If $D^1\cup_f D^1=S^1$?

Question

Suppose $f\colon S^0\to S^0$, so we can form the attaching space $D^1\cup_f D^1$. Is my intuition correct that this space is just $S^1$? Since $S^0=\{1,-1\}$, $f$ is either the identity, or swaps the points. If it's the identity, then I think it's clear that we have $S^1$. If $f$ swaps the points, then the copies of $D^1=[-1,1]$ are attached at opposite points, but I think after you untangle them, it's still $S^1$.

Answer 1

Yes, you get the circle either way. A way to confirm this formally is to write down a homeomorphism between the results of "normal" and "twisted" gluing. The homeomorphism is the identity on the first copy of $D^1$, and $x\mapsto -x$ on the second copy.