6
None.
MAT/03 Geometry.
Oral exam.
The aim of the course is to provide an introduction to the main themes of algebraic topology: constructing algebraic invariants for the study of geometric objects such as polyhedra and topological manifolds. The most important results in these areas will be discussed, illustrating the main proof and problem-solving techniques.
Exact and quasi-exact sequences of groups and homomorphisms. Graded groups. Categories and functors. Homotopy between continuous maps and between topological spaces. Relative homotopy. Retracts and deformation retracts. Contractible spaces. Review of path-connectedness and of the fundamental group of a pointed space. Simply connected spaces. Simplicial complexes. Oriented complexes. Polyhedra, triangulations and subdivisions. Simplicial maps. Simplicial approximation theorem. Chain complexes. Chain maps and chain homotopy. Homology of a chain complex. The simplicial and singular homology functors. Isomorphism between the simplicial homology and the singular homology of a polyhedron. Homotopy invariance of the homology functor. Dimension axiom for homology. Relative homology. Exact homology sequence. The Mayer-Vietoris sequence and its application to computing the homology groups of spheres. Excision. Relationship between the fundamental group and the one-dimensional homology group. Brouwer’s fixed point theorem and invariance of dimension theorem. Homomorphisms induced in homology by particular maps on the sphere. Tangent vector fields on an odd-dimensional sphere. Degree of a map of a sphere into itself and its consequences. Euler-Poincaré characteristic of a polyhedron and its topological invariance. Topological manifolds with and without boundary: local dimension and dimension, homogeneity properties. Orientable and non-orientable surfaces. Notable surfaces: sphere, torus, projective plane, Möbius strip, Klein bottle. Connected sum. Statement of Radò’s theorem. Particular properties of the triangulation of an orientable surface. Classification theorem for closed connected surfaces, showing that in this case orientability and the Euler characteristic are complete invariants.
At the end of the course, students must demonstrate that they
Knowledge and skills acquired on the course topics, the student’s presentation skills and command of language, the ability to apply the knowledge acquired to the solution of simple problems, the ability to support a discussion with examples and counterexamples, and command of the mathematical tools used in the course.
