AI 中文总结
本文引入融合2-范畴的3-群分次扩张，构造了4维流形的非平凡量子同伦不变量，推广了Douglas-Reutter不变量，且在该语境下推广了Pachner定理。
AI 中文摘要
我们引入融合2-范畴的3-群分次扩张，其中3-群由2-交叉模建模。基于此，我们导出配备到同伦3-型的映射同伦类的4维流形的态和不变量，等价于4维流形上的平坦3-丛的不变量。该不变量非平凡，推广了4维流形的Douglas-Reutter不变量。为构建此不变量，我们通过三角剖分的σ-着色编码到3-群σ的分类空间的映射同伦类，并在此语境下推广了Pachner定理。
英文摘要
We introduce 3-group-graded extensions of fusion 2-categories, where 3-groups are modeled by 2-crossed modules. From this data, we derive a state-sum invariant of 4-manifolds equipped with a homotopy class of maps to a homotopy 3-type, or equivalently of flat 3-bundles over 4-manifolds. This invariant is nontrivial and generalizes the Douglas-Reutter invariant of 4-manifolds. To construct our invariant, we encode homotopy classes of maps to the classifying space of a 3-group $σ$ via $σ$-colorings of triangulations, and we generalize Pachner's theorem in this context.
Comments46 pages
