A simplicial complex $S$ is collapsible if there is a sequence of elementary collapses that bring $S$ down to a single point; I'll denote this as $S \searrow \{pt\}$. I am wondering about a similar problem (the dual problem, in some ways).

If I have a simplicial complex $S \subset \Delta^n$, where $\Delta^n$ is the $n$-simplex, and there are elementary collapses bringing $\Delta^n$ to $S$, i.e. $\Delta^n \searrow S$, then $S$ must be contractible. Is the converse true: if $S$ is contractible, then is it true that $\Delta^n \searrow S$? How about if $S$ is collapsible?

  • $\begingroup$ It's certainly true that $\Delta^n \nearrow\searrow S$, since the Whitehead group of the trivial group is trivial. $\endgroup$ – Jim Conant Aug 11 '15 at 5:14
  • $\begingroup$ If I am not mistaken, S is collapsible if and only if S is simple homotopy equivalent to a point, whereas S is contractible if and only if it is homotopy equivalent to point. And homotopy equivalence is known not to be equivalent to simple homotopy equivalence. $\endgroup$ – user43326 Aug 11 '15 at 5:52
  • $\begingroup$ @JimConant I'm not quite sure what you mean by $\Delta^n \nearrow \searrow S$, and I'm not familiar with Whitehead groups yet. Can you expand on your comment please? $\endgroup$ – GraduateStudent Aug 11 '15 at 7:08
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    $\begingroup$ @user132617: I mean there is a series of simple expansions and contractions. (This is called simple homotopy equivalence.) Whether or not homotopy equivalence is the same as simple homotopy equivalence depends on the Whitehead group of the fundamental group. For the trivial group, the Whitehead group is trivial, although this does require a calculation. $\endgroup$ – Jim Conant Aug 11 '15 at 23:52
  • $\begingroup$ @user43326: h.e. and s.h.e. are the same for some fundamental groups, including the trivial group. $\endgroup$ – Jim Conant Aug 11 '15 at 23:53

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