纤维积的欧拉示性数

Euler Characteristic of Fiber Product

摘要: 本文主要研究了两个不同流形$M_1,M_2$ 上Morse函数纤维积空间$C\left(f,g\right)$ 的欧拉示性数$\chi \left(C\left(f,g\right)\right)$ 。通过建立适用于两个流形的欧拉示性数分解公式并结合Morse理论中水平集的欧拉示性数表达式。本文导出了$\chi \left(C\left(f,g\right)\right)$ 的统一表达式。该公式涵盖了临界值对齐与不对齐的情形。研究结果表明，纤维积的欧拉示性数由两个Morse函数的临界点指数通过行列式型的交错和决定，体现了横截映射拓扑与Morse理论之间的深刻联系。

Abstract: This paper primarily investigates the Euler characteristic $\chi \left(C\left(f,g\right)\right)$ of the fibre product space $C\left(f,g\right)$ of Morse functions on two distinct manifolds $M_1$ and $M_2$ . By establishing a decomposition formula for the Euler characteristic applicable to both manifolds and combining it with the expression for the Euler characteristic of level sets in Morse theory, this paper derives a unified expression for $\chi \left(C\left(f,g\right)\right)$ . This formula encompasses both critical point alignment and non-alignment scenarios. The findings reveal that the Euler characteristic of the fibre product is determined by the critical point indices of the two Morse functions through a determinant-type alternating sum, reflecting a profound connection between the topology of cross-maps and Morse theory.