Exponent Problems in Homotopy Theory

0.1. The Moore conjecture and the Barratt conjecture. The fundamental problem in homotopy theory is how to determine homotopy groups. People found that it is very difficult to compute homotopy groups of finite complexes. So then we intend to know what we can say about homotopy groups, that is, how to study properties of homotopy groups instead of explicit calculations. Two famous conjectures on homotopy groups are so-called the Moore conjecture and the Barratt conjecture. The Moore conjecture is as follows: Moore conjecture: Let X be a simply connected finite complex. Assume that the Betti number βn = dimHn(ΩX;Q) has a polynomial growth on n. Then the p-torsion components of π∗(X) has bounded exponent, that is, there exists s >> 0 such that p · (p− torsion of π∗(X)) = 0. The Barratt conjecture is as follows: Barratt conjecture: Let X be a suspension such that the degree map [p] : X → X is null homotopic. Then p · π∗(X) = 0. In the Moore conjecture, the assumption that the rational homology of ΩX has polynomial growth is necessary. For instance, if X = S ∨ S, then any p-torsion component of π∗(X) does not have a bounded exponent. This assumption ⇒ X can