A Solution to the Arf-Kervaire Invariant Problem

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

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. 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. There were The Arf invariant Browder’s theorem

several unsuccessful attempts to solve it in the 1970s. Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.3 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. There were The Arf invariant Browder’s theorem

several unsuccessful attempts to solve it in the 1970s. They Spectral sequences The Adams spectral were all aimed at proving the opposite of what we have proved. sequence The Adams-Novikov spectral sequence

Our strategy

1.3 We now know that the world of homotopy theory is different from what they had envisioned. Barratt and Mahowald called the possible nonexistence of the θj for large j the Doomsday Hypothesis.

After 1980, the problem faded into the background because it was thought to be too hard. Our proof is two giant steps away from anything that was attempted in the 70s.

Main Theorem 0 The Arf-Kervaire elements θj ∈ π2j+1−2(S ) do not exist for j ≥ 7.

The θj in the theorem is the name given to a hypothetical manifold or map between spheres for which the Arf-Kervaire invariant is nontrivial. It has long been known that such things can exist only in dimensions that are 2 less than a power of 2.

A solution to the Our main result (continued) Arf-Kervaire invariant problem

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

1.4 Barratt and Mahowald called the possible nonexistence of the θj for large j the Doomsday Hypothesis.

After 1980, the problem faded into the background because it was thought to be too hard. Our proof is two giant steps away from anything that was attempted in the 70s.

Main Theorem 0 The Arf-Kervaire elements θj ∈ π2j+1−2(S ) do not exist for j ≥ 7.

A solution to the Our main result (continued) Arf-Kervaire invariant problem

Some homotopy theorists, most notably Mark Mahowald, Mike Hill University of Virginia Mike Hopkins Harvard University conjectured the opposite of what we have proved, and had Doug Ravenel University of Rochester derived numerous consequences about homotopy groups of spheres. We now know that the world of homotopy theory is different from what they had envisioned.

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

1.4 After 1980, the problem faded into the background because it was thought to be too hard. Our proof is two giant steps away from anything that was attempted in the 70s.

Main Theorem 0 The Arf-Kervaire elements θj ∈ π2j+1−2(S ) do not exist for j ≥ 7.

A solution to the Our main result (continued) Arf-Kervaire invariant problem

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

1.4 Our proof is two giant steps away from anything that was attempted in the 70s.

Main Theorem 0 The Arf-Kervaire elements θj ∈ π2j+1−2(S ) do not exist for j ≥ 7.

A solution to the Our main result (continued) Arf-Kervaire invariant problem

Background and After 1980, the problem faded into the background because it history was thought to be too hard. 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

1.4 Main Theorem 0 The Arf-Kervaire elements θj ∈ π2j+1−2(S ) do not exist for j ≥ 7.

A solution to the Our main result (continued) Arf-Kervaire invariant problem

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.4 The θj in the theorem is the name given to a hypothetical manifold or map between spheres for which the Arf-Kervaire invariant is nontrivial. It has long been known that such things can exist only in dimensions that are 2 less than a power of 2.

A solution to the Our main result (continued) Arf-Kervaire invariant problem

Background and After 1980, the problem faded into the background because it history was thought to be too hard. Our proof is two giant steps away Exotic spheres The Pontrjagin-Thom from anything that was attempted in the 70s. construction The J-homomorphism The use of surgery The Hirzebruch signature Main Theorem theorem 0 The Arf invariant The Arf-Kervaire elements θj ∈ π2j+1−2(S ) do not exist for Browder’s theorem j ≥ 7. Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.4 It has long been known that such things can exist only in dimensions that are 2 less than a power of 2.

A solution to the Our main result (continued) Arf-Kervaire invariant problem

manifold or map between spheres for which the Arf-Kervaire Our strategy invariant is nontrivial.

1.4 A solution to the Our main result (continued) Arf-Kervaire invariant problem

manifold or map between spheres for which the Arf-Kervaire Our strategy invariant is nontrivial. It has long been known that such things can exist only in dimensions that are 2 less than a power of 2.

1.4 These maps were constructed by Hopf in 1930. Their advent marked the beginning of modern homotopy theory.

Here is Hopf’s definition of the map η : S3 → S2. • Think of S3 as the unit sphere in a 2-dimensional complex vector space C2. • Think of S2 as the one-point compactification of the complex numbers, C ∪ {∞}. 3 • For (z1, z2) ∈ S , define z1/z2 for z2 6= 0 η(z1, z2) = ∞ for z2 = 0

• Maps ν : S7 → S4 and σ : S15 → S8 can be deﬁned in a similar way using quaternions and Cayley numbers or octonions.

A solution to the Our main result (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia θ1, θ2 and θ3 are the squares of the Hopf maps η ∈ π1, ν ∈ π3 Mike Hopkins Harvard University Doug Ravenel University of Rochester and σ ∈ π7.

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

1.5 Their advent marked the beginning of modern homotopy theory.

• Maps ν : S7 → S4 and σ : S15 → S8 can be deﬁned in a similar way using quaternions and Cayley numbers or octonions.

A solution to the Our main result (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia θ1, θ2 and θ3 are the squares of the Hopf maps η ∈ π1, ν ∈ π3 Mike Hopkins Harvard University Doug Ravenel University of Rochester and σ ∈ π7. These maps were constructed by Hopf in 1930.

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

1.5 Here is Hopf’s definition of the map η : S3 → S2. • Think of S3 as the unit sphere in a 2-dimensional complex vector space C2. • Think of S2 as the one-point compactification of the complex numbers, C ∪ {∞}. 3 • For (z1, z2) ∈ S , define z1/z2 for z2 6= 0 η(z1, z2) = ∞ for z2 = 0

• Maps ν : S7 → S4 and σ : S15 → S8 can be deﬁned in a similar way using quaternions and Cayley numbers or octonions.

A solution to the Our main result (continued) Arf-Kervaire invariant problem

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

1.5 • Think of S3 as the unit sphere in a 2-dimensional complex vector space C2. • Think of S2 as the one-point compactiﬁcation of the complex numbers, C ∪ {∞}. 3 • For (z1, z2) ∈ S , deﬁne z1/z2 for z2 6= 0 η(z1, z2) = ∞ for z2 = 0

• Maps ν : S7 → S4 and σ : S15 → S8 can be deﬁned in a similar way using quaternions and Cayley numbers or octonions.

A solution to the Our main result (continued) Arf-Kervaire invariant problem

Here is Hopf’s deﬁnition of the map η : S3 → S2.

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

1.5 • Think of S2 as the one-point compactiﬁcation of the complex numbers, C ∪ {∞}. 3 • For (z1, z2) ∈ S , deﬁne z1/z2 for z2 6= 0 η(z1, z2) = ∞ for z2 = 0

• Maps ν : S7 → S4 and σ : S15 → S8 can be deﬁned in a similar way using quaternions and Cayley numbers or octonions.

A solution to the Our main result (continued) Arf-Kervaire invariant problem

Here is Hopf’s deﬁnition of the map η : S3 → S2. • Think of S3 as the unit sphere in a 2-dimensional complex 2 vector space C . 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

1.5 3 • For (z1, z2) ∈ S , deﬁne z1/z2 for z2 6= 0 η(z1, z2) = ∞ for z2 = 0

• Maps ν : S7 → S4 and σ : S15 → S8 can be deﬁned in a similar way using quaternions and Cayley numbers or octonions.

