等变同伦交错与持久Whitehead定理
Equivariant Homotopy Interleaving and Persistent Whitehead Theorem
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中文总结 AI 辅助
本文针对有限群作用的持久空间，定义了$G$-同伦交错距离并证明其普适性，建立了等变持久Whitehead定理与神经引理，最终导出等变弱大数定律。
中文摘要 AI 辅助
设$G$为有限群。本文发展了持久$G$-空间的等变持久同伦理论的一些结果。我们引入了$G$-稳定和$G$-同伦不变距离的概念，并定义了$G$-同伦交错距离$d^G_{HI}$，它是Blumberg和Lesnick的同伦交错距离的等变类比。我们证明了该距离$d^G_{HI}$是$G$-稳定的和$G$-同伦不变的，并通过证明它支配任何此类距离来建立其普适性。我们进一步证明了等变持久Whitehead定理的一个版本，该定理将等变持久同伦群的交错与持久$G$-空间的$G$-同伦交错联系起来。我们还证明了持久$G$-空间的等变好覆盖的等变持久神经引理。作为推论，我们获得了过滤的等变弱大数定律，为具有有限对称群的随机过滤提供了一个同伦论一致性结果。
英文摘要
Let $G$ be a finite group. In this article, we develop some results of equivariant persistent homotopy theory for persistent $G$-spaces. We introduce the notions of $G$-stable and $G$-homotopy invariant distances and define the $G$-homotopy interleaving distance, an equivariant analogue $d^G_{HI}$ of the homotopy interleaving distance of Blumberg and Lesnick. We prove that this distance $d^G_{HI}$ is $G$-stable and $G$-homotopy invariant and establish its universality by showing that it dominates any such distance. We further prove a version of equivariant persistent Whitehead theorem relating interleavings of equivariant persistent homotopy groups to $G$-homotopy interleavings of persistent $G$-spaces. We also prove an equivariant persistent nerve lemma for equivariant good covers of persistent $G$-spaces. As a consequence, we obtain an equivariant weak law of large numbers for filtrations, providing a homotopy-theoretic consistency result for random filtrations equipped with finite group of symmetries.
发表机构
- Indian Institute of Technology Kanpur（坎普尔印度理工学院）
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