$S^3$ is isomorphic to $SO(3)$, which is a real Lie group and therefore is a differentiable and oriented real 3-fold, which has the double cover by $SU(2)$ which is the universal covering of $SO(3)$ or exponential of $su(2)$. Higher dimensional terms come from Bott's periodicity theorem. This is the QFT explanation.
In other words, thanks to the complex structure, we have a triangulation (approximation by CW (cell) complex attaching several n-dimensinal cell $e^n$ to a point set ${0}$, and Eilenberg-Steenrod axioms of singular homology of integral coefficient [with torsion module]). Then the textbook of Chern-Weil theory of characteristic class or some classic foliation (Postnikov tower of fibration) can tell you that there is a $E_2$ spectral sequence of double complex [see Bott-Tu GTM82, P.251-252] which is computable by exact sequence. This is the algebraic topology answer (non-simply connected space).