Differential Topology

Gereon Quick is Professor of Mathematics at NTNU. His research explores homotopy theory across algebra, geometry, and number theory. He studied and held academic appointments in Cambridge, Paris, Münster, Princeton, and Harvard. His work has been supported by research grants from Germany, Norway, and the USA.

Introduction; 1. A brief introduction to topological spaces; 2. Smooth manifolds; 3. The inverse function theorem, immersions and embeddings; 4. Submersions and regular values; 5. Transversality; 6. Abstract smooth manifolds; 7. Whitney's embedding theorems; 8. Smooth homotopy; 9. Manifolds with boundary; 10. Brouwer fixed point theorem; 11. The Brouwer degree modulo 2; 12. Tubular neighbourhoods and transversality; 13. Intersection theory modulo 2; 14. Orientation; 15. The integer-valued Brouwer degree; 16. Pontryagin construction and Hopf's degree theorem; 17. Vector fields and the Poincaré-Hopf index theorem; Appendix. Solutions to selected exercises; References; Index.