ABSTRACT
A topological space X has both homology and homotopy groups as invariants. The homology groups arose from the study of the Betti numbers of a space. Another invariant, the fundamental group π1(X ) of a space X was also well known. It depends on a choice of a “base point” xo∈X. Its elements are loops; that is, they are continuous maps f : I X of an interval I onto the space X, where both endpoints 0 and 1 of the interval are mapped into the base point. Two paths f and f ´count as equal when the first can be continuously deformed into the second. If g : I X is a second such path, the product fg is defined by following first f, then g; using the definition of equality, one sees that the product is associative, but not necessarily commutative. For example, if the space is a plane with the points a and b deleted, the path from the base point looping once around a does not commute with the path running once around b. The fundamental group π1(X ) of a space X is closely related to its one-dimensional homology group H1(X, Z ), with integral coefficients Z. This homology group is abelian and is, in fact, isomorphic to the abelianization of π1(X ); that is, the group π1(X ) divided by the subgroup of all products of commutators (the commutator of paths f and g is the path fg f –1g –1). In this way, the fundamental group determines the one-dimensional homology.
