TY - JOUR

T1 - Knot concordance and homology cobordism

AU - Cochran, Tim D.

AU - Franklin, Bridget D.

AU - Hedden, Matthew

AU - Horn, Peter D.

PY - 2013

Y1 - 2013

N2 - We consider the question: "If the zero-framed surgeries on two oriented knots in S3 are ℤ-homology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant?" We show that this question has a negative answer in the smooth category, even for topologically slice knots. To show this we first prove that the zero-framed surgery on K is ℤ-homology cobordant to the zero-framed surgery on many of its winding number one satellites P(K). Then we prove that in many cases the τ and s-invariants of K and P(K) differ. Consequently neither τ nor s is an invariant of the smooth homology cobordism class of the zero-framed surgery. We also show that a natural rational version of this question has a negative answer in both the opological and smooth categories by proving similar results for K and its (p, 1)-cables.

AB - We consider the question: "If the zero-framed surgeries on two oriented knots in S3 are ℤ-homology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant?" We show that this question has a negative answer in the smooth category, even for topologically slice knots. To show this we first prove that the zero-framed surgery on K is ℤ-homology cobordant to the zero-framed surgery on many of its winding number one satellites P(K). Then we prove that in many cases the τ and s-invariants of K and P(K) differ. Consequently neither τ nor s is an invariant of the smooth homology cobordism class of the zero-framed surgery. We also show that a natural rational version of this question has a negative answer in both the opological and smooth categories by proving similar results for K and its (p, 1)-cables.

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U2 - 10.1090/S0002-9939-2013-11471-1

DO - 10.1090/S0002-9939-2013-11471-1

M3 - Article

AN - SCOPUS:84875473399

SN - 0002-9939

VL - 141

SP - 2193

EP - 2208

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 6

ER -