I want to prove that $\pi_1 (GL(n,\mathbb{C})) \simeq \mathbb{Z}$. To do so, i alredy proved that $GL(n,\mathbb{C})$ and $U(n)$ have same homotopy type and that there is and homeomorphism between $U(n)$ and $SU(n) \times S^1$. But i'm a bit lost in proving that $SU(n)$ is simply connected. I know that there is a proof using fibrations and long exact sequences of homotopy groups, but i don't know much about either yet. My current knowledge is fundamental groups, the Van Kampen Theorem and covering spaces.
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Isn't SU(2) homeomorphic to a 3-dimensional sphere, and therefore simply connected? – Andreas Blass Feb 23 '24 at 19:49
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Yes it is, but i fail to see how to prove that SU(n) is simply connected for all n – Ggstal Feb 23 '24 at 19:52
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See https://math.stackexchange.com/questions/370837/sun-is-simply-connected-proof-without-fibrations-n2 for a proof using Morse theory. – Moishe Kohan Feb 24 '24 at 01:11