Two knots are homology concordant if they are smoothly concordant in a homology cobordism. The group ${\widehat{\mathcal{C}}}_{\mathbb{Z}}$ (respectively, ${\mathcal{C}}_{\mathbb{Z}}$) was previously defined as the set of knots in homology spheres that bounds homology balls (respectively, in $S^3$), modulo homology concordance. We prove ${\widehat{\mathcal{C}}}_{\mathbb{Z}}/{\mathcal{C}}_{\mathbb{Z}}$ contains a ${\mathbb{Z}}^{\infty }$ subgroup. We construct our family of examples by applying the filtered mapping cone formula to L-space knots, and prove linear independence with the help of the connected knot complex.

中文翻译：

---

同调一致性和无限秩自由子群

如果两个结在同调协调一致中平滑一致，则它们是同调一致的。群组${\widehat{\mathcal{C}}}_{\mathbb{Z}}$（分别，${\mathcal{C}}_{\mathbb{Z}}$) 以前被定义为同源球中的一组结，它们限制同源球（分别在${\mathrm{小号}}^3$)，模同调一致性。我们证明${\widehat{\mathcal{C}}}_{\mathbb{Z}}/{\mathcal{C}}_{\mathbb{Z}}$包含一个${\mathbb{Z}}^{\infty }$子群。我们通过将过滤后的映射锥公式应用于 L 空间结来构建我们的示例系列，并在连接结复合体的帮助下证明线性独立性。