# Homotopy of complex 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 complex projective space

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## Contents

## Statement

This article describes the homotopy groups, including the set of path components , the fundamental group , and the higher homotopy groups of .

### Case

For , is the one-point set. Hence, all its homotopy groups are the trivial group. The set of path components is a one-point set and can be considered the trivial group.

### Case

For , (a homeomorphism), i.e., it is the 2-sphere. Its homotopy groups are hence the same as those of the 2-sphere. Specifically, they are as follows:

- is a one-point set.
- is the trivial group.
- , i.e., it is isomorphic to the group of integers, with the identity map being the generator.
- , i.e., it is isomorphic to the group of integers, with the map being the Hopf fibration.
- .

Higher homotopy groups are the same as those of the 2-sphere.

### Case of higher

For this case, we use the fiber bundle of sphere over projective space with fiber . We get the following long exact sequence of homotopy of a Serre fibration:

For , is trivial. Thus we get the following:

- Case : is a one-point space.
- Case : We get is trivial.
- Case : We get ..
- Case : We get that is the trivial group.
- Case : We get that .
- Case : We get that .