Cascades and perturbed Morse–Bott functions
Cascades and perturbed Morse–Bott functions
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级联和扰动 Morse-Bott 函数
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
D. Hurtubise
中科院分区：
文献类型：
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作者：
A. Banyaga;D. Hurtubise
让fW M ！R是有限维闭光滑流形M上的Morse‐Bott函数。在M和Morse - small函数上选择合适的黎曼度规fjW Cj！R在临界子流形Cj上，可以构造一个莫尔斯链复形，其边界算子由级联计数[16]来定义。类似的数据，也包括一个参数“> 0”，它可以缩放莫尔斯-小函数fj，可以用来定义莫尔斯-小函数f的显式扰动到莫尔斯-小函数h“wm !”R (3;6)。本文证明了h”的Morse - small - Witten链复形与用级联定义的任何足够小的“>0”的Morse链复形是相同的。也就是说，两个链配合物具有相同的生成器，并且它们的边界算子是相同的（取决于符号的选择）。因此，莫尔斯同调定理表明，fW M！R是奇异同构的H . miz /。57 r70;37d05, 37d15, 58e05
Let fW M ! R be a Morse‐Bott function on a finite-dimensional closed smooth manifold M . Choosing an appropriate Riemannian metric on M and Morse‐Smale functions fjW Cj! R on the critical submanifolds Cj , one can construct a Morse chain complex whose boundary operator is defined by counting cascades [16]. Similar data, which also includes a parameter " > 0 that scales the Morse‐Smale functions fj , can be used to define an explicit perturbation of the Morse‐Bott function f to a Morse‐Smale function h"W M! R [3; 6]. In this paper we show that the Morse‐ Smale‐Witten chain complex of h" is the same as the Morse chain complex defined using cascades for any ">0 sufficiently small. That is, the two chain complexes have the same generators, and their boundary operators are the same (up to a choice of sign). Thus, the Morse Homology Theorem implies that the homology of the cascade chain complex of fW M! R is isomorphic to the singular homology H .MIZ/. 57R70; 37D05, 37D15, 58E05