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Stacked spheres

A triangulation of a 2-dimensional sphere is called a stacked sphere if it is obtained inductively from the boundary of a 3-simplex by deleting a 2-face (triangle) $T$ adding a new vertex $v$ and three new triangles formed by $v$ and the edges of $T$. Stacked spheres are very nice and simple triangulations of $S^2$ (related, for example, to Apollonian circle packing). The definition extends to higher dimension.

The question

Q1: Show (in an elementary way) that if $M$ is a triangulated simply connected 3-manifold such that all links of vertices of $M$ are stacked spheres then $M$ is a triangulation of a sphere.

Remark: This follows, of course, from the 3-dim Poincare conjecture proved by Perelman. It would be nice to find a direct argument.

Q2: Characterize 3-manifolds that admit a triangulations with all links being stacked spheres.

You can get such manifolds starting from a stacked 3-dimensional sphere and "glue" a pair of 3-faces. Possibly, topologically, these are the only examples. (I don't know how to prove it even based on Poincare conjecture.)

Two pieces of background

  1. In dimension greater than 3 it can be shown (a paper of mine from 1987) that simply connected spheres with "stacked" links are themselves stacked spheres (This is not the case for $d=3$), and also to show that without any assumptions all such triangulations are obtained from stacked spheres by the above operation.

  2. While the condition "all links are stacked" poses (apparently) severe restrictions there are examples of 2-dimensional links that allow triangulations of arbitrary 3-manifolds. Cooper and Thurston gave (1988) such an example with five types of links.

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    $\begingroup$ Regarding Q2, I think I found a triangulation of S^2xS^1 which admits a triangulation with links of vertices being stacked 2-spheres. $\endgroup$
    – Ian Agol
    Commented Dec 2, 2021 at 21:44
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    $\begingroup$ Ian, I think that if you start with a stacked triangulation of $S^3$ and merge two far-apart 3-face (and delete them) this might be $S^2 \times S^1$ and you can have more complicated handle bodies like that when you merge more pairs. The question is if those are the only 3-manifolds that admit triangulations with stacked links. $\endgroup$
    – Gil Kalai
    Commented Dec 3, 2021 at 1:09
  • $\begingroup$ why can’t you merge like this in higher dimensions? $\endgroup$
    – Ian Agol
    Commented Dec 3, 2021 at 2:17
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    $\begingroup$ You can, and except for the spheres all these manifolds are not simply connected and have this very simple (handle-like) form (I dont remember how they are called). (This is what I meant by saying "such triangulations are obtained from stacked spheres by the above operation.") But for d=3 I dont know if also other manifolds admit triangulations with linked spheres. $\endgroup$
    – Gil Kalai
    Commented Dec 3, 2021 at 11:50

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