Homotopy and the fundamental group

Which of these spaces are simply connected? (Choose all that apply.)

Under the isomorphism $\pi_1(S^1, 1) \cong \mathbb{Z}$ of Theorem 6, what integer corresponds to the class of the loop $\gamma(s) = e^{-6\pi i s}$?

In the proof that the degree is well defined, we lift a path homotopy $F$ from $\gamma$ to $\gamma'$ to $\tilde F$, and then claim that $t \mapsto \tilde F(1, t)$ is constant. Why is it constant?

Brouwer's theorem says every continuous map $D^2 \to D^2$ has a fixed point. For which of these spaces is there a continuous map to itself with no fixed point? (Choose all that apply.)

The closed orientable surface of genus 2 is an octagon with sides glued by the word $a b a^{-1} b^{-1} c d c^{-1} d^{-1}$. What does the Seifert–van Kampen theorem give for its fundamental group?

The lesson's Lean theorem circle_not_retract takes a simply connected space X, maps i : C(AddCircle 1, X) and r : C(X, AddCircle 1), and a hypothesis h, and proves False. Which hypothesis h makes it the no-retraction theorem?

Implement path lifting for the angle. Write winding_number(points) for a list of non-zero complex numbers points describing a closed loop around $0$ (the last point equals the first). Add up the changes of angle between consecutive points, each taken in $(-\pi, \pi]$, and return the total divided by $2\pi$, rounded to the nearest integer.

How many connected 2-sheeted covering spaces does the figure eight $S^1 \vee S^1$ have, up to isomorphism?

In the plane with two holes, let $a$ and $b$ be loops going once around the first and the second hole. True or false: the loop $a b a^{-1} b^{-1}$, which winds $0$ times around each hole, is null-homotopic.