2023/02/14 by Morgan Opie, Opie, Morgan
Computer Science · Mathematics · #55P47 #55P99 #55Q05 #55R25 #55R35 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2302.06963
openalex publication_date 2023/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℂP3 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 ℂP5. 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→ 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.