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J. GE
J. GE
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Let $(X, d)$ be a closed geodesic space which is also a closed manifold. Suppose $d$ is a local $CAT(0)$ metric. Here is the question:

Is it true that $X$ admits a triangulation? (No requiring the triangulation to be PL).

An example of Davis and Januszkiewicz showed that there is a non-triangulable aspherical 4-manifold. Let $P^3$ be the Poincare Homology Sphere. They firstly constructed an aspherical 4 dimensional homology manifold via Gromov's hyperbolization, then replace the neighborhood around non-manifold point (the cone over $P^3$) by Freedman's 4-dimensional contractible manifold $W$ with the boundary $P^3$. It is the second step that the $CAT(0)$ condition is lost and I don't know anything about possible geometric properties of the mysterious $W$.

Let $(X, d)$ be a closed geodesic space which is also a manifold. Suppose $d$ is a local $CAT(0)$ metric. Here is the question:

Is it true that $X$ admits a triangulation? (No requiring the triangulation to be PL).

An example of Davis and Januszkiewicz showed that there is a non-triangulable aspherical 4-manifold. Let $P^3$ be the Poincare Homology Sphere. They firstly constructed an aspherical 4 dimensional homology manifold via Gromov's hyperbolization, then replace the neighborhood around non-manifold point (the cone over $P^3$) by Freedman's 4-dimensional contractible manifold $W$ with the boundary $P^3$. It is the second step that the $CAT(0)$ condition is lost and I don't know anything about possible geometric properties of the mysterious $W$.

Let $(X, d)$ be a geodesic space which is also a closed manifold. Suppose $d$ is a local $CAT(0)$ metric. Here is the question:

Is it true that $X$ admits a triangulation? (No requiring the triangulation to be PL).

An example of Davis and Januszkiewicz showed that there is a non-triangulable aspherical 4-manifold. Let $P^3$ be the Poincare Homology Sphere. They firstly constructed an aspherical 4 dimensional homology manifold via Gromov's hyperbolization, then replace the neighborhood around non-manifold point (the cone over $P^3$) by Freedman's 4-dimensional contractible manifold $W$ with the boundary $P^3$. It is the second step that the $CAT(0)$ condition is lost and I don't know anything about possible geometric properties of the mysterious $W$.

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J. GE
J. GE
  • 2.6k
  • 14
  • 26
Source Link
J. GE
J. GE
  • 2.6k
  • 14
  • 26

Is every geodesic space with non-positively curved metric triangulable?

Let $(X, d)$ be a closed geodesic space which is also a manifold. Suppose $d$ is a local $CAT(0)$ metric. Here is the question:

Is it true that $X$ admits a triangulation? (No requiring the triangulation to be PL).

An example of Davis and Januszkiewicz showed that there is a non-triangulable aspherical 4-manifold. Let $P^3$ be the Poincare Homology Sphere. They firstly constructed an aspherical 4 dimensional homology manifold via Gromov's hyperbolization, then replace the neighborhood around non-manifold point (the cone over $P^3$) by Freedman's 4-dimensional contractible manifold $W$ with the boundary $P^3$. It is the second step that the $CAT(0)$ condition is lost and I don't know anything about possible geometric properties of the mysterious $W$.