论文标题
持久性和度量图的收缩
Contractions in persistence and metric graphs
论文作者
论文摘要
我们证明，从空间$ x $从其子空间$ a $ a $ a $ a $ a $ a $ a $ a嵌入到$ x $的持续图中的持续图中的$ x $ $ x $的$ 1 $ -lipschitz撤回（收缩）的存在。作为一种工具，我们引入了持久模块的紧密注射，作为诱导上述嵌入的地图。我们表明，收缩始终存在于公制图中的最短循环中，并猜想平面公制图中的收缩存在于最短同源性基础的所有循环中。 主要感兴趣的是对大地空空间中环的收缩。这些充当理想的圆形坐标。此外，正如Adamaszek和Adams的定理描述了$ s^1 $的持久图的模式一样，收缩的$ x \ s^1 $表示相同的模式在$ x $的持久图中出现。
We prove that the existence of a $1$-Lipschitz retraction (a contraction) from a space $X$ onto its subspace $A$ implies the persistence diagram of $A$ embeds into the persistence diagram of $X$. As a tool we introduce tight injections of persistence modules as maps inducing the said embeddings. We show contractions always exist onto shortest loops in metric graphs and conjecture on existence of contractions in planar metric graphs onto all loops of a shortest homology basis. Of primary interest are contractions onto loops in geodesic spaces. These act as ideal circular coordinates. Furthermore, as the Theorem of Adamaszek and Adams describes the pattern of persistence diagram of $S^1$, a contraction $X \to S^1$ implies the same pattern appears in persistence diagram of $X$.
