Higher Math: Geometry and Topology

Using the fact that $\operatorname{SL}_n \mathbb{C}$ is connected and simply connected for all $n\geq1$ (see e.g. Fulton–Harris, Section 23.1, Proposition 23.1), together with the short exact sequence $$1 \to \operatorname{SL}_n \mathbb{C} \to \operatorname{GL}_n \mathbb{C} \xrightarrow{\det} \mathbb{C}^\ast \to 1,$$ one can show that $\pi_1(\det)\colon \pi_1(\operatorname{GL}_n \mathbb{C}) \to \pi_1(\mathbb{C}^\ast) = \mathbb{Z}$ is an isomorphism (of abelian fundamental groups, so the base point doesn't matter).