Course

MATH-225: Topology II - fundamental groups

Lectures in this course (101)

Delves into cell attachment, exploring its implications for different cell dimensions.

Covers the attachment of a 2-cell to a space and explores the concept of the attachment application f.

Explores the attachment of a 1-cell to a space and the conditions for points to belong to the same connected component.

Explores the classification of surfaces based on polygonal presentations and the identification of sides and vertices.

Explains the fundamental group of a surface, focusing on polygonal presentation, cell structure, and vertex identification.

Covers the concept of connected sums in RP2 and how to compute them.

Explores the connected sum of surfaces, focusing on torus and RP2, highlighting the resulting homeomorphism with the sphere.

Covers the classification of surfaces using polygonal presentations and homeomorphism.

Explores abélianization of fundamental groups in commutative spaces with examples and proofs.

Explores abelianization examples, homomorphisms, and isomorphisms in group theory.

Explores the Euler characteristic of surfaces and homotopy properties.

Covers definitions, examples, and properties of coatings in the context of group theory.

Covers the properties of coatings, including trivializing opens and automorphisms.

Focuses on proving the closure of Z by showing that its complement is open.

Covers the concept of morphisms of coverings and emphasizes the recognition of automorphisms.

Covers the concept of path lifting and unique path lifting property.

Explains homotopy lifting, paths, proofs, constant paths, discrete spaces, and monomorphisms.

Explores totally discontinuous group actions on sets and their mathematical implications.

Discusses the unique bearing property of paths and solving lifting problems effectively.

Explores the continuity of mappings, paths between points, and centers of openings.