gritty widget
is it correct if i choose alpha = 0.5 and beta = 2 in the third question?
feral copper
Have you tried making a drawing?
So I guess that $\alpha=1/\sqrt{5}$ and $\beta=\sqrt{2}$
gentle ospreyBOT
Matplotlib
gritty widget
that what i get with the first usual norm on R²
feral copper
Ah sorry the norm-1 not norm-2
Yeah that's correct
Although I have a feeling this is more suited for #linear-algebra
gritty widget
Although I have a feeling this is more suited for <#540211747613704221>
sorry
and thank you
odd flame
what is the extension property being referred to here
paper wedge
the extension property is that there exists such map such that when u restrict it to the abelian group u get the original map
extension property is just saying that this exists and justified
hidden crag
the extension property is that there exists such map such that when u restrict i...
This only makes sense when talking about substructures of some sort
What’s being used here is that you can extend any set map from the basis of your free abelian group into the other group to a homomorphism
Like when you define a linear map between VS on Basis elements and extend
odd flame
so we define a map (call it phi) from the basis in C to B
this induces a homomorphism from C to B?
well that's just repeating what timo said
unreal stratus
ye
odd flame
🥔
unreal stratus
a map of [blah]s from a free [blah] F into a [blah] G is equivalently a map of sets from a basis of F into (the set underlying) G
where [blah] = vector space / group / abelian group / module etc
coarse night
Free functor
odd flame
Functor? I hardly know her!
coarse night
you will
undone valley
I was trying to better understand how to calculate the genus of this surfaces
my idea was acting via surgery removing the two crossing bands in the middle
I should obtain what looks like a torus on the right and a sphere with a disk removed on the left but I'm not sure I'm doing the smartest thing to evaluate it
I don't like the behaviour of boundary components
feral copper
It's very crude, but it could be working if you have a lower bound for the genus. Maybe look at the degree of the Alexander polynomial?
Oh wait you mean the surface given by this band presentation, not the link genus?
You could start by doing some handle slides to have a presentation in a standard form
And then the homology shouldn't be too difficult to compute
Actually no, if you put it in a standard form, you can directly see what surface it is (by standard form I mean disks connected by bands without overlapping of the bands)
Screw that, it's even easier x')
There's essentially one way of attaching a 1-handle, so you can remove all the twists and stuff and slide to your heart's content
Count the number of 1- and 0- handles and you have the whole homology
undone valley
yeah it's what i wanted to do but tried to be even more lazy with the surgery argument ahah
thank u anyway
undone valley
the idea of surgery was to have already 2 surfaces in standard form but I will try just to work with isotopy on this surfaces
thanks again
feral copper
Im' just saying that you don't even need to bother with this machinery 😉
void gazelle
Hi, guys, why in this proof, this statement implies that X_H is a covering space?
feral copper
How did they construct the universal covering space? It seems like it's a space of paths. My guess is you should simply use the definition of both \tilde{X} and of a covering map and it will follow
feral copper
Surely you can prove that the map \tilde{X} -> \tilde{X}/~ is a covering map
unreal stratus
I believe these are just subgroups of F({a,b})
Like the <...> just mean "the subgroup generated by"
tawdry valve
in this context too, Hatcher is referring to the subgroup of $F({a,b}) = \pi_1(S^1 \vee S^1, pt)$ which is $p_*(\pi_1(X,x))$
feral copper
They are the covering spaces corresponding to that subgroup of F_2
gentle ospreyBOT
Joseph
unreal stratus
Indeed
feral copper
No
unreal stratus
Nah fam subgroups of da fundamental group
feral copper
**Joseph**
This is this
unreal stratus
Corresponding to le covers
feral copper
Recall the 1:1 correspondance between conjugacy classes of subgroups of \pi_1 and between equivalence classes of connected covering spaces
It's this (bonus: the index of the subgroup is the index of the covering)
nocturne basalt
hatcher talks about the galois correspondence in the same chapter
feral copper
The group of deck transf has a generator: describe this map
(it's a Z/2)
So it is just asking for the involution
E.g. for the first line of examples, it's a rotation by 180°
(rotation around the center which is not in the space, you rotate the figure, you know what I mean :P)
The non-trivial element swaps points pairwise, not just the two lifts of the basepoint
For 3 and 4 the group of Deck transforms is not a Z/2
So it's not an involution
It's a Z/3 yes
(three lifts for the basepoint)
Should be a map of order three
I would say it's not so clear
I'm confident one should be and the other not, but you'd have to check the conjugates of the subgroup
(I mean if I were to write the exercise I'd put one normal and the other not)
Hey actually yeah they're the same space xD
It's just that he chose two different lifts for the basepoint
Yes
feral copper
Recall the 1:1 correspondance between conjugacy classes of subgroups of \pi_1 an...
