OXFORD C3.1 ALGEBRAIC TOPOLOGY 2019-2020
Prof. Alexander F. Ritter, Associate Professor, University of Oxford.
LECTURE NOTES AND EXERCISES
♦ Lecture notes, 1 page per side (version 35, Jan 2020)
♦ Lecture notes, 2 pages per side (version 35, Jan 2020)
♦ Exercise Sheets:
Introduction to category theory. Why functors are useful. Invariance of dimension. Brower fixed point theorem.
Graded abelian groups. Chain complexes of free Abelian groups and their homology. Short exact sequences, induced long exact sequence. Delta complexes and their homology. Euler characteristic.
Singular homology of topological spaces. Relative homology and the Five Lemma. Homotopy invariance. Locality theorem, excision theorem. Mayer-Vietoris Sequence. Equivalence of simplicial, cellular and singular homology.
Degree of a self-map of a sphere. Cell complexes and cellular homology. Application: the hairy ball theorem.
Cohomology of spaces and the Universal Coefficient Theorem. Cup products. Künneth Theorem (without proof).
Topological manifolds and orientability. The fundamental class of an orientable closed manifold. Degree of a map between manifolds of the same dimension. Poincaré Duality, Lefschetz Duality. Alexander Duality. Jordan curve theorem.
Prof. Alexander F. Ritter. Contact
Associate Professor in topology, Mathematical Institute, Oxford.
The Roger Penrose Fellow and Tutor, Wadham College, Oxford.