A solution to the Our main result (continued) Arf-Kervaire invariant problem

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.5 • Maps ν : S7 → S4 and σ : S15 → S8 can be deﬁned in a similar way using quaternions and Cayley numbers or octonions.

A solution to the Our main result (continued) Arf-Kervaire invariant problem

Here is Hopf’s definition of the map η : S3 → S2. • Think of S3 as the unit sphere in a 2-dimensional complex 2 vector space C . Background and history • Think of S2 as the one-point compactification of the Exotic spheres The Pontrjagin-Thom complex numbers, C ∪ {∞}. construction The J-homomorphism 3 The use of surgery • For (z1, z2) ∈ S , define The Hirzebruch signature theorem The Arf invariant z1/z2 for z2 6= 0 Browder’s theorem η(z1, z2) = ∞ for z2 = 0 Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.5 A solution to the Our main result (continued) Arf-Kervaire invariant problem

1.5 Main Theorem 0 The Arf-Kervaire elements θj ∈ π2j+1−2(S ) do not exist for j ≥ 7.

• θ4 ∈ π30 and θ5 ∈ π62 were constructed in the ’60s and ’70s. The latter is the subject of a paper by Barratt-Jones-Mahowald published in 1984.

• The status of θ6 ∈ π126 is still open.

A solution to the Our main result (continued) 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

1.6 • θ4 ∈ π30 and θ5 ∈ π62 were constructed in the ’60s and ’70s. The latter is the subject of a paper by Barratt-Jones-Mahowald published in 1984.

• The status of θ6 ∈ π126 is still open.

A solution to the Our main result (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Main Theorem 0 The Arf-Kervaire elements θj ∈ π2j+1−2(S ) do not exist for j ≥ 7. 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

1.6 • The status of θ6 ∈ π126 is still open.

A solution to the Our main result (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Main Theorem 0 The Arf-Kervaire elements θj ∈ π2j+1−2(S ) do not exist for j ≥ 7. Background and history • θ4 ∈ π30 and θ5 ∈ π62 were constructed in the ’60s and Exotic spheres The Pontrjagin-Thom ’70s. The latter is the subject of a paper by construction The J-homomorphism Barratt-Jones-Mahowald published in 1984. 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

1.6 A solution to the Our main result (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.6 Milnor’s Theorem (1956) Existence of exotic spheres. There are manifolds homeomorphic to S7 but not diffeomorphic to it.

Kervaire’s Theorem (1960) Existence of nonsmoothable manifolds. There is a 10-dimensional topological manifold with no differentiable structure. These theorems are opposite sides of the same coin.

A solution to the The work of Kervaire and Milnor Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Fifty years ago the topological community was startled by two results.

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

1.7 Kervaire’s Theorem (1960) Existence of nonsmoothable manifolds. There is a 10-dimensional topological manifold with no differentiable structure. These theorems are opposite sides of the same coin.

A solution to the The work of Kervaire and Milnor Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.7 These theorems are opposite sides of the same coin.

A solution to the The work of Kervaire and Milnor Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Fifty years ago the topological community was startled by two results. Milnor’s Theorem (1956) Existence of exotic spheres. There are manifolds 7 homeomorphic to S but not diffeomorphic to it. Background and history Exotic spheres The Pontrjagin-Thom Kervaire’s Theorem (1960) construction The J-homomorphism Existence of nonsmoothable manifolds. There is a The use of surgery The Hirzebruch signature 10-dimensional topological manifold with no differentiable theorem The Arf invariant structure. Browder’s theorem Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.7 A solution to the The work of Kervaire and Milnor Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Our strategy

1.7 Let Θk denote the group (under connected sum) of diffeomorphism classes of smooth manifolds Σk homotopy equivalent to Sk .

By the Poincaré Conjecture in dimensions ≥ 5 (proved by Smale in 1962), homotopy equivalence here implies homeomorphism. (The Poincaré Conjecture in dimensions 3 and 4 was not solved until much later. Kervaire-Milnor did not treat those cases.)

Such a Σk , when embedded in Euclidean space, has a (nonunique) framing on its normal bundle and thus represents framed an element in the framed cobordism group Ωk . By the framed Pontrjagin-Thom construction, Ωk is isomorphic to the 0 stable k-stem πk (S ).

A solution to the The classiﬁcation of exotic spheres 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

1.8 By the Poincaré Conjecture in dimensions ≥ 5 (proved by Smale in 1962), homotopy equivalence here implies homeomorphism. (The Poincaré Conjecture in dimensions 3 and 4 was not solved until much later. Kervaire-Milnor did not treat those cases.)

A solution to the The classiﬁcation of exotic spheres Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester Let Θk denote the group (under connected sum) of diffeomorphism classes of smooth manifolds Σk homotopy equivalent to Sk .

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

1.8 (The Poincaré Conjecture in dimensions 3 and 4 was not solved until much later. Kervaire-Milnor did not treat those cases.)

A solution to the The classiﬁcation of exotic spheres Arf-Kervaire invariant problem

By the Poincaré Conjecture in dimensions ≥ 5 (proved by Smale in 1962), homotopy equivalence here implies homeomorphism. 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

1.8 Such a Σk , when embedded in Euclidean space, has a (nonunique) framing on its normal bundle and thus represents framed an element in the framed cobordism group Ωk . By the framed Pontrjagin-Thom construction, Ωk is isomorphic to the 0 stable k-stem πk (S ).

A solution to the The classiﬁcation of exotic spheres Arf-Kervaire invariant problem

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.8 By the framed Pontrjagin-Thom construction, Ωk is isomorphic to the 0 stable k-stem πk (S ).

A solution to the The classiﬁcation of exotic spheres Arf-Kervaire invariant problem

By the Poincaré Conjecture in dimensions ≥ 5 (proved by Smale in 1962), homotopy equivalence here implies homeomorphism. (The Poincaré Conjecture in dimensions 3 Background and history and 4 was not solved until much later. Kervaire-Milnor did not Exotic spheres The Pontrjagin-Thom treat those cases.) construction The J-homomorphism The use of surgery k The Hirzebruch signature Such a Σ , when embedded in Euclidean space, has a theorem (nonunique) framing on its normal bundle and thus represents The Arf invariant Browder’s theorem framed an element in the framed cobordism group Ωk . Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.8 A solution to the The classiﬁcation of exotic spheres Arf-Kervaire invariant problem

By the Poincaré Conjecture in dimensions ≥ 5 (proved by Smale in 1962), homotopy equivalence here implies homeomorphism. (The Poincaré Conjecture in dimensions 3 Background and history and 4 was not solved until much later. Kervaire-Milnor did not Exotic spheres The Pontrjagin-Thom treat those cases.) construction The J-homomorphism The use of surgery k The Hirzebruch signature Such a Σ , when embedded in Euclidean space, has a theorem (nonunique) framing on its normal bundle and thus represents The Arf invariant Browder’s theorem framed an element in the framed cobordism group Ωk . By the Spectral sequences The Adams spectral Pontrjagin-Thom construction, Ωframed is isomorphic to the sequence k The Adams-Novikov 0 spectral sequence stable k-stem πk (S ). Our strategy

1.8 The framing gives a projection

f (V , ∂V ) M / (Dn, ∂Dn) / Dn/∂Dn =∼ Sn

We can extend this map to Rn+k and its one-point compactiﬁcation Sn+k by sending everything outside of V to the base point (or point at ∞) in Sn.

