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Cyclic branched cover of a smooth proper disk

Definition

Assume AC. Let $K \subset S^{3}$ be a smooth oriented knot bounding a smooth proper disk $D \subset B^{4}$ , and let $q \geq 2$ . Write $X_{D}$ for the compact disk exterior and let the positive meridian induce $π_{1} (X_{D}) \to H_{1} (X_{D}; Z) ≅ Z \to Z / q$ . Let $X_{D, q}$ be the connected cover classified by the kernel of this map. Attach $D^{2} \times D^{2}$ to $X_{D, q}$ along the connected cover of $D^{2} \times S^{1}$ using the normal-disk map $z \mapsto z^{q}$ . The resulting compact oriented $4$ -manifold $W_{D, q}$ is the $q$ -fold cyclic cover of $B^{4}$ branched along $D$ ; its boundary is $Σ_{q} (K)$ , and the local branched model is $(x, z) \mapsto (x, z^{q})$ .

Definition references

The positive-meridian cyclic branched cover of the boundary knot

The disk exterior and meridian

The meridian induces the epimorphism to the infinite cyclic group

Charles Livingston and Swatee Naik, Introduction to Knot Concordance, §7.8, pp. 142–145

Axiom of Choice

Depends on

Used by

Dependency tree · two levels

21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

A. J. Casson and C. McA. Gordon, Cobordism of Classical Knots (1986), with appendix by P. M. Gilmer (standard reference, not scraped)