实代数曲线模空间的拓扑
On the topology of moduli spaces of real algebraic curves
摘要：
我们证明了与所有类型的实代数曲线相关的映射类群都是虚对偶群。我们还得到了实代数曲线模空间的轨道同伦群的一些结果。我们实现这些结果通过定义一个新的复杂的不一定可定向曲面的边界称为$\mathcal{A}\mathcal{B}\mathcal{C}$ -复杂。该复合体编码了基本简单弧、边界分量和基本简单闭曲线的合痕类的相交模式。我们表明，这个复杂的同伦等价于一个无限的楔形球，它的尺寸是依赖于表面的拓扑型。
英文摘要：
We show that mapping class groups associated to all types of real algebraic curves are virtual duality groups. We also deduce some results about the orbifold homotopy groups of the moduli spaces of real algebraic curves. We achieve these results by defining a new complex associated to a not necessarily orientable surface with boundary called the $\mathcal{A}\mathcal{B}\mathcal{C}$-complex. This complex encodes the intersection patterns of isotopy classes of essential simple arcs, boundary components, and essential simple closed curves. We show that this complex is homotopy equivalent to an infinite wedge of spheres; its dimension is dependent on the topological type of the surface.
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