New look at Milnor Spheres

Abstract: This paper investigates the nature of exotic spheres from two novel perspectives, addressing the fundamental question: ``What makes an exotic sphere exotic?'' First, we explore spherical T-duality in the context of $\mathrm{S}^3$-bundles over $\mathrm{S}^4$. We demonstrate that the distinct smooth structures of exotic spheres $\Sigma^7$ correspond to spherical T-dual pairs for 16 out of the 28 diffeomorphism classes of smooth structures on seven-dimensional homotopy spheres. Furthermore, we show that spherical T-dual pairs can be encoded in $\star$-diagrams. These diagrams impose equivalences that constrain their $\mathrm{S}^3$-invariant geometries. Second, we introduce homotopy Hopf manifolds $\Sigma^7 \times \mathrm{S}^1$ and examine their complex structures using logarithmic transformations. This analysis links their smooth structures to the algebraic and topological invariants of Fano orbifolds. Additionally, we relate these constructions to the Homological Mirror Symmetry program and propose connections to the theory of topological modular forms, aiming to unify geometric topology, elliptic cohomology, and modular forms.