Introduction to Characteristic Classes

Algebraic Topology · Axiom Academy

LESSON Introduction to Characteristic Classes Exploring the fundamental invariants of vector bundles through Stiefel-Whitney, Chern, and Euler classes 1. Vector Bundles and Their Classification A vector bundle over a base space B is a continuous family of vector spaces parametrized by points in B . The bundle is characterized by its total space E , a projection map π: E → B , and a typical fiber that is a vector space. A topological space E (total space) A continuous surjection π: E → B Local triviality: each point has a neighborhood U where π -1 (U) ≅ U × ℝ n 2. Stiefel-Whitney Classes for Real Bundles The Stiefel-Whitney classes are the fundamental characteristic classes for real vector bundles, taking values in mod 2 cohomology. They detect subtle properties like orientability and the existence of nowhere-zero sections. w(E ⊕ F) = w(E) ∪ w(F) (Whitney sum formula) Naturality: f*w i (E) = w i (f*E) 3. Chern Classes for Complex Bundles Chern classes are the characteristic classes for complex vector bundles, taking values in integral cohomology. They are finer invariants than Stiefel-Whitney classes and play a central role in complex geometry. c 0 (E) = 1 and c i (E) = 0 for i > r c(E ⊕ F) = c(E) ∪ c(F) (Whitney sum formula) c 1 (L ⊗ L') = c 1 (L) + c 1 (L') for line bundles 4. The Euler Class and Euler Characteristic Connection

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