Because again, this
feral copper
Wish I was taught branched covers rather than discovering it all by myself x')
unreal stratus
Nothing
feral copper
You gonna do intersection theory?
unreal stratus
I would be destitute
feral copper
Maybe they're just the same subgroup?
unreal stratus
I have revised everything else except the covering spaces lol
feral copper
Just different set of generators
Have you tried Tietze transformations to go from one to the other?
Very nice article
unreal stratus
Tietze transformations are in our course but given basically 0 application lol
feral copper
Tietze transformations are in our course but given basically 0 application lol
They're amazing but actually they're not, it's just common sense made rigorous
But the thing is (unless I'm wrong): it's not known whether same group <=> presentations are Tietze equivalent
unreal stratus
Ye sure
feral copper
(unless you allow for an infinite sequence of transformations)
Yes it should be a 3-cycle
But I must say, I have trouble drawing it (remember it mustn't have any fixed points)
Here, we can even say that the subgroups 3 and 4 are the same (not just isomorphic or conjugate), because the covering equivalence is the identity map of the space
The deck transform generator for 1 or 2 was the rotation by 180°, this is what I mean by drawing it 🙂
This isn't the same space, right?
The covering space is the subgroup, right?
The deck group is the quotient by that subgroup
(roughly)
Which makes me think that the group of Deck transformations of 3 and 4 is weird
I think I'm too tired and I'm missing something, but that subgroup doesn't seem normal to me
Yes, I'd say it's the quotient by the normal closure then?
Idk
Doesn't Hatcher say it?
golden gust
But the thing is (unless I'm wrong): it's not known whether same group <=> prese...
does tietze equivalent not mean "related by a finite sequence of tietze moves"?
feral copper
does tietze equivalent not mean "related by a finite sequence of tietze moves"?
I remember asking myself that question, and the Wikipedia page is very sketchy about this: https://math.stackexchange.com/q/3893972/259363
You need an infinite number indeed, and the definition in Wikipedia seems to say a finite number only
golden gust
ahh sorry the theorem I'm remembering is for finite presentations
feral copper
Ok, the grroup of deck transf is the fundamental group of the quotient \tilde{X} by the action of the subgroup
feral copper
ahh sorry the theorem I'm remembering is for finite presentations
I'm not sure it's even true for finite presentations, there's always something fishy you need to be careful about in group presentations!
golden gust
it's a theorem that any two finite presentations of a group are related by finitely many tietze moves
feral copper
Tietze also showed this in his paper? Okay then, thanks for correcting me!
golden gust
I don't know the literature but I assume so, because the theorem is attributed to him in my notes
unreal stratus
Helo
golden gust
me too just looking at c_1
golden gust
but I'm looking at their definition as natural transformations
golden gust
$\mathrm{Vect}_n(-) \to H^q(-;G)$ for fixed $n$
gentle ospreyBOT
(m+p)akka
feral copper
c_1 is all that matters
w2 would like to have a word with you
unreal stratus
For complex things 😭
unreal stratus
$\mathrm{Vect}_n(-) \to H^q(-;G)$ for fixed $n$
Ye sure
feral copper
Okay they're kinda the same, but still xD
golden gust
but why do we need to fix the n? because I'd like to define c1 in this way in my essay but I want it to be defined on $\mathrm{Vect}(-)$
golden gust
idk this is general from some notes but take G=Z
gentle ospreyBOT
(m+p)akka
golden gust
which definition of what
this is from some notes
and eg hatcher does it (the total chern class) in more words like "an assignment to each vector bundle..."
