# N-sphere is simply connected for n greater than 1

## Contents

## Statement

Suppose is a natural number greater than 1. Then, the -sphere is a Simply connected space (?). In other words, it is a Path-connected space (?) and its Fundamental group (?) is the trivial group.

## Related facts

### Stronger facts

## Facts used

- Union of two simply connected open subsets with path-connected intersection is simply connected, which is a corollary of the Seifert-van Kampen theorem.
- Suspension of path-connected space is simply connected (this actually follows from the previous fact, but the explanation it offers is in some ways more direct and intuitive).

## Proof

### Proof using fact (1)

We try to show that the -sphere satisfies the conditions for fact (1). Denote by and any two antipodal (diametrically opposite) points of . Then, define and as . We obtain . We find that:

No. | Assertion | Uses ? | Justification |
---|---|---|---|

1 | Both and are open | No | Since is a manifold, points are closed, so their complements are open subsets. |

2 | Both and are simply connected | No | In fact, via the stereographic projection relative to the point , is homeomorphic to , which is contractible and hence simply connected. Similarly, via the stereographic projection relative to the point , is homeomorphic to , which is contractible, and hence simply connected. |

3 | The intersection is nonempty and path-connected | Yes | In fact, it is homeomorphic to . Since , , so is path-connected, hence is path-connected. |

Thus, we see that the conditions to apply fact (1) are met, and we obtain that has trivial fundamental group.

### Proof using fact (2)

For this, we note that is the suspension of . For , , so is a path-connected space. Hence, by fact (2), is simply connected.

Note that this proof is actually the same as the proof using fact (1), but is stated in a different conceptual language that perhaps sheds more light.