# Homotopy of real projective space

From Topospaces

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 , the fundamental group , and all the higher homotopy groups.

### The case

The space 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

In the case we get is homeomorphic to the circle . We have is the one-point space (the trivial group), is the group of integers, and is the trivial group for .

### The case of higher

For , has the -sphere as its double cover and universal cover. Thus, for and . The problem of computing the homotopy of real projective space therefore reduces to the problem of computing the homotopy of spheres.

Hence:

- is the one-point space.
- is the cyclic group:Z2, i.e., .
- is the trivial group for .
- is isomorphic to , the group of integers.
- is isomorphic to , the group of integers.
- is a finite group for .