Gutierrez-Sotomayor奇异流的同伦消去理论
Homotopical Cancellation Theory for Gutierrez-Sotomayor Singular Flows
摘要：
在这篇文章中，我们提出了一个动力同伦抵消理论的Gutierrez-Sotomayor奇异流$φ$，GS-流，奇异表面$M$。该理论推广了光滑动力系统莫尔斯复形的经典理论以及相应的非退化奇异性抵消理论。这是通过定义$(M,φ)$的GS链复形并计算其谱序列$(E^r,d^r)$来实现的.随着$r$的增加，会发生代数取消，导致$E^r$中的模块变得微不足道。本文的主要定理将谱序列中的这些代数消去与奇异曲面$M_r$上的GS流$φ_r $的族$\{M_r,φ_r\}$联系起来，所有这些流都具有与$M$相同的同伦类型。令人惊讶的元素在这些结果是，动力同伦取消GS-奇点的流量$φ_r$是在协调与代数取消模块在$E^r$的相关的频谱序列。此外，谱序列的收敛对应于一个GS-流$φ_{\bar{r}}$上的$M_{\bar{r}}$，对于某些$\bar{r}$，与属性，$φ_{\bar{r}}$承认没有进一步的动态同伦取消GS-奇点。
英文摘要：
In this article, we present a dynamical homotopical cancellation theory for Gutierrez-Sotomayor singular flows $φ$, GS-flows, on singular surfaces $M$. This theory generalizes the classical theory of Morse complexes of smooth dynamical systems together with the corresponding cancellation theory for non-degenerate singularities. This is accomplished by defining a GS-chain complex for $(M,φ)$ and computing its spectral sequence $(E^r,d^r)$. As $r$ increases, algebraic cancellations occur, causing modules in $E^r$ to become trivial. The main theorems herein relate these algebraic cancellations within the spectral sequence to a family $\{M_r,φ_r\}$ of GS-flows $φ_r $ on singular surfaces $M_r$, all of which have the same homotopy type as $M$. The surprising element in these results is that the dynamical homotopical cancellation of GS-singularities of the flows $φ_r$ are in consonance with the algebraic cancellation of the modules in $E^r$ of its associated spectral sequence. Also, the convergence of the spectral sequence corresponds to a GS-flow $φ_{\bar{r}}$ on $M_{\bar{r}}$, for some $\bar{r}$, with the property that $φ_{\bar{r}}$ admits no further dynamical homotopical cancellation of GS-singularities.
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