The space of linear endomorphisms of $\mathbb{R}^2$, which we'll denote as $\mathbb{R}^{2 \times 2}$, is a 4-dimensional space. As such, it is not visualizable by a normal human mind. However, there are some relevant quotients/subspaces of this space which are 3-dimensional (albeit curved) or lower, which can help gain insight into the behavior of this space. For example, by projecting $\mathbb{R}^{2 \times 2}$ into $\mathbb{R}^{3}$ by fixing one of the matrix's parameter, and looking at the behavior of a given submanifold in this space, one can gain some insight. Additionally, making this static parameter vary can help us gain an "MRI" visualization for the behavior of $\mathbb{R}^{4}$, interpreted as $\mathbb{R}^{2 \times 2}$.
If we take take for example the submanifold of singular matrices, we get the equation $ad - bc = 0$ which defines a 3D submanifold (something we can ascertain by seeing it as a (degree 2, homogeneous) polynomial equation with 4 variables in $\mathbb{R}^{4}$, so a curved hyperspace), which we'll call $S$, and which acts as a separating space between the space of matrices with positive determinant, and those with negative determinant. If my (limited) knowledge of algebraic topology serves me, the quotient $\mathbb{R}^{2 \times 2} / S$ is something homeomorphic to $\mathbb{R}$ (ie, $\mathbb{R}^{2 \times 2} \to \mathbb{R}^{2 \times 2} / S$ maps each matrix to its determinant).
Is there/do you know of some form of similar geometric intuition for other famous submanifolds of $\mathbb{R}^{2 \times 2}$ (such as the Lie groups) that allow ways to visualize these spaces' behavior ? Are there similar examples on this issue that you have found insightful in other manifolds/dimensions than $\mathbb{R}^{2 \times 2}$ ? How would you go about resolving quotients of this kind in general, both algebraically and geometrically ?
NB: I understand the intuition behind a quotient space (reducing the points of the denominator manifold to a single point at 0, and having the rest of the topological space follow continuously from this transformation), but I don't know how to calculate one directly, nor how to extract meaningful geometric information from such a calculation in general. Are there any textbooks/resources you'd recommend, ones that introduce relevant tools for this problem both algebraically and geometrically, in a digestible fashion ?
Thank you for your time.
Maybe it's not exactly what you are looking for, but real projective plane can be seen as a smooth submanifold of $\mathbb{R}^4$ and it manifests a topological phenomenon that CANNOT happen for smooth submanifold of $\mathbb{R}^3$ or $\mathbb{R}^2$ .
If you take a loop around the "hole" of a punctured plane $\mathbb{R}^2-0$, you cannot shrink it into a point continously if you want to "remain into the space". Clearly this doesn't change if you travel on the loop twice (you still cannot shrink it!).
Surprisingly for submanifolds of $\mathbb{R}^n$, where $n\geq 4$ it can be the case that if you travel twice on an "unshrinkable" loop you obtain a "shrinkable" one!
For the projective plane this ultimate boils down to the fact that its first homology group is $\mathbb{Z}_2$.