Mark Mahowald’s work on the homotopy groups of spheres

In this paper we attempt to survey some of the ideas Mark Mahowald has contributed to the study of the homotopy of spheres. Of course, this represents just a portion of Mahowald’s work; some other aspects are described elsewhere in this volume. Even within the restricted area of the homotopy of spheres, this survey can only touch on some of Mahowald’s most seminal contributions, and will leave aside many of his ideas on the subject. On the other hand we will try to set the stage upon which Mahowald has acted, so we give brief reviews of certain parts of homotopy theory in Sections 1 and 4. This includes the Image of J, the EHP-sequence, and the Adams spectral sequence. Of course we will not attempt an exhaustive survey of the relevant history of homotopy theory; for more information, the reader may look at G. W. Whitehead’s “Fifty years of homotopy theory ” [88] or at Chapter One of the second author’s book [76]. We will base our account on a discussion of three of Mahowald’s most influ-ential papers: The Metastable Homotopy of Sn (1967), A new infinite family i