Algebraic K-Theory: K0
The Grothendieck group K0 from vector bundles over spheres — visualizing stable equivalence
Add Bundle
Rank:
2
Twist k (in π_1(GL_n)):
0 — trivial
1 — Möbius
2 — Hopf
-1 — anti
Add Bundle to K0
Add Trivial ε^1
Stabilize (+ trivial)
Clear All
Bundle Pairs (K0 elements)
No bundles yet
K0(S²) ≅ ℤ ⊕ ℤ
Theory
K0(X) = Grothendieck group
of Vect(X) (formal differences)
[E] − [F] ≈ [E'] − [F']
iff E⊕G ≅ E'⊕G for some G
K0(S²) ≅ ℤ² via
rank ⊕ first Chern class
K̃0(S²) ≅ ℤ — Hopf bundle
η generates reduced K-theory