What 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
