A Trek through 20th Century Mathematics (3/8) – Topology, Homotopy and Poincaré’s Conjecture

I want to go to South America because they have beautiful landscapes i’ve never been to South America but I’m told that they have nice landscapes at least big mountains, the biggest on Earth actually I was testing you guys, the biggest mountains on Earth are in Asia if you didn’t know that I suck at geography but you guys seem to suck more than I do! Imagine I’ve just arrived in south america and i
wanted to do a hike in these mountains I’ve just arrived so I cannot go very high in elevation because I would get sick, so I can do only hikes up to a certain elevation so say I cannot go above this darker area okay so i do the hike i’m here well i don’t want to get lost
so what i’m going to do is take a cable and leave one end at the initial point in case I get lost I can always go back so I leave one end of the cable where I start and then I go for a hike so i’ll walk all around the mountain enjoying the beautiful view and then i come back, and when i come back, I the other end of the cable with me and I find the end of the cable that I’d left so I have to ends of the cable now. I can tie them together this gives me a loop so now I have a loop here and i want to take a look back so I can try to pull on the loop to get it back but it would get stuck by the mountain I wouldn’t be able to retract the loop because of the mountain in the middle so the loop would be non-retractable now this idea of a loop seems like a very simple idea but it says something very fundamental
about the topological properties of this surface [of the Earth] i cannot be retract the loop i could if i went above the mountain but i can’t and that says that there’s in a very topological sense a hole which corresponds to the mountain now this sort of loops have been studied by French mathematician Henri Poincaré it has really revolutionized the way we think about these spaces it’s one of the greatest ideas of the 20th century this torus is actually a deformation of the weird shape i showed earlier and the connectivity of this torus is the same as the connectivity of the weird surface i showed earlier on this torus you can imagine doing hikes for instance, you’d follow the blue loop and get back to the initial point now if you tried to retract the loop you wouldn’t be able to do so this blue loop is fundamentally non retractable similarly the red loop is also non retractable what’s amusing is that the blue loop and the red loop are not the same kind of non-retractability because you cannot deformed the blue loop to obtain the red loop they’re very different and there are also other loops you can do like you can do the blue loop and then the red loop, it would give you another loop which is very different from the 2 others the set of all these loops that you can do really defines a connectivity of this shape it’s what Poincaré called the fundamental group of the surface in this case it’s a little complicated so Poincaré faced the simplest shape in terms of these loops which would be the sphere if you go around the sphere, you can do any loop the loop you did would always be retractable spaces like that whose loops are always retractable they’re called simply connected the sphere is simply connected what Poincaré proved is that if you take any surface which is bounded in space and has no edge so in a nice-looking surface if it is simply connected then it’s because it is a deformation of a
sphere it’s very much like a sphere so basically in a topological sense all these shapes which are simply connected all the surfaces which are simply connected are more or less the sphere this idea of loops can be very simply generalized to higher dimensions Poincaré then looked at higher dimensions, so, after two, three he looked at dimension 3 so he imagined a 3-dimension sphere kind of hard to imagine and he wondered if any three [simply connected] dimensional space which is
bounded and has no edge is a deformation of the sphere or not i guess he tried but he eventually wrote “this problem would take us too far” it’s kind of saying, it’s too difficult for me, and, for that matter, it’s also too difficult for anyone who would try i call it self-confidence but you could also call it arrogance he’s French so that’s understandable and so he kind of stopped here because
dimension 3 seemed too difficult and what’s amusing with that is that the problem gets simpler in very high dimensions so I’m told… so it was actually first proven for dimensions six and higher in the nineteen fifties then for dimension 5 and then dimension 4 and dimension three was like the hardest one it was actually one of seven problems of
the millenium by the Clay institute of mathematics it’s also the only conjecture of these millenium problems which has been proven it was proven in 2004 by a mathematician named Perelman so that’s very nice, because it kind of tells us that we now can understand any surface, any volume, any space of finite dimension, at least in a topological sense i want to take us further by further i don’t mean infinity because infinity is rather similar to that to go further we will now look at dimensions which are not whole numbers these shapes which have dimensions which are not whole numbers are called fractals