Extending Fibrations of the 3-Torus and Applications to Torus Surgery in 4-Manifolds

2025/12/19 by Nicholas Meyer, Meyer, Nicholas
Mathematics · #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology

paper · doi:10.48550/arxiv.2512.17595

Abstract

Suppose that W and W' are smooth, compact, and oriented 4-manifolds that are either diffeomorphic to S¹ times the exterior E_Y(K) of a fibered knot K in a closed, connected, orientable 3-manifold Y, or are diffeomorphic to Σ_g,1 bundles over the 2-torus with monodromy fixing the boundary of the fiber pointwise. If f: ∂ W' → ∂ W is an orientation-preserving diffeomorphism of the 3-torus boundaries, we have that X = W ∪_f W' is a closed, oriented 4-manifold that fibers over S¹. In particular, if W' = T² × D² and W = S¹ × E_Y(K), then our result shows that the result of doing torus surgery in S¹× Y along S¹ × K is a 4-manifold that fibers over S¹. Furthermore, we extend work of Zentner by showing that the result of torus surgery along S¹ times the unknot U in S¹ × S³ is diffeomorphic to S¹ times a lens space.

Citations

• A note on Stein fillings of contact manifolds
• Variations on Fintushel-Stern Knot Surgery on 4-manifolds
• Knots, Links, and 4-Manifolds

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• On manifolds with multiple lens space filings
• 3-manifolds and 4-dimensional surgery
• Embedding Seifert Fibred 3-Manifolds and Sol 3 -Manifolds in 4-Space
• Bordered manifolds with torus boundary and the link surgery formula
• Seifert fibered four-manifolds with nonzero Seiberg-Witten invariant