Rank-preserving additions for topological vector bundles, after a construction of Horrocks
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913996607586304 |
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| author | Opie, Morgan |
| author_facet | Opie, Morgan |
| contents | We produce group structures on certain sets of topological vector bundles of fixed rank. In particular, we put a group structure on complex rank $2$ bundles on $\mathbb{C}P^3$ with fixed first Chern class. We show that this binary operation coincides with a construction on locally free sheaves due to Horrocks, provided Horrocks' construction is defined. Using similar ideas, we give group structures on certain sets of rank $3$ bundles on $\mathbb{C}P^5$.
These groups arise from the study of relative infinite loop space structures on truncated diagrams. Specifically, we show that the $(2n-2)$-truncation of an $n$-connective map $X\to Y$ with a section is a highly structured group object over the $(2n-2)$-truncation of $Y$. Applying these results to classifying spaces yields the group structures of interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_06963 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rank-preserving additions for topological vector bundles, after a construction of Horrocks Opie, Morgan Algebraic Topology 55R25, 55R35, 55Q05, 55P99, 55P47, We produce group structures on certain sets of topological vector bundles of fixed rank. In particular, we put a group structure on complex rank $2$ bundles on $\mathbb{C}P^3$ with fixed first Chern class. We show that this binary operation coincides with a construction on locally free sheaves due to Horrocks, provided Horrocks' construction is defined. Using similar ideas, we give group structures on certain sets of rank $3$ bundles on $\mathbb{C}P^5$. These groups arise from the study of relative infinite loop space structures on truncated diagrams. Specifically, we show that the $(2n-2)$-truncation of an $n$-connective map $X\to Y$ with a section is a highly structured group object over the $(2n-2)$-truncation of $Y$. Applying these results to classifying spaces yields the group structures of interest. |
| title | Rank-preserving additions for topological vector bundles, after a construction of Horrocks |
| topic | Algebraic Topology 55R25, 55R35, 55Q05, 55P99, 55P47, |
| url | https://arxiv.org/abs/2302.06963 |