Homology concordance and knot Floer homology

Abstract

We study the homology concordance group of knots in integer homology three-spheres which bound integer homology four-balls. Using knot Floer homology, we construct an infinite number of $\mathbb{Z}$-valued, linearly independent homology concordance homomorphisms which vanish for knots coming from $S^3$. This shows that the homology concordance group modulo knots coming from $S^3$ contains an infinite-rank summand. The techniques used here generalize the classification program established in previous papers regarding the local equivalence group of knot Floer complexes over $\mathbb{F}[U,V]/(UV)$. Our results extend this approach to complexes defined over a broader class of rings.

arXiv comment65 pages; 3 figures. Removed some speculative sections for readability. Final version; to appear in Mathematische Annalen

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