Identify all three edges of a single 2-simplex does not produce a Delta-complex structure

Question

Why this does not produce a Delta-complex structure?

Answer 1

One condition is that each restriction of a map $\sigma_\alpha$ to a face should coincide with one of the $\sigma_\beta$ of one dimension lower.

This comes with an implicit restriction that the face maps should respect the orientation induced by the ordering of the vertices, as opposed to the orientation induced by a manifold with boundary on its boundary.

In other words, just by looking at the picture you should be able to tell which vertex is $0$, which is $1$ and which is $2$, for each $2$-simplex. You can do that on the picture on the right, but not on that on the left.

Answer 2

The restriction on the face maps $d_{i}^{n}\colon S_{n+1}\rightarrow S_{n}$ in delta complexes is as follows:

$d_{i}^{n}\circ d_{j}^{n+1}=d_{j-1}^{n}\circ d_{i}^{n+1}$

In the case of a 2-simplex $s$ this implies the following identities between vertices:

$d_{0} \circ d_{1}(s) = d_{0}\circ d_{0}(s)$

$d_{0} \circ d_{2}(s) = d_{1}\circ d_{0}(s)$

$d_{1} \circ d_{2}(s) = d_{1}\circ d_{1}(s)$

In plain English, when building a 2-simplex edge by edge, you can pick the 0th edge freely, but with the 1st edge, you are restricted to picking it so that it starts from the same vertex as the 0th edge starts, and finally, you are forced to pick the 2nd edge so that it starts from where the 0th edge ends and ends where the 1st edge ends.

This leads to a picture like this:

...so if we want the edge orientations to agree with the gluing instructions, we must subdivide the triangle.

However, the formal definition doesn't seem forbid identifying the edges. That just means that there's just one edge that forms a loop, because the identities between vertices require its ends to be the same vertex. That also means that it can't be oriented. So, as a construction, a 2-simplex with identified edges is possible in a Δ-set, but it's geometric realization is rather meaningless, and collapses to a single point.