Simply-connected 4-manifolds with a given boundary

Abstract

Let M M be a closed, oriented, connected 3 3 -manifold. For each bilinear, symmetric pairing ( Z n , L ) ({{\mathbf {Z}}^n},\,L) , our goal is to calculate the set V L ( M ) {\mathcal {V}_L}(M) of all oriented homeomorphism types of compact, 1 1 -connected, oriented 4 4 -manifolds with boundary M M and intersection pairing isomorphic to ( Z n , L ) ({{\mathbf {Z}}^n},\,L) . For each pair ( Z n , L ) ({{\mathbf {Z}}^n},\,L) which presents H ∗ ( M ) {H_ \ast }(M) , we construct a double coset space B L t ( M ) B_L^t(M) and a function c L t : V L ( M ) → B L t ( M ) c_L^t:{\mathcal {V}_L}(M) \to B_L^t(M) . The set B L t ( M ) B_L^t(M) is the quotient of the group of all link-pairing preserving isomorphisms of the torsion subgroup of H 1 ( M ) {H_1}(M) by two naturally occuring subgroups. When ( Z n , L ) ({{\mathbf {Z}}^n},\,L) is an even pairing, we construct another double coset space B ^ L ( M ) {\hat B_L}(M) , a function c ^ L : V L ( M ) → B ^ L ( M ) {\hat c_L}:{\mathcal {V}_L}(M) \to {\hat B_L}(M) and a projection p 2 : B ^ L ( M ) → B L t ( M ) {p_2}:{\hat B_L}(M) \to B_L^t(M) such that p 2 ⋅ c ^ L = c L t {p_2} \cdot {\hat c_L} = c_L^t . Our main result states that when ( Z n , L ) ({{\mathbf {Z}}^n},\,L) is even the function c ^ L {\hat c_L} is injective, as is the function c L t × Δ : V L ( M ) → B L t ( M ) × Z / 2 c_L^t \times \Delta :{\mathcal {V}_L}(M) \to B_L^t(M) \times {\mathbf {Z}}/2 when ( Z n , L ) ({{\mathbf {Z}}^n},\,L) is odd. Here Δ \Delta is a Kirby-Siebenmann obstruction to smoothing. It follows that the sets V L ( M ) {\mathcal {V}_L}(M) are finite and of an order bounded above by a constant depending only on H 1 ( M ) {H_1}(M) . We also show that when H 1 ( M ; Q ) ≅ 0 {H_1}(M;{\mathbf {Q}}) \cong 0 and ( Z n , L ) ({{\mathbf {Z}}^n},\,L) is even, c L t = p 2 ⋅ c ^ L c_L^t = {p_2} \cdot {\hat c_L} is injective. It seems likely that via the functions c L t × Δ c_L^t \times \Delta and c ^ L {\hat c_L} , the sets B L t ( M ) × Z / 2 B_L^t(M) \times {\mathbf {Z}}/2 and B ^ L ( M ) {\hat B_L}(M) calculate V L ( M ) {\mathcal {V}_L}(M) when ( Z n , L ) ({{\mathbf {Z}}^n},\,L) is respectively odd and even. We verify this in several cases, most notably when H 1 ( M ) {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 1 -connected 4 4 -manifolds extending a given homeomorphism of their boundaries. The theory developed is then applied to show that there is an m > 0 m > 0 , depending only on H 1 ( M ) {H_1}(M) , such that for any self-homeomorphism f f of M M , f m {f^m} extends to a self-homeomorphism of any 1 1 -connected, compact 4 4 -manifold with boundary M M .