Here is a list of 100 chapter titles for a book on Algebraic Topology in mathematics, ranging from beginner to advanced topics:
- What is Algebraic Topology?
- Basic Concepts in Topology: Sets, Spaces, and Maps
- Open and Closed Sets: The Building Blocks of Topological Spaces
- Topological Spaces and Continuous Functions
- Homeomorphisms and the Notion of Topological Equivalence
- Metric Spaces and Their Topology
- The Concept of a Basis for a Topology
- Connected and Disconnected Spaces
- Compactness and Its Importance in Topology
- The Hausdorff Condition and Its Significance in Topology
¶ Part 2: Fundamental Concepts and Structures
- The Notion of a Topological Invariant
- Path-Connected Spaces and Fundamental Groups
- The Fundamental Group: Definition and Examples
- Simple Paths and Loops
- Homotopy and Continuous Deformation
- Contractibility and Deformation Retracts
- Fundamental Group of Simple Spaces: Examples
- The Van Kampen Theorem: A Key Tool in Algebraic Topology
- Covering Spaces: Intuition and Applications
- Higher Homotopy Groups and Their Significance
- Introduction to Homology: Basic Definitions and Intuition
- Simplicial Complexes and Their Role in Homology
- Chains, Cycles, and Boundaries in Homology
- The Chain Complex and Boundary Operators
- Computing Homology: An Introduction to Simplicial Homology
- Euler's Polyhedron Formula and Its Consequences
- Singular Homology and Its Generalization
- Homology of Product Spaces
- Exact Sequences and Long Exact Sequences in Homology
- Homology of Free Groups and Abelian Groups
- Introduction to Cohomology: The Dual to Homology
- Cohomology Rings and Their Properties
- Poincaré Duality and Its Implications in Topology
- Cohomology of Simple Spaces
- The Cup Product in Cohomology
- Sheaf Cohomology and Its Applications
- Cohomology with Coefficients: Basic Concepts
- Cohomology of Products and Fiber Bundles
- The Universal Coefficient Theorem in Cohomology
- Singular Cohomology and Poincaré Duality
¶ Part 5: Higher Homotopy and Higher Categories
- Homotopy Groups of a Space: An Overview
- Higher Homotopy Groups and Their Applications
- Higher Category Theory and Algebraic Topology
- Classifying Spaces in Algebraic Topology
- The Postnikov System and Its Role in Homotopy Theory
- The Role of Higher Homotopy Groups in the Classification of Spaces
- Applications of Higher Homotopy Groups in Geometry
- Cohomotopy and Its Connections to Cohomology
- Spectral Sequences: A Tool for Computation in Algebraic Topology
- The Eilenberg-MacLane Spaces and Their Significance
- Introduction to Spectral Sequences
- The Concept of a Filtration in Spectral Sequences
- The Mayer-Vietoris Sequence in Algebraic Topology
- Computing Homology via Spectral Sequences
- The Serre Spectral Sequence and Its Applications
- Advanced Topics in Spectral Sequences
- The Atiyah-Hirzebruch Spectral Sequence
- The Brown-Gersten-Quillen Spectral Sequence
- Applications of Spectral Sequences in Cohomology
- The Use of Spectral Sequences in Fibre Bundle Theory
¶ Part 7: Manifolds and Homotopy Types
- What is a Manifold? An Introduction
- Smooth Manifolds and Differentiable Structures
- Compactness and Orientability of Manifolds
- The Classification of Surfaces
- Orientable and Non-Orientable Manifolds
- The Fundamental Group of Manifolds
- Covering Space Theory for Manifolds
- Manifold Homotopy Types and Their Classification
- The Poincaré Conjecture and Its Resolution
- The Euler Characteristic of a Manifold
- Algebraic Topology in Knot Theory
- Topological Invariants of Manifolds
- The Role of Algebraic Topology in Physics
- Algebraic Topology and Quantum Field Theory
- The Chern-Simons Theory in Topology
- Applications of Algebraic Topology in Robotics
- Persistent Homology and Topological Data Analysis (TDA)
- The Role of Algebraic Topology in Machine Learning
- Topological Quantum Computation and Algebraic Topology
- Algebraic Topology and the Study of Shape in Data
¶ Part 9: Advanced Topics and Further Research
- The Adams Spectral Sequence and Its Applications
- Group Homology and Algebraic Topology
- The Triangulation of Manifolds and its Importance
- Bordism Theory and Its Applications
- Torsion in Homology and Cohomology
- The Stable Homotopy Groups of Spheres
- The Künneth Theorem and Its Applications
- Algebraic Topology and Stable Homotopy Theory
- The Role of Topology in Stable and Unstable Homotopy Theory
- Classifying Spaces and Their Connection to Algebraic Topology
¶ Part 10: Trends and Future Directions in Algebraic Topology
- Recent Developments in Persistent Homology
- The Future of Algebraic Topology in High-Dimensional Data Analysis
- Topological Groups and Their Role in Algebraic Topology
- The Intersection of Algebraic Topology and Geometry
- Recent Advances in Homotopy Theory
- Algebraic Topology and the Study of Singularities
- Topological Methods in the Study of Higher Dimensional Manifolds
- The Future of Algebraic Topology in Theoretical Physics
- Homotopy Theory and Its Impact on the Study of Geometric Structures
- Emerging Applications of Algebraic Topology in Modern Mathematics and Science
This set of chapter titles covers a wide range of topics in Algebraic Topology, progressing from fundamental concepts to more advanced and specialized areas, including both classical and modern developments in the field. The structure allows a reader to develop a deep understanding of the subject, starting from introductory material and progressing through increasingly complex topics and applications.