How can I prove that the horn torus and $\mathbb{T} \cup D_1$ have the same homotopy type?

Question

I'm trying to prove that the horn torus ($W$) defined by rotating the circumference $(x-1)^2+z^2=1, y=0$ around the z axis and $A=A_1 \cup A_2$ where $A_1$ is the torus obtained rotating $(x-2)^2+z^2=1, y=0$ around the z axis and $A_2$ is the disc $x^2+y^2 \le 1, z=0$, have the same homotopy type.

According to Hatcher's Algebraic Topology two spaces $X, Y$ have the same homotopy type if and only if there is another space $Z$ such that $X$ and $Y$ are deformation retracts of it.

So I've been trying to define a 4 dimensional space such that both $W$ and $A$ are deformation retracts of it but I haven't been able to progress any further. I was thinking of considering something like $Z = S^1 \times \mathbb{R}^2$ (parametrizing $S^1$ as ($\cos t, \sin t$)) but when I came to the equations I couldn't find the proper retraction.

What would be a suitable space to prove this? Is this the easiest approach?

Answer 1

You can directly define a homotopy equivalence. Define $f\colon A\to W$ by contracting the disk to a point. The resulting space is clearly homeomorphic to $W$. Define a map $g\colon W\to A$ by mapping a neighborhood of the pinch point onto the disk as follows. A neighborhood of the pinch point looks like an annulus with central circle identified to a point. This is homeomorphic to two disks identified at a point. Map both of these disks by $g$ onto the disk you called $A_2$. Now map the rest of the horned torus onto $A_1$ in the obvious way. That these two maps are homotopy inverse is a good exercise.

A lot of times in topology we just say that crushing a contractible subspace to a point is a homotopy equivalence, but actually it only works for nice spaces where you can generalize this trick I had of mapping a neighborhood of the collapsed point back onto the contractible subspace.

Answer 2

Let's picture the problem by taking the intersection of both spaces with a vertical plane containing the $z$ axis, where $\sigma$ represents the norm of any vector on the $xy$ plane:

The idea described in this answer seems to suggest that a central annulus around the $z$ axis (the one painted in blue) has to be projected on the disk of diameter $1$ in the $xy$ plane and then mapped onto the $D$ disc, while the remaining part of the horn torus (painted in green) would be mapped onto $A_2$, the torus with center at distance $R$ (in the question $R=2$).

While the second mapping is pertinent to the solution, the first is not as it seems to be indicated. The reason is that if we collapse the upper and lower surfaces onto the central disk, we would be joining two separate surfaces onto one and that operation cannot be part of a homotopy because it cannot be reverted: any reversion would need to split the disk back into two surfaces creating a hole that would break continuity.

One way to overcome this difficulty is by first filling the gap between the upper and lower parts of the central annulus and then proceeding with the vertical projection followed by the horizontal scaling:

Note that this filling can be easily accomplished as a horizontal retraction. We can now project+scale and then stretch the green half of the horn torus onto $A_2$ as desired. The reverse transformation can be visualized pictorially this way:

where $D$ is compressed onto the disk of radius $1$, the inner (left) half of the torus (in red) is mapped onto the vertical diameter (also in red) and the outer (right) half (in green) is translated onto the the corresponding half of the horn torus (also in green).

This approach might also inspire a simpler one. I've chosen to describe it because I've written all the formal details and so I know it works.