Explain “homotopy” to me [closed]

I have been struggling with general topology and now, algebraic topology is simply murder. Some people seem to get on alright, but I am not one of them unfortunately.

Please, the answer I need is ideally something very elaborate and extensive, possibly with easy-to-understand examples. Rewriting the definitions with concrete mathematical language and symbols won’t help(those are readily accessible in my lecture notes). I want an explanation of what is happening and which part of the solid definitions are telling me so.

One thing I would like to make clear is, I know the definitions, but I don’t understand them; I can re-iterate them upon request, consult my lecture notes. Issue is I am writing down something I don’t know what it means. It’s like writing ancient Greek. I can remember the shapes of each character, their order and write some sentence down. But that’s it. Can I explain it to someone with my own words? Break it down? Absolutely not. That is why I am here to ask someone who fully understands these ideas to do exactly that for me: break it down. Go in slow motion. Show me the moves and codes behind it.

Here’s the definition I have for homotopy

A homotopy between maps f,g:XY is a map h:X×IY such that

h(x,0)=f(x),h(x,1)=g(x)Y where xX and I=[0,1]. We say maps f,g are homotopic.

It adds that,

A homotopy deforms the map f continuously to g.

So, I am given two sets X,Y whatever they are and two maps f,g that takes an element of X to an element of Y. I don’t know if f,g are bijective, only injective or surjective or whatever. No information on that. Just maps.

And this “homotopy” is a … “map between maps”?
And even if so, what exactly is it doing? All the explanation seems to be done with this single line

h(x,0)=f(x),h(x,1)=g(x)Y

but no, I don’t get what’s happening. So X has a bunch of elements x, and the product space X×I gives me elements of the form {x,t}X×I. Okay. But this map h qualifies as a homotopy as long as the above holds? Then why not just always define h as h(x,t)=f(x) for any t1 and h(x,t)=g(x) for t=1, just as how we might define a piecewise function? Then I can define this “homotopy” on any maps.

Sure, it adds “deforms f CONTINUOUSLY to g” but how is that stated in the definition itself? I see it nowhere.

Here is an example in my notes which didn’t help me understand the definition,

Take X={x} the space with a single element x. Then a map f:XY is the same as and element f(x)Y. A homotopy h:fg:XY is the same as a path h:IY with initial point h(0)=f(x) and terminal point h(1)=g(x)Y. A homotopy h:ff:XY is the same as a closed path h:IY.

Well, first off what is a “path”? Intuition also doesn’t make sense because when mapping one element to another, how can there be different “paths”? x goes to y. Done. It’s not like going from England to Singapore via either Amsterdam or Frankfurt (thus different paths) is it? Unless it’s a map XZY and telling me that xX goes to z1Z then to yY or z2Z and then to yY, that might be different paths from x to y. But here, it is talking about X and Y only.

And why is this ignoring x? It says h:IY? The problem I also have is, this is labelled “example” but it’s not specific at all. “h(0)=f(x) and h(1)=g(x)” over. So? What is this h and how has it been defined? As a map, as a function of some sort? Maybe there are multiple homotopy thinkable, but then what are one or two of them?

It’s like saying f(1)=1 and f(2)=4. Done. Well, to a newbie, maybe secondary school kids, it might be nice to give them an example be it linear f(x)=ax+b or f(x)=x2. To the eyes of the experienced, it might make perfect sense, a specific homotopy might pop out in their minds like popcorns but not in mine.

It’s an utter nightmare. I know this is “abstract” math but can not some more specific-ness be put into it? Pictures and diagrams perhaps?

This is only the tip of the massive massive confusion and dumb-foundedness I am experiencing in this area of study. Maybe once it “clicks” it all goes down like an avalanche but so far it’s nothing but counter-intuitive.

Can someone please make this possible for me to digest? Suggestions to good websites with examples and diagrams and extensive explanations are also welcome. Thank you

Answer

Imagine a malleable object like a rubber band that can occupy different shapes/configurations in space.

Let us call the abstract reference rubber band X (independent of any specific configuration it might take on), and call the space it moves in Y.

To define a particular configuration of the rubber band, one would need to specify where each point in the reference rubber band exists in space. That is, each possible configuration of the rubber band in space corresponds to a map from X to Y.

Maps f,g:XY then correspond to two different configurations of the rubber band in space.

Maps h:X×IY correspond to a time-sequence of configurations that the rubber band takes on. For every point x in the reference rubber band and for every time t in I, the location of that part of the rubber band in space at that time is specified by h(x,t). Visualize h as a movie of the rubber band moving and deforming in space over time.

Now, nothing written so far prevents crazy stuff from happening like the map f taking the rubber band and mapping it to some weird disconnected dust or something like that. Similarly, nothing yet written prevents the rubber band in the movie h to spontaneously teleport from one place to another at every frame, or things like that. The continuity assumptions are minimal assumptions there to prevent such crazy things.

Note that in particular, the continuity assumption still allows the rubber band to pass through itself, or to squish down from a 3d volumetric band to a 2D ring, so long as these processes are done in a continuous manner. If one desires to prevent these cases as well, then more stringent requirements can be used (e.g., isotopy, where the framework is the same but the maps are also required to be injective)

A natural question is, given two continuous configurations f and g of the rubber band, is it possible to continuously move/bend/twist/squish one into the other via a continuous movie h? This is equivalent to asking whether there exists a homotopy h between configurations f and g.

Whether or not two configurations f and g of an object are homotopic depends on both the shape of the object, and the shape of the ambient space it moves in. Consider a loop on the surface of a donut – in some configurations the loop can be continuously collapsed to a point, whereas in other configurations it cannot. Hence the target of study when using homotopy is often to understand the ambient space that the object moves in, rather than the object itself.

Hope this helps with the intuition.

Attribution
Source : Link , Question Author : John Trail , Answer Author : Nick Alger

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