unreal stratus
Oh okay lol I wasn't sure if we were talking about general classes or chern classes
golden gust
I mean I want c1 in the end so if there's something simpler for that I'm 👂
unreal stratus
this is from some notes
Ye this seems p standard
golden gust
but for example how do you talk about c(E (+) F) = c(E) cup c(F) if you fix the ranks of the bundles
unreal stratus
I believe you essentially define char classes to be for specific ranks but then also define "total" classes like this
And similarly for other things like steenrod squares
But hm yeah i've seen people give other definitions e.g. uh
golden gust
oh it seems like this is necessary because you need to fix the group acting on the bundles
unreal stratus
i wouldn't massively worry about it tho lol like there is a unified vibe
golden gust
yeah yeah it's just for what to write down
unreal stratus
this is from some notes
but ye this seems fairly standard
Fair nuff
I am having to omit lots of the "basics" bleak
how much are you talking about bundles etc
golden gust
I used to have a dedicated section talking about basics but basically now I'm just assimilating everything into the sections they're relevant for
golden gust
how much are you talking about bundles etc
not a huge amount
did you figure out the yoneda product on ext
plain raven
Yoneda product on ext is weird
probably deep but i don't understand it lol
Like what does it contribute to our understanding of cohomology?
Jack Duskin develops the Yoneda Ext in his thesis on monads in homological algebra
abstract saffron
unrelated, but when I see monad, I automatically read "monoid in the category of endofunctors"
cedar pebble
Like what does it contribute to our understanding of cohomology?
it's useful to think of cohomology groups as Ext groups in some category, so then the Yoneda product gives you some natural cup product on cohomology
idk you get a surprising amount of mileage in AG for example by going back and forth between cohomology as a geometric thing, and cohomology as Ext
warm quiver
Like what does it contribute to our understanding of cohomology?
Mac Lane's Homology goes pretty deep into that construction, so that's where I would check (like Chapter 8 around page 229)
cedar pebble
there's a fun construction you can do with Ext^2 along the lines of "enhanced bi-extensions" which is related to height pairings that come up in arithmetic geometry, it's a very cute construction
unreal stratus
did you figure out the yoneda product on ext
Oh lol
So I probably didn't tell you abotu this lol but basically I talked to my other advisor (Henriques) and he said that "if you try it you will fail"
Because the papers I was given are way too long to present in a single essay
cedar pebble
If you have a class in Ext^1(M,A) corresponding to an extension 0->A->E_1->M->0 and Ext^1(B,M) corresponding to an extension 0->M->E_2->B->0 Yoneda gives you a class in Ext^2(B,A) by pasting the extensions to get a 2-extension 0->A->E_1->E_2->B->0. Suppose this class in Ext^2 is 0, so these extensions are "compatible" then you can complete the composition E_1->M->E_2 to a square and get a diagram like uhhh
golden gust
So I probably didn't tell you abotu this lol but basically I talked to my other ...
"other" you have two?
also yeah fair enough
unreal stratus
Yeah lol it's cause the main one was in Paris
but tbh not talked to them enough
But yeah anyway I went from doing the image of J to doing Hopf invariant One
and the connection to H-space structures on spheres and division algebras
which tbh is more fun than the vector fields part lol
cedar pebble
$\begin{tikzcd}
& & 0 \arrow[d] & 0 \arrow[d]\
0 \arrow[r] & B \arrow[r] \arrow[d,equals] & E_1 \arrow[r] \arrow[d] & M \arrow[r] \arrow[d] & 0\
0 \arrow[r] & B \arrow[r] & E \arrow[r] \arrow[d] & E_2 \arrow[r] \arrow[d] & 0\
& & A \arrow[r,equals] \arrow[d] & A \arrow[d]\
& & 0 & 0
\end{tikzcd}$
gentle ospreyBOT
nGroupoid
golden gust
Yeah lol it's cause the main one was in Paris
oh wow how did you set that up
cedar pebble
in particular you get an extension 0->B->E->A->0, so a class in Ext^1(A,B)
so this gives you a pairing like
Ext^1(A,M)xExt^1(M,B)->Ext^1(A,B)
under certain conditions like vanishing of Ext^2
it's kinda cute
unreal stratus
oh wow how did you set that up
Oh I mean he is a research fellow here but was away for a couple terms
golden gust
very cool
unreal stratus
No u
Last part is actually easier lol
If I am thinking correctly
Yeah
Just use algebra
I remember having a funny ad hoc method for smth like this
But you can also use uh
A certain functor Grp -> Ab 🤓
||consider abelianisation||
Or the universal properties involved
golden gust
||consider abelianisation||
hahahaha what other functors are there
unreal stratus
No, consider abelianisation
no
Abelianisation of G is G/G'
okay so do you know what the abelianisation of F_m is
If not, guess
lol
ye
okay so can you see this would lead to a surjection Z^m -> Z^n
ye
it's not rly just clear immediately from the nonsense but like
If you have a map G -> H, how is the map G/G' -> H/H' defined
It is clear from this that abelianisation preserves surjections
ye nice
idk the most elementary way to show that there is no surj Z^m -> Z^n lol it'll just be appealing to Q i guess lol
golden gust
"Q, I'm stuck again"
abstract saffron
Oh I mean he is a research fellow here but was away for a couple terms
oh, where was he?
unreal stratus
Orsay
abstract saffron
Lol
unreal stratus
abstract saffron
Is there a chance I know him?
feral copper
idk the most elementary way to show that there is no surj Z^m -> Z^n lol it'll j...
