Enumerating stably trivial vector bundles with higher real $K$-theory
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914707599785984 |
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| author | Chatham, Hood Hu, Yang Opie, Morgan |
| author_facet | Chatham, Hood Hu, Yang Opie, Morgan |
| contents | Given positive integers $r$ and $c$, let $ϕ(r,c)$ denote the number of isomorphism classes of complex rank $r$ topological vector bundles on $\mathbb{CP}^{r+c}$ that are stably trivial. We compute the $p$-adic valuation of the number $ϕ(r,c)$ for all pairs $r$ and $c$ such that $c \leq \operatorname{min}\{r,2p-3\}$. We also give some systematic lower bounds for $p$-divisibility of $ϕ(r,c)$ when $c<2p^2-p-2$, and detect some nontrivial $p$-divisibility for larger $c$. As an additional application of our methods, we find new $p$-torsion in unstable homotopy groups of unitary groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04733 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Enumerating stably trivial vector bundles with higher real $K$-theory Chatham, Hood Hu, Yang Opie, Morgan Algebraic Topology 55R25, 55N20, 55R15 Given positive integers $r$ and $c$, let $ϕ(r,c)$ denote the number of isomorphism classes of complex rank $r$ topological vector bundles on $\mathbb{CP}^{r+c}$ that are stably trivial. We compute the $p$-adic valuation of the number $ϕ(r,c)$ for all pairs $r$ and $c$ such that $c \leq \operatorname{min}\{r,2p-3\}$. We also give some systematic lower bounds for $p$-divisibility of $ϕ(r,c)$ when $c<2p^2-p-2$, and detect some nontrivial $p$-divisibility for larger $c$. As an additional application of our methods, we find new $p$-torsion in unstable homotopy groups of unitary groups. |
| title | Enumerating stably trivial vector bundles with higher real $K$-theory |
| topic | Algebraic Topology 55R25, 55N20, 55R15 |
| url | https://arxiv.org/abs/2403.04733 |