Topological K-Theory

graph TD D1["Def: Vector bundle\nE→X, locally trivial"] D2["Def: K⁰(X)\nGrothendieck of bundles"] D3["Def: K⁻¹(X)\nK⁰(ΣX) suspension"] D4["Def: Reduced K̃\nK̃⁰ = coker K⁰(pt)→K⁰(X)"] T1["Thm: K⁰(X) ring\n⊕,⊗ on bundles"] T2["Thm: Swan\nbundles ↔ proj C(X)-mods"] T3["Thm: Bott periodicity\nK⁻ⁿ≅K⁻ⁿ⁻²"] T4["Thm: Thom isomorphism\nK(E)≅K(X) for spin^c"] T5["Thm: Atiyah–Hirzebruch\nspectral sequence"] D1 --> D2 D2 --> D3 D3 --> D4 D2 --> T1 T1 --> T2 D3 --> T3 T3 --> T4 classDef definition fill:#b197fc,color:#fff classDef theorem fill:#51cf66,color:#fff class D1,D2,D3,D4 definition class T1,T2,T3,T4,T5 theorem