Motivic Homotopy Groups of Spheres and Free Summands of Stably Free Modules
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917320307245056 |
|---|---|
| author | Gant, Sebastian Williams, Ben |
| author_facet | Gant, Sebastian Williams, Ben |
| contents | Working over an algebraically closed field $k$ of characteristic $0$, we show that the motivic stable homotopy groups of the sphere spectrum can be determined entirely from the motivic homotopy groups of the $p$-completed sphere spectra and the motivic cohomology of the ground field, except possibly for the $0$ and $-1$-stems. Using this, we show that the complex realization maps from the motivic homotopy groups to the classical stable homotopy groups are isomorphisms in a range of bidegrees. We apply this to deduce that complex realization also induces isomorphisms on unstable homotopy groups for Stiefel varieties $V_r(\mathbb{A}^n_k)$ in a range of bidegrees. We use this to determine when the projection map $V_r(\mathbb{A}^n_k) \to V_1(\mathbb{A}^n_k)$ admits a right inverse, settling the question of when the universal stably-free module of type $(n,n-1)$ admits a free summand of given rank. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10033 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Motivic Homotopy Groups of Spheres and Free Summands of Stably Free Modules Gant, Sebastian Williams, Ben Algebraic Topology Commutative Algebra 14F42 (Primary), 13C10, 14J60, 57R22 (Secondary) Working over an algebraically closed field $k$ of characteristic $0$, we show that the motivic stable homotopy groups of the sphere spectrum can be determined entirely from the motivic homotopy groups of the $p$-completed sphere spectra and the motivic cohomology of the ground field, except possibly for the $0$ and $-1$-stems. Using this, we show that the complex realization maps from the motivic homotopy groups to the classical stable homotopy groups are isomorphisms in a range of bidegrees. We apply this to deduce that complex realization also induces isomorphisms on unstable homotopy groups for Stiefel varieties $V_r(\mathbb{A}^n_k)$ in a range of bidegrees. We use this to determine when the projection map $V_r(\mathbb{A}^n_k) \to V_1(\mathbb{A}^n_k)$ admits a right inverse, settling the question of when the universal stably-free module of type $(n,n-1)$ admits a free summand of given rank. |
| title | Motivic Homotopy Groups of Spheres and Free Summands of Stably Free Modules |
| topic | Algebraic Topology Commutative Algebra 14F42 (Primary), 13C10, 14J60, 57R22 (Secondary) |
| url | https://arxiv.org/abs/2510.10033 |