Why not using a Z-basis?
unreal stratus
Idk how I am to know aha
unreal stratus
Why not using a **Z**-basis?
Like appealing to Z having IBN?
feral copper
What's IBN?
feral copper
Or you could just tensor by whatever field you want 
But that's... that's cheating it uses Q xD
abstract saffron
If you have a map G -> H, how is the map G/G' -> H/H' defined
butterfly lemma, maybe?
unreal stratus
but ye probably most elementary is saying a surj Z^m -> Z^n corresponds to a m x n (or n x m ig lol depends on conventions) of rank n which is impossible by viewing it as a matrix in Q lol or by just computations with the adjugate and stuff
unreal stratus
butterfly lemma, maybe?
Lol was like socratic
The poitn is any homomorphism G -> H sends G' -> H' and then apply first iso to G -> H -> H/H'
abstract saffron
but ye probably most elementary is saying a surj Z^m -> Z^n corresponds to a m x...
Another way (since I hate explicit computation) can be looking at the kernel
Z^m has a nice representation, iirc, since it's finitely generated
well, the kernel I mean. Also, shouldn't this be in #groups-rings-fields ?
unreal stratus
G \ X is X mod the relation ~ generated by x ~ xg
feral copper
You need the action to be free too
Otherwise the number of preimages is non constant
(i.e. it's a branched cover)
The preimage of a point is the orbit of that point under the action
Maybe try to show that in this case, any x has a neighborhood U such that: if g and h satisfy g(U)=h(U), then g=h
silver umbra
if i have a chart (phi, U) on a manifold such that phi(U) = B(0,r)
can i extend this chart to a map on cl(U)
such that phi(cl(U)) = cl(B(0,r))
and if it isn't possible for arbitrary charts, is it possible if we just take U to always be a regular coordiante ball or smth
coarse night
you may not always be able to extend the chart for example take S² with stereographic projection
silver umbra
sure, so it doesnt work for arbitrary charts
but its always possible to take a neighborhood of a point in a manifold that's a regular coordinate ball, right
so if we take a regular coordinate ball, does the claim work out?
coarse night
You can compose that map with a diffeo with unit ball
supple sable
I am working through hatcher and he says 1D CW complexes are called Graphs and in example 1.22 he says a maximal tree is a contractible subgraph that contains all vertices. This leaves me to expect algebraic topology has application within graph theory (existance of certain subgraphs) is this true?
novel acorn
I am working through hatcher and he says 1D CW complexes are called Graphs and i...
Algebraic topology was born out of graph theory
In some sense
A lot of people attribute the birth of topology/algtop to Euler's analysis of the bridges of Konigsberg
solemn oar
I am working through hatcher and he says 1D CW complexes are called Graphs and i...
Not readily. Topological invariants can detect loops, in some cases they can detect valence numbers. But in all cases I can imagine, it is the calculation of algebro-topological invariants that takes graph theoretical input, not the other way around.
void gazelle
Hi, guys, by the definition of delta-complex structure on X, i feel like we decompose X into the simplices, but how exactly this decomposition is? Do we just decompose X as interior of these simplices?
nocturne basalt
Hi, guys, by the definition of delta-complex structure on X, i feel like we deco...
well it's the collection of maps
void gazelle
yes, but the image of of each map is part of X
feral copper
So a probably dumb question, but: if a surface is null-homologous with coefficients in Z, this means that it bounds a 3-manifold. But what about being null-homologous with Z/2 coefficients? How should I think about it?
umbral panther
Bounds an unoriented 3-manifold
feral copper
Ah it's just that? xD
fading vale
Yes
You can make this stronger with a characteristic classes argument
an n-manifold M is cobordant to 0 in the unoriented cobordism group iff all its steifel whitney numbers vanish
and the same holds true in the oriented case if all the steifel whitney and pontryagin numbers vanish
coarse night
what would be a good reference to read these up?
feral copper
No I meant the embedded surface is null-homologous in the ambiant manifold, not the surface being cobordant zero
feral copper
what would be a good reference to read these up?
