As we all know, the fundmental group of the unit circle is $\mathbb{Z}$. On the other hand, the unit circle could be view as the one-dimensional unitary matrix groups.
So, I wonder what fundmental group of the $n$-dimensional unitary matrix groups is in general.
My first idea follows the proof of the case $n=1$ where the path from $\left[0,1\right)$ to $\mathbb{T}$ could be lift to the map $\left[0,1\right)$ to $\mathbb{R}$. But I do not know how to lift for general case.
My second ideal is to apply some techniques from algebraic topology, but I have no idea about it.
