# Why is $SU(2)$ diffeomorphic to $S^3$?

As the title,my question is why $$SU(2)$$ is diffeomorphic to $$S^3$$ ? How can I structure a diffeomorphic map between $$SU(2)$$ and $$S^3$$?

Thank you very much!

• Write out the elements of $SU(2)$. They satisfy the equation of the sphere Nov 2, 2018 at 4:30
• I can not understand what you mean,please give sufficient evidence. Nov 2, 2018 at 4:45

By definition, we have that $$\displaystyle SU(2)=\left\{\begin{pmatrix}a &-\overline{b}\\ b &\overline{a}\end{pmatrix}: a, b\in\mathbb{C}, |a|^2+|b|^2=1\right\}.$$

Since $$\mathbb{R}^4\cong\mathbb{C}^2$$, we may think of $$S^3$$ as $$\displaystyle S^3=\{(a,b)\in\mathbb{C}^2: |a|^2+|b|^2=1\}$$.

Define a map from $$S^3$$to $$SU(2)$$ as: \begin{aligned}F: S^3&\to SU(2)\\ F(a,b)&=\begin{pmatrix}a &-\overline{b}\\ b &\overline{a}\end{pmatrix}. \end{aligned}

Can you show that $$F$$ is well defined, $$F$$ is injective, and $$F$$ is surjective (i.e. an isomorphism)? After that, you should show that $$F$$ and $$F^{-1}$$ are smooth (or $$C^{\infty}$$). (Hint: Show $$\displaystyle SU(2)$$ is a submanifold of of the set of 2 x 2 matrices with complex entries, or better yet $$\mathbb{R}^8$$. What can you say about $$F$$ and its inverse restricted to these submanifolds?)

• Hi. This answer is very clear but could you maybe help me understanad why $SU(2)$ is a submanifold of $\mathbb{R}^8 \cong \mathbb{C}^4 \cong M_2(\mathbb{C})$?
– user661541
Dec 18, 2019 at 14:23
• Does the condition $|a|^2+|b|^2=1$ not reduce the degrees of freedom by 1, to 3. Afterall, the lie algebra of SU(2) has 3 degrees of freedom:$x\sigma_x+y\sigma_y+z\sigma_z$. Wouldn't SU(2) then map better to just a normal three-dimensional sphere? Jan 30, 2020 at 23:16

Every $$SU(2)$$ matrix is a linear combination of Pauli matrices $$\sigma_1, \sigma_2, \sigma_3$$ and the unit matrix I, which is a base of the associated Lie algebra of the SU2 group. So every $$SU(2)$$ matrix $$X$$ can be written a $$X=x_1\sigma_1 +x_2\sigma_2 +x_3\sigma_3 +x_4.I$$ and the condition that $$det{X} = 1$$ gives $$x_1^2 + x_2^2 +x_3^2 +x_4^2 = 1$$ which is the $$S^3$$ sphere. And every point on that sphere represents an $$SU(2)$$ matrix

• You seem to be confusing $\operatorname{SU}(2)$ with its algebra $\mathfrak{su}(2)$; besides, you can very easily check that any Pauli matrix $\sigma_j$ has a determinant $\det \sigma_j = -1$, so they do not even belong to $\operatorname{SU}(2)$. Aug 27 at 17:52