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From the comments:

1. Suppose that $T$ is a triangulation. If all vertex links are PL $(n-1)$-dimensional spheres then the realisation space of $T$ is a PL manifold and thus a topological manifold. (In the compact case this is equivalent to the usual definition.)

2. There are triangulations of topological manifolds (in fact, of $S^5$) that do not have this property. Examples come from the double suspension theorem of Cannon and also Edwards.

Furthermore:

3. There are triangulations of topological manifolds that have no PL structure. This is (essentially) the failure of the Hauptvermutung.

4. There are closed topological manifolds that do not admit any triangulation. This is the failure of the Triangulation Conjecture, and is obtained by Casson in dimension four and to Manolescu in all higher dimensions.

From the comments:

1. Suppose that $T$ is a triangulation. If all vertex links are PL $(n-1)$-dimensional spheres then the realisation space of $T$ is a PL manifold and thus a topological manifold. (In the compact case this is equivalent to the usual definition.)

2. There are triangulations of topological manifolds (in fact, of $S^5$) that do not have this property. Examples come from the double suspension theorem of Cannon and also Edwards.

Furthermore:

3. There are triangulations of topological manifolds that have no PL structure. This is (essentially) the failure of the Hauptvermutung.

4. There are closed topological manifolds that do not admit any triangulation. This is the failure of the Triangulation Conjecture, and is obtained by Casson in dimension four and by Manolescu in all higher dimensions.

From the comments:

1. Suppose that $T$ is a triangulation. If all vertex links are PL $(n-1)$-dimensional spheres then the realisation space of $T$ is a PL manifold and thus a topological manifold. (In the compact case this is equivalent to the usual definition.)

2. There are triangulations of topological manifolds (in fact, of $S^5$) that do not have this property. Examples come from the double suspension theorem of Cannon and also Edwards.

Furthermore:

3. There are triangulations of topological manifolds that have no PL structure. This is (essentially) the failure of the Hauptvermutung.

4. There are closed topological manifolds that do not admit any triangulation. This is the failure of the Triangulation Conjecture, due to Casson in dimension four and to Manolescu in all higher dimensions.

From the comments:

1. Suppose that $T$ is a triangulation. If all vertex links are PL $(n-1)$-dimensional spheres then the realisation space of $T$ is a PL manifold and thus a topological manifold. (In the compact case this is equivalent to the usual definition.)

2. There are triangulations of topological manifolds (in fact, of $S^5$) that do not have this property. Examples come from the double suspension theorem of Cannon and also Edwards.

Furthermore:

3. There are triangulations of topological manifolds that have no PL structure. This is (essentially) the failure of the Hauptvermutung.

4. There are closed topological manifolds that do not admit any triangulation. This is the failure of the Triangulation Conjecture, and is obtained by Casson in dimension four and to Manolescu in all higher dimensions.

From the comments:

1. Suppose that $T$ is a triangulation. If all vertex links are PL $(n-1)$-dimensional spheres then the realisation space of $T$ is a PL manifold and thus a topological manifold. (In the compact case this is equivalent to the usual definition.)

2. There are triangulations of topological manifolds (in fact, of $S^5$) that do not have this property. Examples come from the double suspension theorem of Cannon and also Edwards.

Furthermore:

3. There are triangulations of topological manifolds that have no PL structure. This is (essentially) the failure of the Hauptvermutung.

4. There are topological manifolds that do not admit any triangulation. This is the failure of the Triangulation Conjecture, due to Casson in dimension four and to Manolescu in general.

From the comments:

1. Suppose that $T$ is a triangulation. If all vertex links are PL $(n-1)$-dimensional spheres then the realisation space of $T$ is a PL manifold and thus a topological manifold. (In the compact case this is equivalent to the usual definition.)

2. There are triangulations of topological manifolds (in fact, of $S^5$) that do not have this property. Examples come from the double suspension theorem of Cannon and also Edwards.

Furthermore:

3. There are triangulations of topological manifolds that have no PL structure. This is (essentially) the failure of the Hauptvermutung.

4. There are closed topological manifolds that do not admit any triangulation. This is the failure of the Triangulation Conjecture, due to Casson in dimension four and to Manolescu in all higher dimensions.