I know Milnor does it in Characteristic Classes
coarse night
The topology of fibre bundle one?
feral copper
The Stiefel--Whitney classes/numbers and cobordism invariants
coarse night
What should I google😞
hardy robin
hello! can someone help me please ?
i want to show that On(Q) is dense in On(R)
umbral panther
No I meant the embedded surface is null-homologous in the ambiant manifold, not ...
You shouldn’t ask for embedding
coarse night
Thanks
feral copper
You shouldn’t ask for embedding
Wdym?
I'm looking at the homology class of the surface in the 4-manifold
feral copper
hello! can someone help me please ?
i want to show that On(Q) is dense in On(R)
Maybe look at the decomposition of an orthogonal matrix into Jordan form and try to approximate any real 2x2 block by a rational 2x2 one?
Not the Jordan form sorry, the one where you have 2x2 rotation matrices on the diagonal
umbral panther
I'm looking at the homology class of the surface in the 4-manifold
Homlogy isn’t about manifolds. A 2d chain is automatically a manifold. A 3d chain isn’t. You can resolve the singularity, but I don’t think you can do it inside the 4 manifold. If you crossed with R3 first there would be room
hardy robin
Maybe look at the decomposition of an orthogonal matrix into Jordan form and try...
oh i didn't know this, i will look out for it
thanks!
plain raven
Hi, guys, by the definition of delta-complex structure on X, i feel like we deco...
yes, that's correct
However, these interiors are not open in X
void gazelle
Thanks, will not open be a problem?
coarse night
It's been 5 minutes I've started reading Stiefel Whittney class. It's defined using axioms but why is SW class of a trivial bundle 0 for i>1?
feral copper
It's part of the definition, you wouldn't want the trivial bundle to have non-zero invariants
coarse night
It's not one of the axioms tho. Does it follow from any trivial bundle is just ℝ ⊕ℝ...⊕ℝ and we know how it's defined for direct sums?
umbral panther
Yes, that’s the definition of trivial bundle
There are lots of definitions [of sw classes]. If you have a rule for direct sums, that should do it. But maybe you have a different definition
buoyant dew
From hatcher. Any help with this? I don't see how the homeomorphism of quotients induces the isomorphism Hn-1 (∆n-1, d∆n-1)→ Hn-1(d∆n,lambda)
Oh that just follows from the previous proposition lol
Damn hatcher words even the most trivial statement like a riddle
Worst math book in existence
feral copper
I also dislike Hatcher, and get disliked for that x')
coarse night
Same
feral copper
Let's found the anti-Hatcher gang
coarse night
🤝
feral copper
I mean, even the typesetting is ugly and doesn't invite you to have a pleasant read...
It's way too blocky
coarse night
Non intuitive intuitive arguments
It waved off the compression lemma like its trivial
Still salty about that
hidden crag
Mandatory Hatcher defense: nice pictures
pseudo coral
does anyone have hatcher handy
feral copper
Mandatory Hatcher defense: nice pictures
Yet not enough pictures
pseudo coral
and wanna help me understand a step in a proof in the appendix on CW complexes (which i have no clue what they are yet i have to present appendix A.1)
pseudo coral
Yet not enough pictures
yeah hatcher could use a bit more pictures specially with the cw complex stuff
novel acorn
hatcher is bad at describing pictures imo
Other stuff he is good at
feral copper
A Hatcher defense I'll never argue with is its open-access policy
That's an amazing thing to do
solemn oar
Worst math book in existence
This statement is too based for this server
novel acorn
Good examples
uhhh debatable
A lot of good ones
And some very very very bad ones
feral copper
Nah I wouldn't say very very bad
But I mean, I'm a guy who thinks there's never enough examples, and never enough pictures
I'd put a picture on every page if I was to write a book
hidden crag
I’ve never encountered really bad ones but I haven’t read the whole thing
pseudo coral
I'd put a picture on every page if I was to write a book
how could u do so with group theory and ring theory lol
novel acorn
Nah I wouldn't say very very bad
How would you describe his fundamental group of torus knots example then
feral copper
I don't care about that I'm a topologist 
novel acorn
feral copper
(not that I don't care, but it's just it's not what I'm discussing!)
