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 influential papers: The Metastable Homotopy of S (1967), A new infinite family in 2π S ∗ (1977), and The Image of J in the EHP sequence (1982). One of Mahowald’s jokes is that in his world there are only two primes: 2, and the “infinite prime.” We will always work localized at 2, unless obviously otherwise. Both authors are happy to have this occasion to thank Mark for the many exciting and fruitful interactions we have had with him. This manuscript is missing some arrows, on pages 9, 11, 14, and 23.