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Bing's house is an example of contractible 2-complex in $\mathbb{R}^3$. One may think that it is a surface without boundary that has two types of singularities: tripod curves — curves where three pieces of surface come together and quadrapod vertices — vertices where 4 curves and 6 pieces of surface come together. Formally speaking, Bing's house has the same local structure as the 2-skeleton of 4-cube.

The singularities of Bing's house look like this:

1-skeleton of Bing's house

It has 2 quadrapod vertices connected by 4 tripod curves (pair of loops and a pair of parallel edges). Bing's house is this graph with 3 discs attached.

Is there a simpler design? (Or another reasonably simple design)

I am interested in surfaces with the same type singularities.

(Motivated by this question.)

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  • $\begingroup$ If Q is the number of quadrapod vertices and T is the number of tripod curves then 2Q=T necessarily, because 6Q-3T will be the Euler characteristic of the boundary of the two-dimensional stratum. $\endgroup$
    – Tom Goodwillie
    Commented Feb 15, 2023 at 12:34
  • $\begingroup$ (This was assuming that none of the curves is a circle, I mean that each of the curves ends at two quadrapod vertices (which might be equal).) $\endgroup$
    – Tom Goodwillie
    Commented Feb 15, 2023 at 12:37
  • $\begingroup$ @TomGoodwillie well it is simply because all vertices have degree 2, right? $\endgroup$
    – Anton Petrunin
    Commented Feb 15, 2023 at 16:32
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    $\begingroup$ You are probably aware of the dunce hat? $\endgroup$
    – Ian Agol
    Commented Feb 15, 2023 at 21:37
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    $\begingroup$ I believe that the dunce hat does not have the desired vertex link about its unique “vertex” (point of “order” greater than three). That is, the dunce cap is not a special spine. $\endgroup$
    – Sam Nead
    Commented Feb 15, 2023 at 21:53

1 Answer 1

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The “abalone shell” is the simplest contractible, non-collapsible two-complex embedding in three-space. This is because it has a single “vertex”. For a picture, see Figure 3 of Matveev’s article.

By the way, Bing’s house has two vertices, not four (as suggested by your figure). See Figure 4 of Matveev’s article.

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    $\begingroup$ right, thank you, I will correct my question $\endgroup$
    – Anton Petrunin
    Commented Feb 16, 2023 at 6:10

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