Homotopy of spheres

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This article describes the value (and the process used to compute it) of some homotopy invariant(s) for a topological space or family of topological spaces. The invariant is homotopy group and the topological space/family is sphere
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This article gives the key facts about the homotopy groups of spheres.


For n = 0

\pi_0(S^0) is a two-point set. If we think of S^0 as a group, \pi_0(S^0) gets the same group structure, namely the structure of the cyclic group of order two. For all k > 0, \pi_k(S^0) is the trivial group.

For n = 1

\pi_0(S^1) is a one-point set (or trivial group, if we choose to use S^1's group structure to induce a group structure on it). \pi_1(S^1) \cong \mathbb{Z}, and \pi_k(S^1) is trivial for all k \ge 2.

For higher n

We have that:

  • \pi_0(S^n) is a one-point set (which we can interpret as a trivial group in some cases).
  • \pi_k(S^n) is the trivial group for 0 < k < n.
  • \pi_n(S^n) \cong \mathbb{Z}.
  • \pi_3(S^2) \cong \mathbb{Z}.
  • \pi_k(S^n) is a finite group for k > n and k \ne 2n - 1.


The case n = 0

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The case n = 1

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For higher n

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