带整数系数的实Heegaard Floer同调
Real Heegaard Floer homology with integral coefficients
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中文总结 AI 辅助
本文将Guth-Manolescu的实Heegaard Floer理论升级为整数系数的ℤ/2-分次不变量，给出相对分次与提升存在的充要障碍，其障碍理论对Floer理论有独立价值。
中文摘要 AI 辅助
我们研究将Guth-Manolescu的实Heegaard Floer理论升级为在整数环ℤ上定义的ℤ/2-分次不变量的问题，具体针对分支链各分支带有一个基点的hat版本。我们仅利用带对合的三维流形相关的纯同调信息，给出相对ℤ/2-分次和ℤ提升存在的充要障碍。当对应障碍消失时，我们研究这类ℤ提升的集合，其先验仅为在挠子作用下的不变量。该障碍理论在更广泛的各类Floer理论中可能具有独立意义，尤其在Lagrangian Floer理论中。
英文摘要
We treat the question of upgrading Guth-Manolescu's real Heegaard Floer theory to a $\mathbb Z/2$-graded invariant defined over $\mathbb Z$, specifically for the hat version with one basepoint on each component of the branching link. We provide if-and-only-if obstructions to the existence of relative $\mathbb Z/2$-gradings and $\mathbb Z$-lifts, in terms of purely homological information associated to the 3-manifold with involution. When the corresponding obstruction vanishes, we study the set of such $\mathbb Z$-lifts, which a priori is only an invariant up to a torsor action. The obstruction theory could be of independent interest in various Floer theories more broadly, particularly Lagrangian Floer theory.