˜ n+k n The resulting map fM : S → S represents an element in the n homotopy group πn+k (S ), which for large n is isomorphic to the stable k-stem πk . Pontrjagin showed that a cobordism ˜ ˜ between M1 and M2 leads to a homotopy between fM1 and fM2 .

A solution to the The Pontrjagin-Thom construction Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester A k-dimensional framed manifold M (such as Σk ) can be embedded in a Euclidean space Rn+k in such a way that it has a tubular neighborhood V homeomorphic to M × Dn.

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

1.9 We can extend this map to Rn+k and its one-point compactiﬁcation Sn+k by sending everything outside of V to the base point (or point at ∞) in Sn.

A solution to the The Pontrjagin-Thom construction Arf-Kervaire invariant problem

f (V , ∂V ) M / (Dn, ∂Dn) / Dn/∂Dn =∼ Sn 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

1.9 ˜ n+k n The resulting map fM : S → S represents an element in the n homotopy group πn+k (S ), which for large n is isomorphic to the stable k-stem πk . Pontrjagin showed that a cobordism ˜ ˜ between M1 and M2 leads to a homotopy between fM1 and fM2 .

A solution to the The Pontrjagin-Thom construction Arf-Kervaire invariant problem

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.9 Pontrjagin showed that a cobordism ˜ ˜ between M1 and M2 leads to a homotopy between fM1 and fM2 .

A solution to the The Pontrjagin-Thom construction Arf-Kervaire invariant problem

f (V , ∂V ) M / (Dn, ∂Dn) / Dn/∂Dn =∼ Sn Background and history n+k Exotic spheres We can extend this map to R and its one-point The Pontrjagin-Thom n+k construction compactiﬁcation S by sending everything outside of V to The J-homomorphism the base point (or point at ∞) in Sn. The use of surgery The Hirzebruch signature theorem The Arf invariant ˜ n+k n The resulting map fM : S → S represents an element in the Browder’s theorem n Spectral sequences homotopy group π + (S ), which for large n is isomorphic to n k The Adams spectral the stable k-stem π . sequence k The Adams-Novikov spectral sequence

Our strategy

1.9 A solution to the The Pontrjagin-Thom construction Arf-Kervaire invariant problem

1.9 • Any map Sn+k → Sn is homotopic to one associated to a framed k-manifold in this way. • Any homotopy between two such maps is induced by a cobordism.

This means that πk is isomorphic to the cobordism group of framed k-manifolds.

Thom used transversality to prove similar theorems in which the framing is replaced by a weaker structure on the normal bundle of a manifold M. This is the basis of modern cobordism theory. He won the Fields Medal for this work in 1958.

A solution to the The Pontrjagin-Thom construction (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

He also showed the converse:

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

1.10 • Any homotopy between two such maps is induced by a cobordism.

This means that πk is isomorphic to the cobordism group of framed k-manifolds.

A solution to the The Pontrjagin-Thom construction (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

He also showed the converse: • Any map Sn+k → Sn is homotopic to one associated to a framed k-manifold in this way.

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

1.10 This means that πk is isomorphic to the cobordism group of framed k-manifolds.

A solution to the The Pontrjagin-Thom construction (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

He also showed the converse: • Any map Sn+k → Sn is homotopic to one associated to a framed k-manifold in this way. • Any homotopy between two such maps is induced by a cobordism. 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

1.10 Thom used transversality to prove similar theorems in which the framing is replaced by a weaker structure on the normal bundle of a manifold M. This is the basis of modern cobordism theory. He won the Fields Medal for this work in 1958.

A solution to the The Pontrjagin-Thom construction (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.10 This is the basis of modern cobordism theory. He won the Fields Medal for this work in 1958.

A solution to the The Pontrjagin-Thom construction (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

He also showed the converse: • Any map Sn+k → Sn is homotopic to one associated to a framed k-manifold in this way. • Any homotopy between two such maps is induced by a cobordism. Background and history This means that πk is isomorphic to the cobordism group of Exotic spheres The Pontrjagin-Thom framed k-manifolds. construction The J-homomorphism The use of surgery Thom used transversality to prove similar theorems in which The Hirzebruch signature theorem the framing is replaced by a weaker structure on the normal The Arf invariant bundle of a manifold M. Browder’s theorem Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.10 He won the Fields Medal for this work in 1958.

A solution to the The Pontrjagin-Thom construction (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

He also showed the converse: • Any map Sn+k → Sn is homotopic to one associated to a framed k-manifold in this way. • Any homotopy between two such maps is induced by a cobordism. Background and history This means that πk is isomorphic to the cobordism group of Exotic spheres The Pontrjagin-Thom framed k-manifolds. construction The J-homomorphism The use of surgery Thom used transversality to prove similar theorems in which The Hirzebruch signature theorem the framing is replaced by a weaker structure on the normal The Arf invariant bundle of a manifold M. This is the basis of modern cobordism Browder’s theorem Spectral sequences theory. The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.10 A solution to the The Pontrjagin-Thom construction (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

He also showed the converse: • Any map Sn+k → Sn is homotopic to one associated to a framed k-manifold in this way. • Any homotopy between two such maps is induced by a cobordism. Background and history This means that πk is isomorphic to the cobordism group of Exotic spheres The Pontrjagin-Thom framed k-manifolds. construction The J-homomorphism The use of surgery Thom used transversality to prove similar theorems in which The Hirzebruch signature theorem the framing is replaced by a weaker structure on the normal The Arf invariant bundle of a manifold M. This is the basis of modern cobordism Browder’s theorem Spectral sequences theory. He won the Fields Medal for this work in 1958. The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.10 When Mk is a sphere (or a homotopy sphere) we get an element in πk (SO(n)). This leads to the deﬁnition of the Hopf-Whitehead J-homomorphism

n J : πk (SO(n)) → πn+k (S ).

Both of these groups are independent of n when n is large, so we get 0 J : πk (SO) → πk (S ). The group of the left and its image on the right are known for all k.

A solution to the The classiﬁcation of exotic spheres (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Two framings on a framed k-manifold Mk ⊂ Rn+k differ by a map Mk → SO(n), the special orthogonal group of n × n real matrices.

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

1.11 This leads to the deﬁnition of the Hopf-Whitehead J-homomorphism

n J : πk (SO(n)) → πn+k (S ).

Both of these groups are independent of n when n is large, so we get 0 J : πk (SO) → πk (S ). The group of the left and its image on the right are known for all k.

A solution to the The classiﬁcation of exotic spheres (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Two framings on a framed k-manifold Mk ⊂ Rn+k differ by a map Mk → SO(n), the special orthogonal group of n × n real matrices. When Mk is a sphere (or a homotopy sphere) we get an element in πk (SO(n)).

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

1.11 Both of these groups are independent of n when n is large, so we get 0 J : πk (SO) → πk (S ). The group of the left and its image on the right are known for all k.

