We know that $$\pi_1(SU(N))=0, \tag{1}$$ $$\pi_1(PSU(N))=\pi_1(SU(N)/(\mathbb{Z}/N))=\pi_0(\mathbb{Z}/N)=\mathbb{Z}/N.\tag{2}$$
Also that $$\pi_1(U(N))=\pi_1(\frac{SU(N)\times U(1)}{\mathbb{Z}/N})=\mathbb{Z}.\tag{3}$$ $$\pi_1(U(1))=\mathbb{Z}.\tag{4}$$
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My question is that how do I visualize the $\pi_1(U(N))=\mathbb{Z}'$ versus the $\pi_1(U(1))=\mathbb{Z}$? Is that the intervals of $\pi_1(U(N))=\mathbb{Z}'$ $N$ times larger or smaller than $\pi_1(U(1))=\mathbb{Z}$? How to visualize the ratios?
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We learn from the long exact sequence of homotopy groups that $$ \pi_1(\mathbb{Z}/N)=0 \to \pi_1(SU(N)\times U(1))=\mathbb{Z} \to \pi_1(U(N)) \to \pi_0(\mathbb{Z}/N)=\mathbb{Z}/N \to \pi_0(SU(N)\times U(1))=0 \to \dots $$ So we have $$0 \to \mathbb{Z} \to \pi_1(U(N)) \to\mathbb{Z}/N \to 0.$$ So it seems $$ \frac{\pi_1(U(N))}{\pi_1(SU(N)\times U(1))} =\frac{\pi_1(U(N))}{\pi_1( U(1))}= \frac{\mathbb{Z}'}{\mathbb{Z}}=\mathbb{Z}/N=\frac{\mathbb{Z}}{N\mathbb{Z}}=\frac{\mathbb{Z}/N}{\mathbb{Z}}. $$