Let $SU(2)=\{U\in M(2,\mathbb C):U^*U=I_2,\det U=1\}$ and $SO(3)=\{A\in M(3,\mathbb R):A^\top A=I_3,\det A=1\}$ be the standard matrix Lie groups, where $I_n$ denotes the $n\times n$ identity matrix. There exists a surjective smooth Lie [group homomorphism](/page/Group%20Homomorphism)
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\begin{align*}
\Phi: SU(2) \to SO(3)
\end{align*}
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whose kernel is $\{I_2,-I_2\}$. Consequently, the induced map