[论文解读] Coboundary and Cosystolic Expansion Without Dependence on Dimension or Degree

Yotam Dikstein, Irit Dinur|arXiv (Cornell University)|Apr 4, 2023

Geometric and Algebraic Topology被引用 2

一句话总结

本文為高維擴展圖（包括同構幾何格的序複形及LSV/KO複形）建立了與維度和次數無關的共胞腔與上胞腔擴展界。透過引入一種新型的色彩限制技術及局部到整體定理的譜證明，作者獲得與環境維度、次數和係數群無關的絕對擴展常數，進而得到更強的拓撲重疊與覆蓋穩定性界。

ABSTRACT

We give new bounds on the cosystolic expansion constants of several families of high dimensional expanders, and the known coboundary expansion constants of order complexes of homogeneous geometric lattices, including the spherical building of $SL_n(F_q)$. The improvement applies to the high dimensional expanders constructed by Lubotzky, Samuels and Vishne, and by Kaufman and Oppenheim. Our new expansion constants do not depend on the degree of the complex nor on its dimension, nor on the group of coefficients. This implies improved bounds on Gromov's topological overlap constant, and on Dinur and Meshulam's cover stability, which may have applications for agreement testing. In comparison, existing bounds decay exponentially with the ambient dimension (for spherical buildings) and in addition decay linearly with the degree (for all known bounded-degree high dimensional expanders). Our results are based on several new techniques: * We develop a new "color-restriction" technique which enables proving dimension-free expansion by restricting a multi-partite complex to small random subsets of its color classes. * We give a new "spectral" proof for Evra and Kaufman's local-to-global theorem, deriving better bounds and getting rid of the dependence on the degree. This theorem bounds the cosystolic expansion of a complex using coboundary expansion and spectral expansion of the links. * We derive absolute bounds on the coboundary expansion of the spherical building (and any order complex of a homogeneous geometric lattice) by constructing a novel family of very short cones.

研究动机与目标

• 消除高維擴展圖共胞腔與上胞腔擴展常數對維度與次數的依賴。
• 利用與維度無關的擴展改進格羅莫夫拓撲重疊常數與迪努爾-梅舒拉姆覆蓋穩定性的界。
• 發展能產生與環境維度、次數和係數群無關的絕對擴展界之技術。
• 確立已知有界次數的高維擴展圖（如LSV與KO複形）可獲得改進的擴展與穩定性保證。
• 證明球面建築與其他同構格的序複形可透過新型短圓錐獲得絕對上胞腔擴展界。

提出的方法

• 引入一種新的「色彩限制」技術，透過將多分圖限制於隨機色彩類子集來證明與維度無關的擴展。
• 提供埃夫拉與考夫曼之局部到整體定理的新型譜證明，消除次數依賴並改善界。
• 構造一類新型極短圓錐，以推導同構幾何格序複形的絕對上胞腔擴展界。
• 應用改進的擴展界，推導格羅莫夫拓撲重疊常數與覆蓋穩定性的更緊界。
• 利用局部到整體定理，將鏈的局部譜擴展與全局共胞腔擴展關聯，並改善其量化依賴關係。
• 利用幾何格序複形的結構，構造顯式鏈並分析其在上胞腔映射下的最小權重。

实验结果

研究问题

• RQ1 共胞腔擴展常數能否在不依賴高維複形之維度與次數的情況下被界定？
• RQ2 局部到整體定理在共胞腔擴展中能否以改進的次數與維度依賴關係重新證明？
• RQ3 同構幾何格的序複形（如SLn(Fq)的球面建築）是否具有絕對上胞腔擴展界？
• RQ4 改進的共胞腔擴展界能否帶來拓撲重疊與覆蓋穩定性更強的保證？
• RQ5 複形的哪些結構性質能促成短圓錐的構造，進而產生絕對擴展界？

主要发现

• 本文確立LSV與KO複形的共胞腔擴展常數 $ h_k(X, \bbF_2) $ 有下界 $ \exp(-O(k^6 \log k)) $，且與維度和次數無關。
• 對於 $ \mathrm{SL}_n(\bbF_q) $ 的球面建築，作者透過一類新型短圓錐推導出與維度和係數群無關的絕對上胞腔擴展界。
• LSV複形的拓撲重疊常數改進為 $ c = \exp(-O(k^7 \log k)) - \varepsilon \cdot \exp(O(k^7 \log k)) $，其中 $ \varepsilon = 1/|X_n(0)| $，且與次數和維度無關。
• KO複形的2-骨架被證明具有 $ \Omega(1) $-拓撲重疊，顯著優於以往的次數依賴界。
• LSV與KO複形的覆蓋穩定性被證明有下界絕對常數 $ c > 0 $，且與群和集合大小無關。
• 作者實現了局部到整體定理的譜證明，完全消除次數依賴，改善先前界，並消除了高維擴展理論中的一個關鍵瓶頸。