Quantitative asphericalization and Gromov's linear bordism problem
Abstract
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 L2 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.
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BibTeX
@misc{lim2026quantitative,
title = {{Quantitative asphericalization and Gromov's linear bordism problem}},
author = {Geunho Lim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25116},
eprint = {2609.25116},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25116},
}
RIS
TY - UNPB
TI - Quantitative asphericalization and Gromov's linear bordism problem
AU - Geunho Lim
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.25116
UR - https://arxiv.org/abs/2609.25116
AB - 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.
ER -