Topology

We find a wealth of ideas about spaces and how they are connected in this mathematical discipline. It is often said that topology started as a focused discipline with the work of Henri Poincare right before and after the end of the nineteenth century. Its interface with geometry is significantly blurred. To get a feeling for what this discipline is about, we recommend the booklets by Alexandroff [1] and by Prasolov [2]. We categorize ecommendations into five subareas. We find that in many cases, the early literature is more intuitive and more accessible, while the modern literature is more streamlined and comprehensive.

General or Point-Set Topology.   This field deals with the foundations of topology based on axioms that capture connectivity without talking about geometric distance. Legend has it that General Topology fought in the First World War and did not return. A classic text on the subject is Kelley [3].

Combinatorial Topology.   This is the precursor of Algebraic Topology. Interest in the mathematics community decreased about 40 years ago, when it became clear that there are aspects of general manifolds that cannot be captured in the piecewise-linear category. The fundamental methods developed in this branch of topology are extremely attractive for computations. We recommend the reissuing of an early trilogy by Alexandrov [4] and the book by Giblin [5].

Algebraic Topology.   Combinatorial studies in topology lead naturally to algebraic structures as discrete representations of continuous spaces. The algebraisation of topology initiated by Emmy Noether has led to a field dominated by the algebraists. We recommend Seifert and Threlfall [6] as an early book on the subject that illustrates the initially auxiliary role of algebra in studying topology. We also recommend Munkres [7], which reflects the modern view, in which algebra grew into a comprehensive framework in which topology is represented.

Differential Topology.   The assumption of smoothness limits the generality of topological phenomena and brings out important topological structures. A beautiful introduction to the subject is Milnor's booklet [8]. An important subfield is Morse Theory, which has its roots in the variational analysis of continuous optimization. An intuitive introduction to that subject is [9].

Knot Theory.   The study of knots exploits methods from many other disciplines. We recommend the book by Adams [10] as a very readable text that also talks about applications, in particular DNA and other biological and chemical structures. Knots in three-dimensional Euclidean space are related to three-manifolds, which arise by excising tubular neighborhoods. We recommend the book by Thurston [11], which explains three-manifolds by stressing the interplay between the geometric and the topological viewpoint.


References:
[1] P. S. Alexandrov. Elementary Concepts of Topology. Dover, New York, 1961.
[2] V.V. Prasolov. Intuitive Topology. American Mathematical Society, 1995.
[3] J. L. Kelley. General Topology. Springer, New York, 1991.
[4] P. S. Alexandrov. Combinatorial Topology. Dover, Mineola NY, 1998.
[5] P. J. Giblin. Graphs, Surfaces, and Homology: An Introduction to Algebraic Topology. Chapman and Hall, London, 1977.
[6] H. Seifert and W. Threlfall. A Textbook of Topology. Academic Press, Boston, 1980.
[7] J. R. Munkres. Elements of Algebraic Topology. Addison-Wesley, Redwood City CA, 1984.
[8] J. W. Milnor. Topology from the Differentiable Viewpoint. Princeton University Press, Princeton NJ, 1965.
[9] Y. Matsumoto. An Introduction to Morse Theory. American Mathematical Society, Providence RI, 2002.
[10] C. C. Adams. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. W. H. Freeman, New York, 1994.
[11] W.P. Thurston. Three-Dimensional Geometry and Topology. Princeton University Press, Princeton NJ, 1997.