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I am looking for basic information about the following idea:

(I) Consider a square. By collapsing two adjacent vertices, we obtain a triangle.

(II) Consider a three-dimensional cube. By collapsing a single face to a line, we obtain a prism with triangular base. Next, by collapsing a line longitudinally, we obtain a pyramix with square base. Next, by contracting two adjacent vertices of the square base, we obtain a tetrahedron.

How can this can be generalized to higher dimensions? It is obvious that we can transform a hypercube to a simplex by a sequence of contractions. But not all contractions of faces produce a polyhedron (with flat faces), and many contractions produce the same object up to congruence. Since the idea is so basic, I hope somebody else has already written about it.

Hence I look for information, in particular about the following questions:

  1. Which contractions actually produce polyhedra, and which class of polyhedra is actually produced?
  2. Since different contractions produce congruent polyhedra, how can we relate (sequences of contractions) and congruence relations?
  3. Do these operations have an analogon on the symmetry groups of the polyhedra?

Of course, it would be best to have a book reference on this topic.

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Perhaps this paper addresses some of your questions?

Nevo, Eran. "Higher minors and Van Kampen's obstruction." Math. Scand. 101 (2007), no. 2, 161–176. (arXiv preprint.)

Nevo proves this:

Thm1.4

This generalizes an $\mathbb{R}^3$ result of Dey et al., "Topology preserving edge contractions" (PDF download.)


               


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See Chapter 6 in Ziegler's Lectures on Polytopes.

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  • $\begingroup$ Could you be more specific? I do not have access to a copy of that book (not yet, that is). $\endgroup$ – shuhalo Sep 3 '15 at 14:00

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