Different Representations of Dunce Cap

Question

I am having a bit of confusion concerning the Dunce Cap while studying simplicial homology, hope someone can help!

Given a solid triangle with vertices $a, b, c$ I usually see the Dunce Cap defined as the quotient on the edges as follows: $[a,b], [a,c]$ and $[b,c]$ are identified, using that orientation on the edges. However, the "n-fold Dunce Cap" is usually defined where all the edges are identified like a cycle:

My understanding is that the $n$-fold Dunce Cap has $\mathbb{Z}/n\mathbb{Z} = H_1$. But that doesn't make sense to me in the usual case, since the classical Dunce Cap is known to be contractible so should have trivial $H_1$ - which means it doesn't make sense in the case $n = 3$.

Can someone tell me what I'm messing up? Are those two identification schemes on the triangle the same space or not? And is it accurate that $\mathbb{Z}/n\mathbb{Z} = H_1$ for the $n$-fold Dunce Cap?

In general, does it matter what order we apply to the edges at the start for any $n$? Or as long as we're identifying all edges/vertices of some $n$-gon will we always get the same thing?

Answer 1

Everything you say looks right to me. In both of the spaces you've described, a cell structure exists with one 2-cell, one 1-cell, and one 0-cell.

• In the classical dunce cap, the boundary of the 2-cell is $1+1-1=1$ times the generator of $C_1$, so the first homology vanishes.
• In the 3-fold dunce cap, the boundary of the 2-cell is $1+1+1=3$ times the generator of $C_1$, so the first homology is $\mathbb{Z}/3\mathbb{Z}$.

Evidently these are different spaces.