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Exotic and localized homology concordance classes
Definition
Let knots be smooth oriented knots in the standard oriented S³ and let I=[0,1]. Let C_sm be their smooth concordance classes, where equivalence is witnessed by a smooth proper annulus in S³×I. Let C_top be their locally flat topological concordance classes in that product. Let C_ex be their exotic-smooth concordance classes: equivalence is witnessed by a smooth proper annulus in a compact smooth oriented 4-manifold W together with an orientation-preserving homeomorphism of pairs (W,∂W)≅(S³×I,S³×{0,1}) matching the endpoint knots. The annulus is oriented so that its boundary is −K₀ at the lower end and K₁ at the upper end, and it has product collars at both ends agreeing with the ambient boundary collars. For a nonzero integer n set R_n=Z[1/|n|]. Let C^{1/n} be the classes under equivalence witnessed by a smooth proper annulus in a compact smooth oriented 4-manifold W with boundary the two standard endpoint 3-spheres such that each boundary inclusion induces an isomorphism H_i(S³;R_n)→H_i(W;R_n) for every i. The annulus is oriented so that its boundary is −K₀ at the lower end and K₁ at the upper end, with product collars at both ends agreeing with the ambient boundary collars. A knot is pseudo-slice if it bounds a smooth proper disk in a compact smooth oriented 4-manifold homeomorphic to B⁴, with its oriented boundary identified with the standard oriented S³. A knot is Z[1/|n|]-slice if it bounds a smooth proper disk in a compact smooth oriented 4-manifold B with its oriented boundary identified with the standard oriented S³ and with H_i(B;R_n)≅H_i(B⁴;R_n) for every i; such a B is an R_n-homology 4-ball. These are sets of equivalence classes; this definition alone asserts no group structure.
Depends on
Used by
- Satellite operators on concordance classes Definition
- Satellite operators preserve concordance Lemma
- The CDR pattern zero-surgery homology cobordism Lemma
- Weak injectivity and auxiliary-pattern reduction Lemma
- Smooth satellite filtrations with topologically slice images Open problem
- Cochran–Davis–Ray satellite injectivity Theorem
- Infinite rank for iterated winding-zero satellites Theorem
- The CDR concordance categories are abelian groups Theorem
Dependency tree · two levels
4 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
- Tim D. Cochran, Christopher W. Davis, and Arunima Ray, “Injectivity of satellite operators in knot concordance” (standard reference, not scraped)
- Tim D. Cochran, Bridget D. Franklin, Matthew Hedden, and Peter D. Horn, “Knot concordance and homology cobordism” (standard reference, not scraped)