Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Real-global equivariant Segal--Becker splitting, explicit Brauer induction, and global Adams operations

Stefan Schwede

Abstract

We prove a splitting result in global equivariant homotopy theory that is a simultaneous refinement of the Segal--Becker splitting and its `Real' and equivariant generalizations, and of the explicit Brauer induction of Boltje and Symonds. We show that the morphism of ultra-commutative Real-global ring spectra from $Σ^\infty_+ B_{\text{gl}}U(1)$ to the Real-global K-theory spectrum that classifies the tautological Real $U(1)$-representation admits a section on underlying Real-global infinite loop spaces. We prove that this global Segal--Becker splitting induces the classical Segal--Becker splittings on equivariant cohomology theories, and that it induces the Boltje--Symonds explicit Brauer induction on equivariant homotopy groups. As an application we rigidify the unstable Adams operations in Real-equivariant K-theory to global self-maps of the Real-global space $\mathbf{BUP}$.

A Real-global equivariant Segal--Becker splitting, explicit Brauer induction, and global Adams operations

Abstract

We prove a splitting result in global equivariant homotopy theory that is a simultaneous refinement of the Segal--Becker splitting and its `Real' and equivariant generalizations, and of the explicit Brauer induction of Boltje and Symonds. We show that the morphism of ultra-commutative Real-global ring spectra from to the Real-global K-theory spectrum that classifies the tautological Real -representation admits a section on underlying Real-global infinite loop spaces. We prove that this global Segal--Becker splitting induces the classical Segal--Becker splittings on equivariant cohomology theories, and that it induces the Boltje--Symonds explicit Brauer induction on equivariant homotopy groups. As an application we rigidify the unstable Adams operations in Real-equivariant K-theory to global self-maps of the Real-global space .
Paper Structure (11 sections, 41 theorems, 397 equations)

This paper contains 11 sections, 41 theorems, 397 equations.

Table of Contents

  1. The C-global stable splitting of U
  2. Relations among equivariant stable homotopy classes in U
  3. The interplay of the global splitting and the eigenspace morphism
  4. The global Segal--Becker splitting
  5. The G-equivariant Segal--Becker splitting and explicit Brauer induction
  6. Global Adams operations
  7. Some C-global homotopy theory
  8. Real-global K-theory
  9. Unstable Real-global K-theory
  10. Connective Real-global K-theory
  11. Periodic Real-global K-theory

Key Result

Proposition 1.11

For every $k\geq 1$, the relation $\varrho_*(\tau_k)=0$ holds in $\pi_{\mathop{\mathrm{ad}}\nolimits(k)}^{\tilde{U}(k)}({\mathbb S})$.

Theorems & Definitions (92)

  • Proposition 1.11
  • proof
  • Theorem 1.16
  • proof
  • Theorem 1.21
  • proof
  • Theorem 2.3
  • proof
  • Proposition 2.8
  • proof
  • ...and 82 more