关于以AR同调球为边界的自旋$4$-流形的相交形式
On intersection forms of spin $4$-manifolds with AR homology sphere boundary
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中文总结 AI 辅助
本文基于Dai--Sasahira--Stoffregen的工作，利用Pin(2)-等变映射约束和Lin的κo-不变量，推导出以AR同调球为边界的自旋4-流形相交形式的约束，并应用于纽结界曲面亏格下界和11/8-猜想的分解障碍。
中文摘要 AI 辅助
本文在Dai--Sasahira--Stoffregen工作的基础上，通过两种方法推导了以AR同调球为边界的紧致、光滑、不定号、自旋$4$-流形的相交形式所满足的约束。首先，我们利用Hopkins--Lin--Shi--Xu在证明$10/8 + 4$-不等式时建立的表示球之间$\noperatorname{Pin}(2)$-等变映射存在性的约束。其次，我们计算AR同调球的Lin的$\kappa o$-不变量。结合Lin的相对$10/8$-不等式版本，该计算产生的约束不同于基于Hopkins--Lin--Shi--Xu的约束。作为应用，我们获得了以AR同调球为双分支覆盖的纽结在各种$4$-流形中所界曲面最小亏格的新下界，并提供了与$11/8$-猜想相关的光滑$4$-流形分解的障碍。
英文摘要
In this paper, building on the work of Dai--Sasahira--Stoffregen, we derive constraints on the intersection forms of smooth, compact, indefinite, spin $4$-manifolds bounded by AR homology spheres using two methods. First, we use the constraints on the existence of $\operatorname{Pin}(2)$-equivariant maps between representation spheres, established by Hopkins--Lin--Shi--Xu in their proof of the $10/8 + 4$-inequality. Second, we compute Lin's $κo$-invariant for AR homology spheres. Combined with Lin's version of the relative $10/8$-inequality, this computation yields constraints different from those based on Hopkins--Lin--Shi--Xu. As applications, we obtain new lower bounds on the minimal genus of surfaces bounded by knots whose double branched covers are AR homology spheres in various $4$-manifolds, and provide an obstruction to smooth $4$-manifold decompositions related to the $11/8$-conjecture.
