Homotopy of real projective space

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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 real projective space
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Statement

This article describes the homotopy groups of the real projective space. This includes the set of path components \pi_0, the fundamental group \pi_1, and all the higher homotopy groups.

The case n = 0

The space \mathbb{P}^0(\R) is a one-point space and all its homotopy groups are trivial groups, and the set of path components is a one-point space.

The case n = 1

In the case n = 1 we get \mathbb{P}^1(\R) is homeomorphic to the circle S^1. We have \pi_0(S^1) is the one-point space (the trivial group), \pi_1(S^1) \cong \mathbb{Z} is the group of integers, and \pi_k(S^1) is the trivial group for k > 1.

The case of higher n

For n > 1, \mathbb{P}^n(\R) has the n-sphere S^n as its double cover and universal cover. Thus, \pi_k(\mathbb{P}^n(\R)) \cong \pi_k(S^n) for k > 1 and \pi_1(\mathbb{P}^n(\R))\cong \mathbb{Z}/2\mathbb{Z}. The problem of computing the homotopy of real projective space therefore reduces to the problem of computing the homotopy of spheres.

Hence:

  • \pi_0(\mathbb{P}^n(\R)) is the one-point space.
  • \pi_1(\mathbb{P}^n(\R)) is the cyclic group:Z2, i.e., \mathbb{Z}/2\mathbb{Z}.
  • \pi_k(\mathbb{P}^n(\R)) is the trivial group for 1 < k < n.
  • \pi_n(\mathbb{P}^n(\R)) is isomorphic to \mathbb{Z}, the group of integers.
  • \pi_3(\mathbb{P}^2(\R)) is isomorphic to \mathbb{Z}, the group of integers.
  • \pi_k(\mathbb{P}^n(\R)) \cong \pi_k(S^n) is a finite group for k > n, k \ne 2n - 1.