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I am trying to digest the meanings of the corks from the both:

  • algebraic topology

and

  • geometry topology

perspectives.

Studying corks is important for understanding the exotic phenomenon of 4-manifolds. A definition of cork can be (a generalized form):

Definition (Cork):

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.

My question:

(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
  • 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?)

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    $\begingroup$ some of Akubulut's earlier papers has very nice intuition. It's all about how you attach a cancelling 1-2 pair where you are now attaching the 2 handle along some attaching circle which is homotopic(not isotopic) to the original one. Key word: positve whitehead multiple of a knot. $\endgroup$ – Anubhav Mukherjee Mar 4 at 2:43

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