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3 Attempting to make the statement more accruate.

A lot of the properties of $SU(n)$ and $U(n)$ can be seen summarised in the "commutative diagramme" below, viewed as fibrations. In particular, the diffeomorphisms for $U(1)$ and $SU(2)$ to spheres falls out from it, but fails for higher dimensions. But you can still see various fibrations, as people above mentioned.

\begin{array}{ccccc} SU(n-1) & \to & U(n-1) & \to & S^1 \\ \downarrow & & \downarrow & & \downarrow\\ SU(n) & \to & U(n) & \to & S^1\\ \downarrow & & \downarrow & & \downarrow\\ S^{2n-1} & \to & S^{2n-1} & \to & {*} \end{array}

2 Fixed array.

A lot of the properties of $SU(n)$ and $U(n)$ can be seen in the commutative diagramme below. In particular, the diffeomorphisms for $U(1)$ and $SU(2)$ to spheres falls out from it, but fails for higher dimensions. But you can still see various fibrations, as people above mentioned.

\begin{array}{ccccc} SU(n-1) & \to & U(n-1) & \to & S^1 \\ \downarrow & & \downarrow & & \downarrow\ downarrow\\ SU(n) & \to & U(n) & \to & S^1S^1\\ \downarrow & & \downarrow & & \downarrow\ downarrow\\ S^{2n-1} & \to & S^{2n-1} & \to & {*} \end{array}

1

A lot of the properties of $SU(n)$ and $U(n)$ can be seen in the commutative diagramme below. In particular, the diffeomorphisms for $U(1)$ and $SU(2)$ to spheres falls out from it, but fails for higher dimensions. But you can still see various fibrations, as people above mentioned.

\begin{array}{ccccc} SU(n-1) & \to & U(n-1) & \to & S^1\ \downarrow & & \downarrow & & \downarrow\ SU(n) & \to & U(n) & \to & S^1\ \downarrow & & \downarrow & & \downarrow\ S^{2n-1} & \to & S^{2n-1} & \to & {*} \end{array}