# Extending a triangulation from a manifold boundary to the interior

Let $$M$$ be a (second-countable topological) compact connected manifold-with-boundary. Suppose $$\partial M$$ has a triangulation. Does there exist a triangulation of $$M$$ which extends the triangulation on $$\partial M$$? If not, what additional conditions would be required for this to hold?

For background, I would like to extend the result that this is true when $$M$$ is the closed ball in $$\mathbb{R}^n$$. Here the triangulation is extended by adding simplex-cones from the center of the ball to the boundary-simplices.

• No: For instance take the E8 manifold and remove an open ball from it. – Moishe Kohan Jun 25 at 20:32
• Thanks, should have thought that myself. I wonder if it suffices to require $M$ triangulable? – kaba Jun 25 at 21:45
• I am sure it is false as well. – Moishe Kohan Jun 25 at 22:55