We end our discussion with certain remarks about generalization of Euler’s formula for manifolds.
The idea of manifolds was implicit in the works of Riemann, however it seems that it appeared first in a fully developed form in Poincare’s works.
A manifold is a topological space X, which is locally Euclidean, i.e.,, each point p in X is contained in an open set in X which is topologically equivalent to Rn for some fixed n. a manifold is called if it has no boundary. For a large class of manifolds (e.g. smooth manifolds) cell division makes sense, by allowing higher dimensional analogue of triangles, called simplexes.
It was Poincare who proved the following far-reaching generalization of Euler’s formula:
Given a closed manifold X, the alternating sum f0-f1+f2-… of the numbers fi of i-dimensional faces of any cell division of X is independent of cell division and is thus a topological invariant e(X) of X.
In fact Poincare improved the above formula by introducing finer invariants bi(X), called Betti numbers as e(X) = b0(X) – b1(X) + b2(X) – b3(X) +…where bi(X) is the rank of Hi(X), which themselves form a sequence of finitely generated abelian groups, called the homolog groups naturally associated to X.
Thus the Euler characteristic theory all began with counting and indeed the interaction of topology-combinatorics remains an important area in mathematical research till date.