On pg 133 of Roman's Introduction to Algebraic Topology it is stated that one requires at least 14 triangles in any triangulation of the torus.

Admittedly, I do not have a very good understanding of triangualations.

From what I understand, the following seems like a perfectly valid triangulation of the torus:


What the mistake in this?

  • $\begingroup$ The 123 and 341 triangles on the left have separate edges (the vertical lines on the left) that share both end points 1 and 3. IIRC that is not allowed in a simplicial complex. $\endgroup$ Aug 23 '15 at 17:04

The NE and SW blocks both contain triangles with vertices 1,3,4. But these intersect in just the edge 13 and the vertex 4, while the intersection of two simplices in a triangulation must be a "face" of each (which might be the empty simplex), not a union of two or more simplices.


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