Fundamental Group | Research & Encyclopedia Articles

Fundamental Group

Henri Poincare (1854-1912) discovered the fundamental group (also known as the Poincare group or the first homotopy group) of a manifold (or more generally, any topological space) and used it to classify manifolds. A manifold is a topological space every point of which has a neighborhood which looks like (i.e. is homeomorphic to) n-dimensional real space R^n for some number n. For example, n-dimensional real space, circles, spheres, the surface of a donut, the universe and the complement of a knot (a knot is the mathematical analogue of a loop of string with its ends glued together) in three dimensional space are all manifolds.

If p is a point in a manifold M and c and c' are paths (i.e. continuous maps of [0,1] into M) which begin and end at p then Poincare considered them to be equivalent (i.e. in the same homotopy class) if one could be continuously deformed into the other. Precisely, they are equivalent if there exists a continuous function from the unit square (i.e. [0,1] x [0,1]) in R^2 to M such that c is that path traced out by F(0,t) as t goes from 0 to 1, c' is that path traced out by F(1,t) as t goes from 0 to 1 and F(t,0) = F(t,1) = p for all t in [0,1]. If d is another path that begins and ends at p then the path which first traces out d and then traces out c is denoted by cd and is called the concatenation of c and d. It can be proved that if c and c' are equivalent then cd and c'd are equivalent. Also the path which traces out c backwards is denoted by c^-1. It has the property that cc^-1 is equivalent to the path that stays at p. The set of equivalence classes of paths (which begin and ends at p) of M with the operation of concatenation is called the fundamental group of M at p and is usually denoted by Pi_1(M,p).

The fundamental group of the real line or sphere (based at any point p) consists of a single equivalence class since any path on either space can be continuously deformed to a point. The fundamental group of the circle, however, is isomorphic to Z the group of integers by the function which assigns to the class of any path c in the circle the number of times it winds areound the circle clockwise minus the number of times it winds around the circle counterclockwise.

If p and p' are points in M and there is a path d from p to p' in M then Pi_1(M,p) is isomorphic to Pi_1(M,p') by the map which assigns to the class of a path c which begins and ends at p to the class of the path which first traces out d backwards then traces out c and the traces out d forwards. Hence if M is path-connected (i.e. there exists a path between any two points of M) then the fundamental group of M is unambiguously defined up to isomorphism. Any continuous map f from a manifold M to a manifold N induces a function denoted by f_* from Pi_1(M,p) to Pi_1(N,f(p)) that assigns the class of the path c to the class of the path f composed with c. If the map f is a homeomorphism (i.e. it is a one-to-one correspondence which preserves the topology of the spaces) then f_* is an isomorphism. Thus fundamental groups can be used to study the topological properties of manifolds.

Poincare conjectured that if two manifolds have isomorphic fundamental groups and some other conditions are satisfied (the manifolds must be closed and have the same homology) then they must be homeomorphic. This was first disproved by James Alexander (1888-1971) of Princeton who found specific three dimensional manifolds satisfying the hypothesis of the conjecture but not the conclusion. However if one assumes in addition to the above hypothesis that the fundamental groups of the manifolds contain only one equivalence class then the problem is still open (for the three dimensional case). It is known as Poincare's conjecture. In 1989, Cameron Gordon and John Luecke proved that if K and K' are knots in R^3 and the fundamental group of the complement of K is isomorphic to the fundamental group of the complement of K' then the two complements are homeomorphic and in fact K can be continuously deformed into K' without intersecting itself in the process. This result resolved one of the longest standing conjectures in knot theory.