A solution to the Lecture 1 Arf-Kervaire invariant problem Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester A solution to the Arf-Kervaire invariant problem Instituto Superior Técnico Lisbon May 5, 2009 Background and history Exotic spheres The Pontrjagin-Thom construction The J-homomorphism The use of surgery The Hirzebruch signature theorem Mike Hill The Arf invariant University of Virginia Browder’s theorem Spectral sequences Mike Hopkins The Adams spectral sequence Harvard University The Adams-Novikov Doug Ravenel spectral sequence Our strategy University of Rochester 1.1 The school at Northwestern is as fertile as manure, full of deep insights, some rather obscure. Mark loves those damn thetas like a sister or brother, and if you don’t like one proof, he’ll give you another. A solution to the A poem by Frank Adams written in the 70s Arf-Kervaire invariant problem Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester Background and history Exotic spheres The Pontrjagin-Thom construction The J-homomorphism The use of surgery The Hirzebruch signature theorem The Arf invariant Browder’s theorem Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence Our strategy 1.2 full of deep insights, some rather obscure. Mark loves those damn thetas like a sister or brother, and if you don’t like one proof, he’ll give you another. A solution to the A poem by Frank Adams written in the 70s Arf-Kervaire invariant problem Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester The school at Northwestern is as fertile as manure, Background and history Exotic spheres The Pontrjagin-Thom construction The J-homomorphism The use of surgery The Hirzebruch signature theorem The Arf invariant Browder’s theorem Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence Our strategy 1.2 Mark loves those damn thetas like a sister or brother, and if you don’t like one proof, he’ll give you another. A solution to the A poem by Frank Adams written in the 70s Arf-Kervaire invariant problem Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester The school at Northwestern is as fertile as manure, full of deep insights, Background and history some rather obscure. Exotic spheres The Pontrjagin-Thom construction The J-homomorphism The use of surgery The Hirzebruch signature theorem The Arf invariant Browder’s theorem Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence Our strategy 1.2 and if you don’t like one proof, he’ll give you another. A solution to the A poem by Frank Adams written in the 70s Arf-Kervaire invariant problem Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester The school at Northwestern is as fertile as manure, full of deep insights, Background and history some rather obscure. Exotic spheres The Pontrjagin-Thom construction The J-homomorphism The use of surgery The Hirzebruch signature theorem Mark loves those damn thetas The Arf invariant like a sister or brother, Browder’s theorem Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence Our strategy 1.2 A solution to the A poem by Frank Adams written in the 70s Arf-Kervaire invariant problem Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester The school at Northwestern is as fertile as manure, full of deep insights, Background and history some rather obscure. Exotic spheres The Pontrjagin-Thom construction The J-homomorphism The use of surgery The Hirzebruch signature theorem Mark loves those damn thetas The Arf invariant like a sister or brother, Browder’s theorem Spectral sequences and if you don’t like one proof, The Adams spectral sequence he’ll give you another. The Adams-Novikov spectral sequence Our strategy 1.2 • It says that a certain algebraically defined invariant Φ(M) (the Arf-Kervaire invariant, to be defined later) on certain manifolds M is always zero. • It says that a related invariant (having to do with secondary cohomology operations) defined on maps between high dimensional spheres is always zero. The question answered by our theorem is nearly 50 years old. It is known as the Arf-Kervaire invariant problem. There were several unsuccessful attempts to solve it in the 1970s. They were all aimed at proving the opposite of what we have proved. A solution to the Our main result Arf-Kervaire invariant problem Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester Our main theorem can be stated in two different but equivalent ways: Background and history Exotic spheres The Pontrjagin-Thom construction The J-homomorphism The use of surgery The Hirzebruch signature theorem The Arf invariant Browder’s theorem Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence Our strategy 1.3 • It says that a related invariant (having to do with secondary cohomology operations) defined on maps between high dimensional spheres is always zero. The question answered by our theorem is nearly 50 years old. It is known as the Arf-Kervaire invariant problem. There were several unsuccessful attempts to solve it in the 1970s. They were all aimed at proving the opposite of what we have proved. A solution to the Our main result Arf-Kervaire invariant problem Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester Our main theorem can be stated in two different but equivalent ways: • It says that a certain algebraically defined invariant Φ(M) (the Arf-Kervaire invariant, to be defined later) on certain manifolds M is always zero. Background and history Exotic spheres The Pontrjagin-Thom construction The J-homomorphism The use of surgery The Hirzebruch signature theorem The Arf invariant Browder’s theorem Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence Our strategy 1.3 The question answered by our theorem is nearly 50 years old. It is known as the Arf-Kervaire invariant problem. There were several unsuccessful attempts to solve it in the 1970s. They were all aimed at proving the opposite of what we have proved. A solution to the Our main result Arf-Kervaire invariant problem Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester Our main theorem can be stated in two different but equivalent ways: • It says that a certain algebraically defined invariant Φ(M) (the Arf-Kervaire invariant, to be defined later) on certain manifolds M is always zero. Background and • It says that a related invariant (having to do with history Exotic spheres secondary cohomology operations) defined on maps The Pontrjagin-Thom construction between high dimensional spheres is always zero. The J-homomorphism The use of surgery The Hirzebruch signature theorem The Arf invariant Browder’s theorem Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence Our strategy 1.3 It is known as the Arf-Kervaire invariant problem. There were several unsuccessful attempts to solve it in the 1970s. They were all aimed at proving the opposite of what we have proved. A solution to the Our main result Arf-Kervaire invariant problem Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester Our main theorem can be stated in two different but equivalent ways: • It says that a certain algebraically defined invariant Φ(M) (the Arf-Kervaire invariant, to be defined later) on certain manifolds M is always zero. Background and • It says that a related invariant (having to do with history Exotic spheres secondary cohomology operations) defined on maps The Pontrjagin-Thom construction between high dimensional spheres is always zero. The J-homomorphism The use of surgery The Hirzebruch signature The question answered by our theorem is nearly 50 years old. theorem The Arf invariant Browder’s theorem Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence Our strategy 1.3 There were several unsuccessful attempts to solve it in the 1970s. They were all aimed at proving the opposite of what we have proved. A solution to the Our main result Arf-Kervaire invariant problem Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester Our main theorem can be stated in two different but equivalent ways: • It says that a certain algebraically defined invariant Φ(M) (the Arf-Kervaire invariant, to be defined later) on certain manifolds M is always zero. Background and • It says that a related invariant (having to do with history Exotic spheres secondary cohomology operations) defined on maps The Pontrjagin-Thom construction between high dimensional spheres is always zero. The J-homomorphism The use of surgery The Hirzebruch signature The question answered by our theorem is nearly 50 years old. theorem It is known as the Arf-Kervaire invariant problem. The Arf invariant Browder’s theorem Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence Our strategy 1.3 They were all aimed at proving the opposite of what we have proved. A solution to the Our main result Arf-Kervaire invariant problem Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester Our main theorem can be stated in two different but equivalent ways: • It says that a certain algebraically defined invariant Φ(M) (the Arf-Kervaire invariant, to be defined later) on certain manifolds M is always zero. Background and • It says that a related invariant (having to do with history Exotic spheres secondary cohomology operations) defined on maps The Pontrjagin-Thom construction between high dimensional spheres is always zero.