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.
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