Exotic Heat PDE's.II

TL;DR

This paper explores exotic heat equations using PDE algebraic topology to identify exotic spheres, providing a novel approach to understanding global solutions and topological properties of manifolds.

Contribution

It introduces a new methodology applying PDE algebraic topology to characterize exotic spheres and analyze global solutions of exotic heat equations.

Findings

Identification of exotic spheres via PDE algebraic topology

Application of exotic heat equations to prove the Poincaré conjecture in higher dimensions

Characterization of global solutions using the geometry of PDEs

Abstract

Exotic heat equations that allow to prove the Poincar\'e conjecture and its generalizations to any dimension are considered. The methodology used is the PDE's algebraic topology, introduced by A. Pr\'astaro in the geometry of PDE's, in order to characterize global solutions. In particular it is shown that this theory allows us to identify $n$-dimensional {\em exotic spheres}, i.e., homotopy spheres that are homeomorphic, but not diffeomorphic to the standard $S^n$.