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Enumerating stably trivial vector bundles with higher real $K$-theory

Hood Chatham, Yang Hu, Morgan Opie

Abstract

Given positive integers $r$ and $c$, let $φ(r,c)$ denote the number of isomorphism classes of complex rank $r$ topological vector bundles on $\mathbb{CP}^{r+c}$ that are stably trivial. We compute the $p$-adic valuation of the number $φ(r,c)$ for all pairs $r$ and $c$ such that $c \leq \operatorname{min}\{r,2p-3\}$. We also give some systematic lower bounds for $p$-divisibility of $φ(r,c)$ when $c<2p^2-p-2$, and detect some nontrivial $p$-divisibility for larger $c$. As an additional application of our methods, we find new $p$-torsion in unstable homotopy groups of unitary groups.

Enumerating stably trivial vector bundles with higher real $K$-theory

Abstract

Given positive integers and , let denote the number of isomorphism classes of complex rank topological vector bundles on that are stably trivial. We compute the -adic valuation of the number for all pairs and such that . We also give some systematic lower bounds for -divisibility of when , and detect some nontrivial -divisibility for larger . As an additional application of our methods, we find new -torsion in unstable homotopy groups of unitary groups.
Paper Structure (27 sections, 49 theorems, 140 equations)

This paper contains 27 sections, 49 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Methods
  4. Paper outline
  5. Conventions
  6. Acknowledgements
  7. Background: Weiss calculus, completion, and higher real K-theories
  8. Weiss calculus and metastable vector bundles
  9. Stable homotopy theory completed at p
  10. Higher real K-theories of height p−1 at a prime p
  11. EO-modules and algebraic EO-theory
  12. First calculations
  13. Calculations at corank p−1
  14. Calculations at corank at most 2p−3
  15. Corank c=2p−2
  16. ...and 12 more sections

Key Result

Theorem 1.3

Let $n> 2$ be an integer. Let ${\mathop{\mathrm{Vect}}\nolimits }_r^0(\mathbb{CP}_{}^{n})$ denote the set of isomorphism classes of stably trivial rank $r$ bundles on $\mathbb{CP}_{}^{n}$. Let $\mathbb{CP}_{r}^{n}$ denote the cofiber of the skeletal inclusion map $\mathbb{CP}_{}^{r-1}\to \mathbb{CP} where ${{\pi_0}{\mathbb S}\text{-}\!\operatorname{Mod}}\left(- ,- \right)$ denotes stable homotopy

Theorems & Definitions (89)

  • Remark 1.2
  • Theorem 1.3: Theorem 2.1, Hu; see \ref{['thm:Yang_main']}
  • Definition 1.4
  • Remark 1.5
  • Theorem 1.7: See \ref{['cor:bundle_ct1']}
  • Theorem 1.8: See \ref{['prop:surjective']}
  • Corollary 1.9: See \ref{['cor:p2']}
  • Proposition 1.10: See \ref{['cor:KO_ah', 'example:bel-shim']}
  • Proposition 1.11: See \ref{['cor:exists_Xi_filtration']}
  • Theorem 2.1: Arone02, Theorem 2 and Theorem 4
  • ...and 79 more