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Motivic Steenrod algebra and Thom obstructions without desingularization

  • Jiahao Hu
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Abstract

We give a desingularization-free proof of Voevodsky's identification of bistable mod-ℓ motivic cohomology operations with the motivic Steenrod algebra over fields of characteristic zero. This supplies the direct argument anticipated by Voevodsky in his study of motivic Eilenberg--MacLane spaces. As an application, we study Thom's topological obstructions to desingularization, which arose in his work on the Steenrod realization problem. Without invoking desingularization, we prove that the primary and all higher Thom obstructions vanish on complex algebraic cycles. The proof uses the motivic bidegrees of the successive k-invariants in the Brown--Peterson tower to show that every lifting obstruction vanishes.

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Cite

BibTeXRIS
BibTeX
@misc{hu2026motivic,
  title = {{Motivic Steenrod algebra and Thom obstructions without desingularization}},
  author = {Jiahao Hu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14128},
  eprint = {2609.14128},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14128},
}
RIS
TY  - UNPB
TI  - Motivic Steenrod algebra and Thom obstructions without desingularization
AU  - Jiahao Hu
PY  - 2026
DA  - 2026/09/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.14128
UR  - https://arxiv.org/abs/2609.14128
AB  - We give a desingularization-free proof of Voevodsky's identification of bistable mod-ℓ motivic cohomology operations with the motivic Steenrod algebra over fields of characteristic zero. This supplies the direct argument anticipated by Voevodsky in his study of motivic Eilenberg--MacLane spaces. As an application, we study Thom's topological obstructions to desingularization, which arose in his work on the Steenrod realization problem. Without invoking desingularization, we prove that the primary and all higher Thom obstructions vanish on complex algebraic cycles. The proof uses the motivic bidegrees of the successive k-invariants in the Brown--Peterson tower to show that every lifting obstruction vanishes.
ER  -