## Bing’s House

So there’s this example of a 2-complex that’s contractible, but not obviously so. Well actually, once you see it, it’s not too hard to see. Bing’s house with two rooms.

Blah. It’s not so apparent what’s going on. It’s a 2-complex, so let’s draw in the relevant 1-complex.

You can see two vertices and four edges. The surfaces of this 2-complex are all disks, and they make threefold incidences to the edges. The two loop edges bound disks, but they don’t show up since they’re the same translucent color as everything else. And all the corners can be somewhat misleading… Here’s a slicker picture with those two disks colored.

There has recently been a few words about it at MathOverflow where it’s pointed out that the contractiblity of Bing’s house is explained in Hatcher’s text and Cohen’s text. In this post, let’s see how this contraction works.

The easiest way to describe the contraction is by going from a point to Bing’s house. Since a point is homotopic to a ball, we’ll start there. Here’s a ball in it’s delicious tater tot form, $D^2 \times I$.

Now we’ll start pushing into the ball from the top and bottom, and eventually smush it into our two rooms. All the action will be in color.

And there you go. In this last one above we’ve smushed our tater tot into a thickening of Bing’s house.

You know how instead of doing a handshake, you could grab each other’s wrists? And your thumbs almost but don’t quite reach your fingers? Bing’s glove for two hands would fit snugly. Then you could have a close and friendly game of knife fight.

Here’s that thickening (still with greyish sides) surrounding Bing’s house and the 1-complex.

Here’s that thickening (all bluish) surrounding the 1-complex without the rest of Bing’s house.

Here are a few pics of all the moving parts at once together with the 1-complex, just ‘cuz the look neato.

In another post we’ll take a look at something much more subtle. Bing’s house actually deformation retracts to a point. Franciscus asked about this, and perhaps it’s what was meant to be asked on MathOverflow. But it’ll take a bit of time to draw it. The trick is to first see how the thickened Bing’s house scrunches down to a point without first expanding it to a tater tot.

~ by Ken Baker on March 25, 2010.

### 14 Responses to “Bing’s House”

1. This is wrong. This is not the required example, as it is not contractible. Indeed, the fundamental group of the complement in R^3 is not trivial (the loops around the tunnels are not contractible in the coplement). To get the correct version of this example the tunnels have to be tangent to the external wall (each tunnel has to share with the external wall a vertical interval from the floor to the ceiling (these two intervals are half as long as the height of the whole “house”). With the correct tunnels, the house is indeed a retract of its convex hall, which is topologically a cube (or a solid ball).

Regards,

Włodzimierz Holsztyński

• No, it’s fine. Instead of having the tubes tangent to the sides, I have them pulled in but with an added wall in each chamber that obstructs loops around the tubes in the complement. Admittedly these interior walls are hard to see. You can make out only the faintest hint of their presence in the first picture. The second picture shows the “necessary” 1-complex whose upper and lower black rectangles bound these walls. To make this more apparent, I included a third picture with these two walls colored red and green. I just now flipped that third picture so it coincides better with the first two.

The version of Bing’s house you mention is also in the literature, but for illustrative purposes I don’t care for the 4-fold incidences it requires.

I guess this must not have been so clear in the post. Thanks for prompting me to emphasize it.

2. Sorry for misinterpreting your (correct) version of the example. Indeed, I thought that those extra walls appear in some pictures only as auxiliary.

Let me mention that the this “2-room house” is one of the examples of a 2-dimensional contractible polyhedron, which has no free (“sharp”) edge, and that’s why it is claimed that the contraction is not obvious. It is not clear how to start a contraction. Another famous example of this type, which even has its separate advantages, is the “Borsuk’s horn”, known in the West as “Scottish hat”.

3. Neat. Alas, rather brief searches on google and mathscinet didn’t obviously bring up Borsuk’s Horn or the Scottish Hat. (Okay, there were many Scottish Hats.) Would you mind pointing out a reference or describing one?

I hadn’t really given much attention to contractible complexes before. For everyone else, go for a poke around.

4. Wow!! Thanks so much for doing this! I eagerly await footage of the deformation retraction!

5. I haven’t heard the term “Borsuk’s Horn”: is this the same as the “Dunce Hat”?
http://en.wikipedia.org/wiki/Dunce_hat_(topology)

6. Bing’s house: It’s difficult to figure out how to start the contraction,
or it’s just difficult to actually draw it…?

• Well a bit of both. Once you figure out how to start it, it takes a bit to figure out how to satisfactorily draw it so that others may understand. And then you need time to do so.

• Obviously there are no drawings around, not even descriptions on how to start. So I wonder: nobody solved the first step — or some people did but prefer to keep the secrect?

7. [...] deformation retraction of Bing’s house. Last time we looked at how Bing’s house with two rooms is contractible. Been so long you might not [...]

8. [...] fácil de ver del cubo macizo a la casa de Bing que podéis ver por ejemplo en el blog de Sketches of topology. El cubo macizo es por supuesto contráctil pues e puede deformar a un punto de manera incluso [...]

9. This is simply wrong; there are no two ways about it. The text claims that the top image is the house with two rooms, but it is not even contractible.

(The fact that some later images are in fact of the house with two rooms is not made clear at all. Rather they are described as images that are helpful to see why the house with two rooms is contractible.)

• Well, it is the house with two rooms and it is contractible. I acknowledged that the necessary dividers in the very first image are difficult — almost impossible — to see. The next two images are there to help clarify where they are, or at least should be.

As for the contractibility, you may care to recall that technical something is contractible if it is homotopy equivalent to a point. The house with two rooms is a 2-complex that we’ve embedded in R^3. A regular neighborhood of an embedded 2-complex is homotopic to that 2-complex. That sequence of pictures from the tater-tot through the red, orange, yellow, green, blue deformations exhibits a homotopy from the ball to a regular neighborhood of the house with two rooms. And hence a homotopy equivalence from the ball to the house with two rooms. Since the ball is homotopic to a point, so is the house with two rooms.