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We produce group structures on certain sets of topological vector bundles offixed rank. In particular, we put a group structure on complex rank $2$ bundleson $\mathbb{C}P^3$ with fixed first Chern class. We show that this binaryoperation coincides with a construction on locally free sheaves due toHorrocks, provided Horrocks' construction is defined. Using similar ideas, wegive 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 structureson 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 objectover the $(2n-2)$-truncation of $Y$. Applying these results to classifyingspaces yields the group structures of interest.

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