Complex rank 3 vector bundles on complex projective 5-space

Abstract

This work concerns two aspects of the study of complex rank $3$ topological vector bundles on complex projective five-space. The first aim is to classify such bundles: to give complete, computable algebraic invariants. This is a nontrivial project because classical invariants like Chern classes do not uniquely determine such bundles. Our classification strategy is motivated by the classical results of Atiyah and Rees for complex rank $2$ topological bundles on $\mathbb{C}P^3$: Atiyah and Rees show that these bundles are determined by their Chern classes, together with an additional $\mathbb{Z}/2$-invariant which should be understood as arising from a twisted version of real topological $K$-theory. We re-examine Atiyah and Rees' approach and excise specialized geometry which does not generalize to higher rank bundles or higher-dimensional spaces. From this perspective, certain algebraic analogies emerge that allow for a similar approach to complex rank $3$ topological bundles on $\CP^5$. Indeed, we show that complex rank $3$ topological vector bundles on $\mathbb{C}P^5$ are determined by their Chern classes together with a $\mathbb{Z}/3$-valued invariant, which arises from a universal invariant valued in twisted, $3$-local $tmf$-cohomology (where $tmf$ is the spectrum of topological modular forms). The second aim of this thesis is to concretely construct interesting topological bundles of complex rank $3$ on $\mathbb{C}P^5$. While our classification results predict the number of complex rank $3$ topological bundles on $\CP^5$ with prescribed Chern classes, and offer an invariant to distinguish bundles with the same Chern classes, they are inexplicit. Ideally, one would like a procedure to realize all topological equivalence classes by concrete bundles. A prototypical construction is given by results of Horrocks, which produce algebraic representatives for all topological equivalence classes of complex rank $2$ bundles on $\mathbb{C}P^3$. Due to the complexity of higher-codimension subschemes of complex projective spaces, this algebraic construction does not admit an obvious generalization. However, we show that it can be reinterpreted through a homotopical lens, where it admits a broadly applicable generalization. In a particular application of this machinery, we produce a homotopical method for constructing new rank $3$ bundles on $\mathbb{C}P^5$, using an explicit construction on classifying spaces. We show that iteratively applying this construction to rank $3$ sums of line bundles produces new rank $3$ bundles that are not themselves sums of line bundles, demonstrating that our methods allow for the construction of interesting bundles from simple ones.