Exoticism on topological spaces and alternative physics: Physics on exotic spheres as toy models

The spaces (sets equipped with some structures) involved in a mathematical structure appearing in a physical theory has been discussed. The spaces are called spaces of theory. Some structures which may be involved in a space of theory (such as topological, algebraic, geometrical, and differential structures) have been introduced and their roles in the formulation of a mathematical structure in a physical theory have been discussed. The equivalence of two mathematical structures appearing in a physical theory and the equivalence of two spaces of theory have been introduced. The possibility that a topological space admits several differential structures and that those structures may not be equivalent has also been exposed. Some implications of the existence of exoticisms of topological spaces on the formulations of physical theory have been discussed. Some explicit or concrete example of exotic spheres having constructed have been exposed and analysed. Some relative regularities on Milnor spheres have also been discussed.