A solution to the The classiﬁcation of exotic spheres (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Background and n history J : πk (SO(n)) → πn+k (S ). 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

1.11 The group of the left and its image on the right are known for all k.

A solution to the The classiﬁcation of exotic spheres (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.11 A solution to the The classiﬁcation of exotic spheres (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Background and n history J : πk (SO(n)) → πn+k (S ). Exotic spheres The Pontrjagin-Thom construction Both of these groups are independent of n when n is large, so The J-homomorphism The use of surgery we get The Hirzebruch signature 0 theorem J : πk (SO) → πk (S ). The Arf invariant Browder’s theorem The group of the left and its image on the right are known for all Spectral sequences The Adams spectral k. sequence The Adams-Novikov spectral sequence

Our strategy

1.11 It sends an exotic sphere Σk to its framed cobordism class, modulo the indeterminacy related to the nonuniqueness of the framing. Kervaire-Milnor denote this homomorphism by p0 in their Lemma 4.5.

An element in the kernel of τk is represented by an exotic sphere Σk bounding a framed manifold Mk+1. Milnor’s original Σ7 was such an example, bounding a D4-bundle over S4, which can be framed.

An element in the cokernel of τk is a framed k-manifold which is not framed cobordant to a sphere.

A solution to the The classiﬁcation of exotic spheres (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester Hence we have a homomorphism

0 τk :Θk → cokerk J = πk (S )/im J

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

1.12 Kervaire-Milnor denote this homomorphism by p0 in their Lemma 4.5.

An element in the kernel of τk is represented by an exotic sphere Σk bounding a framed manifold Mk+1. Milnor’s original Σ7 was such an example, bounding a D4-bundle over S4, which can be framed.

An element in the cokernel of τk is a framed k-manifold which is not framed cobordant to a sphere.

A solution to the The classiﬁcation of exotic spheres (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester Hence we have a homomorphism

0 τk :Θk → cokerk J = πk (S )/im J

It sends an exotic sphere Σk to its framed cobordism class, modulo the indeterminacy related to the nonuniqueness of the framing. 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

1.12 An element in the kernel of τk is represented by an exotic sphere Σk bounding a framed manifold Mk+1. Milnor’s original Σ7 was such an example, bounding a D4-bundle over S4, which can be framed.

An element in the cokernel of τk is a framed k-manifold which is not framed cobordant to a sphere.

A solution to the The classiﬁcation of exotic spheres (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester Hence we have a homomorphism

0 τk :Θk → cokerk J = πk (S )/im J

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.12 Milnor’s original Σ7 was such an example, bounding a D4-bundle over S4, which can be framed.

An element in the cokernel of τk is a framed k-manifold which is not framed cobordant to a sphere.

A solution to the The classiﬁcation of exotic spheres (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester Hence we have a homomorphism

0 τk :Θk → cokerk J = πk (S )/im J

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.12 An element in the cokernel of τk is a framed k-manifold which is not framed cobordant to a sphere.

A solution to the The classiﬁcation of exotic spheres (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester Hence we have a homomorphism

0 τk :Θk → cokerk J = πk (S )/im J

It sends an exotic sphere Σk to its framed cobordism class, modulo the indeterminacy related to the nonuniqueness of the framing. Kervaire-Milnor denote this homomorphism by p0 in Background and their Lemma 4.5. history Exotic spheres The Pontrjagin-Thom construction An element in the kernel of τk is represented by an exotic The J-homomorphism k k+1 The use of surgery sphere Σ bounding a framed manifold M . Milnor’s original The Hirzebruch signature 7 4 4 theorem Σ was such an example, bounding a D -bundle over S , The Arf invariant which can be framed. Browder’s theorem Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.12 A solution to the The classiﬁcation of exotic spheres (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester Hence we have a homomorphism

0 τk :Θk → cokerk J = πk (S )/im J

1.12 There are framings on S1 × S1, S3 × S3 and S7 × S7 which are not framed cobordant to spheres.

Let η : S3 → S2 be the Hopf map and consider the composite

Ση η S4 / S3 / S2.

The preimage of a typical point in S2 is an exotically framed torus S1 × S1 in S4.

A solution to the The classiﬁcation of exotic spheres (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester We are studying the homomorphism

0 τk :Θk → cokerk J = πk (S )/im J

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

1.13 Let η : S3 → S2 be the Hopf map and consider the composite

Ση η S4 / S3 / S2.

The preimage of a typical point in S2 is an exotically framed torus S1 × S1 in S4.

A solution to the The classiﬁcation of exotic spheres (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester We are studying the homomorphism

0 τk :Θk → cokerk J = πk (S )/im J

There are framings on S1 × S1, S3 × S3 and S7 × S7 which are Background and not framed cobordant to spheres. 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

1.13 The preimage of a typical point in S2 is an exotically framed torus S1 × S1 in S4.

A solution to the The classiﬁcation of exotic spheres (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester We are studying the homomorphism

0 τk :Θk → cokerk J = πk (S )/im J

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.13 A solution to the The classiﬁcation of exotic spheres (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester We are studying the homomorphism

0 τk :Θk → cokerk J = πk (S )/im J

There are framings on S1 × S1, S3 × S3 and S7 × S7 which are Background and not framed cobordant to spheres. history Exotic spheres The Pontrjagin-Thom Let η : S3 → S2 be the Hopf map and consider the composite construction The J-homomorphism The use of surgery Ση η The Hirzebruch signature 4 3 2 theorem S / S / S . The Arf invariant Browder’s theorem 2 Spectral sequences The preimage of a typical point in S is an exotically framed The Adams spectral 1 1 4 sequence torus S × S in S . The Adams-Novikov spectral sequence

Our strategy

1.13 By using surgery (which was originally invented for this purpose!) one can convert M to another framed manifold in the same cobordism class which is roughly n/2-connected, without disturbing the boundary.

When n is odd, we can surger Mn into a sphere Σn or a disk Dn, whose boundary must be an ordinary sphere Sn−1.

This implies τk :Θk → cokerk J is onto when k is odd and one-to-one when k is even.

A solution to the The use of surgery Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Let Mn be a framed manifold, either closed or bounded by a sphere Σn−1.

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

1.14 When n is odd, we can surger Mn into a sphere Σn or a disk Dn, whose boundary must be an ordinary sphere Sn−1.

This implies τk :Θk → cokerk J is onto when k is odd and one-to-one when k is even.

A solution to the The use of surgery Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Let Mn be a framed manifold, either closed or bounded by a sphere Σn−1.

By using surgery (which was originally invented for this purpose!) one can convert M to another framed manifold in the same cobordism class which is roughly n/2-connected, without Background and history disturbing the boundary. 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

1.14 This implies τk :Θk → cokerk J is onto when k is odd and one-to-one when k is even.

A solution to the The use of surgery Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Let Mn be a framed manifold, either closed or bounded by a sphere Σn−1.

By using surgery (which was originally invented for this purpose!) one can convert M to another framed manifold in the same cobordism class which is roughly n/2-connected, without Background and history disturbing the boundary. Exotic spheres The Pontrjagin-Thom construction n n The J-homomorphism When n is odd, we can surger M into a sphere Σ or a disk The use of surgery n n−1 The Hirzebruch signature D , whose boundary must be an ordinary sphere S . theorem The Arf invariant Browder’s theorem

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.14 A solution to the The use of surgery Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Let Mn be a framed manifold, either closed or bounded by a sphere Σn−1.

