Two integer homology spheres, \(Y_0, Y_1\) are said to be homology cobordant if there exists an oriented 4-manifold \(W\) such that \(\partial W = -Y_0 \sqcup Y_1\), and the inclusions \(Y_i \rightarrow W\) induce isomorphisms in homology. Two knots \(K_0 \subset Y_0\), \(K_1 \subset Y_1\) are homology concordant if there exists a homology cobordism \(W\) between \(Y_0\) and \(Y_1\), and a smoothly embedded annulus \(A \subset W\) such that \(\partial A = -K_0 \sqcup K_1\), generalizing the concept of concordance between two knots in \(S^3\).
This paper addresses a structural question for the group of homology concordances for knots in homology spheres (the group operation here is the connected sum of the manifolds and the knots). In particular, if \(\mathscr{C}_{\mathbb{Z}}\) denotes the group of knots in \(S^3\), modulo homology concordance, and if \(\widehat{\mathscr{C}}_{\mathbb{Z}}\) denotes the group of knots \(K \subset Y\) where \(Y\) is a homology 3-sphere that bounds a homology 4-ball, modulo homology concordance, then the main theorem (Theorem 1.1) of the paper states that the subgroup \(\mathscr{C}_{\mathbb{Z}} \subset \widehat{\mathscr{C}}_{\mathbb{Z}}\) is of infinite index, precisely, \(\widehat{\mathscr{C}}_{\mathbb{Z}} / \mathscr{C}_{\mathbb{Z}}\) is infinitely generated, and contains a subgroup isomorphic to \(\mathbb{Z}\).
The authors provide an infinite family of knots contained in homology spheres \((K_j, Y_j) \in \widehat{\mathscr{C}}_{\mathbb{Z}}\) for which a Heegaard Floer theoretic invariant \(\theta(Y_j,K_j)\) is unbounded, which proves the infinite generation result. They provide a knot \((Y,K)\) such that \(Y\) is a homology sphere bounding a homology 4-ball, such that \(K\) is not homology concordant to any knot in \(S^3\), and such that the same holds for \((\#_n(Y,K))\), which provides them with the \(\mathbb{Z}\) summand in the quotient as mentioned above.
The crux of the arguments lies in finding obstructions leveraging several Heegaard Floer theoretic concordance invariants for knots in \(S^3\), which satisfy similar properties and relations among them under homology concordance as concordance of knots in \(S^3\).
The Heegaard Floer homology toolbox comes with various invariants which the authors make use of in this paper, such as a homology concordance invariant for closed, orientable 3-manifolds, the \(d\)-invariant, and several knot concordance invariants such as \(\tau, \varepsilon, \Upsilon\). Section 4 of the paper is devoted to showing that these are indeed homology concordance invariants and to proving their various properties. The authors then make use of a filtered version of the Heegaard Floer mapping cone formula, which lets one find the knot Floer homology of the core of the surgery in the closed 3-manifold, obtained by doing a surgery on a knot in \(S^3\), using the knot Floer chain complex associated with the knot. In Sections 2, 3 and 6, they make use of the non-filtered and the filtered versions of the mapping cone formulas to get the desired results.