TheHoTTGame/1FundamentalGroup/pi1S1.md

Fundamental Group of S¹

Prerequisites :

• circle

• loop space

• have useless maps S¹ → ℤ and ℤ → S¹

• we make helix from ℤ ≡ ℤ

Motivating Steps : 0. After sufficient pondering, you guess that loops are determined by how many times they go around

1. Count number of times loops go around using ℤ by setting the single loop to +1.
2. You can make a comparison maps between loop space and ℤ but can't yet show an equivalence
3. You realize you need to use the definition of S¹, so you go from base ≡ base ≃ Z to base ≡ x ≃ Bundle x. i.e. You try to make the equivalence over S¹

Random thought : to prove a = b, we realise that a = f(x0) and b = g(x0) for x0 : X, and instead show (x : X) → f x = g x. This turns out to be easier since we now get access to the recursor of X.