Metastable complex vector bundles over complex projective spaces

TL;DR

This paper uses Weiss calculus to explicitly compute the counts of certain topological complex vector bundles with vanishing Chern classes over complex projective spaces, revealing intricate periodic patterns.

Contribution

It introduces a novel application of Weiss calculus to determine exact numbers of topological bundles over complex projective spaces, providing explicit formulas and patterns.

Findings

Number of rank n-2 bundles with vanishing Chern classes over CP^n for n>3

Number of rank n-1 bundles with vanishing Chern classes over CP^n for n>2

Identifies periodic patterns in bundle counts based on n modulo 24

Abstract

We apply Weiss calculus to compute the number of topological complex vector bundles of rank $n-2$ with vanishing Chern classes over $\mathbb{C}P^n$ for $n>3$, as given by the list $1, 1, 12, 2, 1, 3, 2, 2, 3, 1, 4, 6, 1, 1, 6, 2, 1, 3, 4, 2, 3, 1, 2, 6$, where the $i$-th entry in this list is the number of such bundles whenever $n$ is congruent to $i$ modulo $24$, starting with $i = 0$. Similarly, the number of rank $n-1$ bundles with vanishing Chern classes over $\mathbb{C}P^n$ for $n>2$ is $2$ when $n$ is odd and $1$ when $n$ is even.