Homology concordance and knot Floer homology
Let \(Y\), \(Y'\) be integer homology spheres that are homology null-cobordant, and \(K \hookrightarrow Y\), \(K' \hookrightarrow Y'\) be knots. These knots are said to be \textit{homology concordant} if there is a homology cobordism \(W\) from \(Y\) to \(Y'\) and an annulus, smoothly embedded in \(W\), cobounding \(K\) and \(K'\).\N\NThe group \(\widehat{\mathcal{C}_{\mathbb{Z}}}\) consists of homology concordance classes of pairs \((Y, K )\), with the group operation given by connected sum (of both \(3\)-manifold and knot). This group is a natural generalization of the classical concordance group. One can also consider the subgroup of \(\widehat{\mathcal{C}_{\mathbb{Z}}}\) consisting of classes with a representative \((S^3, K)\); denote this subgroup \(\mathcal{C}_{\mathbb{Z}}\).\N\NThe main result of the paper under review is the construction of an infinite family of homomorphisms from \(\widehat{\mathcal{C}_{\mathbb{Z}}}\) to \(\mathbb{Z}\). An immediate corollary of this result is that the \(\mathbb{Z}^{\infty}\)-subgroup of \(\widehat{\mathcal{C}_{\mathbb{Z}}}\) identified by \textit{H. Zhou} [J. Topol. 14, No. 4, 1369--1395 (2021; Zbl 1540.57009)] is in fact a summand. The main result is established using knot Floer homology.\N\NFurther results include a relation between the homomorphisms of this paper and the \(\tau\) invariant, constraining the value of the homomorphisms in Seifert fibre spaces, and various genus bounds.
scientific article; zbMATH DE number 7940280
scientific article