# (Soft) Why do our visual understandings of homotopy equivalence correspond to the definitions?

This may be too soft, and it may be a confusion only about formalism. But seeing why the homotopies equivalences I visualize in my head and that people describe are the same thing as the definition of a homotopy.

I can see why my visualization of a homotopy of maps is the same as the definition, since the definition basically says: "maps are homotopic is there is a movie taking one to the other." But, I have an issue with homotopy equivalences, since it seems I make the maps between the spaces the "movie" and not the homotopy between these maps and the identity.

When people describe why two spaces are homotopic, they seem to construct the same kind of "movie" that deforms one space into the other.

Take the following example. A sphere with a line from the North Pole to the south through the sphere. I think that this should be homotopy equivalent to $S^1 \vee S^2$. In my head, I visualize slowly moving the endpoints of the line through the sphere to one another, and once they reach each other, I have a sphere and a circle that touch in one point.

But, the definition of a homotopy equivalence is a map $f: X \to Y$ such that there is a map $g: Y \to X$, with $g f$ and $fg$ homotopic as maps to the identity.

If I try to see how my visualization would correspond to this, it would seem that the end result if the map $f$, and perhaps doing it backwards is the map $g$. But, what am I, and it seems most people I talk to, doing in the intermediate stages?

Please ask for clarification.

• You are right to be suspicious of the movies. They generally give an accurate result if your spaces are CW complexes, but they can lead you astray in general. [A favorite example of mine is that if you take two copies of the Hawaiian Earring space connected by a line segment, contracting the line segment is not a homotopy equivalence.] In reality, you should be considering the image of one space deforming inside the other space. – Cheerful Parsnip Apr 5 '18 at 21:44
• I've no idea what you mean by "the intermediate stages"? But you need the homotopies between the maps in both directions to ensure that you aren't playing naughty tricks like contracting a sphere to a point. – Rob Arthan Apr 5 '18 at 21:58
• I think your intuitive view is rather that of the composite map $g\circ f$ being deformed into $id_X$ by some homotopy $H_t:X\times I\rightarrow I$. The slice of the "movie" at time $t_0$ is the image in $X$ of the map $H_{t_0}:X\rightarrow X$. My personal opinion is that this is not a particularly point of view; really when you are visualising deforming, say, $S^2\cup I$ into $S^1\vee S^2$, you are visualising them as deformations of spaces embedded into $\mathbb{R}^3$. This isn't really possible when discussing the homotopy type of, say, $Map(X,Y)$. – Tyrone Apr 6 '18 at 13:18
• For example, how does one 'visualise' the homotopy equivalence $Map(X,S^1)\simeq S^1$ for a simply connected space $X$? – Tyrone Apr 6 '18 at 13:23