Revisiting the Cohen-Jones-Segal construction in Morse-Bott theory

Ciprian Mircea Bonciocat
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Abstract

In 1995, Cohen, Jones and Segal proposed a method of upgrading any given Floer homology to a stable homotopy-valued invariant. For a generic pseudo-gradient Morse-Bott flow on a closed smooth manifold $M$, we rigorously construct the conjectural stable normal framings, which are an essential ingredient in their construction, and give a rigorous proof that the resulting stable homotopy type recovers $\Sigma^\infty_+ M$. We further show that one can recover Thom spectra $M^E$ for all reduced $KO$-theory classes $E$ on $M$, by using slightly modified stable normal framings. Our paper also includes a construction of the smooth corner structure on compactified moduli spaces of broken flow lines with free endpoint, a formal construction of Piunikhin-Salamon-Schwarz type continuation maps, and a way to relax the stable normal framing condition to orientability in orthogonal spectra.
重新审视莫尔斯-波特理论中的科恩-琼斯-西格尔结构
1995 年,科恩(Cohen)、琼斯(Jones)和西格尔(Segal)提出了一种将任何给定的弗洛尔同调提升为稳定同调值不变式的方法。对于封闭光滑流形 $M$ 上的一般伪梯度莫尔斯-波特流,我们严格地构造了猜想的稳定法线框架(这是其构造中的一个基本要素),并给出了严格的证明,即所得到的稳定同调类型恢复了 $\Sigma^\infty_+M$。我们还进一步证明,通过使用稍加修改的稳定正则框架,我们可以为 $M$ 上所有还原的 $KO$ 理论类 $E$ 恢复 Thom spectra $M^E$。我们的论文还包括对具有自由端点的断裂流线的紧凑模空间上的光滑角结构的构造、对皮乌尼金-萨拉蒙-施瓦茨类型延续映射的正式构造,以及将稳定法线框架条件放宽到正交谱中的定向性的方法。
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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