Geodesic
Rank-preserving additions for topological vector bundles, after a construction of Horrocks
Opie, Morgan P1
1Department of Mathematics, UCLA, Los Angeles, CA, United States
Algebraic and Geometric Topology, Tome 25 (2025) no. 4, pp. 2451-2476
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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 the latter 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.

DOI : 10.2140/agt.2025.25.2451
Keywords: vector bundles, topological bundles, Horrocks' construction, locally free sheaves, rank 2 bundles, rank 3 bundles, projective spaces

Opie, Morgan P  1

1 Department of Mathematics, UCLA, Los Angeles, CA, United States 
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Opie, Morgan P. Rank-preserving additions for topological vector bundles, after a construction of Horrocks. Algebraic and Geometric Topology, Tome 25 (2025) no. 4, pp. 2451-2476. doi: 10.2140/agt.2025.25.2451

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