The deformation retraction of Bing’s house.
I found it tricky to satisfactorily sketch the full thing, but what I’ve got should do. I’ll show off the key move here, give one collection of pics of the entire deformation, and then point y’all to the Flickr for other viewing angles. If any of y’all other modelers/illustrators/mathematicians make a slicker version, let me know. I’d be happy to share it here.
So what makes deformation retractions of 2-complexes easy — or at least gives you a place to start — is the existence of a “free edge”, an edge of the 1-complex of valence 1, a 1-cell run along just once by the 2-cells. Bing’s house doesn’t have any. The edges of its 1-complex shown in black (as in the previous post, though I’ve drawn the house blue and more rounded here) all have valence 3.
Let’s focus on those little loops bounding disks that form interior walls upstairs and downstairs. Here’s an upstairs one. If we were just contracting, we could begin something like this:
Smush the disk down to make a fissure and then pull it apart. Now we see a free edge and can start pushing it in. That’d be great for the deformation retraction… but for a deformation retraction, everything has to stay within the original complex. The pulling apart (well, the smushing too) takes it off its original structure.
Since a deformation retraction is a homotopy, we can retain the memory of the topology of the house as we start squishing it through its own walls. So when we smush the disk, we can get as much slack from the tubes to its sides to just pull it down through the disk. I’ll show this slightly separated like a fissure again, but the sides of the new cleft should actually occupy the same space where the original disk was. Then keeping these in along the disk we can start pushing in a hole… the image of the house under the homotopy at this stage now has a free edge.
Here’s the same sequence drawn on top of the original complex (left) and within a thin neighborhood of the original complex (right).
So we do this to both disks to get some free edges and then we start pushing.
Here’s a sequence of the deformation retract from one view point shown both in “see through” and “solid construction”. Together you should get the gist of the entire deformation. Check out the Flickr set for other views.
|Quick view change…|
Of course I was hoping it would go something more like this.