On homology spheres of the trivial local equivalence class

Date

2026-01-01

Author

Lee, Jaewon
Şavk, Oğuz

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In the homology cobordism group 3Z, 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 I along with the homomorphism h : 3Z → 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 I originate from 3Z. In this paper, by employing the 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 Pin(2)-equivariant Seiberg–Witten Floer stable homotopy type.

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https://hdl.handle.net/11511/119848

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Department of Mathematics, Article