Topological invariant

Any property of a topological space that is invariant under homeomorphisms.

If a set $X$ is equipped with some structure which gives rise to a unique topology and hence makes $X$ into a topological space, then by a topological invariant of the set $X$ is understood a property of this topological space induced by the given structure on $X$. For example, one talks of connectedness (cf. Connected space) of a metric space, or of simple connectedness of a given differentiable manifold, when one has in mind the corresponding property of the topological space whose topology is that induced by the given metric or differential-geometric structure on the set $X$.

From the very beginning, in topology a great deal of attention was paid to so-called numerical invariants (besides the simplest topological invariants, such as connectedness, compactness, etc.). Initially these were defined mainly for polyhedra. The most important of these were the dimension and the Betti numbers (cf. Betti number). For closed surfaces, the genus of a surface had been studied even earlier. This was now expressed in terms of the first Betti number. Later a great deal of importance was attached to topological invariants which are groups, and later to still other algebraic structures such as, for example, the Betti groups or homology groups of different dimensions (cf. Homology group), whose ranks are the Betti numbers; the fundamental group, a generalization of which to arbitrary dimension are the homotopy groups (cf. Homotopy group); and also the intersection ring of a manifold, soon to be replaced by the cohomology ring of Alexander and Kolmogorov, which is more general and more convenient for applications, being defined not only for polyhedra but also for an extremely wide class of topological spaces.

In the case of polyhedra important topological invariants are often, indeed principally, defined as properties of a simplicial complex which is a triangulation of the given polyhedron. Such definitions required a proof of an invariance theorem, asserting that the corresponding property does not change on passing from one triangulation of a given polyhedron to another triangulation of the same polyhedron or of a homeomorphic polyhedron.