论文标题
三种口味的弦拓扑
String topology in three flavours
论文作者
论文摘要
我们从几何和代数的角度描述了两个主要的弦拓扑操作，即Chas-Sullivan产品和Goresky-Hhingston Coproduct。几何结构使用Thom-Pontrjagin相交理论，而代数结构则以Hochschild同源性为例。我们通过几何交点对镜头空间上的产品和相关的计算进行计算，并推断出相关的区分3维透镜空间。在代数上，我们描述了这些操作在泰特 - 霍奇柴尔德综合体上共同定义的结构。我们使用有理同义理论方法来勾勒出简单连接的流形和真实系数的几何定义和代数定义之间的等效性，从而强调了配置空间的作用。最后，我们在代数和几何上研究操作的不变特性。
We describe two major string topology operations, the Chas-Sullivan product and the Goresky-Hingston coproduct, from geometric and algebraic perspectives. The geometric construction uses Thom-Pontrjagin intersection theory while the algebraic construction is phrased in terms of Hochschild homology. We give computations of products and coproducts on lens spaces via geometric intersection, and deduce that the coproduct distinguishes 3-dimensional lens spaces. Algebraically, we describe the structure these operations define together on the Tate-Hochschild complex. We use rational homotopy theory methods to sketch the equivalence between the geometric and algebraic definitions for simply connected manifolds and real coefficients, emphasizing the role of configuration spaces. Finally, we study invariance properties of the operations, both algebraically and geometrically.
