From a noncomputability result to new interesting definitions and computability results

Abstract

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.

We explain that a solution can be found for the computing problem of the homotopy groups. The something is a K-theory group. We obtain in this way a new understanding of the algebraic K-theory groups and positive results about their computability.

At first we recall in a simplified way the necessary topology definitions so that even nontopologists should be able to follow this talk with interest.