Rank-preserving additions for topological vector bundles, after a construction of Horrocks

Opie, Morgan P

doi:10.2140/agt.2025.25.2451

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Rank-preserving additions for topological vector bundles, after a construction of Horrocks

Morgan P Opie

Algebraic & Geometric Topology 25 (2025) 2451–2476

DOI: 10.2140/agt.2025.25.2451

Abstract

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 $ℂ P^{3}$ with fixed first Chern class. We show that this binary operation coincides with a construction on locally free sheaves due to Horrocks, provided the latter is defined. Using similar ideas, we give group structures on certain sets of rank $3$ bundles on $ℂ P^{5}$ .

These groups arise from the study of relative infinite loop space structures on truncated diagrams. Specifically, we show that the $(2 n - 2)$ -truncation of an $n$ -connective map $X \to Y$ with a section is a highly structured group object over the $(2 n - 2)$ -truncation of $Y$ . Applying these results to classifying spaces yields the group structures of interest.

Keywords

vector bundles, topological bundles, Horrocks' construction, locally free sheaves, rank 2 bundles, rank 3 bundles, projective spaces

Mathematical Subject Classification

Primary: 55P99, 55R25, 55R35

Secondary: 55P47, 55Q05

References

Publication

Received: 6 November 2023

Revised: 30 November 2023

Accepted: 27 December 2023

Published: 11 August 2025

Authors

Morgan P Opie
Department of Mathematics UCLA Los Angeles, CA United States
https://www.math.ucla.edu/~mopie/

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Volume 25, issue 4 (2025)