Topological manifold

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

Introduces differential forms on manifolds, covering tangent bundles and intersection pairings.

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

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

Delves into topological entropy in compact manifolds and Reeb flows, emphasizing the forcing of entropy via cylindrical contact homology.

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

Explores Ruelle resonances for geodesic flows on non-compact manifolds, emphasizing the role of smoothness and eigenvalues in the dynamics.

Covers the topology of Riemann surfaces, focusing on orientation and orientability.

Explores the dynamics of singular Riemann surface foliations near singular points and their relation to compact manifolds.

Demonstrates how to obtain homotopy equivalences between different spaces using the homotopy extension property.