Filtered instanton Floer homology and the homology cobordism group

For any $s\in [-\mathrm{\infty },0]$ and oriented homology 3-sphere $Y$, we introduce a homology cobordism invariant $r_s(Y)\in (0,\mathrm{\infty }]$. 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 Fr{\o}yshov invariant negative, then all positive $1\mathrm{/}n$-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 $[0,\mathrm{\infty }]$. Moreover, we compute an approximate value of $r_s$ for the hyperbolic 3-manifold obtained by $1\mathrm{/}2$-surgery along the mirror of the knot $5_2$.