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, .
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.