## Akbulut’s corks and $$h$$-cobordisms of smooth, simply connected 4-manifolds.(English)Zbl 0868.57031

Being given a 5-dimensional smooth $$h$$-cobordism $$M^5$$ between two simply connected closed 4-manifolds, $$M_0$$ and $$M_1$$, this paper shows that there is a sub-$$h$$-cobordism $$A^5 \subset M^5$$ between $$A_0 \subset M_0$$ and $$A_1 \subset M_1$$ satisfying (1) $$A_0$$, hence also $$A$$ and $$A_1$$, are compact contractible manifolds, and (2) $$M\text{-int} A$$ is a product $$h$$-cobordism, i.e. $$(M_0\text{-int} A_0) \times [0,1]$$. Further, $$A$$ can be chosen so that:
(A) $$M$$-$$A$$ (hence each $$M_i$$-$$A_i)$$ is simply connected.
(B) $$A$$ is diffeomorphic to the ball.
(C) $$A_0 \times I$$ and $$A_1 \times I$$ are diffeomorphic to the ball.
(D) $$A_0$$ is diffeomorphic to $$A_1$$ by a diffeomorphism which is an involution on $$\partial A_0= \partial A_1$$.
As a corollary, any homotopy 4-sphere is obtained from $$S^4$$ by cutting out a contractible 4-manifold $$A_0$$ from $$S^4$$ and gluing it back with an involution of $$\partial A_0$$. The author uses the term ‘Akbulut’s corks’ to refer to the ball $$A$$, a nontrivial $$h$$-cobordism, the point being that every $$h$$-cobordism is obtained from a product $$h$$-cobordism by taking out a cork and putting it back with a twist.

### MSC:

 57R80 $$h$$- and $$s$$-cobordism 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010)

### Keywords:

$$h$$-cobordism; 4-manifolds; homotopy 4-sphere
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