Filtered instanton Floer homology and the homology cobordism group
Fintushel-Stern and Furuta developed orbifold Yang-Mills gauge theory and proved that infinitely many Brieskorn 3-spheres are linearly independent in the (3-dimensional) homology cobordism group. In this work, by translating their work into the words of filtered instanton Floer homology, we introduce a family of real-valued homology cobordism invariants r_s(Y). The values of r_s(Y) are contained in the set of critical values of SU(2) Chern-Simons functionals, and we have a negative definite cobordism inequality and connected sum formula, which give several new results for homology 3-spheres. As one of colloraries, we give infinitely many homology 3-spheres which cannot bound any definite 4-manifold. As another collorary, we show that if the 1-surgery of a knot has negative Froyshov invariant, 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(Y) for the 1/2-surgery of the knot 5_2 in the Rolfsen’s table.
This is a joint work with Yuta Nozaki and Kouki Sato.