Lectures related to Lifting Criterion

Fundamental Groups

Explores fundamental groups, homotopy classes, and coverings in connected manifolds.

Group Actions and Fundamental Groups

Delves into group actions, coverings, fundamental groups, and homomorphisms in topological spaces.

Homology of Riemann Surfaces

Explores the homology of Riemann surfaces, including singular homology and the standard n-simplex.

Base B for X: Connected Sets and Open Bases

Explores the concept of a base B for a topological space X and the properties of connected sets and open bases.

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.

Topology: Fundamental Groups and Surfaces

Discusses fundamental groups, surfaces, and their topological properties in detail.

Topology: Homotopy and Cone Attachments

Discusses homotopy and cone attachments in topology, emphasizing their significance in understanding connected components and fundamental groups.

Pushouts in Group Theory: Universal Properties Explained

Covers the construction and universal properties of pushouts in group theory.

Modular curves: Riemann surfaces and transition maps

Covers modular curves as compact Riemann surfaces, explaining their topology, construction of holomorphic charts, and properties.

Topology: Fundamental Groups and Applications

Provides an overview of fundamental groups in topology and their applications, focusing on the Seifert-van Kampen theorem and its implications for computing fundamental groups.

Topology: Separation Criteria and Quotient Spaces

Discusses separation criteria and quotient spaces in topology, emphasizing their applications and theoretical foundations.

Open Mapping Theorem

Explains the Open Mapping Theorem for holomorphic maps between Riemann surfaces.

Local Homeomorphisms and Coverings

Covers the concepts of local homeomorphisms and coverings in manifolds, emphasizing the conditions under which a map is considered a local homeomorphism or a covering.

Closed Curves and Topological Spaces

Explores closed curves in topological spaces, emphasizing their properties and significance in mathematics.

Relative Homotopy Groups

Covers relative homotopy groups, establishing long exact sequences and defining boundary homomorphisms.

Universal Covering Log in to Mediaspace to watch this video

Explores the concept of a universal cover of a topological space and the necessary conditions for a space to have one.

Manifolds: Charts and Compatibility Log in to Mediaspace to watch this video

Covers manifolds, charts, compatibility, and submanifolds with smooth analytic equations.

Categorical Perspective: Group Actions Log in to Mediaspace to watch this video

Generalizes group actions beyond sets, providing a comprehensive framework for various mathematical contexts.

Induced Homomorphisms on Relative Homology Groups Log in to Mediaspace to watch this video

Covers induced homomorphisms on relative homology groups and their properties.

Dimension Theory: Topological Space and Closed Subsets Log in to Mediaspace to watch this video

Explores dimension theory in topological spaces and closed subsets, including height interpretation and additivity.