Question : $C$ is a pure, full-dimensional polytopal complex(a special case of a regular cell complex) in $\mathbb{R}^d$. I know that the boundary of the underlying set is a PL-sphere. Is it true that $C$ is a PL-ball?

Definitions for this question:

2. Polytopal complex is a finite nonempty collection of convex polytopes in $\mathbb{R}^d$ that contains all faces of its polytopes, and such that the intersection of two of its polytopes is a face of each of them.
3. Dimension of the complex is the largest dimension of a polytope in the complex.
4. A complex $C$ is pure if each of its faces is contained in a face of dimension $dim(C)$.
5. Underlying set is the union of its faces.
6. PL stands for piecewise-linear
7. PL-ball is something that is PL-homemorphic to a simplex.
8. PL-sphere is something that is PL-homeomorphic to a boundary of a simplex.

(If the answer is yes for the above question) Further Question : Now we look at a non-full dimensional pure polytopal complex $C$. We define the "boundary" of $C$ to be the set of points where there does not exist a neighborhood that is PL-homemorphic to $\mathbb{R}^{dim(C)}$. Then if the "boundary" of $C$ is a PL-sphere, is $C$ a PL-ball?

P.S. Please give a comment if some parts of the question is not suitably defined. Also, I have no idea how hard this is, so please comment if you know it is obvious or is really hard.

Question : $C$ is a pure, full-dimensional polytopal complex(a special case of a regular cell complex) in $\mathbb{R}^d$. I know that the boundary of the underlying set is a PL-sphere. Is it true that $C$ is a PL-ball?

Definitions for this question:

2. Polytopal complex is a finite nonempty collection of convex polytopes in $\mathbb{R}^d$ that contains all faces of its polytopes, and such that the intersection of two of its polytopes is a face of each of them.
3. Dimension of the complex is the largest dimension of a polytope in the complex.
4. A complex $C$ is pure if each of its faces is contained in a face of dimension $dim(C)$.
5. Underlying set is the union of its faces.
6. PL stands for piecewise-linear
7. PL-k-ball is something that is PL-homemorphic to a simplex of $k$ dimensions.
8. PL-(k-1)-sphere is something that is PL-homeomorphic to a boundary of a simplex of $k$ dimension.

(If the answer is yes for the above question) Further Question : Now we look at a non-full dimensional pure polytopal complex $C$. We define the "boundary" of $C$ to be the set of points where there does not exist a neighborhood that is PL-homemorphic to $\mathbb{R}^{dim(C)}$. Then if the "boundary" of $C$ is a PL-(dim(C)-1)-sphere, is $C$ a PL-dim(C)-ball?

P.S. Please give a comment if some parts of the question is not suitably defined. Also, I have no idea how hard this is, so please comment if you know it is obvious or is really hard.

9/19 : Edited so it includes the dimension restriction.

Question : $C$ is a pure, full-dimensional polytopal complex(a special case of a regular cell complex) in $\mathbb{R}^d$. I know that the boundary of the underlying set is a PL-sphere. Is it true that $C$ is a PL-ball?

Definitions for this question:

2. Polytopal complex is a finite nonempty collection of convex polytopes in $\mathbb{R}^d$ that contains all faces of its polytopes, and such that the intersection of two of its polytopes is a face of each of them.
3. Dimension of the complex is the largest dimension of a polytope in the complex.
4. A complex $C$ is pure if each of its faces is contained in a face of dimension $dim(C)$.
5. Underlying set is the union of its faces.
6. PL stands for piecewise-linear
7. PL-ball is something that is PL-homemorphic to a simplex.
8. PL-sphere is something that is PL-homeomorphic to a boundary of a simplex.

(If the answer is yes for the above question) Further Question : Now we look at a non-full dimensional pure polytopal complex $C$. We define the "boundary" of $C$ to be the set of points where there does not exist a neighborhood that is PL-homemorphic to $\mathbb{R}^{dim(C)}$. Then if the "boundary" of $C$ is a PL-sphere, is $C$ a PL-ball?

P.S. Please give a comment if some parts of the question is not suitably defined.

Question : $C$ is a pure, full-dimensional polytopal complex(a special case of a regular cell complex) in $\mathbb{R}^d$. I know that the boundary of the underlying set is a PL-sphere. Is it true that $C$ is a PL-ball?

Definitions for this question:

2. Polytopal complex is a finite nonempty collection of convex polytopes in $\mathbb{R}^d$ that contains all faces of its polytopes, and such that the intersection of two of its polytopes is a face of each of them.
3. Dimension of the complex is the largest dimension of a polytope in the complex.
4. A complex $C$ is pure if each of its faces is contained in a face of dimension $dim(C)$.
5. Underlying set is the union of its faces.
6. PL stands for piecewise-linear
7. PL-ball is something that is PL-homemorphic to a simplex.
8. PL-sphere is something that is PL-homeomorphic to a boundary of a simplex.

(If the answer is yes for the above question) Further Question : Now we look at a non-full dimensional pure polytopal complex $C$. We define the "boundary" of $C$ to be the set of points where there does not exist a neighborhood that is PL-homemorphic to $\mathbb{R}^{dim(C)}$. Then if the "boundary" of $C$ is a PL-sphere, is $C$ a PL-ball?

P.S. Please give a comment if some parts of the question is not suitably defined. Also, I have no idea how hard this is, so please comment if you know it is obvious or is really hard.