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**A fake compact contractible 4-manifold.**
*(English)*
Zbl 0839.57015

Two smooth contractible 4-manifolds \(V,W\) are constructed with the same boundary so that they are homeomorphic but are not diffeomorphic relative to the boundary. The differential structure on \(V\) arises from that of \(W\) as the pullback under a self-homeomorphism \(F\) on \(W\) which extends a diffeomorphism \(f\) of the boundary. That \(f\) does not extend to a diffeomorphism plays a key role and is related to the fact that a knot \(\alpha\) in \(\partial W\) is not slice in \(W\). \(W\) is a Mazur manifold formed by attaching a 2-handle to \(S^1 \times B^3\) and \(\alpha\) is the loop given by \(S^1 \times 1 \subset S^1 \times S^2 = \partial (S^1 \times B^3)\). E. C. Zeeman [Topology 2, 341-358 (1963; Zbl 0116.40801)] raised the question as to whether \(\alpha\) is slice, and this paper finally settles this question negatively.

This can be contrasted with an earlier result of the author and R. Kirby [Mich. Math. J. 26, 259-284 (1979; Zbl 0443.57011)] that \(\alpha\) is slice in another smooth contractible manifold with the same boundary. The manifold \(W\) arises as a piece of two homotopy equivalent closed manifolds \(M \#\) and \(\widetilde M\). If \(f\) were to extend to a diffeomorphism, these manifolds would be diffeomorphic. \(\widetilde M\) is shown to be diffeomorphic to (3)\(\# (20)\), and thus has vanishing Donaldson polynomials by S. K. Donaldson’s connected sum theorem [Topology 29, 257-315 (1990; Zbl 0715.57007)].

However, the fact that \(M\) is a homotopy \(K3\) surface containing an embedded \(\Sigma (2, 3, 7)\) is used to show that there is a nonvanishing Donaldson polynomial for \(M \#\) by an argument due to R. Fintushel and R. Stern [J. Differ. Geom. 34, 255-265 (1991; Zbl 0754.14021)].

A key role in the proof is played by Kirby calculus constructions of diffeomorphisms of various 3- and 4-manifolds. There is also a description of a nontrivial \(h\)-cobordism between \(V\) and \(W\) constructed with an algebraically cancelling pair of two- and three-handles to the interior of \(W\). In a related paper appearing immediately following this one in the same journal [ibid., 357-361 (1991; Zbl 0839.57016), see the review below] the author uses similar constructions to form two homeomorphic 4-manifolds \(Q_1\), \(Q_2\) which are homotopy equivalent to \(\text{int} (B^4)\), have the same boundary, but are not diffeomorphic to each other.

This can be contrasted with an earlier result of the author and R. Kirby [Mich. Math. J. 26, 259-284 (1979; Zbl 0443.57011)] that \(\alpha\) is slice in another smooth contractible manifold with the same boundary. The manifold \(W\) arises as a piece of two homotopy equivalent closed manifolds \(M \#\) and \(\widetilde M\). If \(f\) were to extend to a diffeomorphism, these manifolds would be diffeomorphic. \(\widetilde M\) is shown to be diffeomorphic to (3)\(\# (20)\), and thus has vanishing Donaldson polynomials by S. K. Donaldson’s connected sum theorem [Topology 29, 257-315 (1990; Zbl 0715.57007)].

However, the fact that \(M\) is a homotopy \(K3\) surface containing an embedded \(\Sigma (2, 3, 7)\) is used to show that there is a nonvanishing Donaldson polynomial for \(M \#\) by an argument due to R. Fintushel and R. Stern [J. Differ. Geom. 34, 255-265 (1991; Zbl 0754.14021)].

A key role in the proof is played by Kirby calculus constructions of diffeomorphisms of various 3- and 4-manifolds. There is also a description of a nontrivial \(h\)-cobordism between \(V\) and \(W\) constructed with an algebraically cancelling pair of two- and three-handles to the interior of \(W\). In a related paper appearing immediately following this one in the same journal [ibid., 357-361 (1991; Zbl 0839.57016), see the review below] the author uses similar constructions to form two homeomorphic 4-manifolds \(Q_1\), \(Q_2\) which are homotopy equivalent to \(\text{int} (B^4)\), have the same boundary, but are not diffeomorphic to each other.

### MSC:

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |

57R80 | \(h\)- and \(s\)-cobordism |