论文标题
Gutierrez-Sotomayor单数流的同型取消理论
Homotopical Cancellation Theory for Gutierrez-Sotomayor Singular Flows
论文作者
论文摘要
在本文中，我们介绍了Gutierrez-Sotomayor奇异流量$φ$，GS-FLOWS的动力同义取消理论，在单个表面上$ M $。该理论概括了平滑动力学系统的摩尔斯音乐复合物的经典理论，以及针对非分类奇异性的相应取消理论。这是通过定义$（m，φ）$的GS链复合物并计算其光谱序列$（E^r，d^r）$来实现的。随着$ r $的增加，代数取消发生，导致模块$ e^r $变得微不足道。本文的主要定理将光谱序列中的这些代数取消与gs-flows $ \ {m_r，φ_r\} $ gs-flows $φ_r$的$ \ {m_r，φ_r\} $相关的这些代数取消，在单数表面$ m_r $上，所有这些都具有与$ m $相同的同型。这些结果中令人惊讶的元素是，$φ_R$的gs-singularition的动态同位素取消与其相关光谱序列的$ e^r $的代数取消相吻合。 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.
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.
