I’ve been working on a couple of articles talking about homology, which is an interesting (but difficult) topic in algebraic topology. While I was writing, I used a metaphor with a technique that’s used in homotopy, and realized that while I’ve referred to it obliquely, I’ve never actually talked about homotopy.
When we talked about homeomorphisms, we talked about how two spaces are homeomorphic (aka topologically equivalent) if and only if one can be continuously deformed into the other – that is, roughly speaking, transformed by bending, twisting, stretching, or squashing, so long as nothing gets torn.
Homotopy is a formal equivalent of homeomorphism for functions between topological spaces, rather than between the spaces themselves. Two continuous functions f and g are homotopic if and only if f can be continuously transformed into g.
The neat thing about the formal definition of homotopy is that it finally gives us a strong formal handle on what this continuous deformation stuff means in strictly formal terms.
So, let’s dive in and hit the formalism.
Suppose we’ve got two topological spaces, S and T, and two continuous functions f,g:S→T. A homotopy is a function h which associates every value in the unit interval [0,1] with a function from S to T. So we can treat h as a function from S×[0,1]→T, where ∀x:h(x,0)=f(x) and h(x,1)=g(x). For any given value x, then, h(x,·) is a curve from f(x) to g(x).
Thus – expressed simply, the homotopy is a function that precisely describes the transformation between the two homotopical functions. Homotopy defines an equivalence relation between continuous functions: continuous functions between topological spaces are topologically equivalent if there is a homotopy between them. (This paragraph originally included an extremely confusing typo – in the first sentence, I repeatedly wrote “homology” where I meant “homotopy”. Thanks to commenter elspi for the catch!)
We can also define a type of homotopy equivalence between topological spaces. Suppose again that we have two topological spaces S and T. S and T are homotopically equivalent if there are continuous functions f:S→T and g:T→S where gºf is homotopic to the identity function for T, 1T, and fºg is homotopic to the identity function for S, 1S. The functions f and g are called homotopy equivalences.
This gives us a nice way of really formalizing the idea of continuous deformation of spaces in homeomorphism – every homeomorphism is also a homotopy equivalence. But it’s not both ways – there are homotopy equivalences that are not homeomorphisms.
The reason why is interesting: if you look at our homotopy definition, the equivalence is based on a continuous deformations – including contraction. So, for example, a ball is not homeomorphic to a point – but it is homotopically equivalent. The contraction all the way from the ball to the point doesn’t violate anything about the homotopical equivalence. In fact, there’s a special name for the set of topological spaces that are homotopically equivalent to a single point: they’re called contractible spaces. (Originally, I erroneously wrote “sphere” instead of “ball” in this paragraph. I can’t even blame it on a typo – I just screwed up. Thanks to commenter John Armstrong for the catch.
Addendum: Commenter elspi mentioned another wonderful example of a homotopy that isn’t a homeomorphism, and I thought it was a good enough example that I wish I’d included it in the original post, so I’m promoting it here. The mobius band is homotopically equivalent to a circle – compact the band down to a line, and the twist “disappears” and you’ve got a circle. But it’s pretty obvious that the mobius is not homeomorphic to a circle!. Thanks again, elspi – great example!
/me brandishes his Notices
The sphere can’t be homotopy equivalent to the point. If it were homotopy equivalent the two would have isomorphic homologies, and they don’t. Are you thinking of the ball? The ball is contractible, the sphere isn’t.
Watch the typos. You mean homotopic not homologic.
You should have $h in hom(Sx[0,1], T)$ (If the catagory is Top, then you want h to be continuous).
John is right about the sphere. It is not homotopy equivalent to a point (both the second homology and the second homotopy groups are $Z$)
R^n is homotopy equivalent to a point though, and a mobuis band is homotopy equivalent to a circle.
Mmmmmm, now we’re talking! This lies pretty darned close to my own research area, and is an area I’m very fond of.
Particularily great fun can be had once you take the topological intuition and go do something completely different with it. In this vein, John Baez has in one of the older This Weeks Finds discussed homotopies between different proofs of the same theorem, with a pretty explicitly given example of the stages inbetween; and you can also look at homotopies between algebraic theories and similar funky uses…
John:
Yes, you’re right – I meant ball, not sphere. I’ll correct it when I get the chance.
elspi:
Oy, I can’t believe I did that! Thanks for catching it; that’s a terrible type that could confuse the living daylights out of people! I guess I’ve got homology on the brain from trying to figure out how to write about it. Anyway, it’s corrected. Thanks again.
Also, thanks for the Mobius example – would you believe I actually didn’t know about that one? It’s obvious in retrospect, but either it wasn’t mentioned in my text, or I somehow skipped over it. It’s a really beautiful example of the difference between homotopy equivalence and homeomorphism.
The nicer thing with the Mobius band is the fact that a regular band is also homotopy equiv. to a circle and thus you have a homotopy equiv. between the Mobius and the untwisted band. Now, it’s pretty clear that a circle is not homeo. to the Mobius (for example, by the “remove two points argument). The fact that a Mobius and a band aren’t homeo. is a bit trickier, but they are still homotopy equiv.
To check on my understanding as I hope my mistakes gets corrected, and to connect to Mark’s reference to isotopy in the earlier post; the oracle of Wikipedia sez that an isotopy is a homotopy where the continuous functions are homeomorphisms. That would make it possible to connect the ball and point by a homotopy but not by an isotopy, I believe.
In an earlier comment I said that mathematicians (people who do mathematics and no more) usually use sets of points instead of intervals. This works as a good case in point. Even though you talk about ‘continuous’ deformation, you still end up using functions which map discrete points (or members) to other discrete points (albeit contained in an interval), as opposed to mapping continuous interval(s) into continuous interval(s), or continuous interval(s) into discrete point(s).
Great! Something I’ve been waiting to find an intuitive explanation of for awhile.
So what is this “cohomology” I keep hearing about?
Bob:
This post is about homo*topy*, not homo*logy*. We’ll get to cohomology when we get to homology – which is a lot more confusing that homotopy.
Am I right that the next step in this story is to impose a group structure on the classes of homotopic paths on a surface, and thus classify that surface? If so, please please cover that at some point – all I can remember about it is that it was very cool!
Homotopic loops sharing a common base point, and it doesn’t entirely classify the surface, but yeah, that’s one place this is headed.
Thanks for a very interesting and timely post, Mark.
Your posts about homeomorphisms and homotopy are bringing me back to something I’ve been wondering about for a long time–would these be useful techniques in modeling developmental and evolutionary anatomical transformations?
The reason I ask is that getting back into mathematics (presumably a lot of topology, if my intuition is correct) to the degree it would take me to learn this (and learn it to be able to do it *right*, not just Dembskian hand-waving and post-hoc question-begging) would be substantial at this point. I’m willing to do it if it would provide a rigorous and useful underpinning to my comparative anatomical information system (a mathematics of comparative anatomy, as it were).
On the other hand, if trying to misapply these mathematical tools to anatomy is the fast track to Cranksville–not so much, then. So if you have an opinion on the suitability of trying to apply these concepts in the very different area of comparative anatomy, I’d be most interested in hearing what you think.
Thanks!