Let $M$ be the open Mobius band (a $2$-manifold without boundary) embedded in $\Bbb R^3$, and let $\nu$ and $\tau$ denote the normal bundle and tangent bundle of $M$, respectively. How do we know that $w_1(\nu)$ is nonzero?
My attempt: By the Whitney duality theorem, $w(\nu)=w(\tau)^{-1}=w(M)^{-1}$. Since $M$ is homotopy equivalent to $S^1$, $w(M)=w(S^1)=1$, so $w(\nu)=1$. Something must be wrong but I can't see it.