Homotopy of real projective space

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 $$k > n, k \ne 2n - 1$$.