The fundamental group

Abstract

Abstract The fundamental group is an algebraic label attached to a topological space. Naturally enough, it is a group and is constructed using paths in the space, continuous analogues of the sequences of edges constituting paths in a graph. The more convoluted is the shape of the space, the more variety there will be among the paths in that space and the larger will be the fundamental group. The fundamental group first arose in Poincare’s work on the three-body problem. His idea was to classify the possible evolutions of a system of three bodies moving under mutual gravitational forces as paths in the phase space with coordinates given by the bodies’ positions and velocities, but to regard nearby paths as the same. This led to the notion of deformation of one path into another, now formalized as homotopy of paths. The fundamental group consists of homotopy classes of paths in a space. The notion of homotopy is also crucial in complex analysis: Cauchy’s theorem on contour integration may be phrased as saying that the integrals of a holomorphic function around two homotopic contours are the same (see e.g. Priestley 1990). Further development of the extensive theory of the fundamental group, beyond that covered here, may be found in Armstrong (1983) and Brown (1988). Our principal application will be to knots: any knot yields a group, namely the fundamental group of its complement in 3, and algebraic properties of the group reflect properties of the knot.