论文标题
增强的Bruhat分解和摩尔斯理论
Enhanced Bruhat decomposition and Morse theory
论文作者
论文摘要
如果其所有临界值都截然不同，则摩尔斯函数称为强。给定这样的函数$ f $和一个字段$ \ mathbb {f} $ barannikov构建了$ f $的一些关键点的配对，现在也称为条形码。在每对Barannikov对的情况下，我们自然会关联（签名）$ \ mathbb {f} \ setMinus \ {0 \} $的元素;我们称其为bruhat号码。该论文致力于研究这些Bruhat数字。我们调查了几种情况，其中所有这些数字的产品（有些被提高到电力$ -1 $）独立于$ f $，并将其解释为reidemeister扭转。 我们通过证明通用功能的通用路径必须满足某个方程式mod 2（最初在其他假设下以\ cite {akhm}的形式证明），将结果应用于单参数摩尔斯理论的设置。 在线性代数级别上，我们的构造是通过$ gl（\ Mathbb {f}）$的经典bruhat分解的以下变体来提供的。 unitriangular基质是上三角形的矩阵，其对角线上有1个。考虑$ \ mathbb {f} $上方的所有矩形矩阵，直至左右乘法。增强的Bruhat分解描述了每个等效类别中的规范代表。
Morse function is called strong if all its critical values are pairwise distinct. Given such a function $f$ and a field $\mathbb{F}$ Barannikov constructed a pairing of some of the critical points of $f$, which is now also known as barcode. With every Barannikov pair we naturally associate (up to sign) an element of $\mathbb{F} \setminus \{0\}$; we call it Bruhat number. The paper is devoted to the study of these Bruhat numbers. We investigate several situations where the product of all these numbers (some being raised to the power $-1$) is independent of $f$ and interpret it as a Reidemeister torsion. We apply our results in the setting of one-parameter Morse theory by proving that generic path of functions must satisfy a certain equation mod 2 (this was initially proven in \cite{Akhm} under additional assumptions). On the linear-algebraic level our constructions are served by the following variation of a classical Bruhat decomposition for $GL(\mathbb{F})$. A unitriangular matrix is an upper triangular one with 1's on the diagonal. Consider all rectangular matrices over $\mathbb{F}$ up to left and right multiplication by unitriangular ones. Enhanced Bruhat decomposition describes canonical representative in each equivalence class.
