34 lines
872 B
Markdown
34 lines
872 B
Markdown
Fundamental Group of S¹
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================
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Prerequisites :
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- circle
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- loop space
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- have useless maps S¹ → ℤ and ℤ → S¹
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- we make helix from ℤ ≡ ℤ
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-
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Motivating Steps :
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0. After sufficient pondering, you guess that
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loops are determined by how many times they go around
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1. Count number of times loops go around using ℤ
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by setting the single loop to +1.
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2. You can make a comparison maps between loop space and ℤ
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but can't yet show an equivalence
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3. You realize you need to use the definition of S¹,
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so you go from `base ≡ base ≃ Z` to `base ≡ x ≃ Bundle x`.
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i.e. You try to make the equivalence _over_ S¹
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Random thought :
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to prove `a = b`, we realise that `a = f(x0)`
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and `b = g(x0)` for `x0 : X`,
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and instead show `(x : X) → f x = g x`.
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This turns out to be easier since we now get
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access to the recursor of `X`.
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