Brief Introduction to Topology

Topology is one of four major areas of abstract mathematics: algebra (equations), analysis (limits), foundations (set theory and logic), and topology. The word topology comes from , Greek for place. Broadly speaking, topology is the study of space and continuity. Since topology includes the study of continuous deformations of a space, it is often popularly called rubber sheet geometry. Vector spaces Rⁿ are spaces, but a space can be a finite set of points or an infinite dimensional space of functions. The surfaces of a donut and a sphere appear the same on a small scale, so what properties distinguish them? To distinguish various spaces, notions from algebra or analysis are often introduced. We may associate groups, rings, modules, or other algebraic objects to a topological space in order to precisely describe information. These constructions may even give information back about algebra, e.g., group cohomology, or analysis, e.g., de Rham cohomology.

One important type of space is a manifold, a space which looks like Rⁿ near any point. Studying high-dimensional spaces is important for basic reasons. A many dimensional space may be dictated by a problem and then geometrically studied for insight and understanding. For example, under Newtonian mechanics the state of a particle in ordinary 3-space is determined by its position (3 coordinates) and its velocity (also 3 coordinates), so a system of n particles may be specified by a point in R⁶ⁿ.

The modern study of topology began with Henri Poincaré at the end of the 19th century, who was investigating foundational questions in celestial mechanics. It wields a philosophical power that has made it a dominant idea in 20th century mathematics: it is often the ingredient that allows one to move from local to global results. For example, solutions to differential equations called flows are often restricted to manifolds by preserved quantities. There is a result in 19th century mathematical physics that some flows, called completely integrable, admit n commuting flows (Liouville Theorem). The 20th century version is that the solution is a linear flow on an n-dimensional cylinder (Liouville-Arnold Theorem). The first result tells us about the solution only near a point, but the second result tells us about the whole solution.

At LSU, topologists study a variety of topics such as spaces from algebraic geometry, topological semigroups and ties with mathematical physics. A major topic studied at LSU is the placement problem. This problem is to determine the manner in which a space N can sit inside of a space M. Usually there is some notion of equivalence. If N is the circle and M is R³ then the subject is classical knot theory. A map of the circle into R³ can be the obvious one   we can draw on paper or a complicated one which is knotted and can't be drawn on paper without crossing itself. Studying the placements of circles in 3-dimensional manifolds can inform us about the manifold. Other versions of the placement problem that have been or are presently studied at LSU are hyperplanes in a vector space, graphs in R³, and n-dimensional spheres in (n+2)-dimensional spheres.

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