Topology

1. Specialized Topics
   1. Manifolds
      1. Definition and Examples
         1. Topological Manifolds
         2. Low-Dimensional Manifolds
         3. Exotic Spheres
      2. Differentiable Manifolds
         1. Charts and Atlases
         2. Tangent Spaces
         3. Vector Fields and Differential Forms
         4. Orientability and Oriented Manifolds
      3. Riemannian Manifolds
         1. Riemannian Metrics
         2. Curvature: Scalar, Ricci, and Sectional
         3. Geodesics and the Exponential Map
         4. Connections and Covariant Derivatives
         5. Riemannian Submanifolds
   2. Fiber Bundles
      1. Definition and Examples
         1. Trivial and Non-Trivial Bundles
         2. Vector Bundles
         3. Tangent and Cotangent Bundles
         4. Real and Complex Line Bundles
      2. Principal Bundles
         1. Fiber Bundles with a Group Action
         2. Structure Group and Transition Functions
         3. Connection on a Principal Bundle
         4. Gauge Theory and Applications
      3. Associated Bundles
         1. Construction and Examples
         2. Associated Vector Bundles
         3. Pullback Bundles
   3. Knot Theory
      1. Definition and Examples
         1. Knots and Links
         2. Basic Knots: Trefoil, Figure-Eight
         3. Knot Diagrams and Reidemeister Moves
      2. Knot Invariants
         1. Fundamental Group of the Knot Complement
         2. Knot Polynomials: Jones, Alexander, and HOMFLY
         3. Vassiliev Invariants
         4. Knot Floer Homology
      3. Seifert Surfaces
         1. Definition and Construction
         2. Relationship to Knot Genus
         3. Seifert Matrices
         4. Applications in 3-Manifold Topology
   4. Topological Groups
      1. Definition and Examples
         1. Basic Definitions and Examples: Circle Group, Torus
         2. Topological Vector Spaces as Topological Groups
         3. Compact and Locally Compact Groups
      2. Lie Groups
         1. Surfaces of Lie Group Structure
         2. Lie Algebras and Exponential Map
         3. Representations of Lie Groups
         4. Classical Lie Groups: SU(n), SO(n), SL(n,R)
      3. Group Actions and Orbit Spaces
         1. Examples of Group Actions in Topology
         2. Homogeneous Spaces
         3. Quotient Spaces
         4. Applications in Symmetry and Dynamical Systems