$Spin(N)$ $N>2$ is simple-connected

Question

How to explain that $Spin(N)$ $N>2$ is simple-connected, we already know fundamental group of $SO(N)$ $(N>2 )$is $Z/2$,and $Spin(N)$ is $SO(N)$ nontrivial double covering.

Answer 1

By covering space theory, for nice spaces (the above spaces are nice) if $Y$ is an $n$-sheeted covering of $X$ then $\pi_{1}(Y)$ is an index $n$ subgroup of $\pi_{1}(X)$.

Hence $\pi_{1}(\text{Spin}(N))$ is an index 2 subgroup of $\pi_{1}(SO(N)) = \mathbb{Z}/2\mathbb{Z}$, so $\pi_{1}(\text{Spin}(N))$ has to be trivial.