By using surgery (which was originally invented for this purpose!) one can convert M to another framed manifold in the same cobordism class which is roughly n/2-connected, without Background and history disturbing the boundary. Exotic spheres The Pontrjagin-Thom construction n n The J-homomorphism When n is odd, we can surger M into a sphere Σ or a disk The use of surgery n n−1 The Hirzebruch signature D , whose boundary must be an ordinary sphere S . theorem The Arf invariant Browder’s theorem

This implies τk :Θk → cokerk J is onto when k is odd and Spectral sequences The Adams spectral one-to-one when k is even. sequence The Adams-Novikov spectral sequence

Our strategy

1.14 When m = 2` is even, the cup product gives us a pairing

H2`(M; Z) ⊗ H2`(M; Z) → H4`(M, ∂M; Z)

represented by a symmetric unimodular matrix B with even diagonal entries.

Such matrices have been classiﬁed over the real numbers up to the appropriate equivalence relation. The key invariant is the signature σ(B), the difference between the number of positive and negative eigenvalues over R, which is always divisible by 8.

A solution to the Obstructions in the middle dimension Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester When n = 2m, we can surger our framed manifold M2m into an (m − 1)-connected manifold, but we may not be able to get rid of Hm(M).

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

1.15 Such matrices have been classiﬁed over the real numbers up to the appropriate equivalence relation. The key invariant is the signature σ(B), the difference between the number of positive and negative eigenvalues over R, which is always divisible by 8.

A solution to the Obstructions in the middle dimension Arf-Kervaire invariant problem

When m = 2` is even, the cup product gives us a pairing

H2`(M; Z) ⊗ H2`(M; Z) → H4`(M, ∂M; Z) Background and history Exotic spheres The Pontrjagin-Thom represented by a symmetric unimodular matrix B with even construction The J-homomorphism diagonal entries. 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

1.15 The key invariant is the signature σ(B), the difference between the number of positive and negative eigenvalues over R, which is always divisible by 8.

A solution to the Obstructions in the middle dimension Arf-Kervaire invariant problem

When m = 2` is even, the cup product gives us a pairing

to the appropriate equivalence relation. Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.15 A solution to the Obstructions in the middle dimension Arf-Kervaire invariant problem

When m = 2` is even, the cup product gives us a pairing

to the appropriate equivalence relation. The key invariant is the Spectral sequences The Adams spectral signature σ(B), the difference between the number of positive sequence The Adams-Novikov and negative eigenvalues over R, which is always divisible by 8. spectral sequence

Our strategy

1.15  2 −1 0 0 0 0 0 0   −1 2 −1 0 0 0 0 0     0 −1 2 −1 0 0 0 −1     0 0 −1 2 −1 0 0 0  B =    0 0 0 −1 2 −1 0 0     0 0 0 0 −1 2 −1 0     0 0 0 0 0 −1 2 0  0 0 −1 0 0 0 0 2

A solution to the An interesting matrix Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Here is a symmetric matrix with even diagonal entries and signature 8.

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

1.16 A solution to the An interesting matrix Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Here is a symmetric matrix with even diagonal entries and signature 8.

 2 −1 0 0 0 0 0 0   −1 2 −1 0 0 0 0 0    Background and  0 −1 2 −1 0 0 0 −1  history   Exotic spheres  0 0 −1 2 −1 0 0 0  The Pontrjagin-Thom B =   construction  0 0 0 −1 2 −1 0 0  The J-homomorphism    0 0 0 0 −1 2 −1 0  The use of surgery   The Hirzebruch signature 0 0 0 0 0 −1 2 0 theorem   The Arf invariant 0 0 −1 0 0 0 0 2 Browder’s theorem Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.16 8

1 2 3 4 5 6 7

The nodes on the graph correspond to the rows/columns of the matrix.

Nodes i and j are connected by an edge iff bi,j 6= 0.

A solution to the The Dynkin diagram for E8 Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

The matrix on the previous page is related to the following Dynkin diagram.

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

1.17 The nodes on the graph correspond to the rows/columns of the matrix.

Nodes i and j are connected by an edge iff bi,j 6= 0.

A solution to the The Dynkin diagram for E8 Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

The matrix on the previous page is related to the following Dynkin diagram.

Background and history 1 2 3 4 5 6 7 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

1.17 Nodes i and j are connected by an edge iff bi,j 6= 0.

A solution to the The Dynkin diagram for E8 Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

The matrix on the previous page is related to the following Dynkin diagram.

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.17 A solution to the The Dynkin diagram for E8 Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

The matrix on the previous page is related to the following Dynkin diagram.

Background and history 1 2 3 4 5 6 7 Exotic spheres The Pontrjagin-Thom construction The J-homomorphism The nodes on the graph correspond to the rows/columns of the The use of surgery The Hirzebruch signature matrix. theorem The Arf invariant Browder’s theorem Nodes i and j are connected by an edge iff bi,j 6= 0. Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.17 If M is framed (as in our case), its Pontrjagin numbers and therefore its signature vanish. This means when M is closed we can surger it into a sphere, so τ4` is onto.

If our M4` is bounded by a sphere diffeomorphic to S4`−1, then we can close M by attached a 4`-ball. We get a new manifold N4` that is framed at every point except the center of that ball.

Such a manifold is said to be almost framed.

A solution to the Implications of the Hirzebruch signature theorem Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

The Hirzebruch signature theorem relates the signature of a smooth oriented closed 4`-manifold M to its Pontrjagin numbers.

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

1.18 This means when M is closed we can surger it into a sphere, so τ4` is onto.

If our M4` is bounded by a sphere diffeomorphic to S4`−1, then we can close M by attached a 4`-ball. We get a new manifold N4` that is framed at every point except the center of that ball.

Such a manifold is said to be almost framed.

A solution to the Implications of the Hirzebruch signature theorem Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

The Hirzebruch signature theorem relates the signature of a smooth oriented closed 4`-manifold M to its Pontrjagin numbers.

If M is framed (as in our case), its Pontrjagin numbers and therefore its signature vanish. 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

1.18 If our M4` is bounded by a sphere diffeomorphic to S4`−1, then we can close M by attached a 4`-ball. We get a new manifold N4` that is framed at every point except the center of that ball.

Such a manifold is said to be almost framed.

A solution to the Implications of the Hirzebruch signature theorem Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

The Hirzebruch signature theorem relates the signature of a smooth oriented closed 4`-manifold M to its Pontrjagin numbers.

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.18 Such a manifold is said to be almost framed.

A solution to the Implications of the Hirzebruch signature theorem Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

The Hirzebruch signature theorem relates the signature of a smooth oriented closed 4`-manifold M to its Pontrjagin numbers.

