= Solution
A <Lie group> is a group which is a finite-dimensional smooth Hausdorff second-countable manifold, with smooth multiplication and inversion. For the <special unitary group>, consider the real <vector space>
$$
\mathfrak{su}(n)=\{X\in M_n(\mathbb C):X^*=-X,\ \operatorname{tr}X=0\}.
$$
The diagonal entries of a skew-Hermitian <matrix> supply $n$ real parameters, and the entries above the diagonal supply $2\binom n2$; the trace-zero condition removes one real parameter. Thus $\dim_{\mathbb R}\mathfrak{su}(n)=n^2-1$.
The <matrix> facts used to construct charts are these: the <matrix exponential> is analytic, its derivative at zero is the identity, and it has an analytic inverse <matrix logarithm> on a sufficiently small neighborhood of the identity. On those neighborhoods $\log(e^X)=X$, $e^{\log U}=U$, and $\det(e^X)=e^{\operatorname{tr}X}$. For a unitary <matrix> in that neighborhood, the <spectral theorem> writes $U=W\operatorname{diag}(e^{i\theta_j})W^*$ with principal angles close to zero, and $\log U=W\operatorname{diag}(i\theta_j)W^*$ is skew-Hermitian.
Choose a norm ball of radius $\varepsilon<\pi/n$ small enough to lie in the exponential's inverse domain. If $U$ is unitary, $\det U=1$ and $\|\log U\|<\varepsilon$ in operator norm, then
$$
\operatorname{tr}\log U\in2\pi i\mathbb Z,\qquad
|\operatorname{tr}\log U|\le n\|\log U\|<\pi,
$$
so its trace is zero. Conversely $X\in\mathfrak{su}(n)$ gives $(e^X)^*e^X=I$ and $\det(e^X)=1$. Thus the logarithm identifies an open neighborhood $V$ of $I$ in $SU(n)$, with its <matrix> subspace topology, with the open ball $B_\varepsilon\subset\mathfrak{su}(n)$. Choosing a real linear identification $\mathfrak{su}(n)\cong\mathbb R^{n^2-1}$ turns this into a chart.
For every $A\in SU(n)$ take the translated chart
$$
\boxed{\phi_A:AV\longrightarrow B_\varepsilon,\qquad \phi_A(U)=\log(A^{-1}U)}.
$$
Their overlap maps are $X\mapsto\log(B^{-1}A e^X)$ wherever both charts are defined, so they are real analytic. They cover the group; the <matrix> topology is Hausdorff and second-countable. This constructs the real manifold directly, rather than merely counting equations. <Matrix> multiplication is polynomial in real/imaginary entries and inversion here is $U\mapsto U^*$; in the logarithm charts both are smooth. The <logarithm charts for the special unitary group> therefore prove
$$
\boxed{SU(n)\text{ is a Lie group of real dimension }n^2-1}.
$$
For $n=1$ it is the one-point group of dimension zero.
For $n=2$, let the first column be $(a,b)$, of unit norm. Every unit vector orthogonal to it is a scalar of modulus one times $(-\bar b,\bar a)$. The <determinant> of the resulting <matrix> is that scalar, so <determinant> one fixes it to $1$. Hence
$$
\boxed{\Phi:S^3\subset\mathbb C^2\longrightarrow SU(2),\qquad
(a,b)\longmapsto\begin{pmatrix}a&-\bar b\\b&\bar a\end{pmatrix}}
$$
is bijective, with inverse given by taking the first column. The forward map is real polynomial in the four coordinates; composing with the local logarithm charts verifies smoothness into the constructed manifold. The inverse is a smooth <matrix>-coordinate projection and, in the usual embedded sphere charts, is smooth into $S^3$. Thus $\boxed{SU(2)\cong S^3\text{ by a diffeomorphism}}$, the explicit realization of <SU(2) as the three-sphere>.
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