Example of CW complex not $\Delta$-complex, and $\Delta$-complex not Simplicial complex.

Question

I have been searching for an example of a topological space $X$, which admits a CW complex structure, but no $\Delta$-complex structure. Notice that $S^2$ is not a valid example for what I want. I know that $S^2$ can be constructed as a CW complex with one 0-cell and one 2-cell, and that this is not a $\Delta$-complex structure. But there is a $\Delta$-complex structure for $S^2$, with one 0-cell, one 1-cell and two 2-cells.

I have the same question for the comparison of $\Delta$-complexes and simplicial complexes. One example that I found is $S^1$, but as in the previous case, this doesn't work. Even though the traditional construction with one 0-cell and one 1-cell is a $\Delta$-complex and not a simplicial complex, you can still construct $S^1$ as a simplicial complex with three 0-cells and three 1-cells.

So if you have any example of a space that admits one structure but not the other, or if there is some general result comparing this three structures, I would be really happy if you share this with me.

Thanks

Answer 1

Every $\Delta$-complex is homeomorphic to a simplicial complex: barycentrically subdivide the cells of a $\Delta$-complex twice, and every new small triangle will be completely determined by its vertices, and so we have a simplicial complex. See also this MSE answer.

Not every CW complex is homeomorphic to a simplicial complex. See this MathOverflow question and its answers. There, Evan Jenkins mentions the book Cellular Structures in Topology by Fritsch and Piccinini, which contains the following example in section 3.4. I've made a graph in Desmos 3D for easier visualization, with screenshots below.

The CW complex has five 0-cells (blue), five 1-cells (red), and one 2-cell (purple). The 1-skeleton as a graph is isomorphic to a square with an extra edge hanging off, but the attaching map of the 2-cell is infinitely wiggly, specifically on that extra edge.

The proof that this is not triangulizable (i.e. homeomorphic to a simplicial complex) goes roughly like this:

• It's compact so any triangulation would have to be finite.
• No point on the $z$-axis in interior to a 2-simplex.
• At each local maximum level of the surface, the point of the on $z$-axis must be a 0-simplex: if it were interior to a 1-simplex, then other nearby points would have homeomorphic neighborhoods to that local-maximum-level point. But this is not the case: neighborhoods of the local-maximum-level points ("tops of folds") do not look like neighborhoods of their close neighbors.
• There are infinitely many such maximum levels, so infinitely many 0-simplices, so there's no finite triangulation.

In the images below, note that the rough texture moving along the $x$-axis is just from limited rendering; the only real "roughness" is the waviness from the wavy constant-$x$ cross-sections, and the "pinching" at the $z$-axis.