Mod-two Thom class of the Möbius line bundle

The generator of $H^1([-1,1],\{-1,1\};\mathbb{Z})$ is the pair connector of the endpoint class $(0,1)$ modulo diagonal constants. The Möbius clutching map is reflection $t\mapsto -t$, which swaps the endpoint coordinates. It sends $(0,1)$ to $(1,0)$, and $(1,0)=-(0,1)$ modulo the diagonal $(1,1)$. Thus its integral orientation monodromy is $-1$.

A global integral generator would have to return to itself after the base loop, while step 1.1 returns its negative. Since a generator of the free module $\mathbb{Z}$ is nonzero, this is impossible. Modulo two, $-1=1$ and the same transition fixes the unique nonzero generator, giving the canonical ${\mathbb{F}}_2$ orientation of [F1].

Apply [F2] with rank one and $R={\mathbb{F}}_2$. It produces a unique normalized degree-one Thom class and the displayed shift for every $k$. The two fiber endpoints, the base-loop start/end, the zero vector, degree-zero unit and zero classes are explicit above. The base and fibers are nonempty, and the coefficient rings are fixed, so empty and zero-ring cases are inapplicable. The monodromy calculation is finite and choice-free; AC is used exactly through [A1] in [F2].