PreprintCC BY 4.0arXiv
Pullback metrics and homological obstructions to BLD remetrization of branched covers
Abstract
We characterize admissible BLD remetrizations of branched self-covers of spheres by linear local contractibility of their canonical pullback length metrics. A relative homology obstruction involving the total degree of an inverse image component yields uniform porosity of branch values. Explicit degree-two cusp covers of 𝕊n, n≥3, have uniformly porous branch values of Hausdorff dimension n−2 but admit no compatible Ahlfors n-regular, linearly locally contractible BLD source metric. Short homologically essential loops provide the obstruction, showing that porosity is necessary but insufficient for admissibility. We also prove quasisymmetric invariance and sharp degree dependence of porosity for sphere-to-sphere covers in dimension two.
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BibTeX
@misc{zhong2026pullback,
title = {{Pullback metrics and homological obstructions to BLD remetrization of branched covers}},
author = {Deguang Zhong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01087},
eprint = {2610.01087},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01087},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Pullback metrics and homological obstructions to BLD remetrization of branched covers
AU - Deguang Zhong
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01087
UR - https://arxiv.org/abs/2610.01087
AB - We characterize admissible BLD remetrizations of branched self-covers of spheres by linear local contractibility of their canonical pullback length metrics. A relative homology obstruction involving the total degree of an inverse image component yields uniform porosity of branch values. Explicit degree-two cusp covers of 𝕊^n, n≥3, have uniformly porous branch values of Hausdorff dimension n−2 but admit no compatible Ahlfors n-regular, linearly locally contractible BLD source metric. Short homologically essential loops provide the obstruction, showing that porosity is necessary but insufficient for admissibility. We also prove quasisymmetric invariance and sharp degree dependence of porosity for sphere-to-sphere covers in dimension two.
ER -