发表机构
Saitama University（埼玉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为Lefschetz复形建立离散Morse-Bott理论，通过四种保持余数级数的移动进行约简，计算Morse-Bott不等式系数及Betti数，并证明系数非负性，适用于CW复形。
AI 中文摘要
我们为具有实值关联函数且每个维度上只有有限多个胞腔的Lefschetz复形发展了一套离散Morse-Bott理论。我们的主要结果是一个基于四种称为“移动”的基本运算的约简程序，用于计算与Morse-Bott不等式相关的余数级数$R_t$。这些移动保持$R_t$不变，该程序通过在每个维度上使用有限多次移动，递归地构造一个由两个胞腔组成且Poincaré级数消失的基本块的并集。由此产生的分解通过计数相应维度中的块来计算$R_t$的系数。同一程序还通过计数作为孤立胞腔被移除的胞腔来计算原始复形的Betti数。作为推论，我们得到了$R_t$系数的非负性以及Lefschetz复形上的Morse-Bott不等式。该方法特别适用于CW复形，在其中通过将$R_t$的每个系数与相应维度中基本块的数量对应起来获得非负性。
英文摘要
We develop a discrete Morse-Bott theory for Lefschetz complexes with real-valued incidence functions and finitely many cells in each dimension. Our main result is a reduction procedure, based on four elementary operations called Moves, for computing the remainder series $R_t$ associated with the Morse-Bott inequality. These Moves preserve $R_t$, and the procedure recursively constructs, using finitely many Moves in each dimension, a disjoint union of two-cell elementary blocks with vanishing Poincaré series. The resulting decomposition computes the coefficients of $R_t$ by counting the blocks in the corresponding dimensions. The same procedure also computes the Betti numbers of the original complex by counting the cells removed as isolated cells. As consequences, we obtain the nonnegativity of the coefficients of $R_t$ and the Morse-Bott inequality for Lefschetz complexes. The method applies in particular to CW complexes, where the nonnegativity is obtained by identifying each coefficient of $R_t$ with the number of elementary blocks in the corresponding dimension.
Comments32 pages, 8 figures
