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I'm reading Andersen and Kashaev's A TQFT from quantum Teichmüller theory and the following condition in their definition of admissible oriented triangulated pseudo $3$-manifold confused me:

Definition 10. An oriented triangulated pseudo $3$-manifold $X$ is called admissible if $$S_r(X)\neq \varnothing$$ and ...

Here $S_r(X)$ is the set of based gauge equivalence classes of shape structures on $X$.

Since $S_r(X)$ is a quotient set of $S(X)$ the set of shape structures on $X$, $S_r(X)$ is non-empty if and only if $S(X)$ is non-empty. However, a shape structure on $X$ is defined as the following:

Definition 1. A Shape structure on $X$ is an assignment to each edge of each tetrahedron of $X$ a positive number called the dihedral angle $$\alpha_X\colon \Delta_3^1(X)\to \mathbb{R}_+$$ so that the sum of the three angles at the edges from each vertex of each tetrahedron is $\pi$.

Where, let $\Delta_i(X)$ be the set of $i$-dimensional cells of $X$, $\Delta_3^1(X)$ is defined tautologically by

$$\Delta_3^1(X) = \{(a,b)\mid a\in \Delta_i(X), b\in \Delta_j(a)\},$$

where $\Delta_j(a)$ is taken as if there is not any identification on the boundary of $a$.

It seems to me that, no matter what $X$ is, the constant function mapping everything to $\frac{1}{3}\pi$ is always a shape structure since the sum of any three angles at any three edges would be $\pi$ with this assignment. Therefore $S(X)$ cannot be empty, hence $S_r(X)\neq \varnothing$ is always true.

Am I missing something? Is there any simple example of oriented triangulated pseudo $3$-manifold without any shape structure?

Thanks in advance.

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  • $\begingroup$ In the linked ArXiv paper, Definition 10 is about the second homology group of $X$ (minus the vertices of the triangulation, I think). The definition does not mention $S_r(X)$ directly... $\endgroup$
    – Sam Nead
    Commented Feb 27 at 10:05
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    $\begingroup$ @SamNead Uh sorry, the version I've been reading is actually the published version. I didn't compare it with the version on ArXiv while posting this question. The ArXiv version appears to be a bit outdated compared to the published one according to the date of receipt, and the published version contains graphics that are not on the ArXiv version. I've changed the link to the published version, please have a look. I wonder if they noticed something which made them add the condition that $S_r(X)\neq\varnothing$ before publishing... $\endgroup$
    – Shana
    Commented Feb 27 at 12:02
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    $\begingroup$ @SamNead Moreover, even on the ArXiv version, there is a sentence right before section 1.6 saying that "If $S_r(X)$ (equivalently $S_r(Y)$) is nonempty, then so are $U$ and $U'$. If it is true that $S_r(X)$ is always nonempty, there is no reason to write this sentence... $\endgroup$
    – Shana
    Commented Feb 27 at 12:09
  • $\begingroup$ I've now looked at the published version. Unfortunately, I get stuck even earlier! At the bottom of page 893 they define $P^e$, as the "canonical map from the set of shape structures on $X$ to the set of shape structures on $Y$". But the 3-2 move is only only yields a shape structure on $Y$ if $e$ is balanced in $X$. But that is not a hypothesis on $S(X)$ (as defined in Definition 1). So I am at a loss. Sorry! ..... Perhaps all edges in a shaped triangulation are supposed to be balanced? Is there a gentler paper that refers to this one, and explains the issues? $\endgroup$
    – Sam Nead
    Commented Feb 27 at 12:59
  • $\begingroup$ Oh, and thank you for changing the link to the published version. $\endgroup$
    – Sam Nead
    Commented Feb 27 at 13:01

1 Answer 1

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I think you are correct and the set of shape structures is never empty. There can certainly fail to be angle structures (where there's a condition on the sum of the angles at each edge of the assembled triangulation) but the authors are careful to distinguish these from shape structures, so I think they intended $S_r(X)$ to always be nonempty.

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  • $\begingroup$ Thank you for answering! I'm a bit confused by your last phrase though. By "their definition was intended", did you mean that they deliberately added an assumption that is not theoretically necessary and they might know it but somehow decided to add it anyway? $\endgroup$
    – Shana
    Commented Feb 27 at 13:45
  • $\begingroup$ Sorry, I meant that I think they intended to not put any conditions on shape structures other than the local one on each tetrahedron, so it is intended that the set of shape structures is never empty. $\endgroup$
    – Calvin McPhail-Snyder
    Commented Feb 27 at 16:04
  • $\begingroup$ I see. Thank you! $\endgroup$
    – Shana
    Commented Feb 27 at 16:10

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