From a Noncomputability Result to new interesting Definitions and Computability Results

In this talk we examine the strange situation encountered in algebraic topology: on one hand no general algorithm is able to decide whether some topological space is simply connected; this is an easy consequence of the undecidability of the word problem. On the other hand most of the important results in algebraic topology assume that the spaces under consideration are simply connected! So that one can ask for algorithms that use some method or other, and always compute something, in such a way that if the space given as input is simply connected, then the result obtained is the good one. The problem is to explain what is something in general.