Characteristic Classes
by John W. Milnor, James D. Stasheff · John W. Milnor, James D. Stasheff
Annals of Mathematics Studies volume 76, the standard introduction to vector bundle invariants. Constructs Stiefel-Whitney, Euler, Chern and Pontryagin classes via Grassmannians and the Thom isomorphism, then applies them to cobordism, the Hirzebruch signature theorem and obstruction theory. Assumes singular cohomology.
This link may earn us a small commission at no extra cost to you. Affiliate disclosure
More resources on Characteristic Classes
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
nLab: Characteristic Class
nLab entry presenting characteristic classes abstractly as composition with a universal cocycle on a classifying space, with examples including Chern, Stiefel-Whitney and Pontryagin classes, the Chern character and the Maslov index. Shows how the classical examples fit one categorical definition.
Lecture Course: Characteristic Classes (28 lectures)
Twenty-eight recorded blackboard lectures of roughly ninety minutes each, taught in English. Runs from topological versus differentiable manifolds through Stiefel-Whitney classes, division algebras, Steenrod squares, Wu's formula and the Chern-Simons invariant.
Characteristic Classes and K-Theory (Cambridge Part III lecture notes)
Roughly 100 pages of lecture notes from the Cambridge Part III course of the same name, developing vector bundles, the splitting principle, Stiefel-Whitney, Chern and Pontryagin classes, then topological K-theory and its applications. Written by a researcher whose own field is characteristic classes. The only resource that carries the subject into K-theory and the splitting principle at course length, and it comes with four example sheets on the same page.
Differential Geometry: Connections, Curvature, and Characteristic Classes
Springer Graduate Texts volume 275, a self-contained route from Gaussian curvature and geodesics through connections on vector bundles to Chern-Weil theory and characteristic classes on principal bundles. Assumes manifolds and de Rham cohomology.
Differential Forms in Algebraic Topology
Springer Graduate Texts volume 82. Builds de Rham cohomology, Mayer-Vietoris arguments and spectral sequences, then treats the Euler, Chern and Pontryagin classes via Chern-Weil theory and classifying spaces. 331 pages.