Semifree Cochain Models for Fibrations Revisited
Abstract
We construct cochain models for Serre fibrations over path-connected bases with coefficients in an arbitrary field. Our construction refines Brown's twisted tensor product to produce a differential graded module model over the singular cochain algebra of the base. If the homology of the fibre is finite dimensional in each degree and on which the action of the fundamental group of the base is degreewise nilpotent, then this model is a semifree resolution generated by the cohomology of the fibre. This extends the classical construction for simply connected bases. As applications, we compare the resulting model with arbitrary semifree resolutions, showing that every such resolution recovers the cohomology of the fibre after base change to the basepoint, and obtain an explicit bound on the page at which an associated spectral sequence collapses.
Related
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BibTeX
@misc{guo2026semifree,
title = {{Semifree Cochain Models for Fibrations Revisited}},
author = {Yanning Guo and Ruizhi Huang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07769},
eprint = {2610.07769},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07769},
}
RIS
TY - UNPB
TI - Semifree Cochain Models for Fibrations Revisited
AU - Yanning Guo
AU - Ruizhi Huang
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07769
UR - https://arxiv.org/abs/2610.07769
AB - We construct cochain models for Serre fibrations over path-connected bases with coefficients in an arbitrary field. Our construction refines Brown's twisted tensor product to produce a differential graded module model over the singular cochain algebra of the base. If the homology of the fibre is finite dimensional in each degree and on which the action of the fundamental group of the base is degreewise nilpotent, then this model is a semifree resolution generated by the cohomology of the fibre. This extends the classical construction for simply connected bases. As applications, we compare the resulting model with arbitrary semifree resolutions, showing that every such resolution recovers the cohomology of the fibre after base change to the basepoint, and obtain an explicit bound on the page at which an associated spectral sequence collapses.
ER -