If M is framed (as in our case), its Pontrjagin numbers and therefore its signature vanish. This means when M is closed Background and history we can surger it into a sphere, so τ4` is onto. Exotic spheres The Pontrjagin-Thom construction 4` 4`−1 The J-homomorphism If our M is bounded by a sphere diffeomorphic to S , then The use of surgery The Hirzebruch signature we can close M by attached a 4`-ball. We get a new manifold theorem 4` The Arf invariant N that is framed at every point except the center of that ball. Browder’s theorem

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.18 A solution to the Implications of the Hirzebruch signature theorem Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

The Hirzebruch signature theorem relates the signature of a smooth oriented closed 4`-manifold M to its Pontrjagin numbers.

Spectral sequences The Adams spectral Such a manifold is said to be almost framed. sequence The Adams-Novikov spectral sequence

Our strategy

1.18 For ` = 2, this integer is 224 = 8 · 28.

On the other hand, there is a way to construct a framed 4`-manifold bounded by a sphere Σ4`−1 such that σ(B) is any multiple of 8. This gives us 28 distinct differentiable structures on S7.

The kernel of τ4`−1 is a large cyclic group whose order was determined by Kervaire-Milnor.

A solution to the The signature of an almost framed manifold Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Hirzebruch’s formula implies that our signature σ(N4`) is divisible by a certain integer related to the numerator of a Bernoulli number.

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

1.19 On the other hand, there is a way to construct a framed 4`-manifold bounded by a sphere Σ4`−1 such that σ(B) is any multiple of 8. This gives us 28 distinct differentiable structures on S7.

The kernel of τ4`−1 is a large cyclic group whose order was determined by Kervaire-Milnor.

A solution to the The signature of an almost framed manifold Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Hirzebruch’s formula implies that our signature σ(N4`) is divisible by a certain integer related to the numerator of a Bernoulli number. For ` = 2, this integer is 224 = 8 · 28.

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

1.19 The kernel of τ4`−1 is a large cyclic group whose order was determined by Kervaire-Milnor.

A solution to the The signature of an almost framed manifold Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Hirzebruch’s formula implies that our signature σ(N4`) is divisible by a certain integer related to the numerator of a Bernoulli number. For ` = 2, this integer is 224 = 8 · 28.

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.19 A solution to the The signature of an almost framed manifold Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Hirzebruch’s formula implies that our signature σ(N4`) is divisible by a certain integer related to the numerator of a Bernoulli number. For ` = 2, this integer is 224 = 8 · 28.

On the other hand, there is a way to construct a framed Background and 4`−1 history 4`-manifold bounded by a sphere Σ such that σ(B) is any Exotic spheres The Pontrjagin-Thom multiple of 8. This gives us 28 distinct differentiable structures construction 7 on S . The J-homomorphism The use of surgery The Hirzebruch signature theorem The kernel of τ4`−1 is a large cyclic group whose order was The Arf invariant determined by Kervaire-Milnor. Browder’s theorem Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.19 It is onto when k is odd or divisible by 4. It is one-to-one when k is even, and has a known kernel when k = 4` − 1.

We have not yet discussed the kernel for k = 4` + 1 or the cokernel for k = 4` + 2.

It turns out that the two groups are related. For each `, one is trivial iff the other is Z/2.

A solution to the The one remaining case Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

To recap, we have a homomorphism

τk :Θk → cokerk J

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

1.20 It is one-to-one when k is even, and has a known kernel when k = 4` − 1.

We have not yet discussed the kernel for k = 4` + 1 or the cokernel for k = 4` + 2.

It turns out that the two groups are related. For each `, one is trivial iff the other is Z/2.

A solution to the The one remaining case Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

To recap, we have a homomorphism

τk :Θk → cokerk J

It is onto when k is odd or divisible by 4.

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

1.20 We have not yet discussed the kernel for k = 4` + 1 or the cokernel for k = 4` + 2.

It turns out that the two groups are related. For each `, one is trivial iff the other is Z/2.

A solution to the The one remaining case Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

To recap, we have a homomorphism

τk :Θk → cokerk J

It is onto when k is odd or divisible by 4.

It is one-to-one when k is even, and has a known kernel when Background and = ` − history k 4 1. 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

1.20 It turns out that the two groups are related. For each `, one is trivial iff the other is Z/2.

A solution to the The one remaining case Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

To recap, we have a homomorphism

τk :Θk → cokerk J

It is onto when k is odd or divisible by 4.

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.20 A solution to the The one remaining case Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

To recap, we have a homomorphism

τk :Θk → cokerk J

It is onto when k is odd or divisible by 4.

It is one-to-one when k is even, and has a known kernel when Background and = ` − history k 4 1. Exotic spheres The Pontrjagin-Thom construction We have not yet discussed the kernel for k = 4` + 1 or the The J-homomorphism The use of surgery cokernel for k = 4` + 2. The Hirzebruch signature theorem The Arf invariant It turns out that the two groups are related. For each `, one is Browder’s theorem Spectral sequences trivial iff the other is Z/2. The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.20 We have a pairing

H2`+1(M; Z/2) ⊗ H2`+1(M; Z/2) → H4`+2(M, ∂M; Z/2)

Evaluation on the fundamental class gives us a quadratic form

λ : H2`+1 ⊗ H2`+1 → Z/2.

There is a map (not a homomorphism) µ : H2`+1 → Z/2 such that λ(x, y) = µ(x) + µ(y) + µ(x + y).

A solution to the Framed 4` + 2-manifolds Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester We have a framed 4` + 2-manifold, possibly bounded by a sphere Σ4`+1. We can surger it into a 2`-connected manifold.

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

1.21 Evaluation on the fundamental class gives us a quadratic form

λ : H2`+1 ⊗ H2`+1 → Z/2.

There is a map (not a homomorphism) µ : H2`+1 → Z/2 such that λ(x, y) = µ(x) + µ(y) + µ(x + y).

A solution to the Framed 4` + 2-manifolds Arf-Kervaire invariant problem

H2`+1(M; Z/2) ⊗ H2`+1(M; Z/2) → H4`+2(M, ∂M; Z/2)

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

1.21 There is a map (not a homomorphism) µ : H2`+1 → Z/2 such that λ(x, y) = µ(x) + µ(y) + µ(x + y).

A solution to the Framed 4` + 2-manifolds Arf-Kervaire invariant problem

H2`+1(M; Z/2) ⊗ H2`+1(M; Z/2) → H4`+2(M, ∂M; Z/2)

Background and Evaluation on the fundamental class gives us a quadratic form history Exotic spheres 2`+1 2`+1 The Pontrjagin-Thom λ : H ⊗ H → Z/2. 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

1.21 A solution to the Framed 4` + 2-manifolds Arf-Kervaire invariant problem

H2`+1(M; Z/2) ⊗ H2`+1(M; Z/2) → H4`+2(M, ∂M; Z/2)

Background and Evaluation on the fundamental class gives us a quadratic form history Exotic spheres 2`+1 2`+1 The Pontrjagin-Thom λ : H ⊗ H → Z/2. construction The J-homomorphism The use of surgery 2`+1 The Hirzebruch signature There is a map (not a homomorphism) µ : H → Z/2 such theorem The Arf invariant that Browder’s theorem

λ(x, y) = µ(x) + µ(y) + µ(x + y). Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.21 This value is its Arf invariant Φ(M), which is the obstruction to doing surgery in the middle dimension.

The Arf-Kervaire invariant Φ(M) of a framed (4` + 2)-manifold is deﬁned to be the Arf invariant of its quadratic form.

