I’ve been trying to understand this a lot recently and maybe an explanation from another person would help. Thanks 🙂
(Just explaining what it asks and how to go about learning about it would be greatly appreciated)
In: Mathematics
>**Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.**
So let’s break that down. Imagine you have a super-elastic scrunchie that will contract right down to an infinitesimal point and will also ‘hug’ the surface of anything you wrap it around, but will also stretch as far as you need it to.
Now imagine you have a beach ball. You can put the scrunchie around every part of that beach ball and slide it off just by contracting or expanding it, *without any part of it ever leaving the surface of the ball*. (This is the important part that often gets overlooked.) [It just gets smaller and smaller, and then slips off, like this.](https://upload.wikimedia.org/wikipedia/commons/9/9e/P1S2all.jpg) Boom. Simple.
Now imagine doing it to a cube. Sure, you’ve got some angles now, but your magic scrunchie doesn’t care about that. You can still slide it off in exactly the same way you did to the beach ball.
Now imagine doing that to a table, [like this one](https://www.ikea.com/gb/en/images/products/taerendoe-table__0737362_PE741023_S5.JPG). Tables are weirdly-shaped; they have lots of strange angles and odd sticky-out bits. However, you can still do it. There’s no way of putting that scrunchie around the table that you can’t slip it off, just the way you did towards the beach ball. The table and the beach ball and the cube are what’s called *homeomorphic*: effectively, for the purposes of a branch of mathematics called topology, you can treat them as pretty much the same.
Now imagine doing it to a giant doughnut. Imagine that some naughty little monkey has cut and resewn your magic scrunchie *through the hole* in the middle of the doughnut, [like the red line here](https://upload.wikimedia.org/wikipedia/commons/5/54/Torus_cycles.png). Well, you’re screwed now; you can’t slip your scrunchie out of the hole without re-cutting it. (This isn’t the case with the ball, the cube or the table, by the way. No matter how you cut or re-attach the scrunchie, it’ll still work.) The doughnut *isn’t* homeomorphic to the cube, ball or table. It is, however, homeomorphic to a coffee mug; even though the mug has the big divot in it to hold coffee, it also has a handle that you can thread your scrunchie through and re-sew it, trapping it forever. However, you don’t even need to cut it. Imagine looping it around the widest part of the doughnut, [like the purple line here](https://upload.wikimedia.org/wikipedia/commons/5/54/Torus_cycles.png). You can slip it loose, but you *can’t* contract it down to a single point without some part of the scrunchie leaving the surface of the doughnut.
Your breakfast doughnut and the mug that holds your morning coffee are pretty much the same, to a topologist, but the table you’re sitting on probably isn’t — unless it has a single hole in it.
Let’s go back to the original conjecture:
>**Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.**
‘Simply connected’ means that, if you have your magical scrunchie, you can pull it loose from every possible position on the object without cutting it.
Homeomorphic means ‘We can pretend it’s the same shape as far as topology is concerned even if it looks different; mathematically, it checks out’.
[Wikipedia article](https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture). If you look at the top of the article, it states the conjecture with each word hyperlinked to explain what it means. I’d use that as a way to “learn about it”.
It basically says that any “[blob](https://upload.wikimedia.org/wikipedia/commons/thumb/5/52/Ricci_flow.png/200px-Ricci_flow.png)” or 3D surface that doesn’t have any holes in it, or any edges, is equivalent to a sphere, from a mathematical point of view.
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