A classification of complex rank 3 vector bundles on complex projective 5-space
Morgan Opie

TL;DR
This paper classifies complex rank 3 vector bundles on complex projective 5-space, revealing that Chern classes alone do not distinguish all bundles and introducing a new invariant based on topological modular forms.
Contribution
It provides a complete classification of rank 3 bundles on CP^5 satisfying the Schwarzenberger condition and introduces a novel invariant to distinguish bundles with identical Chern classes.
Findings
Number of isomorphism classes depends on divisibility of Chern classes by 3.
Chern classes are incomplete invariants for these bundles.
A new Z/3-valued invariant distinguishes bundles with same Chern classes.
Abstract
Given integers , there is a complex rank topological bundle on with -th Chern class equal to if and only if satisfy the Schwarzenberger condition. Provided that the Schwarzenberger condition is satisfied, we prove that the number of isomorphism classes of rank bundles on with is equal to if and are both divisible by and equal to otherwise. This shows that Chern classes are incomplete invariants of topological rank bundles on . To address this problem, we produce a universal class in the -cohomology of a Thom spectrum related to , where denotes topological modular forms localized at . From this class and orientation data, we construct a -valued invariant of the bundles of interest and prove that our invariant separates…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Geometry and complex manifolds · Advanced Algebra and Geometry
