Differential topology

What is the degree of the map $f(z) = \bar z^{\,4}$ from the unit circle $S^1 \subset \C$ to itself?

Find the index at the origin of the planar vector field $v(x, y) = (x^2 - y^2, \, -2xy)$.

Which of these closed manifolds admit a smooth vector field with no zeros? (Choose all that apply.)

Let $h : T^2 \to \R$ be the height function of an upright torus. Which statement is correct?

The rotation of the annulus $A = \{1 \le |z| \le 2\}$ by the angle $\pi$ is continuous and has no fixed point. Which hypothesis of the Brouwer fixed-point theorem does it violate?

In the proof of the hairy ball theorem we use the homotopy $F(x, t) = \cos(\pi t)\, x + \sin(\pi t)\, v(x)$ from the identity to the antipodal map. Why does the proof need $v(x)$ to be a unit tangent vector at every point?

A closed orientable surface of genus $2$ stands upright, and $h$ is its height function — a Morse function with one maximum, one minimum and some saddle points. How many saddle points does it have?

The formal section proves critical_values_null : μ (f '' {x | (fderiv ℝ f x).det = 0}) = 0 for a differentiable f : E → E. Which informal statement is this?

The degree of a map $S^1 \to S^1$, and the index of a planar zero, are winding numbers. Write winding(points): given a list of non-zero complex numbers $w_0, w_1, \dots, w_{n-1}$ sampled along a closed curve (with $w_n = w_0$), return the number of times the curve winds anticlockwise round $0$, by adding up the changes of argument (each reduced to the range $(-\pi, \pi]$) and dividing by $2\pi$. Round to the nearest integer.