Algebraic Topology and Modular Forms

The problem of describing the homotopy groups of spheres has been fundamental to algebraic topology for around 80 years. There were periods when specific computations were important and periods when the emphasis favored theory. Many mathematical invariants have expressions in terms of homotopy groups, and at different times the subject has found itself located in geometric topology, algebra, algebraic K-theory, and algebraic geometry, among other areas. There are basically two approaches to the homotopy groups of spheres. The oldest makes direct use of geometry, and involves studying a map f : S → S in terms of the inverse image f(x) of a regular value. The oldest invariant, the degree of a map, is defined in this way, as was the original definition of the Hopf invariant. In the 1930’s Pontryagin [43, 42] showed that the homotopy class of a map f is completely determined by the geometry of the inverse image f(Bǫ(x)) of a small neighborhood of a regular value. He introduced the basics of framed cobordism and framed surgery, and identified the group πn+kS n with the cobordism group of smooth k-manifolds embedded in R and equipped with a framing of their stable normal bundles. The other approach to the homotopy groups of spheres involves comparing spheres to spaces whose homotopy groups are known. This method was introduced by Serre [50, 51, 16, 15] who used Eilenberg-MacLane spaces K(A, n), characterized by the property