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Characteristic classes

Real and complex vector bundles, integral characteristic classes, curvature forms, manifold pairings, bordism, signature, PL comparison and exotic spheres, with complete proofs and solved exercises.

Twenty-five chapters · 155 solved exercises · English · editable Markdown

Written and self-checked by GPT-6.1 Sol (OpenAI), Ultra. Human mathematical sources are credited in the chapters. Three marked Roberts adaptations retain CC BY 4.0. Independent review remains separate.

Reading guide · Attribution and component terms · Source-to-results navigation

  1. Vector bundles and their constructions · Markdown
  2. Grassmannians and classifying maps · Markdown
  3. Thom classes and Euler classes · Markdown
  4. The Gysin sequence and projective splitting · Markdown
  5. Steenrod squares and Stiefel–Whitney classes · Markdown
  6. Projective tangent bundles and their obstructions · Markdown
  7. Schubert cells and universal Grassmannian cohomology · Markdown
  8. Chern classes and the integral universal ring · Markdown
  9. Pontryagin classes and oriented universal cohomology · Markdown
  10. Manifold duality, the diagonal and Wu classes · Markdown
  11. Frame fields and primary obstructions · Markdown
  12. Connections, curvature and characteristic forms · Markdown
  13. Characteristic numbers and projective-product independence · Markdown
  14. The oriented cobordism ring · Markdown
  15. Thom spaces, transversality and the Pontryagin–Thom construction · Markdown
  16. Homotopy fibres and the Serre spectral sequence · Markdown
  17. Rational homotopy and the Hurewicz range · Markdown
  18. Rational oriented bordism and projective generators · Markdown
  19. Stiefel–Whitney numbers and unoriented bordism · Markdown
  20. Multiplicative sequences and the signature theorem · Markdown
  21. Odd-prime reduced powers and the Wu classes · Markdown
  22. Combinatorial L-classes and piecewise linear fibres · Markdown
  23. Smooth fibres, triangulation comparison and lens spaces · Markdown
  24. Milnor's exotic seven-spheres · Markdown
  25. Smooth triangulations and the integral PL obstruction · Markdown