New look at Milnor Spheres

S. Donaldson disproved the smooth h-cobordism conjecture in dimension 4 by studying invariants coming from the moduli space of connections on SO (3) -bundles over smooth 4-dimension manifolds X. In this paper, we reverse his point of view in dimension 8. Namely, we consider fibrations coming from diffeomorphisms classes of homotopy Hopf manifolds ⁷¹, i. e. , ⁷ is a homotopy sphere, seeing the bases as moduli spaces for additional structures. Our overarching objective is to establish a correspondence between the smooth invariants of Kervaire--Milnor and categorical invariants. One interprets pairwise non-diffeomorphic smooth structures on ⁷¹ in terms of non-rationality. This paper's contributions are diverse, each carrying its significance. We believe profound connections exist between the themes addressed, which connect hypersurface singularity theory, Homological Mirror Symmetry, Topological Modular forms (tmf), and Spherical T-duality.