Let $r \ge 2$. How to prove that the fundamental group $\pi_1(GL_r(\mathbb{C}))$ of the invertible matrices $GL_r(\mathbb{C})$ over $\mathbb{C}$ equals $\mathbb{Z}$.
What about $\pi_1(GL_r(\mathbb{R}))$?
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See https://math.stackexchange.com/questions/3637/fundamental-group-of-gln-c-is-isomorphic-to-z-how-to-learn-to-prove-facts-lik – peter a g May 07 '18 at 15:34
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And note the linked questions (and other references) in the above - for instance, see https://math.stackexchange.com/questions/214637/fundamental-group-of-gl-n-mathbbr?noredirect=1&lq=1 – peter a g May 07 '18 at 16:09