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Manifolds and surfaces thread! I’m gonna try keep this open to a completely general audience, so I won’t assume anything other than an idea of what a derivative is and a willingness to stop and think.
1/22
1/22
As you'd expect mathematicians care about shapes a whole lot. Manifolds are an incredibly important type of shape, but we need to motivate them a little to see this.
2/22
2/22
Just like we dont really care about the difference between congruent triangles, in different contexts we ignore details about what we're studying to look at general properties. The most common thing we do is look at objects "up to homeomorphism"-ie pretend everything is clay
3/22
3/22
This is topology. We can bend and stretch things as much as we want, but not cut, tear, or glue. So a sphere is not homeomorphic to a doughnut (would need to make a hole), but we consider a coffee cup equivalent to a doughnut. Sometimes we want a little more structure though
4/22
4/22
We only want to look at things that "locally" look like normal space: e.g. we can make a map of any part of the shape, but not necessarily a "good" map of the whole thing. This is like how world maps distort shapes. If we try map the whole world things get messed up, but!
5/22
5/22
If we pick a small area, say a city, we can map the area out without messing it up. The roundedness and weirdness of the entire globe doesn't really matter when you're zoomed in! This is a nice condition to have in general
Shapes like this are called topological manifolds.
6/22
Shapes like this are called topological manifolds.
6/22
A nicer thing to work with sometimes is smooth manifolds- these are just like topological manifolds except we add a structure which lets us take derivatives of functions. Equivalence of smooth manifolds are called diffeomorphisms. A difference is that they care about corners
7/22
7/22
So this gif shows a homeomorphism between the triangle and circle, so they are topologically the same! However they are not diffeomorphic, think about a function around the corners -there's going to be a jump- so the derivative isn't defined.
8/22
8/22
Okay so we can classify all topological and smooth manifolds right? Uhh.. no,, nope. The face we can't leads to most of modern geometry/topology. Let's try anyway though! Some rules: we'll only look at connected manifolds, no reason not to. Plus one technical thing:
9/22
9/22
(Skip this tweet if you want)
We'll only look at "closed" manifolds. These are manifolds which have no boundary and are compact: you can think of this as making sure the shape is finite, but it's not quite right. Also @ mathematicians yeah I'm skipping other conditions idc
10/22
We'll only look at "closed" manifolds. These are manifolds which have no boundary and are compact: you can think of this as making sure the shape is finite, but it's not quite right. Also @ mathematicians yeah I'm skipping other conditions idc
10/22
Now in maths we like to start with the simple cases, so we'll start looking at one dimensional smooth manifolds then move up. In 1-dim things are actually very easy! We can have a circle or an interval (line segment): we just straighten out the kinks in the curve.
11/22
11/22
In two dimensions we get a little spicier: there's 3 building blocks:
-A sphere
-A (hollow) doughnut
-The real projective plane*
Sticking* these together gives us every possible surface. Sticking two doughnuts together gives a doughnut with two holes for example.
12/22
-A sphere
-A (hollow) doughnut
-The real projective plane*
Sticking* these together gives us every possible surface. Sticking two doughnuts together gives a doughnut with two holes for example.
12/22
Okay so firstly by sticking together I mean "cut a disk out of both and stick them together along the boundary" this is the connected sum.
Now what's the real projective plane? It's the space of lines through the origin in R^3... not the easiest to visualise but...
13/22
Now what's the real projective plane? It's the space of lines through the origin in R^3... not the easiest to visualise but...
13/22
It looks something like a disk with its boundary glued to itself. Also the sum of a doughnut and the projective plane gives you the sum of three projective planes, so these three building blocks really are all you can get
14/22
14/22
Okay, three dimensions? I admittedly know less about this case than others but three is where things get rough. There's no tidy set of building blocks here. The Poincaré conjecture (2003) proved "Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere".
15/22
15/22
This case is still a hugely active area of research, 3 dimensions is really interesting, so different tools need their own threads. However you'll notice I've stopped talking about topological vs smooth for all this - what's up with that?
16/22
16/22
Well it turns out in 1,2,3 dimensions topological manifolds are equivalent to smooth ones: If a manifold is topological it has a unique smooth structure. In higher dimensions this.. does not hold.. ever..
17/22
17/22
In 4 dimensional manifolds are wild, we know very very little. As an example there are infinitely many "exotic R^4's". This is a manifold homeomorpic to four dimensional space, but not diffeomorphic. In every other dimension there is only one differential structure on R^4
18/22
18/22
However in dimensions greater than 4 something really weird happens: Things actually become a whole lot more understandable! The extra wiggle room of those extra dimensions lets us do something called surgery to stitch manifolds together and classify them.
19/22
19/22
So that's roughly where things stand now. We mostly understand things in 1,2,3?,n>4 dimensions, but 4-dim is unknown. So what do we do? What I'm working on is exotic manifolds - manifolds with weird smooth structure. This thread motivates a whole lot of directions though.
20/22
20/22
For example algebraic topology uses algebra to understand shapes, Riemannian geometry adds the idea of length and angle to look at curvature, Morse theory adds analysis to the mix (and leads to the classification in n>4), knot theory plays with knots! usually 3-dim.
21/22
21/22
Some futher reading:
-various wikipedia pages, starting on the surfaces page
-Milnor Topology from the Differential Viewpoint
-Weeks The Shape of Space
22/22
-various wikipedia pages, starting on the surfaces page
-Milnor Topology from the Differential Viewpoint
-Weeks The Shape of Space
22/22
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