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Complete RO-Graded Tambara Structure of Hunderlineℤ for cyclic groups

  • Dongming Zhang
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Abstract

We compute the RO-graded Tambara functor of Hunderlineℤ for every finite cyclic group. We introduce a valuation language for homotopy groups: for prime-power cyclic group equivariant, classes are represented by compatible vertex arrays or by edge arrays. In these coordinates we compute products, restrictions, transfers, Weyl actions, and norms. Then we assemble the results into arbitrary finite cyclic group equivariant case. Besides the computation, we also develop the Tate compression theory, which is used in the computation. The Tate compression diagram cut an equivariant spectra into several layer, which the homotopy group can be describe directly by each layers. From it we also give a filtration for π⋆(Hunderlineℤ), compatible with products, restrictions, transfers, Weyl actions, and norms.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026complete,
  title = {{Complete $RO$-Graded Tambara Structure of $H\underline\mathbb Z$ for cyclic groups}},
  author = {Dongming Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37885},
  eprint = {2610.37885},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37885},
}
RIS
TY  - UNPB
TI  - Complete RO-Graded Tambara Structure of Hunderlineℤ for cyclic groups
AU  - Dongming Zhang
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37885
UR  - https://arxiv.org/abs/2610.37885
AB  - We compute the RO-graded Tambara functor of Hunderlineℤ for every finite cyclic group. We introduce a valuation language for homotopy groups: for prime-power cyclic group equivariant, classes are represented by compatible vertex arrays or by edge arrays. In these coordinates we compute products, restrictions, transfers, Weyl actions, and norms. Then we assemble the results into arbitrary finite cyclic group equivariant case. Besides the computation, we also develop the Tate compression theory, which is used in the computation. The Tate compression diagram cut an equivariant spectra into several layer, which the homotopy group can be describe directly by each layers. From it we also give a filtration for π_⋆(Hunderlineℤ), compatible with products, restrictions, transfers, Weyl actions, and norms.
ER  -