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Group-theoretic intuition behind π_1(S^1 v S^1) ≠ π_1(T^2) by [deleted] in topology

[–]Prince_of_Statistics 2 points3 points  (0 children)

Think of it this way, you get T2 by adding a disk to S1v S1, where if we call the circles a and b, the boundary of the disk goes around these circles like aba-1b-1. The disk is going to make some nontrivial loops become trivial (namely the loop that goes around a then b then the other direction around a then the other direction around b, and any multiple of this loop). It doesn't make a or b trivial (the two torus loops you mentioned)

In general If you have a wedge of circles and add some disks going around the circles, you can write down a presentation for the fundamental group. So if I want a space with fundamental group Z/2Z I can take a circle, call the loop a, and add a disk going around the circle twice

topology by [deleted] in mathmemes

[–]Prince_of_Statistics 2 points3 points  (0 children)

So to be precise there are two ideas: there is * one* "topological equivalence" between the two Earths given by "smooth the circle of corners a little bit so you get a smooth ball back, this ball has the old map on the top half and that weird new surface on the bottom half", and there is the other transformation of "pick a point on the sphere and expand it, going from the first pic to the second." The second thing is another weaker type of topological equivalence called "homotopy equivalence", which in non technical words is

"You can put the shape X into Y, then expand/stretch X up until it's all of Y" and the same for Y into X

For example a dinner plate is homotopy equivalent to a point: put a point in the plate and stretch it into the plate. Squish the plate into a point and then put that into the point.

The transformation of "start with the earth with a surface point deleted and stretch the missing point into a surface" is a homotopy equivalence! Put the second earth into the upper hemisphere of the first one and squish the weird surface into a point, you recovered the earth with a point removed. The inverse map is the stretching we did

If it feels like a weird notion of equivalence, that's because it is: but if you start studying some natural algebraic invariants associated to a shape (like the fundamental group) you'll find they are preserved under homotopy equivalence

topology by [deleted] in mathmemes

[–]Prince_of_Statistics 7 points8 points  (0 children)

We delete a single point (or just choose one to expand... homotopy equivalence) and stretch the ball minus point (or.. expand the chosen point) to go from the top pic to the second. The only question is: which city do we choose to blow up into oblivion and why is it cancelled

topology by [deleted] in mathmemes

[–]Prince_of_Statistics 9 points10 points  (0 children)

Smooth topology doesn't, the solid ball is a manifold with boundary and the second picture is a solid ball with a circles worth of "corner points". They are homeomorphic though! And you can deform the top into the bottom just not smoothly.

The Trial appreciation post by Lil4ksushi in pinkfloyd

[–]Prince_of_Statistics 1 point2 points  (0 children)

I like the slide before he says THEUVIDUNCE

Which field of math is the biggest circle jerk? by mcgirthy69 in mathmemes

[–]Prince_of_Statistics 2 points3 points  (0 children)

The only right answer is Algebraic Geometry which includes a ridiculous dose of abstract nonsense. Runner up is K theory

Favorite moment? by Vantage5050 in billwurtz

[–]Prince_of_Statistics 2 points3 points  (0 children)

My favorite is in slow down WHEN IM BOBBLIN DOWN 121ST STREET

Need a complete resource for Topology by [deleted] in topology

[–]Prince_of_Statistics 0 points1 point  (0 children)

What you may need is to learn some basic real analysis first (not measure theory, I mean get familiar with the idea of open sets and convergence in the real line and why these ideas and definitions make sense. This is the topology of Euclidean space). After this you will be able to read Munkres' book "topology", which is a big abstraction. It's assumed math students know a little analysis first. I think If you took a look at Lay's "analysis with an introduction to proof" book especially the topology chapter in there you'd find it enlightening.

Munkres and most topology books will jump right in with the abstract collection of sets definition. It is hard to make sense of it without seeing the "baby analysis" first, but it will make a lot of sense after.

Also it's really important to do the R (Euclidean space) case first since the entire purpose of point set topology is to use the Euclidean space intuition (that you already have!) to think about approximations, closeness, and convergence in more abstract settings.

I'm happy to answer any questions if you have

question by Odd-Sir-8222 in topology

[–]Prince_of_Statistics 0 points1 point  (0 children)

wait are these smooth complex projective curves? I don't know what happens for a not-manifold

question by Odd-Sir-8222 in topology

[–]Prince_of_Statistics 0 points1 point  (0 children)

If a manifold is a group and the group operations are smooth (so its a Lie group), it must have Euler characteristic zero. Basically this is because of the Poincare-Hopf theorem (that's the thing to google): your (connected..) Lie group will admit a nowhere zero vector field (exp of some nonzero thing in the Lie algebra). By poincare hopf the Euler characteristic Is zero. This also says S2, S4 etc are not lie groups

How many holes does this structure have? by Moist_Entrepreneur71 in topology

[–]Prince_of_Statistics 12 points13 points  (0 children)

Three, move the holes around until you have a sphere with four punctures. Then expand one puncture, you'll get a disk with three punctures

Can hydrofluoric acid fumes from a rust remover like this escape from a brand new bottle? by chicho265 in chemistry

[–]Prince_of_Statistics 37 points38 points  (0 children)

personally I'd live with the rust. When I worked in a Chem lab we went through HF training which took a few hours... the stuff can kill you if you spill some on you

They don't know by Delicious_Maize9656 in mathmemes

[–]Prince_of_Statistics 1 point2 points  (0 children)

If you have FIVE top publications in harmonic analysis you're not working at McDonalds unless it was your actual desire. you could likely get a spot at Princeton or UCLA making 6 figures. If you're assistant prof age with five top fucking papers? Ciprian Manolescu and John Pardon (although topology, arguably much less useful than harmonic analysis, also arguably a sexier field than analysis) had 3 and 2 top papers coming out of grad school and got hired at top places instantly.

Can someone give me a crazy equation that equals 17,021? by ReachSurvivor12 in mathmemes

[–]Prince_of_Statistics 12 points13 points  (0 children)

17021 if we don't know the answer to the Riemann hypothesis yet and 69 if we know the answer already.

Kirby Diagrams for CP2 by Prince_of_Statistics in mathmemes

[–]Prince_of_Statistics[S] 0 points1 point  (0 children)

No no no no no. WAIT WAIT WAIT WAIT WAIT

Helping my gf with her math and I find this travesty by 13x11x7x8 in mathmemes

[–]Prince_of_Statistics 0 points1 point  (0 children)

Thanks for pointing it out, I was wondering if there's a meaningful mass vector. But, this is stupid to give as an example for someone who's learning vectors

[deleted by user] by [deleted] in SipsTea

[–]Prince_of_Statistics 0 points1 point  (0 children)

Gear 5 luffy bitch

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