On homology spheres of the trivial local equivalence class

It remains an open question whether nontrivial linear dependences exist among Seifert fibered spheres in the homology cobordism group. In this talk, we consider an infinite family of homology spheres that form the trivial local equivalence class in involutive Heegaard Floer theory, thereby potentially yielding nontrivial dependences between Seifert fibered spheres in the homology cobordism group. We then focus on a survey of filtered instanton Floer theory. Finally, as an application, we prove that these candidates are linearly independent. In particular, we compare the invariants from the two theories: involutive Heegaard Floer theory and filtered instanton Floer theory. If time permits, we also compare them with Pin(2)-equivariant Seiberg-Witten Floer theory. This is joint work with Jaewon Lee (KAIST, Korea).