Homotopy group

Given a based space $(𝑋, 𝑥 0)$ and $𝑛 \in ℕ$ , we define an $𝑛$ -dimensional loop with base $𝑥 0$ to be a continuous function $𝛼 : (𝕀 𝑛, 𝜕 𝕀 𝑛) \to (𝑋, 𝑥 0)$ , i.e. mapping the boundary of the unit hypercube to the basepoint. Given loops $𝛼, 𝛽 \in 𝖳 𝗈 𝗉 • ((𝕀 𝑛, 𝜕 𝕀 𝑛), (𝑋, 𝑥 0))$ we define their concatenation as

𝛼 ⊙ 𝛽 (𝑡 1, \dots, 𝑡 𝑛) = {𝛽 (2 𝑡 1, 𝑡 2, \dots, 𝑡 𝑛) 𝑡 1 \in [0, 12] 𝛼 (2 𝑡 1 - 1, 𝑡 2, \dots, 𝑡 𝑛) 𝑡 1 \in [12, 1]

The $𝑛$ th homotopy group $𝜋 𝑛 (𝑋, 𝑥 0)$ is the set of homotopy classes of such loops with the concatenation operation, i.e. as a set $𝜋 𝑛 (𝑋, 𝑥 0) = 𝗁 𝖳 𝗈 𝗉 • ((𝕀 𝑛, 𝜕 𝕀 𝑛), (𝑋, 𝑥 0))$ . #m/def/homotopy Equivalently, $𝜋 𝑛 (𝑋, 𝑥 0) = 𝗁 𝖳 𝗈 𝗉 • ((𝕊 𝑛, 1), (𝑋, 𝑥 0))$ .

Properties

$𝑛 = 1$ gives the fundamental group
Non-fundamental homotopy groups are abelian
Non-fundamental homotopy groups of a circle are trivial

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