Browder’s work on the Arf-Kervaire invariant problem Mike Hill Mike Hopkins Browder’s work on Arf-Kervaire Doug Ravenel invariant problem Panorama of Topology A Conference in Honor of William Browder May 10, 2012 Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester 1.1 Browder’s Theorem (1969) The Kervaire invariant of a smooth framed (4m + 2)-manifold M can be nontrivial only if m = 2j−1 − 1 for some j > 0. This 2 happens iff the element hj is a permanent cycle in the Adams spectral sequence. Browder’s work on Browder’s theorem and its impact the Arf-Kervaire invariant problem Mike Hill Mike Hopkins In 1969 Browder proved a remarkable theorem about the Doug Ravenel Kervaire invariant. 1.2 Browder’s Theorem (1969) The Kervaire invariant of a smooth framed (4m + 2)-manifold M can be nontrivial only if m = 2j−1 − 1 for some j > 0. This 2 happens iff the element hj is a permanent cycle in the Adams spectral sequence. Browder’s work on Browder’s theorem and its impact the Arf-Kervaire invariant problem Mike Hill Mike Hopkins In 1969 Browder proved a remarkable theorem about the Doug Ravenel Kervaire invariant. 1.2 This 2 happens iff the element hj is a permanent cycle in the Adams spectral sequence. Browder’s work on Browder’s theorem and its impact the Arf-Kervaire invariant problem Mike Hill Mike Hopkins In 1969 Browder proved a remarkable theorem about the Doug Ravenel Kervaire invariant. Browder’s Theorem (1969) The Kervaire invariant of a smooth framed (4m + 2)-manifold M can be nontrivial only if m = 2j−1 − 1 for some j > 0. 1.2 Browder’s work on Browder’s theorem and its impact the Arf-Kervaire invariant problem Mike Hill Mike Hopkins In 1969 Browder proved a remarkable theorem about the Doug Ravenel Kervaire invariant. Browder’s Theorem (1969) The Kervaire invariant of a smooth framed (4m + 2)-manifold M can be nontrivial only if m = 2j−1 − 1 for some j > 0. This 2 happens iff the element hj is a permanent cycle in the Adams spectral sequence. 1.2 Browder’s work on Browder’s theorem and its impact (continued) the Arf-Kervaire invariant problem Browder’s Theorem (1969) Mike Hill Mike Hopkins The Kervaire invariant of a smooth framed (4m + 2)-manifold Doug Ravenel M can be nontrivial only if m = 2j−1 − 1 for some j > 0. This 2 happens iff the element hj is a permanent cycle in the Adams spectral sequence. 1.3 Browder’s work on Browder’s theorem and its impact (continued) the Arf-Kervaire invariant problem Browder’s Theorem (1969) Mike Hill Mike Hopkins The Kervaire invariant of a smooth framed (4m + 2)-manifold Doug Ravenel M can be nontrivial only if m = 2j−1 − 1 for some j > 0. This 2 happens iff the element hj is a permanent cycle in the Adams spectral sequence. 1.3 Browder’s work on Browder’s theorem and its impact (continued) the Arf-Kervaire invariant problem Browder’s Theorem (1969) Mike Hill Mike Hopkins The Kervaire invariant of a smooth framed (4m + 2)-manifold Doug Ravenel M can be nontrivial only if m = 2j−1 − 1 for some j > 0. This 2 happens iff the element hj is a permanent cycle in the Adams spectral sequence. 1.4 This result established a link between surgery theory, specifically an unanswered question in the Kervaire-Milnor classification of exotic spheres, and stable homotopy theory, specifically the Adams spectral sequence. This connection made the problem of constructing a smooth framed manifold with nontrivial Kervaire invariant in dimension 2j+1 − 2 a cause celebre in algebraic topology throughout the 1970s. For 40 years it was the definitive theorem on this subject. Browder’s work on Browder’s theorem and its impact (continued) the Arf-Kervaire invariant problem Mike Hill Mike Hopkins Doug Ravenel Browder’s Theorem (1969) The Kervaire invariant of a smooth framed (4m + 2)-manifold M can be nontrivial only if m = 2j−1 − 1 for some j > 0. This 2 happens iff the element hj is a permanent cycle in the Adams spectral sequence. 1.5 and stable homotopy theory, specifically the Adams spectral sequence. This connection made the problem of constructing a smooth framed manifold with nontrivial Kervaire invariant in dimension 2j+1 − 2 a cause celebre in algebraic topology throughout the 1970s. For 40 years it was the definitive theorem on this subject. Browder’s work on Browder’s theorem and its impact (continued) the Arf-Kervaire invariant problem Mike Hill Mike Hopkins Doug Ravenel Browder’s Theorem (1969) The Kervaire invariant of a smooth framed (4m + 2)-manifold M can be nontrivial only if m = 2j−1 − 1 for some j > 0. This 2 happens iff the element hj is a permanent cycle in the Adams spectral sequence. This result established a link between surgery