I know that past a certain point, one should graduate from the view that homology/homotopy groups "count holes" in any realistic, grounded, real-life meaning of the word "hole". However, I still want to see if there is any visual intuition to be had regarding the Poincare homology sphere, which is sort of the first example of Is there a non-simply-connected space with trivial first homology group? (also the first example in this MO thread).
I did come across https://ben300694.github.io/pdfs/2020/2020-01-29_Slides_Some_faces_of_the_Poincare_Homology_Sphere_(Bonn,_IMPRS_Seminar).pdf, which presents some nice but difficult (for me) to make-sense-of pictures
or another picture here
https://mathoverflow.net/questions/353814/understanding-fundamental-group-of-poincare-homology-sphere.
I searched up "visualization of poincare homology sphere", and there was only one Youtube video https://www.youtube.com/watch?v=e9ogYSd7CtU&ab_channel=VisualMath.
I'm wondering if there's any way of using this 3D animated visualization, or even a sequence of 2D pictures like above, to get an intuition for why the Poincare homology sphere has "no holes" according to homology, but "holes" according to homotopy; in other words I'm wondering if I can see, somehow with my own eyes, some "non-contractible path/cycle that is also the boundary of some submanifold" (which I think is the intuitive picture of what trivial homology but non-trivial homotopy means?)
P.S. compare also the state of affairs to that of the Hopf fibration; both involve counterintuitive phenomenon happening in dimensions at the very edge of what we can visualize, but there have been many visualizations/expositions of the Hopf fibration over the years, such as this video of Niles Johnson 11 years ago (https://www.youtube.com/watch?v=AKotMPGFJYk&ab_channel=NilesJohnson), this paper of David Lyons linked on Niles's website https://nilesjohnson.net/hopf-articles/Lyons_Elem-intro-Hopf-fibration.pdf (also on Arxiv), this more recent "mainstream" video https://www.youtube.com/watch?v=nsHcKO7HvFY&ab_channel=Poppro, and also this absolutely wonderful interactive version by Samuel Li https://samuelj.li/hopf-fibration/.
See also these valiant attempts at visualization the Milnor exotic $7$-sphere by Niles Johnson: https://www.youtube.com/watch?v=II-maE5HEj0&ab_channel=NilesJohnson.