Filtered instanton Floer homology and the homology cobordism group

• Published in: arXiv.org 2022-01
• Main Authors: Nozaki, Yuta, Sato, Kouki, Taniguchi, Masaki
• Format: Article
• Language: English
• Subjects: Homology; Invariants; Surgery
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1905.04001

For any \(s \in [-\infty, 0] \) and oriented homology 3-sphere \(Y\), we introduce a homology cobordism invariant \(r_s(Y)\in (0,\infty]\). The values \(\{r_s(Y)\}\) are included in the critical values of the \(SU(2)\)-Chern-Simons functional of \(Y\), and we show a negative definite cobordism inequality and a connected sum formula for \(r_s\). As applications, we obtain several new results on the homology cobordism group. First, we give infinitely many homology 3-spheres which cannot bound any definite 4-manifold. Next, we show that if the 1-surgery of \(S^3\) along a knot has the Frøyshov invariant negative, then all positive \(1/n\)-surgeries along the knot are linearly independent in the homology cobordism group. In another direction, we use \(\{r_s\}\) to define a filtration on the homology cobordism group which is parametrized by \([0,\infty]\). Moreover, we compute an approximate value of \(r_s\) for the hyperbolic 3-manifold obtained by \(1/2\)-surgery along the mirror of the knot \(5_2\).