A manifold $M$ is simply connected if for every pair of 1-cubes $c_1,c_2: [0,1]\rightarrow M$ with
$c_1(0) = c_1(1) = c_2(0) = c_2(1) = t$
there is a 2-cube $b$ such that
1) $b(1,0) = c_1$ and $b(1,1) = c_2$
2) for all $p$ in $[0,1]$, $b(2,0)(p) = b(2,1)(p) = t$
My question is, if I'm given a manifold M,
for example choose $M = S^2 = \{(x,y,z)\in \mathbb{R}^3: x^2 + y^2 + z^2 = 1\}$
how do I prove that it is simply connected?
Also, how would you prove that a manifold is not simply connected?