# N-sphere is (n-1)-connected

From Topospaces

## Statement

Suppose is a natural number (i.e., ). Then, the -Sphere (?) is -connected. In other words:

- For , the -sphere, better known as the circle, is a path-connected space.
- For , the -sphere is a Path-connected space (?) and Simply connected space (?) and all its homotopy groups up to the homotopy group are trivial. In other words, are all trivial.

By the Hurewicz theorem, this is equivalent (for ) to the assertion that is simply connected and the first homology groups are trivial.