Understanding exotic spheres from the viewpoint of global singularity theory of smooth maps

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In a recent preprint (arXiv: http://arxiv.org/abs/2009.05928), we studied the existence and construction problems for special generic maps of rational homology spheres. The novelty of our approach is to consider the torsion subgroup of the integral homology of such manifolds. We showed that if a rational homology sphere of odd dimension n = 2k + 1 > 4 admits a special generic map into a Euclidean space of dimension < n, then the cardinality of its integral homology group of degree k is a square. On the one hand, we showed that any square can can be realized in our homological condition. On the other hand, there are examples of rational homology spheres that do not satisfy our homological condition. Our results paved the way to a subsequent project, in which we study special generic maps of highly connected manifolds in terms of the linking form, which is a torsion analog of the intersection form. In another project, we developed a new approach to intersection spaces that is based on Sullivan's PL polynomial differential forms. Our result implies uniqueness of the rational cohomology ring of intersection spaces. This result is a new discovery in the research field, and we published the case of isolated singularities. In ongoing work, we generalize our approach along the construction of Agustin and Fernandez de Bobadilla to a class of singular spaces of arbitrary stratification depth including toric varieties. Moreover, in joint work with T. Essig, we are finalizing a preprint about the construction of a fundamental class for intersection spaces in stratification depth two.