Filtered instanton Floer homology and the homology cobordism group.

For any $s \in \mathbb{R}_{\leq 0} \cup \{-\infty\}$ and oriented homology $3$-sphere $Y$ , we introduce a homology cobordism invariant $r_s(Y )$ whose value is in $ \mathbb{R}_{>0} \cup \{\infty \}$. The values $\{r_s (Y )\}$ are contained in the critical values of the Chern-Simons functional of $Y$ , and we have a negative definite cobordism inequality for any $s$. Moreover, for the case of $r_0$, we give a connected sum formula, which gives several new results on the homology cobordism group. As one of such results, we give infinitely many homology $3$-spheres which cannot bound any definite $4$-manifold. As another result, we show that if the $1$-surgery of a knot has the Froyshov invariant negative, then all positive $1/n$-surgeries of the knot are linearly independent in the homology cobordism group. Moreover, as a hyperbolic example, we compute an approximate value of $r_s$ for the $1/2$-surgery of the mirror of the knot $5_2$ in Rolfsen's knot table.