A solution to the The Arf invariant Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

This map µ : H2`+1(M; Z/2) → Z/2 is either 1 most of the time or 0 most of the time.

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

1.22 The Arf-Kervaire invariant Φ(M) of a framed (4` + 2)-manifold is deﬁned to be the Arf invariant of its quadratic form.

A solution to the The Arf invariant Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

This map µ : H2`+1(M; Z/2) → Z/2 is either 1 most of the time or 0 most of the time.

This value is its Arf invariant Φ(M), which is the obstruction to Background and history doing surgery in the middle dimension. 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

1.22 A solution to the The Arf invariant Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

This map µ : H2`+1(M; Z/2) → Z/2 is either 1 most of the time or 0 most of the time.

This value is its Arf invariant Φ(M), which is the obstruction to Background and history doing surgery in the middle dimension. Exotic spheres The Pontrjagin-Thom construction The J-homomorphism The Arf-Kervaire invariant Φ(M) of a framed (4` + 2)-manifold The use of surgery The Hirzebruch signature is deﬁned to be the Arf invariant of its quadratic form. theorem The Arf invariant Browder’s theorem

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.22 If there is, then τ4`+2 has a cokernel of order 2 and τ4`+1 is one-to-one.

If there is not, then τ4`+1 has a kernel of order 2 and τ4`+2 is onto.

Kervaire answered the question in the negative for ` = 2. He constructed a framed 10-manifold bounded by an exotic 9-sphere. By coning off its boundary, he got his nonsmoothable closed topological 10-manifold.

A solution to the The Arf-Kervaire invariant question Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Is there a closed framed (4` + 2)-manifold with nontrivial Arf-Kervaire invariant?

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

1.23 If there is not, then τ4`+1 has a kernel of order 2 and τ4`+2 is onto.

A solution to the The Arf-Kervaire invariant question Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Is there a closed framed (4` + 2)-manifold with nontrivial Arf-Kervaire invariant?

If there is, then τ4`+2 has a cokernel of order 2 and τ4`+1 is one-to-one.

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

1.23 Kervaire answered the question in the negative for ` = 2. He constructed a framed 10-manifold bounded by an exotic 9-sphere. By coning off its boundary, he got his nonsmoothable closed topological 10-manifold.

A solution to the The Arf-Kervaire invariant question Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Is there a closed framed (4` + 2)-manifold with nontrivial Arf-Kervaire invariant?

If there is, then τ4`+2 has a cokernel of order 2 and τ4`+1 is one-to-one.

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.23 He constructed a framed 10-manifold bounded by an exotic 9-sphere. By coning off its boundary, he got his nonsmoothable closed topological 10-manifold.

A solution to the The Arf-Kervaire invariant question Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Is there a closed framed (4` + 2)-manifold with nontrivial Arf-Kervaire invariant?

If there is, then τ4`+2 has a cokernel of order 2 and τ4`+1 is one-to-one.

Background and history If there is not, then τ4`+1 has a kernel of order 2 and τ4`+2 is Exotic spheres The Pontrjagin-Thom onto. construction The J-homomorphism The use of surgery The Hirzebruch signature Kervaire answered the question in the negative for ` = 2. theorem The Arf invariant Browder’s theorem

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.23 By coning off its boundary, he got his nonsmoothable closed topological 10-manifold.

A solution to the The Arf-Kervaire invariant question Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Is there a closed framed (4` + 2)-manifold with nontrivial Arf-Kervaire invariant?

If there is, then τ4`+2 has a cokernel of order 2 and τ4`+1 is one-to-one.

9-sphere. Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.23 A solution to the The Arf-Kervaire invariant question Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Is there a closed framed (4` + 2)-manifold with nontrivial Arf-Kervaire invariant?

If there is, then τ4`+2 has a cokernel of order 2 and τ4`+1 is one-to-one.

9-sphere. By coning off its boundary, he got his nonsmoothable Spectral sequences The Adams spectral closed topological 10-manifold. sequence The Adams-Novikov spectral sequence

Our strategy

1.23 Browder’s Theorem (1969) Relation to the Adams spectral sequence. A framed (4` + 2)-manifold with nontrivial Arf-Kervaire invariant can exist only when ` = 2j−1 − 1 for some integer j. In that case it exists iff the Adams spectral sequence element

2 2,2j+1 2,2j+1 hj ∈ E2 = ExtA (Z/2, Z/2) is a permanent cycle.

A solution to the Enter stable homotopy theory Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Algebraic topologists attacked this question vigorously in the 1960s. The best result was the following.

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

1.24 In that case it exists iff the Adams spectral sequence element

2 2,2j+1 2,2j+1 hj ∈ E2 = ExtA (Z/2, Z/2) is a permanent cycle.

A solution to the Enter stable homotopy theory Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Algebraic topologists attacked this question vigorously in the 1960s. The best result was the following.

Browder’s Theorem (1969) Relation to the Adams spectral sequence. A framed Background and (4` + 2)-manifold with nontrivial Arf-Kervaire invariant can exist history j−1 only when ` = 2 − 1 for some integer j. 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

1.24 A solution to the Enter stable homotopy theory Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Algebraic topologists attacked this question vigorously in the 1960s. The best result was the following.

Browder’s Theorem (1969) Relation to the Adams spectral sequence. A framed Background and (4` + 2)-manifold with nontrivial Arf-Kervaire invariant can exist history j−1 only when ` = 2 − 1 for some integer j. In that case it exists Exotic spheres The Pontrjagin-Thom iff the Adams spectral sequence element construction The J-homomorphism The use of surgery 2 2,2j+1 2,2j+1 The Hirzebruch signature hj ∈ E = Ext (Z/2, Z/2) theorem 2 A The Arf invariant Browder’s theorem

is a permanent cycle. Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.24 It is deﬁned for all integers j ≥ 0.

2 θj denotes any element in π2j+1−2 that is detected by hj .

Adams showed that hj is a permanent cycle only for 0 ≤ j ≤ 3. These hj represent 2ι (twice the fundamental class) and the Hopf maps η ∈ π1, ν ∈ π3 and σ ∈ π7.

For j ≥ 4 there is a nontrivial differential

2 d2(hj ) = h0hj−1.

A solution to the The classical Adams spectral sequence Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Here A denotes the mod 2 Steenrod algebra and Doug Ravenel University of Rochester

2,2j 1,2j hj ∈ E2 = ExtA (Z/2, Z/2)

j is the element corresponding to Sq2 .

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

1.25 2 θj denotes any element in π2j+1−2 that is detected by hj .

Adams showed that hj is a permanent cycle only for 0 ≤ j ≤ 3. These hj represent 2ι (twice the fundamental class) and the Hopf maps η ∈ π1, ν ∈ π3 and σ ∈ π7.

For j ≥ 4 there is a nontrivial differential

2 d2(hj ) = h0hj−1.

A solution to the The classical Adams spectral sequence Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Here A denotes the mod 2 Steenrod algebra and Doug Ravenel University of Rochester

2,2j 1,2j hj ∈ E2 = ExtA (Z/2, Z/2)

j is the element corresponding to Sq2 . It is deﬁned for all integers j ≥ 0.

