#homotopy

I struggled to come up with a neat thing that it's homotopic to for years. Not continually more like a few hours every few months but still.

I always wondered why in homology/homotopy there is the possibility to allow for other rings than just the whole numbers Z. For the rational numbers Q, there is a cool example:

Imagine an infinite chain of parallel (unit) circles with a starting circle, hanged up on an infinite ray. We are allowed to jump with a kind of portal from each point z on the first circle to the point z^2 on the second circle, which is just the point with double the (counter-clockwise) angle w.r.t. to the ray. From the second to the third circle, we create the same portal with triple the angle instead, and from the kth circle to the (k+1)th circle, we multiply the angle by (k+1). The general map is therefore given by z -> z^{k+1}.

Now, homology is, simplified, the study of holes in an object. Of course, the first circle has one "hole" described by a counter-clock wise rotation, which we call [γ].

If you follow the portal for the whole rotation into the second circle, this loop goes around the second circle twice, which we write by 2[γ].

The number of loops is again tripled by the identification into the third circle, having 3 × 2 [y], where the two comes from the doubling from before.

In general, going along the image from the first circle to the kth sends [γ] to k! [γ], where k! is the kth factorial.

Now here is the neet part: If we want to describe a basic rotation [α] around the kth circle, we just divide by the number of loops in [α] to get a curve that goes around the kth circle once counter-clock wise, which is just k! [γ] by the portal logic (negative number just means clockwise rotation). Dividing by k! gives back [γ], and thus every rotation in one of the circles is just given uniquely by a rational multiple of [γ].

Summarizing, we have that every "loop" in this construction can be identified with a rational number and is therefore (up to isomorphism) just given as Q. ***

So here we have it: A homotopical structure which is visualizable as a rational structure.

Cell-like closed 0-dimensional decompositions of ℝ³ are ℝ&sup4; factors by Robert Edwards and Richard Miller