标题： 关于平凡局部等价类的同调球体 显示英文标题

标题： On homology spheres of the trivial local equivalence class

摘要： 在同调cobordism群$\Theta_{\mathbb{Z}}^3$中，尚不清楚Seifert纤维化球面之间是否存在非平凡的线性相关性。 基于对合Heegaard Floer理论，Hendricks、Manolescu和Zemke引入了局部等价群$\mathfrak{I}$，并具有同态 $h:\Theta_{\mathbb{Z}}^3 \rightarrow \mathfrak{I}$。 并且可以通过Dai和Stoffregen的工作，在$\mathfrak{I}$中找到Seifert纤维化球面像之间的非平凡线性相关性。 因此，有趣的问题是这种在$\mathfrak{I}$中的线性相关性是否源于$\Theta_{\mathbb{Z}}^3$。 在本文中，我们证明这在一般情况下并不成立。 我们证明了在$\mathfrak{I}$中无限多个同调球线性相关，即使在有理同调cobordism群$\Theta_{\mathbb{Q}}^3$中也是线性无关的，这是通过采用Nozaki、Sato和Taniguchi开发的过滤瞬子Floer同调的$r_s$不变量。 此外，这些同调球包括$\operatorname{Pin(2)}$-等变Seiberg--Witten Floer稳定同伦类型的平凡局部等价类的同调球。 显示英文摘要

摘要： In the homology cobordism group $\Theta_{\mathbb{Z}}^3$, it is not known if there is a non-trivial linear dependence between Seifert fibered spheres. Based on involutive Heegaard Floer theory, Hendricks, Manolescu, and Zemke introduced the local equivalence group $\mathfrak{I}$ with the homomorphism $h:\Theta_{\mathbb{Z}}^3 \rightarrow \mathfrak{I}$. And it is possible to find a non-trivial linear dependence between the image of Seifert fibered spheres in $\mathfrak{I}$ using the work of Dai and Stoffregen. Therefore, it is interesting to ask if such a linear dependence in $\mathfrak{I}$ originates from $\Theta_{\mathbb{Z}}^3$. In this paper, we prove that this is not the case in general. We show that infinitely many homology spheres linearly dependent in $\mathfrak{I}$ are linearly independent even in the rational homology cobordism group $\Theta_{\mathbb{Q}}^3$, by employing the $r_s$-invariants from the filtered instanton Floer homology developed by Nozaki, Sato, and Taniguchi. Moreover, these homology spheres include homology spheres of the trivial local equivalence class of $\operatorname{Pin(2)}$-equivariant Seiberg--Witten Floer stable homotopy type.