论文标题
精制的Bobtcheva-Messia不变4维
Refined Bobtcheva-Messia Invariants of 4-Dimensional 2-Handlebodies
论文作者
论文摘要
在本文中，我们完善了我们最近建造的不变式$ 4 $ 2 $ 2 $ - handlebodies，最高可达$ 2 $ - 变形。更确切地说，我们定义了形式$（w，ω）$的成对的不变性，其中$ w $是$ 4 $二维的$ 2 $ 2 $ - handlebody，$ω$是$ h^2（w，\ partial w; g）$的相对同胞学类，而$ g $ $ g $是一个阿贝尔集团。该构建所需的代数输入是单型色带Hopf $ g $ -Coalgebra。我们研究了这些精制的不变性量，以限制量子组$ u = u_q \ mathfrak {sl} _2 $均匀订购的根部$ q $，以及其编织的扩展$ \ tilde {u} = \ tilde {u} = \ tilde {U} $ g = \ mathbb {z}/2 \ mathbb {z} $，我们将它们与原始不变性联系起来。我们根据精致的公式推断了原始不变的公式，从而将Witten-Reshetikhin-Turaev的分裂推广到旋转结构和同胞学类方面。此外，我们确定了与小量子组$ \ bar {u} = \ bar {u} _q \ mathfrak {sl} _2 $的非量子组相关的非不变性。
In this paper we refine our recently constructed invariants of $4$-dimensional $2$-handlebodies up to $2$-deformations. More precisely, we define invariants of pairs of the form $(W,ω)$, where $W$ is a $4$-dimensional $2$-handlebody, $ω$ is a relative cohomology class in $H^2(W,\partial W;G)$, and $G$ is an abelian group. The algebraic input required for this construction is a unimodular ribbon Hopf $G$-coalgebra. We study these refined invariants for the restricted quantum group $U = U_q \mathfrak{sl}_2$ at a root of unity $q$ of even order, and for its braided extension $\tilde{U} = \tilde{U}_q \mathfrak{sl}_2$, which fits in this framework for $G=\mathbb{Z}/2\mathbb{Z}$, and we relate them to our original invariant. We deduce decomposition formulas for the original invariants in terms of the refined ones, generalizing splittings of the Witten-Reshetikhin-Turaev invariants with respect to spin structures and cohomology classes. Moreover, we identify our non-refined invariant associated with the small quantum group $\bar{U} = \bar{U}_q \mathfrak{sl}_2$ at a root of unity $q$ whose order is divisible by 4 with the refined one associated with the restricted quantum group $U$ for the trivial cohomology class $ω=0$.
