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.

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

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