# A user guide to the theory on Corks

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?)

• 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. – Anubhav Mukherjee Mar 4 at 2:43