定量球面化与Gromov线性配边问题
Quantitative asphericalization and Gromov's linear bordism problem
- Kangnam University（康南大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明Gromov线性猜想的PL配边版本对广泛球面化群成立，并利用定量球面化方法将PL (4k-1)-流形的$L^2$ $\rho$-不变量界从超阶乘增长改进为阶乘增长。
AI中文摘要：
我们证明了Gromov线性猜想的一个PL配边版本，该版本适用于所有维度中具有忠实表示的流形上的一类广泛的球面化群。由于每个群都能嵌入到一个球面化群中，因此在允许目标群扩大后，该猜想的配边版本成立。我们的结果同时适用于有向和无向情形。在PL范畴中，不需要有界局部几何。作为应用，我们获得了与万有覆盖相关的PL (4k-1)-流形的Cheeger-Gromov $L^2$ $\rho$-不变量的改进线性界，将先前维度上的超阶乘增长替换为阶乘增长。证明基于一种定量球面化方法，该方法能显式控制所得配边的复杂度。
英文摘要：
We prove a PL bordism version of Gromov's linearity conjecture over a broad class of asphericalization groups for manifolds endowed with faithful representations in all dimensions. Since every group embeds into an asphericalization group, this bordism version of the conjecture holds after allowing the target group to be enlarged. Our results apply in both the oriented and unoriented settings. In the PL category, no bounded local geometry is required. As an application, we obtain improved linear bounds for the Cheeger--Gromov $L^2$ $ρ$-invariants of PL $(4k-1)$-manifolds associated with their universal covers, replacing the previous superfactorial growth in dimension by factorial growth. The proofs are based on a quantitative asphericalization method that provides explicit control of the complexity of the resulting bordisms.
