A classification of complex rank 3 vector bundles on \(\mathbb{C} P^5\) (Q6594397)

The paper under review studies rank \(3\) topological vector bundles on \(\mathbb{C}P^{5}\). While Chern classes are enough to classify such bundles stably, it turns out that they do not suffice to provide unstable classification.\N\NSometimes Chern classes also provide unstable classification, namely for bundles on \(\mathbb{C} P^{n}\) of rank either \(\leq 1\) or \(\geq n\), but if the rank \(r\) satisfies \(1 < r <n\) there is no uniform answer.\N\NThe author summarizes earlier work on this topic. In particular, for rank \(2\) bundles on \(\mathbb{C}P^{3}\) there is a result of Atiyah and Rees which employs in addition to the Chern classes \(c_1\) and \(c_2\) a \(\mathbb{Z}/2\)-valued invariant \(\alpha\) obtained from a certain twisted characteristic class in \(KO\)-theory.\N\NThe main result of the paper is Theorem 1.4, a corollary of Theorems 1.7, 1.8 and 4.9, which are also of interest on their own.\N\NFirstly, Theorem 1.4 states that, given integers \(a_1\), \(a_2\) and \(a_3\) there exists a rank \(3\) bundle over \(\mathbb{C} P^{5}\) with Chern classes \(c_1\), \(c_2\) and \(c_3\) equal to \(a_1\), \(a_2\) and \(a_3\) respectively, if and only if Schwarzenberg identity \(S_5\) stated explicitly in Lemma 2.17 is fulfilled. Secondly, assuming \(S_5\), it says that depending on mod \(3\) properties of \(c_1\) and \(c_2\), there are \(3\) or \(1\) isomorphism classes of rank \(3\) bundles over \(\mathbb{C} P^{5}\) with fixed Chern classes \(c_1\), \(c_2\) and \(c_3\), and that these Chern classes together with a certain \(\mathbb{Z}/3\)-valued invariant \(\rho\) distinguish the isomorphism classes.\N\NTheorem 1.7, which can be viewed as a refinement of a part of Theorem 1.4 states that a certain geometrically defined action of \(\pi_{10} BU(3) \cong \mathbb{Z}/3\) on the set \(\mathcal{V}_{a_1,a_2,a_3}\) of isomorphism classes of rank \(3\) bundles on \(\mathbb{C} P^{5}\) with the Chern classes \(a_1\), \(a_2\) and \(a_3\) is transitive and describes the cases when the action is free and trivial.\N\NTheorem 1.8 states that there is a twisted characteristic class \(\tilde{\rho}\) in \(\mathrm{tmf}_{(3)}\)-cohomology with a certain property that later enables its use in defining the new \(\mathbb{Z}/3\)-valued invariant \(\rho\). The passage from \(\tilde{\rho}\) to the \(\mathbb{Z}/3\)-valued invariant \(\rho\) is described in Theorem 4.9 and can be viewed as ``untwisting'' of \(\tilde{\rho}\).\N\NTheorem 1.7 is proved in Section 2 by studying the set of homotopy classes of maps \([\mathbb{C}P^5, BU(3)]\) using completion techniques, Postnikov-type decompositions, action of the Steenrod algebra, and action of \(\pi_{10} BU(3) \cong \mathbb{Z}/3\) defined via certain pinch map in Construction 1.6.\N\NTheorem 1.8 is proved in Section 3. The use of \(\mathrm{tmf}_{(3)}\)-twisted cohomology classes in analogy with the above mentioned approach of Atiyah and Rees for rank \(2\) bundles over \(\mathbb{C} P^3\) is motivated in an illuminating discussion in the introduction of the article. The proof itself is in the setting of localization at \(3\) and uses explicit calculations of the low-dimensional homotopy groups (type) of the space \(BU(3)\), spectrum \(\mathrm{tmf}_{(3)}\), various Thom spaces, action of the mod \(3\) Steenrod algebra and spectral sequences.\N\NTheorem 4.9 is stated and proved in Section 4. It comprises a careful analysis of the available orientations to obtain the desired ``untwisting''. Examples are given and an outlook is discussed.\N\NIn total, many classical techniques from homotopy theory are used together with available calculations regarding low-dimensional homotopy of \(\mathrm{tmf}_{(3)}\) and combined in a suitable way. The introduction of the paper provides a good summary of the contents, motivation, and ideas used.