I am trying to digest the meanings of the corks from the both:
- algebraic topology
- geometry topology
Studying corks is important for understanding the exotic phenomenon of 4-manifolds. A definition of cork can be (a generalized form):
Let $C$ be a contractible 4-manifold and $t$ a self-diffeomorphism $∂C → ∂C$ on the boundary. If $t$ cannot extend to a map $C → C$ as a diffeomorphism, then $(C, t)$ is called a cork.
Definition (Cork twist):
For a pair of exotic two 4-manifolds $M1$, $M2$ and a smooth embedding $C ֒→ M1$, if $M1(C, t) = M2$, then $(C, t)$ is called a cork for $M1$, $M2$. We call the deformation $M1 → M1(C, t)$ a cork twist.
This definition seems to be weaker than the usual Stein cork.
Definition (Stein twist):
I think the Stein cork follows that --- $C$ is Stein and $t$ satisfies $t^2 =$ id. If the 4-manifold $C$ of a cork $(C, t)$ is Stein, then $(C, t)$ is called a Stein cork.
(1) The above may be from a more geometry topology viewpoint. Do we have similar definitions from a more? algebraic topology viewpoint
(2) What are the intuitions and the background motivations to define the
- Cork twist
- Stein Cork
in the above ways? Why is that a helpful idea?
(Bonus: Are these only useful in 4-manifolds? Or also suitable for other dimensional manifolds?)