Algebraic Topology

The study of intrinsic qualitative aspects of spatial objects (e.g., Surfaces, Spheres, Tori, Circles, Knots, Links, configurationspaces, etc.) that remain invariant under both-directions continuous One-to-One (Homeomorphic)transformations. The discipline of algebraic topology is popularly known as ``Rubber-Sheet Geometry'' and can alsobe viewed as the study of Disconnectivities. Algebraic topology has a great deal of mathematicalmachinery for studying different kinds of Hole structures, and it gets the prefix ``algebraic'' since manyHole structures are represented best by algebraic objects likeGroups and Rings.

A technical way of saying this is that algebraic topology is concerned with Functors from thetopological Category of Groups and Homomorphisms. Here, theFunctors are a kind of filter, and given an ``input'' Space, they spit out something else inreturn. The returned object (usually a Group or Ring) is then a representation of the Hole structureof the Space, in the sense that this algebraic object is a vestige of what the original Space was like (i.e.,much information is lost, but some sort of ``shadow'' of the Space is retained--just enough of a shadow tounderstand some aspect of its Hole-structure, but no more). The idea is that Functors give much simpler objects todeal with. Because Spaces by themselves are very complicated, they are unmanageable without looking atparticular aspects.

Combinatorial Topology is a special type of algebraic topology that uses Combinatorial methods.

References

Dieudonné, J. A History of Algebraic and Differential Topology: 1900-1960. Boston, MA: Birkhäuser, 1989.

• Decreasing Function
• Decreasing Sequence
• Decreasing Series
• Dedekind Cut
• Dedekind Eta Function
• Dedekind Function
• Dedekind Ring
• Dedekind's Axiom
• Dedekind's Problem
• Dedekind Sum
• Deducible
• Deep Theorem
• Defective Matrix
• Defective Number
• Deficiency
• Deficient Number
• Definable Set
• Defined
• Definite Integral
• Degenerate
• Degree
• Degree (Algebraic Surface)
• Degree (Map)
• Degree of Freedom
• Degree (Polynomial)