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Okay homotopy theory thread, or "how topologists tell spaces apart". This is completely general, you don't even need to know what a function is*, and there's cool stuff from the start!
*but knowing that topology looks at shapes that are made of clay helps, sorry
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*but knowing that topology looks at shapes that are made of clay helps, sorry
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You may have seen people say that topology ignores size and angle, so we can stretch and squish, not cut - so how do we tell spaces apart in topology? Let's look at a few different ways and see if we can form a general theory. Some of these ideas I'll pick up in a later thread
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So if you play around with shapes without cutting, tearing, or gluing them, you'll pretty quickly notice that we can't fill in holes/gaps. e.g. a circle is different to a disk, a doughnut (torus) is different to a sphere, two shapes can't be smooshed together into one.
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https://twitter.com/ravenscimaven/status/1350266556891148288?s=20
So we might hope to classify spaces by how many holes they have. Now we need to consider:
-What do we mean by a hole?
-How do we do find holes?
-What topological information does this ignore? Are spaces fully explained by their holes?
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So we might hope to classify spaces by how many holes they have. Now we need to consider:
-What do we mean by a hole?
-How do we do find holes?
-What topological information does this ignore? Are spaces fully explained by their holes?
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Okay first: we're clearly talking about different things by holes depending on context. One way we could decide a hole's type is by looking at what we need to fill it in: a 0-dimensional hole is filled in by a line, a 1-dim hole is filled by a disk, etc. Examples help -
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Wait so 1-dim spheres (circles) have one 1-dim hole, and 2-dim spheres have one 2-dim hole. What if an n-dim sphere captured the idea of an n-dim hole? It definitely works for 0-dim: x²2= 1 has two points as solution, two points have a 0-dim hole between them
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Why not try to use spheres to characterize holes? A common idea in math is to study objects by looking at maps into or out of them instead. So let's look at maps into spaces, in particular maps from n-dimensional spheres to topological spaces. What does this look like?
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We can think of a (continuous) map from space X to Y as placing a copy of X in Y, but squishing it around a bit. e.g. we can map a circle to a plane by drawing it as is, or we could twist it around a bit. Here's some pics:
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Now, what do you notice? There's lots of different ways to map spheres into shapes, but most of them are topologically equivalent: if you draw a circle all twisted up on a page you can deform it into a point while staying on the page. What if there was a hole in the page?
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So the claim is that (up to topological equivalence) these maps characterize the holes of spaces. e.g. for the torus any circle we draw on it can be deformed to a point, unless the circle encloses one of the two holes. Also note we have one of these sets in each dimension.
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So say we can work out the non-contractible maps of each dimension n for a space X, what do we have? First I really should say that each of these sets of equivalence classes of maps forms a group π_n(X) - we can add maps together by doing one then the other. For example-
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In mapping the circle to itself we can wrap it around itself as many times as we want. We can also wrap it clockwise or counterclockwise: we say that π_1(S^1) is the integers. Can you see what happens for the sphere and torus?
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We call these groups the homotopy groups.
Now does knowing these groups classify topological spaces? Unfortunately not. As an example the cylinder and Möbius band have the same homotopy groups but are clearly different, the band has a twist.
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Now does knowing these groups classify topological spaces? Unfortunately not. As an example the cylinder and Möbius band have the same homotopy groups but are clearly different, the band has a twist.
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That sucks, but we do know if any homology groups of two spaces differ they aren't topologically equivalent.
Also, can we compute these groups easily? Unfortunately not. It's actually really hard when n>1, let's see why*:
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*one reason at least
Also, can we compute these groups easily? Unfortunately not. It's actually really hard when n>1, let's see why*:
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*one reason at least
We started by trying to look at n-spheres as having one n-dim hole and nothing else, but that is (un)fortunately not true! There are non-trivial maps from higher dimensional spheres to lower ones dimensional ones. The first example is the Hopf fibration from S^3 to S^2:
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In fact these "higher homotopy groups" aren't uncommon. Higher homotopy groups are a mess, the beauty of mathematics is a lie!
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Well if everything is so complicated why bother?Three reasons:
1)Trying to work with these things has led to a whole lot of amazing math, and we've learned a lot about shapes through it.
2)Ď€_1 is actually not *too* hard to compute and tells us a lot
3)Thinking this way is nice
1)Trying to work with these things has led to a whole lot of amazing math, and we've learned a lot about shapes through it.
2)Ď€_1 is actually not *too* hard to compute and tells us a lot
3)Thinking this way is nice
I'll wrap this thread up with some comments and questions, but will continue the story later.
-See if you can figure out what π_1 should be for some spaces, also we call π_1 the fundamental group.
-Usually we look at maps with a basepoint-
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-See if you can figure out what π_1 should be for some spaces, also we call π_1 the fundamental group.
-Usually we look at maps with a basepoint-
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e.g. we pick a point on the sphere and a point in our space and make sure the point on the sphere maps there. When do you think this matters? hint: think about 0-dimensional holes.
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Also I never said why we should care that we have groups, even though it hurt me. Groups are beautiful and easy to work with. We can classify and play with groups much much easier than spaces, so seeing groups is exciting. Also-
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Groups have a lot of things we can do with them: products, quotients, etc. Spaces also have this: we can take disjoint unions, smash products, quotients etc. One thing that would be amazing is if we could show these operations transfer over using homotopy groups
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e.g. if say, the homotopy groups of the (wedge) sum of two spaces was the (direct) sum of the homology groups. In category terms we are trying to relate limits, colimits etc. between different categories with a collection of functors between them
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Turns out there are lots of results like this, the first people usually learn about is Van Kampen's theorem, but that's for a different thread. Finally there is actually a version of this which is a lot easier to compute: homology groups.
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The core idea goes back to the start: an n-dim hole is a boundary of some n+1 dim space which isn't filled in, so the set of holes is the full set of boundaries minus ones which can be filled in in the space. I plan to write more about homology later.
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(please don't give money if you're struggling!!):
https://ko-fi.com/elhayes
Also please let me know if there's anything else I can do for accessibility
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