On homology spheres of the trivial local equivalence class

Abstract

In the homology cobordism group $\Theta_\mathbb{Z}^3$, it is not known if there are non-trivial linear dependences between Seifert fibered spheres. Based on involutive Heegaard Floer theory, Hendricks, Manolescu, and Zemke introduced the local equivalence group $\mathfrak{I}$ along with the homomorphism $h:\Theta_\mathbb{Z}^3 \rightarrow \mathfrak{I}$. Using the work of Dai and Stoffregen, one can find non-trivial linear dependences between the images of Seifert fibered spheres under $h$. Therefore, it is interesting to ask if such dependences in $\mathfrak{I}$ originate from $\Theta_\mathbb{Z}^3$. In this paper, by employing the $r_s$-invariants from the filtered instanton Floer homology developed by Nozaki, Sato, and Taniguchi, we provide certain conditions to guarantee that such relations are not realized even in the rational homology cobordism group. We also discuss the local equivalence class of the $\operatorname{Pin(2)}$-equivariant Seiberg--Witten Floer stable homotopy type.