On homology spheres of the trivial local equivalence class

In the homology cobordism group normal upper Theta Subscript double struck upper Z Superscript 3 Θ Z 3 ΘZ³ , 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 German upper I I I along with the homomorphism h colon normal upper Theta Subscript double struck upper Z Superscript 3 Baseline right arrow German upper I h : Θ Z 3 → I h\;:\;ΘZ³ → 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 German upper I I I originate from normal upper Theta Subscript double struck upper Z Superscript 3 Θ Z 3 ΘZ³ . In this paper, by employing the r Subscript s r s rₛ -invariants from the filtered instanton Floer homology developed by Nozaki, Sato and Taniguchi, we provide certain conditions to guarantee that such relations are not realised even in the rational homology cobordism group. We also discuss the local equivalence class of the normal upper P normal i normal n left parenthesis 2 right parenthesis P i n