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sniffnoyWhat would happen to homotopy theory if we used a more general notion of homotopy?
Let me make a formal definition: Given topological spaces X and Y and continuous maps f,g:X→Y, we'll say f and g are C-homotopic if there exists a connected space Z with points z0 and z1 and a continuous map h:X×Z→Y such that h(x,z0)=f(x) and h(x,z1).
So, obviously, this is a more inclusive notion than our usual notion of homotopy. We can then talk about C-homotopy equivalence, C-contractibility, C-homotopy groups, etc. And certainly there are maps that are C-homotopic but not homotopic; let Y be connected but not path-connected, and consider two points in Y in different path components as maps from the one-point space.
But can we find less trivial examples of maps that are C-homotopic but not homotopic? What about examples that just straight up are *not* C-homotopic? What about examples of spaces that are C-homotopy equivalent, but not homotopy equivalent, as well as spaces that aren't C-homotopy equivalent at all? (Question I tried unsuccessfully to answer: Is the topologists's sine curve C-contractible?)
Do C-homotopy groups agree with the usual homotopy groups? Do our usual algebraic topology functors respect C-homotopy in addition to just homotopy? (I asked John about this, he suggested that cohomology at least probably should.)
I'm posting this here as idle speculation because really, I don't know topology very well; I don't know enough to try to answer this. (Maybe someone already has. John hadn't heard of such a thing, that much I can say.) I thought of asking MathOverflow... but I was afraid I wouldn't be able to understand any answer I got! So yeah, I'm posting this here.
-Harry
Let me make a formal definition: Given topological spaces X and Y and continuous maps f,g:X→Y, we'll say f and g are C-homotopic if there exists a connected space Z with points z0 and z1 and a continuous map h:X×Z→Y such that h(x,z0)=f(x) and h(x,z1).
So, obviously, this is a more inclusive notion than our usual notion of homotopy. We can then talk about C-homotopy equivalence, C-contractibility, C-homotopy groups, etc. And certainly there are maps that are C-homotopic but not homotopic; let Y be connected but not path-connected, and consider two points in Y in different path components as maps from the one-point space.
But can we find less trivial examples of maps that are C-homotopic but not homotopic? What about examples that just straight up are *not* C-homotopic? What about examples of spaces that are C-homotopy equivalent, but not homotopy equivalent, as well as spaces that aren't C-homotopy equivalent at all? (Question I tried unsuccessfully to answer: Is the topologists's sine curve C-contractible?)
Do C-homotopy groups agree with the usual homotopy groups? Do our usual algebraic topology functors respect C-homotopy in addition to just homotopy? (I asked John about this, he suggested that cohomology at least probably should.)
I'm posting this here as idle speculation because really, I don't know topology very well; I don't know enough to try to answer this. (Maybe someone already has. John hadn't heard of such a thing, that much I can say.) I thought of asking MathOverflow... but I was afraid I wouldn't be able to understand any answer I got! So yeah, I'm posting this here.
-Harry
no subject
Date: 2014-09-28 02:09 am (UTC)His "h-contractible" is exactly what I called "C-contractible". His "h-path-connected" is weird and let's ignore it for a moment.
But, he gives an example which I was also thinking about and I totally forgot to mention on LJ! I was also thinking about the question of
whether [0,1]x[0,1] with the lexicographic order topology was C-contractible; I forgot to mention it above though. And I totally missed that it was because you can just use minimum to contract it! Obviously, that argument generalizes to any bounded linear continuum. And a similar argument works if there's only one endpoint; applying the argument twice shows that any linear continuum is C-contractible.
(Hm, this gives me another idea -- what if instead of using arbitrary connected spaces, we just used linear continua? So more general than
just the unit interval but less general than connected spaces in general. Would that make a difference?)
Then he introduces this weird notion which he calls "h-path-connected", which is, like, path-connected, but instead of using [0,1], he uses
h-contractible spaces. So I guess that generalizes -- you could consider a new definition of homotopy (yet another one!) based not on
connected spaces but h-contractible spaces. Does that even work? Is the wedge sum of two h-contractible spaces h-contractible? I mean I
would expect so but it's not obvious ot me (maybe that's just because I don't know enough topology). I guess if this all works, then you could iterate this construction... yikes...
But that part seems pretty unnatural to me so I mostly want to ignore it. I think connected is better. Or maybe linear continua. :)
(Question: I am just reinventing shape theory? I should probably read about that. I don't think it's the same as this...)