Question about the Lie group $SU(3) \times SU(2) \times U(1)$ and the concept of manifold

Let me begin by answering the simpler question "What is $U(1)$ as a manifold?" Recall that the elements of $U(1)$ are $1\times1$, complex, unitary matrices. If $g$ is a generic $1\times1$, complex matrix, then we can write \begin{align} g = \begin{pmatrix} r e^{i\varphi} \end{pmatrix} \end{align} for real numbers $r$, $\varphi$. A unitary matrix obeys $U^{\dagger}U = 1$, so a $1\times1$ unitary matrix $g$ will have $r=1$. That is \begin{align} g = \begin{pmatrix} e^{i\varphi} \end{pmatrix} \end{align} We see that we can parameterize $U(1)$ by a single real number $\varphi$. Since $g$ only depends on $e^{i\varphi}$, the parameters $\varphi$ and $\varphi + 2\pi$ label the same group element. So we can say that $U(1)$ as a manifold is the same as the space of real numbers $\varphi$ with the identification $\varphi \simeq \varphi + 2\pi$. This space is precisely the circle $\mathbb{T} \simeq S^1$.

We can do basically the same thing to find a parametrization for $SU(2)$. A generic element $g$ of $SU(2)$ can be written in terms of four real numbers $x$, $y$, $z$, $w$ as \begin{align} g = \begin{pmatrix} x + iy & -z + iw\\ z+iw & x - iy \end{pmatrix}, \end{align} where $x^2 + y^2 + z^2 + w^2 = 1$. This latter constraint comes from demanding that $\det g = 1$. The space of points $(x,y,z,w) \in \mathbb{R}^4$ obeying $x^2 + y^2 + z^2 + w^2 = 1$ is the 3-sphere $S^3$. So as a manifold we identify $SU(2)$ with $S^3$.

$SU(3)$ is topologically more complicated than the lower-dimensional cases. Suffice it to say that we can label points on $SU(3)$ by a set of real numbers with some relations between them, as with $U(1)$ and $SU(2)$.

For two groups $G$ and $H$, the direct product $G\times H$ consists of pairs $(g,h)$ of elements $g\in G$, $h\in H$. Similarly the direct product $G_1 \times G_2 \times G_3 \times \ldots$ consists of tuples $(g_1, g_2, g_3, \ldots)$.

So an element of $U(1) \times SU(2) \times SU(3)$ is a tuple $(g_1, g_2, g_3)$ where $g_1 \in U(1)$ is a point on the circle, $g_2 \in SU(2)$ a point on the 3-sphere, and $g_3 \in SU(3)$ a point in $SU(3)$.