https://studiegids.vu.nl/en/courses/2024-2025/XB_0068At the end of this course: -the student understands the introduced concepts such as transversality, degree of a function, intersection number, and is able to use these to prove some fundamental results; -the student understands the meaning of the theorems and knows how to derive themDifferential topology studies differentiable manifolds and differentiable functions from a topological viewpoint. In contrast to differential geometry properties arising from metrics are not studied, and in contrast to the general topology course, the spaces are restricted to nicely behaving manifolds. Studied properties of the manifolds are rather global than local. The classification of coverings can be seen as an alternative formulation of the fundamental theorem of Galois theory, allowing to derive an algebraic theorem from a topological perspective. The following topics will be covered during the course:Embedded manifolds and smooth mapsImmersions and submersions. Preimage theorem.Transversality. Transversality theorem. Transversality homotopy theorem. Extension theorem.Homotopy and stability.Sard's theorem. Manifolds with boundary.Brouwer’s fixed point theorem.Intersection theory mod 2. Boundary theorem. Winding number. Jordan-Brouwer Separation theorem.Borsuk-Ulam theoremIf time permits: Lefschetz fixed-point theorem and Poincare-Hopf index theorem. This is roughly the content of Chapter 1-2 of the accompanying book, where 1.1-1.4 are considered to be reminders and summaries but not presented in detail.Lectures and tutorials (2+2 hours per week)For this course there is a midtermexamination (50%) and a final exam (50%). There will also be a resitexamination.Guilluimin and Pollack: Differential TopologyBachelor Mathematics, year 3Topology.