Are these spaces differential manifolds?

Question

I'm thinking about this spaces.

My doubt arise since the intersection point of the triangles.

Regards!

Answer 1

The main point here is that these are not even topological manifold: there are points (in fact 1 point) which does not have any neighbourhood homeomorphic to an open set of $R^n$. As you guessed the problem is the "intersection point of the triangles" (vague enough but it should be clear what it means.

But even if you removed say the lower triangle there would be still problems: a smooth manifolds does not have "corners". So the upper triangle in the first picture would be a topological manifold but not a smooth one (thinking of the topology and smooth structure as a submanifold of $R^2$). In the second one is not clear whether the boundary is to be included or not. If yes the triangle is not a topological manifold but a topological manifold with boundary; it is not a smooth manifold with boundary as it has corners. The open triangle would be a smooth manifold.