Description
Book SynopsisReflecting the emerging role of topology, this book is uniquely innovative in its cross-disciplinary approach. * Includes topological applications in the scientific fields of physics, chemistry, biology, engineering, and economics. * Applets are available via a related website.
Trade Review"…helpful to a beginning student, especially one who is interested in the connections between topology and the world of applications." (
Mathematical Reviews, 2007k)
"…a welcome addition to what is now a long list of good undergraduate topology books." (CHOICE, August 2007)
"..a celebration of topology and its many applications. I enjoyed reading it and believe that it would be an interesting textbook from which to learn." (MAA Reviews, January 12, 2007)
Table of ContentsPreface.
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.