theory, specifically an unanswered question in the Kervaire-Milnor classification of exotic spheres, 1.5 This connection made the problem of constructing a smooth framed manifold with nontrivial Kervaire invariant in dimension 2j+1 − 2 a cause celebre in algebraic topology throughout the 1970s. For 40 years it was the definitive theorem on this subject. Browder’s work on Browder’s theorem and its impact (continued) the Arf-Kervaire invariant problem Mike Hill Mike Hopkins Doug Ravenel Browder’s Theorem (1969) The Kervaire invariant of a smooth framed (4m + 2)-manifold M can be nontrivial only if m = 2j−1 − 1 for some j > 0. This 2 happens iff the element hj is a permanent cycle in the Adams spectral sequence. This result established a link between surgery theory, specifically an unanswered question in the Kervaire-Milnor classification of exotic spheres, and stable homotopy theory, specifically the Adams spectral sequence. 1.5 For 40 years it was the definitive theorem on this subject. Browder’s work on Browder’s theorem and its impact (continued) the Arf-Kervaire invariant problem Mike Hill Mike Hopkins Doug Ravenel Browder’s Theorem (1969) The Kervaire invariant of a smooth framed (4m + 2)-manifold M can be nontrivial only if m = 2j−1 − 1 for some j > 0. This 2 happens iff the element hj is a permanent cycle in the Adams spectral sequence. This result established a link between surgery theory, specifically an unanswered question in the Kervaire-Milnor classification of exotic spheres, and stable homotopy theory, specifically the Adams spectral sequence. This connection made the problem of constructing a smooth framed manifold with nontrivial Kervaire invariant in dimension 2j+1 − 2 a cause celebre in algebraic topology throughout the 1970s. 1.5 Browder’s work on Browder’s theorem and its impact (continued) the Arf-Kervaire invariant problem Mike Hill Mike Hopkins Doug Ravenel Browder’s Theorem (1969) The Kervaire invariant of a smooth framed (4m + 2)-manifold M can be nontrivial only if m = 2j−1 − 1 for some j > 0. This 2 happens iff the element hj is a permanent cycle in the Adams spectral sequence. This result established a link between surgery theory, specifically an unanswered question in the Kervaire-Milnor classification of exotic spheres, and stable homotopy theory, specifically the Adams spectral sequence. This connection made the problem of constructing a smooth framed manifold with nontrivial Kervaire invariant in dimension 2j+1 − 2 a cause celebre in algebraic topology throughout the 1970s. For 40 years it was the definitive theorem on this subject. 1.5 This would n correspond to an element θj 2 πn+2j+1−2S for large n. Some homotopy theorists, most no- tably Mahowald, speculated about what would happen if θj existed for all j. He derived numerous conse- quences about homotopy groups of spheres. The possible nonexistence of the θj for large j was known as the Doomsday Hypothesis. Mark Mahowald Browder’s work on Browder’s theorem and its impact (continued) the Arf-Kervaire invariant problem Mike Hill Mike Hopkins Doug Ravenel Browder’s theorem says that there is a framed manifold with nontrivial Kervaire invariant in dimension 2j+1 − 2 iff a certain element in the Adams spectral sequence survives. 1.6 Some homotopy theorists, most no- tably Mahowald, speculated about what would happen if θj existed for all j. He derived numerous conse- quences about homotopy groups of spheres. The possible nonexistence of the θj for large j was known as the Doomsday Hypothesis. Mark Mahowald Browder’s work on Browder’s theorem and its impact (continued) the Arf-Kervaire invariant problem Mike Hill Mike Hopkins Doug Ravenel Browder’s theorem says that there is a framed manifold with nontrivial Kervaire invariant in dimension 2j+1 − 2 iff a certain element in the Adams spectral sequence survives. This would n correspond to an element θj 2 πn+2j+1−2S for large n. 1.6 He derived numerous conse- quences about homotopy groups of spheres. The possible nonexistence of the θj for large j was known as the Doomsday Hypothesis. Browder’s work on Browder’s theorem and its impact (continued) the Arf-Kervaire invariant problem Mike Hill Mike Hopkins Doug Ravenel Browder’s theorem says that there is a framed manifold with nontrivial Kervaire invariant in dimension 2j+1 − 2 iff a certain element in the Adams spectral sequence survives. This would n correspond to an element θj 2 πn+2j+1−2S for large n. Some homotopy theorists, most no- tably Mahowald, speculated about what would happen if θj existed for all j. Mark Mahowald 1.6 The possible nonexistence of the θj for large j was known as the Doomsday Hypothesis.