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

TL;DR

This paper constructs group structures on certain topological vector bundles, extending Horrocks' construction, and explores their algebraic and topological properties using infinite loop space techniques.

Contribution

It introduces new group structures on topological vector bundles of fixed rank, linking algebraic constructions with topological and homotopical methods.

Findings

Group structure on complex rank 2 bundles on CP^3 with fixed first Chern class.

Group structures on certain rank 3 bundles on CP^5.

Connection between these groups and relative infinite loop space structures.

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.