Course

MATH-225: Topology II - fundamental groups

Lectures in this course (129)

Seifert van Kampen: demonstration and relationship with rectangles

Covers the demonstration of Seifert van Kampen theorem and the relationship with rectangles and paths.

Sphere and Projective Plane

Explores examples related to the sphere and the projective plane.

Well Pointed Spaces and Wedge

Discusses well pointed spaces, neighborhoods, wedges, examples, and the universal property of the quotient.

Fundamental Group of Wedge

Explores the fundamental group of a wedge and its applications.

Retracte: Fundamental Group and Cell Attachment

Covers the concept of a subspace being a retract of another space and fundamental groups, including examples like contracting the teeth of a necklace.

Cell Reattachment Preparations

Covers the preparatory steps for cell reattachment and applying the Seifert-van Kampen theorem.

Calcul du pi1 de XUCA

Covers the calculation of the fundamental group of XUCA by building forms and defining applications.

Group Actions: Quotients and Homomorphisms

Discusses group actions, quotients, and homomorphisms, emphasizing practical implications for various groups and the construction of complex projective spaces.

Cell Attachment: Large Cell

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

Attachment of a 2-cell

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

Attachment of a 1-cell

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

Polygonal Presentations: Classification of Surfaces

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

The Fundamental Group of a Surface

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

Sum of RP2 Connected Sums

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

Connected Sum of Torus and RP2

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

Surface Classification

Covers the classification of surfaces using polygonal presentations and homeomorphism.

Abélianization: Fundamental Groups

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

Abelianization of Fundamental Group

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

Quotient by a Relation

Covers the concept of quotient space by a relation and its applications.

Euler Characteristic: Surfaces and Homotopy

Explores the Euler characteristic of surfaces and homotopy properties.