Algebra 2

Geometry and Topology

Lecturer, TA
Eitan Sayag

Homotopy, Homotopy type, deformation retracts.
The idea of functors and categories.
The fundamental group of the circle and of the spheres.
Applications: (n=2) Brouwer fixed point theorem , The Borsuk-Ulam Theorem, The fundamental theorem of algebra.
Siefert-van-Kampen Theorem. Applications.
The ABC of surfaces: Torus, Projective plane, Moebuis strip, Klien bottle. What is Oreintability.
Covering spaces.
Higher Homotopy groups.
Classification of Surfaces.
Homology Theory
Definition and baseic properties:
Mayer-Vietoris Sequence.
Jordan Curve theorem.
Brouwer fixed point theorem, invariance of dimension, invariance of domain.
degree of a map.
CW complexes.
Manifolds their Geometry and Toplogy.
Topological and Smooth manifolds. Sard's theorem and approximation with smooth maps.
Tangent spaces. Vector bundles, fiber bundles.
Embedding theorems, immersions.
Applications: Robotics and complexity.
Applications: What is the shape of the universe.