# Digital topology, animal problem, 2-sphere and torus

I have the following question relating digital topology, surfaces, particularly $S^2$ and torus.

Can a body $B$ constructed with cubes (without cavities or tunnels) and with frontier homeomorphic to the $2$-sphere $S^2$ be reduced to a single cube by only deleting cubes while preserving $2$-manifold property (well-composed image) in each cube deletion?

Suppose not, then must exists a body $B$ constructed with cubes with frontier homeomorphic to $S^2$ such that if we remove any cube $c$ from $B$ creating body $B-c$ then $B-c$ has a boundary which is not a $2$-manifold. Such a body $B$ exists?

In Janos Pach animal problem (solved by Akira Nakamura) we have the same body $B$ but only deletions is not sufficient, this can be checked seeing Shermer's examples (See Günter Ziegler's Lectures on Polytopes, Springer, 1995, p.276. and some irreducible animals presented there) or some cube version of the Being's house, also one can notice that in this examples exists some cubes that can be removed creating a hole in $B$ (not preserving homeomorphism). Starting to remove this cubes one can reduce Shermer's examples to a single cube while maintain the $2$-manifold property (well-composed image).

I have Pach's problem, but with the ball-requirement removed and deletions only.

Thanks in advance for any suggestion, reference, commentary or hint.

Nice day!

## 1 Answer

Now that I think of it, my answer was wrong. I was just reproducing examples mentioned by the question poser (Bing's, as seen in Ziegler's book). These examples cannot be reduced keeping the boundary to be a 2-sphere, but the question is whether there are examples that cannot be reduced keeping the boundary a 2-manifold.