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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Simply-connected $ 4$-manifolds with a given boundary

Author: Steven Boyer
Journal: Trans. Amer. Math. Soc. 298 (1986), 331-357
MSC: Primary 57N13; Secondary 57N10
MathSciNet review: 857447
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Abstract: Let $ M$ be a closed, oriented, connected $ 3$-manifold. For each bilinear, symmetric pairing $ ({{\mathbf{Z}}^n},\,L)$, our goal is to calculate the set $ {\mathcal{V}_L}(M)$ of all oriented homeomorphism types of compact, $ 1$-connected, oriented $ 4$-manifolds with boundary $ M$ and intersection pairing isomorphic to $ ({{\mathbf{Z}}^n},\,L)$.

For each pair $ ({{\mathbf{Z}}^n},\,L)$ which presents $ {H_ \ast }(M)$, we construct a double coset space $ B_L^t(M)$ and a function $ c_L^t:{\mathcal{V}_L}(M) \to B_L^t(M)$. The set $ B_L^t(M)$ is the quotient of the group of all link-pairing preserving isomorphisms of the torsion subgroup of $ {H_1}(M)$ by two naturally occuring subgroups.

When $ ({{\mathbf{Z}}^n},\,L)$ is an even pairing, we construct another double coset space $ {\hat B_L}(M)$, a function $ {\hat c_L}:{\mathcal{V}_L}(M) \to {\hat B_L}(M)$ and a projection $ {p_2}:{\hat B_L}(M) \to B_L^t(M)$ such that $ {p_2} \cdot {\hat c_L} = c_L^t$.

Our main result states that when $ ({{\mathbf{Z}}^n},\,L)$ is even the function $ {\hat c_L}$ is injective, as is the function $ c_L^t \times \Delta :{\mathcal{V}_L}(M) \to B_L^t(M) \times {\mathbf{Z}}/2$ when $ ({{\mathbf{Z}}^n},\,L)$ is odd. Here $ \Delta $ is a Kirby-Siebenmann obstruction to smoothing. It follows that the sets $ {\mathcal{V}_L}(M)$ are finite and of an order bounded above by a constant depending only on $ {H_1}(M)$. We also show that when $ {H_1}(M;{\mathbf{Q}}) \cong 0$ and $ ({{\mathbf{Z}}^n},\,L)$ is even, $ c_L^t = {p_2} \cdot {\hat c_L}$ is injective.

It seems likely that via the functions $ c_L^t \times \Delta $ and $ {\hat c_L}$, the sets $ B_L^t(M) \times {\mathbf{Z}}/2$ and $ {\hat B_L}(M)$ calculate $ {\mathcal{V}_L}(M)$ when $ ({{\mathbf{Z}}^n},\,L)$ is respectively odd and even. We verify this in several cases, most notably when $ {H_1}(M)$ is free abelian.

The results above are based on a theorem which gives necessary and sufficient conditions for the existence of a homeomorphism between two $ 1$-connected $ 4$-manifolds extending a given homeomorphism of their boundaries.

The theory developed is then applied to show that there is an $ m > 0$, depending only on $ {H_1}(M)$, such that for any self-homeomorphism $ f$ of $ M$, $ {f^m}$ extends to a self-homeomorphism of any $ 1$-connected, compact $ 4$-manifold with boundary $ M$.

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Article copyright: © Copyright 1986 American Mathematical Society