The Möbius and trivial real lines over the circle

Write $S^0=\{a,b\}$. A clutching map is a pair $(g(a),g(b))$ of nonzero real numbers. A gauge on either interval has boundary values in the same sign component, so the sign of $g(a)g(b{)}^{-1}$ is unchanged. Conversely, multiply by a constant gauge and use paths within ${\mathbb{R}}_{>0}$ or ${\mathbb{R}}_{<0}$ to normalize the pair to $(1,1)$ or $(1,-1)$. Thus [F1] gives at most and at least these two classes.

For $(1,1)$ the two trivial intervals glue their fiber coordinates without a sign change, producing $S^1\times \mathbb{R}$. For $(1,-1)$, cut the circle at one equatorial point; the remaining interval bundle closes by $(0,t)\sim (1,-t)$, which is the Möbius line. These two witnesses realize the normalized classes.

The invariant in step 1.1 has values $+1$ and $-1$ on the two witnesses in step 2.1, so they are not isomorphic; exhaustion in step 1.1 shows there are no others.