Does Hopf-Umlaufsatz Theorem holds for piecewise smooth simple closed curves with cusps at the endpoints?
I checked from various resources, some of them added the condition: without cusps. But from what I see from Loring Tu Differential Geometry, he didn't mention about this condition. So I only want to confirm with this. If we can remove this condition, why ?
You can allow cusps, but there is the question whether to count them with $\pi$ or with $-\pi$. When the interior is circled counterclockwise an outward looking cusp should count $\pi$ and an inward looking cusp $-\pi$.
Imagine an "equilateral triangle" made of circular arcs curving inwards, so that we have three cusps at the vertices. When we go around counterclockwise we obtain $-60^\circ=-{\pi\over3}$ for each circular arc. At the cusps do the following: Replace the peak with a tiny "halfcircle" on the inside. This gives you $+\pi$ at each vertex, so that in total you have $3\bigl(-{\pi\over3}+\pi\bigr)=2\pi$, as the Umlaufsatz predicts.