N-sphere is (n-1)-connected

Statement
Suppose $$n$$ is a natural number (i.e., $$n \ge 1$$). Then, the $$n$$-fact about::sphere is $$(n - 1)$$-connected. In other words:


 * For $$n = 1$$, the $$n$$-sphere, better known as the circle, is a path-connected space.
 * For $$n \ge 2$$, the $$n$$-sphere is a fact about::path-connected space and fact about::simply connected space and all its homotopy groups up to the $$(n - 1)^{th}$$ homotopy group are trivial. In other words, $$\pi_1(S^n), \pi_2(S^n), \dots, \pi_{n-1}(S^n)$$ are all trivial.

By the Hurewicz theorem, this is equivalent (for $$n \ge 2$$) to the assertion that $$S^n$$ is simply connected and the first $$n - 1$$ homology groups are trivial.