<p>I'm looking for the correct name of geometric structures obtained as follows. </p>

<p>2-structures: A collection $X$ of triangles is a $2$-structure. If $X$ is a $2$-structure and $Y$ is obtained from $X$ by gluing two free edges belonging to triangles in $X$, then $Y$ is a $2$-structure. </p>

<p>3-structures: A collection $X$ of tetrahedra is a $3$-structure. If $X$ is a $3$-structure and $Y$ is obtained from $X$ by gluing two free triangles belonging to tetrahedra in $X$, then $Y$ is a $3$-structure. </p>

<p>k-structures: A collection $X$ of $k$-simplices is a $k$-structure. If $X$ is a $k$-structure and $Y$ is obtained from $X$ by gluing two free (k-1)-simplices belonging to $k$-simplices in $X$, then $Y$ is a $(k-1)$-structure. </p>

<p>Alternatively, a $k$-structure can be non-inductively defined as a homogeneous simplicial k-complex $X$, satisfying the following conditions.</p>

<ol>
<li>The intersection of any two k-simplices in X is either empty or a (k-1) simplex. </li>
<li>Each (k-1)-simplex is the intersection of at most two $k$-simplices in $X$.</li>
</ol>

<p><strong>Question:</strong> What is the correct name for $k$-structures? </p>

CC BY-SA 3.0
2017-06-06T00:36:27.430Z
I believe these are $\Delta$-complexes. See https://math.stackexchange.com/questions/1528005/simplicial-complex-vs-delta-complex-vs-cw-complex
Joshua Grochow
CC BY-SA 3.0
2017-06-06T06:53:12.933Z
@JoshuaGrochow thanks for the suggestion. But $\Delta$-complexes are more general than simplicial complexes.  The structures defined above are a special case of simplicial complexes. For instance, gluing two triangles along some edge yields a 2-structure. But gluing two triangles along some vertex does not, even though this is a simplicial complex.
Mateus de Oliveira Oliveira
CC BY-SA 3.0
2017-06-06T09:51:31.420Z
This is a purely mathematical question that doesn't actually involve computer science, so if you do not get a good answer here, you may try math.stackexchange.com or mathoverflow.net .
Emil Jeřábek
CC BY-SA 3.0
2017-06-06T15:00:29.877Z
@EmilJeřábek I was hesitating where to ask it first. I think that here I may have some chance of getting an answer from people working in computational geometry.
Mateus de Oliveira Oliveira
CC BY-SA 3.0
2017-06-06T16:32:40.283Z
Sorry, you're right. As I'm understanding you, your condition (1) is already part of the definition of simplicial complex, so you're basically looking at simplicial complexes with the additional restriction (2), right? If so, then wouldn't an equivalent characterization be "simplicial complexes that are also manifolds"?
Joshua Grochow
CC BY-SA 3.0
2017-06-06T17:50:57.420Z
Nevermind; I think the relationship between your complexes and manifolds is more subtle than I at first thought...
Joshua Grochow
CC BY-SA 3.0
2017-06-07T04:56:41.690Z
The first definition appears to yield simplicial complexes whose dual graphs are trees. In particular, every $(k-2)$-simplex must lie on the boundary. The second definition gives something more general; I've seen these described as _(simplicial) pseudo-manifolds_.
Jeffε
<reference-request><computational-geometry><topology><topological-graph-theory>
Structures obtained by gluing simplices
Mateus de Oliveira Oliveira
# Structures obtained by gluing simplices

(Asked by: username_0 on 2017-06-06)

I'm looking for the correct name of geometric structures obtained as follows.

2-structures: A collection $X$ of triangles is a $2$-structure. If $X$ is a $2$-structure and $Y$ is obtained from $X$ by gluing two free edges belonging to triangles in $X$, then $Y$ is a $2$-structure.

3-structures: A collection $X$ of tetrahedra is a $3$-structure. If $X$ is a $3$-structure and $Y$ is obtained from $X$ by gluing two free triangles belonging to tetrahedra in $X$, then $Y$ is a $3$-structure.

k-structures: A collection $X$ of $k$-simplices is a $k$-structure. If $X$ is a $k$-structure and $Y$ is obtained from $X$ by gluing two free (k-1)-simplices belonging to $k$-simplices in $X$, then $Y$ is a $(k-1)$-structure.

Alternatively, a $k$-structure can be non-inductively defined as a homogeneous simplicial k-complex $X$, satisfying the following conditions.

1. The intersection of any two k-simplices in X is either empty or a (k-1) simplex.
2. Each (k-1)-simplex is the intersection of at most two $k$-simplices in $X$.

**Question:** What is the correct name for $k$-structures?

**Tags:** reference-request, computational-geometry, topology, topological-graph-theory

> **Comments:**
> **username_1**: I believe these are $\Delta$-complexes. See https://math.stackexchange.com/questions/1528005/simplicial-complex-vs-delta-complex-vs-cw-complex  
> **username_0**: @JoshuaGrochow thanks for the suggestion. But $\Delta$-complexes are more general than simplicial complexes.  The structures defined above are a special case of simplicial complexes. For instance, gluing two triangles along some edge yields a 2-structure. But gluing two triangles along some vertex does not, even though this is a simplicial complex.  
> **username_2**: This is a purely mathematical question that doesn't actually involve computer science, so if you do not get a good answer here, you may try math.stackexchange.com or mathoverflow.net .  
> **username_0**: @EmilJeřábek I was hesitating where to ask it first. I think that here I may have some chance of getting an answer from people working in computational geometry.  
> **username_1**: Sorry, you're right. As I'm understanding you, your condition (1) is already part of the definition of simplicial complex, so you're basically looking at simplicial complexes with the additional restriction (2), right? If so, then wouldn't an equivalent characterization be "simplicial complexes that are also manifolds"?  
> **username_1**: Nevermind; I think the relationship between your complexes and manifolds is more subtle than I at first thought...  
> **username_3**: The first definition appears to yield simplicial complexes whose dual graphs are trees. In particular, every $(k-2)$-simplex must lie on the boundary. The second definition gives something more general; I've seen these described as _(simplicial) pseudo-manifolds_.  

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## 0 answers

