Effective homology, a survey.∗

Algebraic topology consists in associating to topological spaces algebraic invariants in order to describe their essential properties. In some (exceptional) cases, it is possible in this way to classify a particular topological space inside a more or less large set of spaces or equivalence classes of spaces, for example up to homotopy equivalence. In the most important cases, a functor is defined, capable of working on some topological spaces to produce an algebraic object. And very frequently, if this functor works on a “finite” topological space, then the result is also a “finite” algebraic object. The meaning of the adjective “finite” is the following : an object is finite if it can be “reasonably” coded on a theoretical or actual machine.