Topology

1. Algebraic Topology
   1. Fundamental Group
      1. Definition and Examples
         1. Basics: Loop Spaces and Homotopy Classes
         2. Calculation in Various Spaces (e.g., circles, spheres, and tori)
      2. Properties of the Fundamental Group
         1. Group Structure
         2. Path Lifting and Homotopy Lifting
      3. Applications of Fundamental Groups
         1. Classification of Covering Spaces
         2. Use in Algebraic Invariants
      4. Covering Spaces
         1. Definition and Basic Properties
         2. Relationship with Fundamental Groups
         3. Universal Covering Spaces
         4. Galois Correspondence
      5. Van Kampen's Theorem
         1. Statement and Applications
         2. Examples of Use
         3. Implications in Complex Spaces
   2. Homology
      1. Simplicial Homology
         1. Simplicial Complexes
         2. Chain Complexes and Boundaries
         3. Homology Groups: Definition and Examples
         4. Calculations with Triangulations
      2. Singular Homology
         1. Definition of Singular Simplices
         2. Functorial Properties
         3. Relative Homology and Mayer-Vietoris Sequence
      3. Homological Algebra Concepts
         1. Chain Maps and Homotopy
         2. Exact Sequences and Long Exact Sequences
      4. Applications of Homology
         1. Algebraic Invariants
         2. Topological Classifications
   3. Cohomology
      1. Cohomology Groups
         1. Definition via Homological Categories
         2. Cup Product and Cohomological Rings
         3. Cohomology with Coefficients
      2. Duality Theorems
         1. Universal Coefficient Theorem
         2. Poincaré Duality in Manifolds
      3. Advanced Concepts in Cohomology
         1. De Rham Cohomology
         2. Čech Cohomology
   4. Homotopy
      1. Homotopy Equivalence
         1. Definition and Examples
         2. Contractible Spaces
      2. Homotopy Groups
         1. Higher Homotopy Groups (π_n)
         2. Properties and Calculations
         3. Relation to Fundamental Group
      3. Applications of Homotopy Theory
         1. Obstructions to Deformation
         2. Homotopy Types in Topological Spaces
      4. Advanced Homotopy Theory
         1. Whitehead Theorem
         2. Hurewicz Theorem
         3. Homotopy Limits and Colimits
   5. Advanced Topics in Algebraic Topology
      1. Spectral Sequences
         1. Definition and Applications
         2. Eilenberg-Moore and Serre Spectral Sequences
      2. K-theory
         1. Vector Bundles and K-Groups
         2. Atiyah-Singer Index Theorem
      3. Bordism Theory
         1. Cobordism Groups
         2. Classifying Spaces and Characteristic Classes