An exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere.
- Naively, I thought that there is no algebraic topological invariant that distinguishing the exotic spheres from each others (?). For example, are there any integration over the local quantities on the differentiable manifolds that can distinguish exotic spheres?
(e.g. I dont think there are any characteristic classes, homotopy or co/homology groups, or co/bordism can distinguish or exotic spheres of the same dimension. Yes?)
The fact 1 seems to have some tension, if not contradicts, to the fact 2. Because the fact 2 of the abelian monoid / group structure seems to hint that there are some algebraic topological quantities as "topological invariants" that can distinguish exotic spheres. Yes or no?
Moreover, there are so called Kervaire, Kervaire-Milnor invariants, Kervaire invariant problem and the Kirby–Siebenmann invariant. Are these quantities as invariants of exotic spheres topologically? In the sense that, we can obtain the topological data (global) from integration over the local quantity? (Analogous to characteristic classes?)
In short, are "invariants" of Exotic spheres more of the quantities of (a) differential or (b) topological?