论文标题
稳定地图与定向表面的同型不变
A homotopy invariant of stable maps to oriented surfaces
论文作者
论文摘要
一组通用地图$ f：m \至f $的f $ dimension $ m $ m \ ge 2 $到定向的表面$ f $是封闭的平滑曲线$σ（f）$。我们研究$σ（f）$的组件数量的均衡。单数集的图像$ f（σ）$通过所谓的棋盘功能继承了规范的本地方向。这样的局部方向产生了累积绕组编号$ω（f）\ in \ frac {1} {2} {2} \ mathbb {z} $ $σ（f）$。当歧管$ m $的尺寸甚至是我们还定义一个不变的$ i（f）$，这是$σ（f）$的零件数量的残留类模量$ 4 $ 4 $，即cusps的数量和两倍的自我交流点$ f（σ）$。 Using the cumulative winding number and the invariant $I(f)$, we show that the parity of the number of connected components of $Σ(f)$ does not change under homotopy of $f$ provided that one of the following conditions is satisfied: (i) the dimension of $M$ is even, (ii) the singular set of the homotopy is an orientable manifold, or (iii) the image of the singular set of the homotopy does not have三重自我讲道点。
The singular set of a generic map $f: M\to F$ of a manifold $M$ of dimension $m\ge 2$ to an oriented surface $F$ is a closed smooth curve $Σ(f)$. We study the parity of the number of components of $Σ(f)$. The image $f(Σ)$ of the singular set inherits canonical local orientations via so-called chessboard functions. Such a local orientation gives rise to the cumulative winding number $ω(f)\in \frac{1}{2}\mathbb{Z}$ of $Σ(f)$. When the dimension of the manifold $M$ is even we also define an invariant $I(f)$ which is the residue class modulo $4$ of the sum of the number of components of $Σ(f)$, the number of cusps, and twice the number of self-intersection points of $f(Σ)$. Using the cumulative winding number and the invariant $I(f)$, we show that the parity of the number of connected components of $Σ(f)$ does not change under homotopy of $f$ provided that one of the following conditions is satisfied: (i) the dimension of $M$ is even, (ii) the singular set of the homotopy is an orientable manifold, or (iii) the image of the singular set of the homotopy does not have triple self-intersection points.
