Determinant classifies loops in complex general linear groups

Continuous Gram–Schmidt on the ordered columns writes every $A\in {\mathrm{GL}}_n(\mathbb{C})$ uniquely as $A=QR$, where $Q\in U(n)$ and $R$ is upper triangular with positive real diagonal. No denominator vanishes because each initial set of columns is independent. The path $Q((1-t)R+tI)$ remains invertible and fixes $U(n)$ pointwise, so it is a deformation retraction of ${\mathrm{GL}}_n(\mathbb{C})$ onto $U(n)$.

The last-column map $SU(n)\to S^{2n-1}$ is locally trivial: near a chosen unit vector, continuous Gram–Schmidt completes that vector together with a fixed nearby frame, and multiplying the first completed vector by the inverse determinant puts the completion in $SU(n)$. Its fiber over the last basis vector is $SU(n-1)$. Thus $SU(n-1)\to SU(n)\to S^{2n-1}$ is a fibration.

Since $SU(1)$ is a point, induct simultaneously that $SU(n)$ is path-connected and simply connected. For $n\ge 2$, the sphere $S^{2n-1}$ is path-connected and has trivial fundamental group by [F3]. The pointed low-degree part of [F2], applied to step 1.2, first carries path-connectedness of the fiber and base to $SU(n)$ and then carries the inductive equality ${\pi }_1(SU(n-1))=0$ and ${\pi }_1(S^{2n-1})=0$ to ${\pi }_1(SU(n))=0$.

Determinant $U(n)\to U(1)$ is a fibration with fiber $SU(n)$ and section $s(z)=\mathrm{diag}(z,1,\dots ,1)$. By [F2] and step 2.1, ${\det }_{\ast }$ is injective on ${\pi }_1$, while the section makes it surjective. Step 1.1 transfers this isomorphism to ${\mathrm{GL}}_n(\mathbb{C})$ and ${\mathbb{C}}^{\times }$.

If a loop $g$ has determinant winding $k$, [F4] says $\det g$ is homotopic to $z\mapsto z^k$. Step 3.1 says that $g$ and $s(z^k)$ represent the same based homotopy class, which is precisely the displayed diagonal loop. Every construction was finite and explicit, so no choice principle was used.