Topology and Its Applications

Introduction.

I. 1 Preliminaries.

1.2 Cardinality.

1. Continuity.

1. 1 Continuity and Open Sets in Rⁿ.

1.2 Continuity and Open Sets in Topological Spaces.

1.3 Metric, Product, and Quotient Topologies.

1.4 Subsets of Topological Spaces.

1.5 Continuous Functions and Topological Equivalence.

1.6 Surfaces.

1.7 Application: Chaos in Dynamical Systems.

1.7.1 History of Chaos.

1.7.2 A Simple Example.

1.7.3 Notions of Chaos.

2. Compactness and Connectedness.

2.1 Closed Bounded Subsets of R.

2.2 Compact Spaces.

2.3 Identification Spaces and Compactness.

2.4 Connectedness and path-connectedness.

2.5 Cantor Sets.

2.6 Application: Compact Sets in Population Dynamics and Fractals.

3. Manifolds and Complexes.

3.1 Manifolds.

3.2 Triangulations.

3.3 Classification of Surfaces.

3.3.1 Gluing Disks.

3.3.2 Planar Models.

3.3.3 Classification of Surfaces.

3.4 Euler Characteristic.

3.5 Topological Groups.

3.6 Group Actions and Orbit Spaces.

3.6.1 Flows on Tori.

3.7 Applications.

3.7.1 Robotic Coordination and Configuration Spaces.

3.7.2 Geometry of Manifolds.

3.7.3 The Topology of the Universe.

4. Homotopy and the Winding Number.

4.1 Homotopy and Paths.

4.2 The Winding Number.

4.3 Degrees of Maps.

4.4 The Brouwer Fixed Point Theorem.

4.5 The Borsuk-Ulam Theorem.

4.6 Vector Fields and the Poincare' Index Theorem.

4.7 Applications I.

4.7.1 The Fundamental Theorem of Algebra.

4.7.2 Sandwiches.

4.7.3 Game Theory and Nash Equilibria.

4.8 Applications 1I: Calculus.

4.8.1 Vector Fields, Path Integrals, and the Winding Number.

4.8.2 Vector Fields on Surfaces.

4.8.3 1ndex Theory for n-Symmetry Fields.

4.9 Index Theory in Computer Graphics.

5. Fundamental Group.

5. I Definition and Basic Properties.

5.2 Homotopy Equivalence and Retracts.

5.3 The Fundamental Group of Spheres and Tori.

5.4 The Seifert-van Kampen Theorem.

5.4.1 Flowers and Surfaces.

5.4.2 The Seifert-van Kampen Theorem.

5.5 Covering spaces.

5.6 Group Actions and Deck Transformations.

5.7 Applications.

5.7.1 Order and Emergent Patterns in Condensed Matter Physics.

6. Homology.

6.1 A-complexes.

6.2 Chains and Boundaries.

6.3 Examples and Computations.

6.4 Singular Homology.

6.5 Homotopy Invariance.

6.6 Brouwer Fixed Point Theorem for Dⁿ.

6.7 Homology and the Fundamental Group.

6.8 Betti Numbers and the Euler Characteristic.

6.9 Computational Homology.

6.9.1 Computing Betti Numbers.

6.9.2 Building a Filtration.

6.9.3 Persistent Homology.

Appendix A: Knot Theory.

Appendix B: Groups.

Appendix C: Perspectives in Topology.

C.1 Point Set Topology.

C.2 Geometric Topology.

C.3 Algebraic Topology.

C.4 Combinatorial Topology.

C.5 Differential Topology.

References.

Bibliography.

Index.