穿越流的具有约束因子的Grassmanian多项式空间
Spaces of polynomials with constrained divisors as Grassmanians for traversing flows
摘要：
本文研究了具有边界的紧致光滑流形$X$上的{ \sf遍历}向量流$v$。对于一个给定的紧致流形$\hat X$，配备了一个遍历向量场$\hat v$，这是{ \sf凸}相对于$\partial\hat X$，我们考虑浸没/嵌入$α: X \to \hat X$，使$\dim X = \dim \hat X$和$α(\partial X)$避免了一个先验选择的相切模式$Θ$的$\hat v$ -轨迹。特别地，对于每个$\hat v$ -轨迹$\hatγ$，我们通过偶数$d$来限制$\hatγ\cap α(\partial X)$的基数。我们称$(\hat X, \hat v)$为$(X, v)$对的{ \sf凸伪包络/包络}。这里的向量字段$v = α^\dagger(\hat v)$是$α$ -从$\hat v$到$X$的转换。对于固定的$(\hat X, \hat v)$，我们在凸伪群/群$α: (X, v) \to (\hat X, \hat v)$之间引入一个等价关系，我们称之为{ \sf拟纯}.拟全同的概念是伪同边与伪合痕的交叉。在拟拓扑$\mathcal{QT}_d(Y, \mathbf cΘ)$的研究中，组合类型避开闭偏序集$Θ$的实一元$d$次多项式空间$\mathcal P_d^{\mathbf cΘ}$扮演经典的Grassmanian角色。我们计算，在同伦理论的条款，涉及$(\hat X, \hat v)$和$\mathcal P_d^{\mathbf cΘ}$，拟的凸矩阵，避免了$Θ$相切模式。我们引入了伪拟序类的特征类，并证明了它们是其拟同伦类的不变量。然后证明了拟序$\mathcal{QT}_d(Y, \mathbf cΘ)$常稳定为$d \to \infty$ .
英文摘要：
We study {\sf traversing} vector flows $v$ on smooth compact manifolds $X$ with boundary. For a given compact manifold $\hat X$, equipped with a traversing vector field $\hat v$ which is {\sf convex} with respect to $\partial\hat X$, we consider submersions/embeddings $α: X \to \hat X$ such that $\dim X = \dim \hat X$ and $α(\partial X)$ avoids a priory chosen tangency patterns $Θ$ to the $\hat v$-trajectories. In particular, for each $\hat v$-trajectory $\hatγ$, we restrict the cardinality of $\hatγ\cap α(\partial X)$ by an even number $d$. We call $(\hat X, \hat v)$ a {\sf convex pseudo-envelop/envelop} of the pair $(X, v)$. Here the vector field $v = α^\dagger(\hat v)$ is the $α$-transfer of $\hat v$ to $X$. For a fixed $(\hat X, \hat v)$, we introduce an equivalence relation among convex pseudo-envelops/ envelops $α: (X, v) \to (\hat X, \hat v)$, which we call a {\sf quasitopy}. The notion of quasitopy is a crossover between bordisms of pseudo-envelops and their pseudo-isotopies. In the study of quasitopies $\mathcal{QT}_d(Y, \mathbf cΘ)$, the spaces $\mathcal P_d^{\mathbf cΘ}$ of real univariate polynomials of degree $d$ with real divisors whose combinatorial types avoid the closed poset $Θ$ play the classical role of Grassmanians. We compute, in the homotopy-theoretical terms that involve $(\hat X, \hat v)$ and $\mathcal P_d^{\mathbf cΘ}$, the quasitopies of convex envelops which avoid the $Θ$-tangency patterns. We introduce characteristic classes of pseudo-envelops and show that they are invariants of their quasitopy classes. Then we prove that the quasitopies $\mathcal{QT}_d(Y, \mathbf cΘ)$ often stabilize, as $d \to \infty$.
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