We all know that determining the homotopy groups of a topological space is one of the central problems in homotopy theory. There is always a deep relation between the homotopy groups and other mathematical invariants which have wide applications in many fields such as geometric topology, algebra, algebraic K-theory and algebraic geometry. Via the view of homotopy theory, we try to approach the homotopy groups from two different sides: (i) Consider the general property of homotopy groups such as homotopy exponent; (ii) Consider the computation of stable homotopy groups of spheres.

This book concerns a central problem in algebraic topology-stable homotopy groups of spheres. Ever since J. P. Serres historic result showed that the homotopy groups of spheres are finitely generated in 1950s, the determination of homotopy groups of spheres has always been the focus of algebraic topologists. Up to now, still much has not been known to us. In our book, we consider the stable part of the homotopy groups of spheres. We will mainly apply two computation tools to detect nontrivial elements of the stable homotopy groups of spheres-Adams spectral sequence and May spectral sequence....