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Enumerating complex rank $n$ vector bundles on $\mathbb CP^{n+1}$

Morgan P. Opie

Abstract

We enumerate complex rank $n$ topological vector bundles on $\mathbb CP^{n+1}$ with prescribed Chern classes. This extends work of Atiyah and Rees in the case $n=2$ and work of Hu in the case that all Chern classes are zero.

Enumerating complex rank $n$ vector bundles on $\mathbb CP^{n+1}$

Abstract

We enumerate complex rank topological vector bundles on with prescribed Chern classes. This extends work of Atiyah and Rees in the case and work of Hu in the case that all Chern classes are zero.

Paper Structure

This paper contains 10 sections, 22 theorems, 46 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Other related work
  3. Outline
  4. Notations and conventions
  5. Acknowledgements
  6. The Postnikov tower for $BU(n)$ through dimension $2n+2$
  7. The Moore--Postnikov tower for $BU(n) \to BU$
  8. The number of representatives for a given stable bundle on $\mathbb CP^{n+1}$ class with vanishing $n$-th Chern class
  9. Extending vector bundles on $\mathbb CP^{n+2}$
  10. The Schwarzenberger conditions

Key Result

Theorem 1.3

Let $n \geq 1$ and let $\vec{a}=(a_1,\ldots ,a_n) \in \mathbb Z^n$. Then:

Figures (2)

  • Figure 1: Multiplicative generators for the $E_2$-page are indicated, as well as the class $c_1\iota_{2n+1}$ since it is significant.
  • Figure 2: Multiplicative generators for the $E_2$-page are indicated, as well as the classes $c_1\iota_9$ and $c_1\iota_9'$.

Theorems & Definitions (38)

  • Definition 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 3.1
  • Definition 3.2
  • Proposition 3.3: MT
  • Lemma 3.4
  • ...and 28 more