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

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 $\mathbb{C}P^3$ 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.