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.

'The material of the book is comparable to that of Milnor's classic treatise, but Quick's exposition is much more detailed and concrete. Students with a basic knowledge of linear algebra and multidimensional calculus will have no difficulty understanding this material. Almost all exercises have detailed solutions, so, apart from being a great companion to lectures, the book is ideal for self-study.' Christopher Denniger, University of Münster

• 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.

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.

'The material of the book is comparable to that of Milnor's classic treatise, but Quick's exposition is much more detailed and concrete. Students with a basic knowledge of linear algebra and multidimensional calculus will have no difficulty understanding this material. Almost all exercises have detailed solutions, so, apart from being a great companion to lectures, the book is ideal for self-study.' Christopher Denniger, University of Münster

• 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.

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.

'The material of the book is comparable to that of Milnor's classic treatise, but Quick's exposition is much more detailed and concrete. Students with a basic knowledge of linear algebra and multidimensional calculus will have no difficulty understanding this material. Almost all exercises have detailed solutions, so, apart from being a great companion to lectures, the book is ideal for self-study.' Christopher Denniger, University of Münster

• 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.