Problem 1.1.5 from Hatcher's

Solution 1:

For the third implication you take your homotopy $S^1\times I\rightarrow X$ that is constant at $S^1\times\{1\}$ and show that this map factors through the quotient of $S^1\times I$ by $S^1\times\{1\}$, which is homeomorphic to $D^2$.

For the second implication, you don't have $D^2\rightarrow X$ is a homeomorphism. The two sphere, $S^2$, has trivial $\pi_1$ but is not homeomorphic to $D^2$. Let $\alpha\in\pi_1(X,x_0)$. Then $\alpha$ has a representative $f_\alpha:S^1\rightarrow X$. By (b), this map extends to a map $D^2\rightarrow X$. This is the same as a map $S^1\times I\rightarrow X$ such that the restriction $S^1\times\{1\}\rightarrow X$ is a constant map. You just have to make sure this homotopy can be based.