# Homotopy of real projective space

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