Abstract
This chapter presents an account of the principal methods that are used to compute the v1-periodic homotopy groups of spheres and many compact simple Lie groups. The two main tools have been J-homology and the unstable Novikov spectral sequence (UNSS). The v1-periodic homotopy groups are important because for spaces such as spheres and compact simple Lie groups they give a significant portion of the actual homotopy groups and yet are often completely calculable. The goal of this chapter is to explain how those calculations can be made.
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