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The generators of the K-groups of the sphere

Hermann Schulz-Baldes, Tom Stoiber

Abstract

This note presents an elementary iterative construction of the generators for the complex $K$-groups $K_i(C(\SM^d))$ of the $d$-dimensional spheres. These generators are explicitly given as the restrictions of Dirac or Weyl Hamiltonians to the unit sphere. Connections to solid state physics are briefly elaborated on.

The generators of the K-groups of the sphere

Abstract

This note presents an elementary iterative construction of the generators for the complex -groups of the -dimensional spheres. These generators are explicitly given as the restrictions of Dirac or Weyl Hamiltonians to the unit sphere. Connections to solid state physics are briefly elaborated on.
Paper Structure (7 sections, 5 theorems, 47 equations)

This paper contains 7 sections, 5 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Complex Clifford algebras
  3. The connecting maps of $K$-theory
  4. Six-term exact sequence of the sphere
  5. Index map of odd-dimensional spheres
  6. Exponential map of even dimensional spheres
  7. Relation to the Fredholm picture

Key Result

Proposition 1

For $d\geq 2$ even, let $\Gamma_1,\ldots,\Gamma_{d+1}$ be a selfadjoint irreducible representation of the Clifford algebra ${\mathbb C}_{d+1}$. A generator of $\widetilde{K}_0(C({\mathbb S}^d))$ is given by the selfadjoint unitary For odd $d$, let $\Gamma_1,\ldots,\Gamma_{d}$ be a selfadjoint irreducible representation of the Clifford algebra ${\mathbb C}_{d}$, a generator of ${K}_1(C({\mathbb S}

Theorems & Definitions (5)

  • Proposition 1
  • Lemma 1
  • Proposition 2: WO or LS1
  • Proposition 3
  • Proposition 4