Let \(X\) be a finite complex and \(z \in H_m(X)\) an integral homology class. One can ask whether there exists a closed, oriented, \(m\)-dimensional manifold \(M\) with fundamental class \([M] \in H_m(M)\) and a continuous map \(f\colon M \to X\) such that \(z = f_*([M])\)?
Interestingly, René Thom was able to show that 1) this is not always the case and 2) it is actually the case if we only want to represent in such a way some multiple \(kz\) of \(z\).
If a homology class \(z\) is representable in such a way, the next question one wants to investigate is whether one can improve the map \(f\) in the representation; concretely, can we assume that \(f\) is an immersion or even an embedding? (For this we now assume that \(X\) itself is a manifold.)
A recent paper by Diarmuid Crowley and Mark Grant (arXiv:2412.15359) gives concrete examples that this is not always possible, i.e. an example where \(f\) can not be made an immersion and an example where it can be made an immersion but not an embedding.
It is interesting that despite this being a classical topic, there were no such examples up to now in the literature.