A Hurewicz Theorem for $RO(C_2)$-graded Equivariant Homology Governed by Vector Fields on Spheres
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , , , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866918523208466432 |
|---|---|
| author | Guo, Manyi Li, Guchuan Lu, Yunze Ma, Sihao Wu, Yuchen Xu, Zhouli Yang, Albert Jinghui Zhang, Shangjie |
| author_facet | Guo, Manyi Li, Guchuan Lu, Yunze Ma, Sihao Wu, Yuchen Xu, Zhouli Yang, Albert Jinghui Zhang, Shangjie |
| contents | We determine the $RO(C_2)$-graded Hurewicz images of the $C_2$-equivariant Eilenberg--MacLane spectra $H\underline{\mathbb F_2}$, $H\underline{\mathbb Z}$ and $H\underline{A}$, where $\underline{\mathbb F_2}$ and $\underline{\mathbb Z}$ denote the constant Mackey functors with values in $\mathbb F_2$ and $\mathbb Z$, respectively, and $\underline A$ denotes the Burnside Mackey functor.
Surprisingly, the answer is closely tied to the problem of vector fields on spheres: the element $\fracθ{ρ^kτ^n}$ in the negative cone of the homotopy groups of $H\underline{\mathbb F_2}$ lies in the Hurewicz image if and only if $S^n$ admits $k$ linearly independent vector fields.
Moreover, using the Generalized Leibniz Rule and the Generalized Mahowald Trick introduced by arXiv:2412.10879, we show that there are nonzero Adams differentials of arbitrary length supported by filtration-$0$ elements in the genuine $C_2$-equivariant Adams spectral sequence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_26334 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Hurewicz Theorem for $RO(C_2)$-graded Equivariant Homology Governed by Vector Fields on Spheres Guo, Manyi Li, Guchuan Lu, Yunze Ma, Sihao Wu, Yuchen Xu, Zhouli Yang, Albert Jinghui Zhang, Shangjie Algebraic Topology We determine the $RO(C_2)$-graded Hurewicz images of the $C_2$-equivariant Eilenberg--MacLane spectra $H\underline{\mathbb F_2}$, $H\underline{\mathbb Z}$ and $H\underline{A}$, where $\underline{\mathbb F_2}$ and $\underline{\mathbb Z}$ denote the constant Mackey functors with values in $\mathbb F_2$ and $\mathbb Z$, respectively, and $\underline A$ denotes the Burnside Mackey functor. Surprisingly, the answer is closely tied to the problem of vector fields on spheres: the element $\fracθ{ρ^kτ^n}$ in the negative cone of the homotopy groups of $H\underline{\mathbb F_2}$ lies in the Hurewicz image if and only if $S^n$ admits $k$ linearly independent vector fields. Moreover, using the Generalized Leibniz Rule and the Generalized Mahowald Trick introduced by arXiv:2412.10879, we show that there are nonzero Adams differentials of arbitrary length supported by filtration-$0$ elements in the genuine $C_2$-equivariant Adams spectral sequence. |
| title | A Hurewicz Theorem for $RO(C_2)$-graded Equivariant Homology Governed by Vector Fields on Spheres |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2605.26334 |