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

Kirby, Rob

Turk. J. Math. 20, No. 1, 85-93 (1996).

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.

Reviewer: R.E.Stong (Charlottesville)

MSC:

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

Keywords:

\(h\)-cobordism; 4-manifolds; homotopy 4-sphere