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00:00 - 18:0018:00 - 23:0023:00 - 00:00

Huy
Huy
23:00
?
0celo7
0celo7
What's the precise definition
Huy
Huy
f and g: X to Y are homotopic if there is a cont function $H: X \times [0,1] \to Y$ such that $H(x,0) = f(x)$ and $H(x,1) = g(x)$
0celo7
0celo7
I know what homotopy means
I mean of mapping class group
Huy
Huy
I just said it above
0celo7
0celo7
I do not know what "up to a homotopy" means.
Huy
Huy
23:01
that means equivalence classes
0celo7
0celo7
OF WHAT
Huy
Huy
of homeomorphisms
which belong to the same class if they are homotopic
0celo7
0celo7
What's your notion of orientation
Huy
Huy
pick an orientation on the surface
usual manifold orientation
0celo7
0celo7
is this a $C^0$ surface?
Huy
Huy
23:02
its the torus dude
0celo7
0celo7
$C^\infty$, $C^\omega$?
What's the regularity
Huy
Huy
0celo7 pls
what happened to u
0celo7
0celo7
what?
Huy
Huy
so do you know what the mcg of the torus is
0celo7
0celo7
how are you defining orientation? homologically?
via top degree forms?
chart overlap Jacobians?
Huy
Huy
23:04
pick your favourite definition
fix a definition for the orientation
and then fix an orientation
0celo7
0celo7
orientation preserving homeomorphism only makes sense with the homological definition
Huy
Huy
are u sure
0celo7
0celo7
yes
Huy
Huy
ok
then pick that one
0celo7
0celo7
how are you going to calculate the others with a non-differentiable map
I don't know how that one works
Huy
Huy
23:05
what
everything is diffeo
0celo7
0celo7
You said homeomorphism
Huy
Huy
yeah but every homeo is homotopic to a diffeo
so who cares
0celo7
0celo7
that's nontrivial (is it even true?)
Huy
Huy
for compact surfaces it's true
that's all i need
0celo7
0celo7
sketch the proof please
Huy
Huy
23:07
lol
not at 1am
maybe tomorrow if you say pls
0celo7
0celo7
Ok, let's see if I can figure it out
for the torus
Huy
Huy
ok
0celo7
0celo7
PSL(2,Z)
Huy
Huy
stop cheating
0celo7
0celo7
how am I cheating?
Huy
Huy
23:08
idk
0celo7
0celo7
I remember it from string theory, sue me
Huy
Huy
good thing that it's wrong
0celo7
0celo7
bull
Huy
Huy
it's just SL
huehue
0celo7
0celo7
oh
you get P when you don't care about orientation
Huy
Huy
23:09
btw
I think the fact that homeos are homotopic/isotopic to diffeos is actually due to Munkres
0celo7
0celo7
If it's true, it will be in Hirsch
Huy
Huy
in his work where he tried to find non diffeomorphic smooth structures on $S^7$
a short article for freshmen
0celo7
0celo7
what?
Huy
Huy
yeah, check "Obstructions to the Smoothing of Piecewise-Differentiable Homeomorphisms" theorem 6.3
u don't know that article ?
0celo7
0celo7
nope
link?
Huy
Huy
23:12
ah
I meant Milnor
sry
it's after 1am
it's time to stop if I start changing names
0celo7
0celo7
liiiiiink please
Huy
Huy
for what
0celo7
0celo7
that article
Huy
Huy
the s^7 business ?
0celo7
0celo7
why is it for freshmen
Huy
Huy
23:14
it was a jok
like ur "phd level algebrist" stuff
just reversed
@0celo7: apparently the homeo homotopic to diffeo also works on 3-manifolds
pretty impressive
00:00 - 18:0018:00 - 23:0023:00 - 00:00

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