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 ${\mathbb{P}}^0(\mathbb{R})$ 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 ${\mathbb{P}}^1(\mathbb{R})$ 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 $n>1$, ${\mathbb{P}}^n(\mathbb{R})$ has the $n$-sphere $S^n$ as its double cover and universal cover. Thus, ${\pi }_k({\mathbb{P}}^n(\mathbb{R}))\cong {\pi }_k(S^n)$ for $k>1$ and ${\pi }_1({\mathbb{P}}^n(\mathbb{R}))\cong \mathbb{Z}/2\mathbb{Z}$. The problem of computing the homotopy of real projective space therefore reduces to the problem of computing the homotopy of spheres.

Hence:

• ${\pi }_0({\mathbb{P}}^n(\mathbb{R}))$ is the one-point space.
• ${\pi }_1({\mathbb{P}}^n(\mathbb{R}))$ is the cyclic group:Z2, i.e., $\mathbb{Z}/2\mathbb{Z}$.
• ${\pi }_k({\mathbb{P}}^n(\mathbb{R}))$ is the trivial group for $1<k<n$.
• ${\pi }_n({\mathbb{P}}^n(\mathbb{R}))$ is isomorphic to $\mathbb{Z}$, the group of integers.
• ${\pi }_3({\mathbb{P}}^2(\mathbb{R}))$ is isomorphic to $\mathbb{Z}$, the group of integers.
• ${\pi }_k({\mathbb{P}}^n(\mathbb{R}))\cong {\pi }_k(S^n)$ is a finite group for $k>n,k\ne 2n-1$.