Filtered instanton Floer homology and the homology cobordism group

We introduce a family of real-valued homology cobordism invariants of homology 3-spheres. The invariants are derived from filtered instanton Floer homology, and those values are critical values of the SU(2)-Chern-Simons functionals. As its application, we produce infinitely many homology 3-spheres that cannot bound either a positive or negative definite 4-manifold. As another application, we show that if the 1-surgery of a knot has the Froyshov invariant negative, then the 1/n-surgeries (n>0) of the knot are linearly independent in the homology cobordism group. This is joint work with Yuta Nozaki and Masaki Taniguchi.