论文标题
几何解释较高的杯子产品，并应用于组合销结构
Geometrically Interpreting Higher Cup Products, and Application to Combinatorial Pin Structures
论文作者
论文摘要
我们为Steenrod的$ \ cup_i $产品提供了对公式的几何解释，为猜想提供了明确的结构。我们从单纯形和分支结构构建每个单纯形内的矢量场特殊框架，使我们能够将$ \ cup_i $的Cochain级公式解释为双蜂窝分解上的广义交叉点。它可以被认为是测量双电池集合之间的相交以及另一个集合的变厚，变化的版本，其中矢量场框架决定了增厚和变化。在简单复合物的双1骨架的附近定义此矢量场框架，使我们能够在三角歧管上结合$ spin $和$ pin^\ pm $结构。我们使用它们来几何解释Gu-Wen/Gaiotto-Kapustin的“ Grassmann积分”，而无需使用Grassmann变量。特别是，我们发现，可以使用我们的矢量字段和$ \ cup_i $的解释，以及双重1-Sketoron的一定三价分辨率，并解释gaiotto-kapustin的“二次完善”属性。这使我们可以将其功能的范围扩展到任意三角剖分，并明确看到其与旋转结构的联系。 Vandermonde矩阵在所有构造中都起着关键作用。
We provide a geometric interpretation of the formulas for Steenrod's $\cup_i$ products, giving an explicit construction for a conjecture of Thorngren. We construct from a simplex and a branching structure a special frame of vector fields inside each simplex that allow us to interpret cochain-level formulas for the $\cup_i$ as a generalized intersection product on the dual cellular decomposition. It can be thought of as measuring the intersection between a collection of dual cells and thickened, shifted version of another collection, where the vector field frame determines the thickening and shifting. Defining this vector field frame in a neighborhood of the dual 1-skeleton of a simplicial complex allows us to combinatorially define $Spin$ and $Pin^\pm$ structures on triangulated manifolds. We use them to geometrically interpret the `Grassmann Integral' of Gu-Wen/Gaiotto-Kapustin, without using Grassmann variables. In particular, we find that the `quadratic refinement' property of Gaiotto-Kapustin can be derived geometrically using our vector fields and interpretation of $\cup_i$, together with a certain trivalent resolution of the dual 1-skeleton. This lets us extend the scope of their function to arbitrary triangulations and explicitly see its connection to spin structures. Vandermonde matrices play a key role in all constructions.
