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.
• $\pi _{2n-1}(\mathbb {P} ^{n}(\mathbb {R} ))$ is an infinite group.
• $\pi _{k}(\mathbb {P} ^{n}(\mathbb {R} ))\cong \pi _{k}(S^{n})$ is a finite group for $k>n,k\neq 2n-1$.