On homology concordance in contractible manifolds and two-bridge links

Zhou, Hugo

doi:10.2140/agt.2026.26.1597

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On homology concordance in contractible manifolds and two-bridge links

Hugo Zhou

Algebraic & Geometric Topology 26 (2026) 1597–1634

DOI: 10.2140/agt.2026.26.1597

Abstract

Let ${\hat{𝒞}}_{ℤ}$ be the group which consists of manifold-knot pairs $(Y, K)$ modulo homology concordance, where $Y$ is an integer homology sphere bounding an integer homology ball, and let $𝒞_{ℤ}$ be the subgroup consisting of pairs $(S^{3}, K)$ . Dai, Hom, Stoffregen and Truong showed that the quotient group ${\hat{𝒞}}_{ℤ} ∕ 𝒞_{ℤ}$ admits a $ℤ^{\infty}$ -summand. In this paper, we improve the result by showing that there exists a family ${(Y, K_{m})}_{m > 1}$ generating a $ℤ^{\infty}$ -summand where $Y$ is the boundary of a smooth contractible $4$ -manifold. In fact, we give a $ℤ$ -count of such families.

The examples are constructed using a family of knots obtained by blowing down a component of a two-bridge link. They are studied in Jonathan Hales’s thesis. Using the algorithm due to Ozsváth, Szabó and Hales we give a classification of the knot Floer homology of a larger family of such knots that might be of independent interest.

Keywords

homology concordance, contractible manifolds, two-bridge links, Heegaard Floer homology, $(1, 1)$ knots

Mathematical Subject Classification

Primary: 57K10, 57K18

References

Publication

Received: 21 July 2023

Revised: 19 October 2024

Accepted: 12 May 2025

Published: 23 May 2026

Authors

Hugo Zhou
Department of Mathematics University of Michigan Ann Arbor, MI United States

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Volume 26, issue 5 (2026)