Algebraic topology

Related lectures (32)

Explores the homotopy theory of chain complexes, focusing on retractions and model category structures.

Explores the long exact sequence in homotopy, emphasizing the importance of sets and groups in the sequence.

Topology: Classification of Surfaces and Fundamental Groups

Discusses the classification of surfaces and their fundamental groups using the Seifert-van Kampen theorem and polygonal presentations.

CW Approximation Theorem

Explores the CW Approximation Theorem, constructing CW complexes from spaces to ensure isomorphism on homology groups.

Simplicial and Singular Homology Equivalence Log in to Mediaspace to watch this video

Demonstrates the equivalence between simplicial and singular homology, proving isomorphisms for finite s-complexes and discussing long exact sequences.

Derived functors: Identity and Homotopy Categories Log in to Mediaspace to watch this video

Explores derived functors in model categories, focusing on identity and homotopy categories.

Homology with coefficients Log in to Mediaspace to watch this video

Covers homology with coefficients, introducing the concept of defining homology groups with respect to arbitrary abelian groups.

Singular Homology: First Properties Log in to Mediaspace to watch this video

Covers the first properties of singular homology and the preservation of decomposition and path-connected components in topological spaces.

Model Categories and Homotopy Theory: Functorial Connections Log in to Mediaspace to watch this video

Covers the relationship between model categories and homotopy categories through functors preserving structural properties.

Zig Zag Lemma Log in to Mediaspace to watch this video

Covers the Zig Zag Lemma and the long exact sequence of relative homology.

Cellular Homology: Applications Log in to Mediaspace to watch this video

Delves into applying cellular homology to compute homology groups and Euler characteristic, showcasing its practical implications.

Homology: Introduction and Applications Log in to Mediaspace to watch this video

Introduces homology as a tool to distinguish spaces in all dimensions and provides insights into its construction and applications.