Preface

The main idea of algebraic topology is to try to use algebraic structures to say something qualitative about topological spaces. Over time one developed many such algebraic invariants. Perhaps the simplest one to define is the so-called fundamental group of a space. Most probably the reader has seen its definition in one form or another. Roughly speaking, one chooses a point in space and then defines some calculus using all possible loops anchored at this point. A clear weakness of such an invariant is that it does not tell us anything beyond the first few dimensions: taking any space and then attaching balls of dimension 3 and higher to that space will not be detected by the fundamental group at all. That in itself can be fixed by introducing higher-dimensional homotopy groups. What is much worse, from the point of view of applied topology, is that not only are these invariants hard to compute, but the famous result of P.S. Novikov, [No55], actually tells us that it is not decidable whether or not the fundamental group is trivial.