feral copper
How would you describe his fundamental group of torus knots example then
Don't remember this one
novel acorn
Idk that is the worst passage in any book I have ever read even the pictures of what he's doing make no sense
feral copper
The fundamental group of a torus is easy though, there's not 100 ways of picturing it, so how does he do it?
unreal stratus
Oh yeah there are so many things in hatcher that never made sense til i looked elsewhere lol
pseudo coral
anhyone have hatcher handy n wanna go through a proof w me lol its in appendix A.1
unreal stratus
which proof
pseudo coral
im trhying to decode his proof of appendix A.1
feral copper
Proof of what?
pseudo coral
compact subspace of CW complex is conatined in finite subcomplex
pseudo coral
im unsure i know what a CW complex even is
feral copper
I hate those God damn lemmas
unreal stratus
hm there are a couple different ways to defeine it lol
pseudo coral
ive been assigned to present it sadly
ive been looking online for a good defn havent found one yet their so. confusing ..
solemn oar
Defining CW complexes can be a pain
feral copper
Worst one is:
If a subspace A contains the k-skeleton of X, then (X,A) is k-connected
feral copper
Defining CW complexes can be a pain
Defining it is fine, it's just proving stuff that sucks
solemn oar
I believe May has one
feral copper
Ah you fixed it, I was about to jump on this and say you don't allow for infinite CW-complexes
gentle ospreyBOT
potato
pseudo coral
and is X the union of all of these
unreal stratus
Hm good point lol
pseudo coral
X_i's
feral copper
and is X the union of all of these
Inductive limit, not just the union
There's a topology compatibility that's required too
But whatever 😛
solemn oar
Or rather, the colimit of the sequence of inclusions.
novel acorn
Ah you fixed it, I was about to jump on this and say you don't allow for infinit...
you do?
unreal stratus
Ye lol
feral copper
Man I put RP^oo before CP^oo what a moron I am
unreal stratus
Though "Infinite CW complex" can also refer to attaching infinitely many cell at any step too right
like R if you just take Z and join up stuff by intervals
novel acorn
I was thinking more of an infinite dimension CW complex
unreal stratus
but ye
pseudo coral
so in the proof
hidden crag
S^inf is not real
feral copper
Though "Infinite CW complex" can also refer to attaching infinitely many cell at...
Yeah, but you really need the inductive limit stuff for the topology of the whole space to be that of the subspaces
hidden crag
Contractible sphere
unreal stratus
S^inf is not real
it's just a point bro
novel acorn
wait so why don't we allow for infinite CW complexes lol
Is there some problems with the topology at the limit?
hidden crag
Hoax
unreal stratus
we usually do allow for infinite CW complexes...
novel acorn
Ah OK
solemn oar
Imagine having more than one representative of each homotopy type.
unreal stratus
But tbf our course only defined finite ones
feral copper
Wait no it's not just a point
novel acorn
I misinterpreted what was being said
gentle ospreyBOT
MyMathYourMath
unreal stratus
It is contractible which was the joke lol
feral copper
Yeah xD
pseudo coral
i dont see how S is closed in X
solemn oar
Analysis peeps be like I'm studying functions pt -> pt like lmao what functions???
feral copper
**MyMathYourMath**
I'm not sure I understand what you mean
pseudo coral
@feral copper its the third line in his proof he claims the set S is closed
solemn oar
If you want to prove S^inf is contractible rigorously the proof is pretty neat too.
unreal stratus
If you want to prove S^inf is contractible rigorously the proof is pretty neat t...
Like the one where you shift stuff along one by one?
I thought that was cute
Ig Whiteheading it would be a bit overkill lol
solemn oar
Like the one where you shift stuff along one by one?
Yeah, and you actually get to play with the CW topology
feral copper
is it because S \cap X^n is closed
Yes exactly
The topology is that induced by the subcomplexes, so any subset is closed iff any intersection with X_n is closed for all n
hidden crag
Yeah okay my way was overkill indeed
feral copper
And by definition S \cap X_n is finite so it is closed
pseudo coral
got it thanks! @feral copper
unreal stratus
Yeah, and you actually get to play with the CW topology
ye cool
Ye ig the other way is just to use S^n -> S^infty inducing an iso on pi_{k < n} right
feral copper
This just means you're weakly contractible
unreal stratus
Ig Whiteheading it would be a bit overkill lol
lol