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Dear experts,

let $T$ be finite tetrahedral complex in flat 3-dimensional euclidean space. Additionally, let $T$ be 'homogeneous' in a sense that each simplex in $T$ is a face of some tetrahedron from $T$. Are there any known sufficient conditions (say in terms of Betti numbers of $T$ and possibly Betti numbers of some other complex derived from $T$) for $T$ to be homeomorphic to a 3D ball?

My understanding is that the condition on Betti numbers of $T$ itself is necessary but not sufficient (consider two tetrahedrons touching at a vertex or an edge, but not sharing any 2-simplex).

Consider another complex $T'$, which is dual to $T$ in the following sense. Each 'vertex' (0-cell) of $T'$ corresponds to a tetrahedron in $T$. Each 'edge' (1-cell) of $T'$ corresponds to an interior (i.e., not lying on the global boundary of $T$) 2-simplex in $T$. Each 2-cell of $T'$ corresponds to an interior 1-simplex in $T$. Each 3-cell of $T'$ corresponds to an interior vertex of $T$. The boundary operation on $T'$ is defined by 'transposing' the boundary operation in $T$. For example, if a vertex in $T$ is part of the boundary of an edge in $T$, then the corresponding 3-cell in $T'$ has corresponding 2-cell as part of its boundary, with the same sign (orientation).

Suppose both $T$ and $T'$ have 1 connected component (that is, corresponding Betti number equals one), and all other Betti numbers of $T$ and $T'$ vanish. Does this imply that $T$ is homeomorphic to a ball? If yes, would you please provide me with a reference?

For example, if $T$ contains one tetrahedron (plus all its faces), $T'$ will consist of a single vertex. For two tetrahedrons touching each other at edge or vertex (but not sharing a 2-simplex), $T'$ will consist of two isolated points.

Another example: suppose one takes a ball, picks two diametrically opposite points on the surface and pushes them inside until they meet. Then the deformed ball (with the two opposite points glued) is "triangulated with tetrahedrons". My understanding is that the obtained tetrahedral complex will have the same Betti numbers as the 3D ball, however 'dual' complex will have 1-cycle which will not be a boundary of any 2-chain.

Update: I'm looking for (reasonably) fast algorithms to check whether a given set of tetrahedrons is or is not homeomorphic to a 3d ball. The tetrahedrons are picked from a tetrahedral mesh of some nice domain. I have been looking into Betti numbers because there exist fast (linear or almost linear complexity) algorithms for computing Betti numbers of simplicial complexes embeddable in $\mathbb{R}^3$, see here. Admittedly, I do not know whether these algorithms will (provably) work for the complex $T'$, which is not necessarily simplicial.

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# Sufficient conditions for a 3D tetrahedral complex to be homeomorphic to a 3D ball

Dear experts,

let $T$ be finite tetrahedral complex in flat 3-dimensional euclidean space. Additionally, let $T$ be 'homogeneous' in a sense that each simplex in $T$ is a face of some tetrahedron from $T$. Are there any known sufficient conditions (say in terms of Betti numbers of $T$ and possibly Betti numbers of some other complex derived from $T$) for $T$ to be homeomorphic to a 3D ball?

My understanding is that the condition on Betti numbers of $T$ itself is necessary but not sufficient (consider two tetrahedrons touching at a vertex or an edge, but not sharing any 2-simplex).

Consider another complex $T'$, which is dual to $T$ in the following sense. Each 'vertex' (0-cell) of $T'$ corresponds to a tetrahedron in $T$. Each 'edge' (1-cell) of $T'$ corresponds to an interior (i.e., not lying on the global boundary of $T$) 2-simplex in $T$. Each 2-cell of $T'$ corresponds to an interior 1-simplex in $T$. Each 3-cell of $T'$ corresponds to an interior vertex of $T$. The boundary operation on $T'$ is defined by 'transposing' the boundary operation in $T$. For example, if a vertex in $T$ is part of the boundary of an edge in $T$, then the corresponding 3-cell in $T'$ has corresponding 2-cell as part of its boundary, with the same sign (orientation).

Suppose both $T$ and $T'$ have 1 connected component (that is, corresponding Betti number equals one), and all other Betti numbers of $T$ and $T'$ vanish. Does this imply that $T$ is homeomorphic to a ball? If yes, would you please provide me with a reference?

For example, if $T$ contains one tetrahedron (plus all its faces), $T'$ will consist of a single vertex. For two tetrahedrons touching each other at edge or vertex (but not sharing a 2-simplex), $T'$ will consist of two isolated points.

Another example: suppose one takes a ball, picks two diametrically opposite points on the surface and pushes them inside until they meet. Then the deformed ball (with the two opposite points glued) is "triangulated with tetrahedrons". My understanding is that the obtained tetrahedral complex will have the same Betti numbers as the 3D ball, however 'dual' complex will have 1-cycle which will not be a boundary of any 2-chain.