Morse homology and equivariance

Erkao Bao, Tyler Lawson
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Abstract

In this paper, we develop methods for calculating equivariant homology from equivariant Morse functions on a closed manifold with the action of a finite group. We show how to alter $G$-equivariant Morse functions to a stable one, where the descending manifold from a critical point $p$ has the same stabilizer group as $p$, giving a better-behaved cell structure on $M$. For an equivariant, stable Morse function, we show that a generic equivariant metric satisfies the Morse--Smale condition. In the process, we give a proof that a generic equivariant function is Morse, and that equivariant, stable Morse functions form a dense subset in the $C^0$-topology within the space of all equivariant functions. Finally, we give an expository account of equivariant homology and cohomology theories, as well as their interaction with Morse theory. We show that any equivariant Morse function gives a filtration of $M$ that induces a Morse spectral sequence, computing the equivariant homology of $M$ from information about how the stabilizer group of a critical point acts on its tangent space. In the case of a stable Morse function, we show that this can be further reduced to a Morse chain complex.
莫尔斯同调与等差数列
在本文中,我们开发了从有限群作用的封闭流形上的等变莫尔斯函数计算等变同调的方法。我们展示了如何将 $G$ 等变莫尔斯函数改变为稳定的等变莫尔斯函数,其中从临界点 $p$ 下降的流形具有与 $p$ 相同的稳定群,从而在 $M$ 上得到更好的单元结构。对于一个等变的稳定莫尔斯函数,我们证明了一般等变度量满足莫尔斯--斯马尔条件。在此过程中,我们证明了一般等变函数是莫尔斯函数,并且等变的、稳定的莫尔斯函数在$C^0$拓扑中构成了所有等变函数空间的密集子集。最后,我们阐述了等变同调理论和同调理论,以及它们与莫尔斯理论的相互作用。我们证明,任何等变莫尔斯函数都会给出一个诱导莫尔斯谱序列的 $M$ 滤波,根据临界点稳定器群如何作用于其切线空间的信息计算 $M$ 的等变同调。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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