Adams filtrations and secondary coherence for fibers of Steenrod operations
Abstract
Let f:Y→∏λΣdλH𝔽2 be a locally finite family of positive-degree stable operations, where Y=H𝔽2 or Hℤ; in the integral case assume dλ≥2. We determine the Adams filtration of its fiber from the minimal generators of im(f*) as a left Steenrod module. An explicit target automorphism separates a minimal subfamily from the redundant coordinates and gives a noncanonical stable splitting. Combining this reduction with the Bruner--Rognes two-extension theorem and an acyclic cancellation gives E3=E∞. In positive homotopy degrees, the filtration-one subgroup is the image of the dual of the induced map on minimal-generator quotients, and the redundant coordinates account for filtration zero. We also specify the zero-stem filtration and its possible cyclic extension of order four. A terminating Adem recursion realizes the splitting for the full positive even-square family and its finite truncations. Separately, for a crossed bimodule and fixed lifts of the differentials of a free primary complex, we construct a class in the third cohomology of an explicit Hom complex. Its vanishing is equivalent to the existence of coherent null-composite tracks with those fixed differential lifts.
Related
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BibTeX
@misc{phuc2026adams,
title = {{Adams filtrations and secondary coherence for fibers of Steenrod operations}},
author = {Dang Vo Phuc},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.18906},
eprint = {2605.18906},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.18906},
}
RIS
TY - UNPB
TI - Adams filtrations and secondary coherence for fibers of Steenrod operations
AU - Dang Vo Phuc
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2605.18906
UR - https://arxiv.org/abs/2605.18906
AB - Let f:Y→∏_λΣ^d_λH𝔽_2 be a locally finite family of positive-degree stable operations, where Y=H𝔽_2 or Hℤ; in the integral case assume d_λ≥2. We determine the Adams filtration of its fiber from the minimal generators of im(f^*) as a left Steenrod module. An explicit target automorphism separates a minimal subfamily from the redundant coordinates and gives a noncanonical stable splitting. Combining this reduction with the Bruner--Rognes two-extension theorem and an acyclic cancellation gives E_3=E_∞. In positive homotopy degrees, the filtration-one subgroup is the image of the dual of the induced map on minimal-generator quotients, and the redundant coordinates account for filtration zero. We also specify the zero-stem filtration and its possible cyclic extension of order four. A terminating Adem recursion realizes the splitting for the full positive even-square family and its finite truncations. Separately, for a crossed bimodule and fixed lifts of the differentials of a free primary complex, we construct a class in the third cohomology of an explicit Hom complex. Its vanishing is equivalent to the existence of coherent null-composite tracks with those fixed differential lifts.
ER -