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

1.25 Adams showed that hj is a permanent cycle only for 0 ≤ j ≤ 3. These hj represent 2ι (twice the fundamental class) and the Hopf maps η ∈ π1, ν ∈ π3 and σ ∈ π7.

For j ≥ 4 there is a nontrivial differential

2 d2(hj ) = h0hj−1.

A solution to the The classical Adams spectral sequence Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Here A denotes the mod 2 Steenrod algebra and Doug Ravenel University of Rochester

2,2j 1,2j hj ∈ E2 = ExtA (Z/2, Z/2)

j is the element corresponding to Sq2 . It is deﬁned for all integers j ≥ 0.

2 Background and + θj denotes any element in π2j 1−2 that is detected by hj . 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

1.25 These hj represent 2ι (twice the fundamental class) and the Hopf maps η ∈ π1, ν ∈ π3 and σ ∈ π7.

For j ≥ 4 there is a nontrivial differential

2 d2(hj ) = h0hj−1.

A solution to the The classical Adams spectral sequence Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Here A denotes the mod 2 Steenrod algebra and Doug Ravenel University of Rochester

2,2j 1,2j hj ∈ E2 = ExtA (Z/2, Z/2)

j is the element corresponding to Sq2 . It is deﬁned for all integers j ≥ 0.

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.25 For j ≥ 4 there is a nontrivial differential

2 d2(hj ) = h0hj−1.

A solution to the The classical Adams spectral sequence Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Here A denotes the mod 2 Steenrod algebra and Doug Ravenel University of Rochester

2,2j 1,2j hj ∈ E2 = ExtA (Z/2, Z/2)

j is the element corresponding to Sq2 . It is deﬁned for all integers j ≥ 0.

2 Background and + θj denotes any element in π2j 1−2 that is detected by hj . history Exotic spheres The Pontrjagin-Thom Adams showed that hj is a permanent cycle only for 0 ≤ j ≤ 3. construction The J-homomorphism These hj represent 2ι (twice the fundamental class) and the The use of surgery The Hirzebruch signature Hopf maps η ∈ π1, ν ∈ π3 and σ ∈ π7. theorem The Arf invariant Browder’s theorem

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.25 A solution to the The classical Adams spectral sequence Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Here A denotes the mod 2 Steenrod algebra and Doug Ravenel University of Rochester

2,2j 1,2j hj ∈ E2 = ExtA (Z/2, Z/2)

j is the element corresponding to Sq2 . It is deﬁned for all integers j ≥ 0.

1.25 12

10 2 P h1 2 8 P h2 Pc0 6 Ph1 Ph2 f 4 d0 e0 0 c0 g c1 2 h2 h0 3 h1 h2 h3 h4 0 ι 0 2 4 6 8 10 12 14 16 18 20

A solution to the The classical Adams spectral sequence (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Here is a picture of the Adams spectral sequence for the prime Doug Ravenel University of Rochester 2 in low dimensions.

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

1.26 A solution to the The classical Adams spectral sequence (continued) Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Here is a picture of the Adams spectral sequence for the prime Doug Ravenel University of Rochester 2 in low dimensions.

10 2 P h1 2 Background and 8 P h2 history Pc0 Exotic spheres The Pontrjagin-Thom construction 6 The J-homomorphism The use of surgery Ph1 The Hirzebruch signature Ph2 f theorem 4 d0 e0 0 c0 g The Arf invariant Browder’s theorem c1 2 Spectral sequences h2 The Adams spectral h0 3 sequence h1 h2 h3 h4 The Adams-Novikov 0 ι spectral sequence 0 2 4 6 8 10 12 14 16 18 20 Our strategy

1.26 The classical Adams spectral sequence of the previous slide is based on ordinary mod 2 cohomology and the Steenrod algebra and was introduced by Adams in 1959.

It is possible to use other cohomology or homology theories for the same purpose.

Complex cobordism theory has proven to be extremely useful. The corresponding spectral sequence was ﬁrst studied by Novikov in 1967 and is known as the Adams-Novikov spectral sequence.

A solution to the Spectral sequences in stable homotopy theory Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester Spectral sequences have been used to study the stable homotopy groups of spheres for the past 50 years.

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

1.27 It is possible to use other cohomology or homology theories for the same purpose.

Complex cobordism theory has proven to be extremely useful. The corresponding spectral sequence was ﬁrst studied by Novikov in 1967 and is known as the Adams-Novikov spectral sequence.

A solution to the Spectral sequences in stable homotopy theory Arf-Kervaire invariant problem

The classical Adams spectral sequence of the previous slide is based on ordinary mod 2 cohomology and the Steenrod algebra and was introduced by Adams in 1959. 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

1.27 Complex cobordism theory has proven to be extremely useful. The corresponding spectral sequence was ﬁrst studied by Novikov in 1967 and is known as the Adams-Novikov spectral sequence.

A solution to the Spectral sequences in stable homotopy theory Arf-Kervaire invariant problem

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.27 The corresponding spectral sequence was ﬁrst studied by Novikov in 1967 and is known as the Adams-Novikov spectral sequence.

A solution to the Spectral sequences in stable homotopy theory Arf-Kervaire invariant problem

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.27 A solution to the Spectral sequences in stable homotopy theory Arf-Kervaire invariant problem

The classical Adams spectral sequence of the previous slide is based on ordinary mod 2 cohomology and the Steenrod algebra and was introduced by Adams in 1959. Background and history Exotic spheres It is possible to use other cohomology or homology theories for The Pontrjagin-Thom construction the same purpose. The J-homomorphism The use of surgery The Hirzebruch signature theorem Complex cobordism theory has proven to be extremely useful. The Arf invariant The corresponding spectral sequence was ﬁrst studied by Browder’s theorem Spectral sequences Novikov in 1967 and is known as the Adams-Novikov spectral The Adams spectral sequence sequence. The Adams-Novikov spectral sequence

Our strategy

1.27 Here is the v1-periodic part.

2 α α◦ α ◦ α ◦ α ◦ α ◦ 1 2/2 3 5 α 7 9 α 0 α4/4 6/3 α 10/3 0 2 4 6 8 10 12 14 8/516 18 20

The box indicates a copy of Z(2). Circles indicate group orders. Black lines indicate α1-multiplication. Red lines indicate differentials. Green lines indicate group extensions.

A solution to the The Adams-Novikov spectral sequence for p = 2 in low Arf-Kervaire invariant dimensions problem Mike Hill University of Virginia Mike Hopkins Harvard University It is helpful to separate it into two parts having to do with Doug Ravenel University of Rochester v1-periodic and v1-torsion elements. These are related to the image and cokernel of J.

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

1.28 The box indicates a copy of Z(2). Circles indicate group orders. Black lines indicate α1-multiplication. Red lines indicate differentials. Green lines indicate group extensions.

Here is the v1-periodic part.

Our strategy

1.28 Circles indicate group orders. Black lines indicate α1-multiplication. Red lines indicate differentials. Green lines indicate group extensions.

Here is the v1-periodic part.

6 Background and history Exotic spheres 4 The Pontrjagin-Thom construction The J-homomorphism The use of surgery 2 The Hirzebruch signature ◦ ◦ ◦ ◦ ◦ theorem α1 α2/2 α3 α5 α7 α9 The Arf invariant 0 α α6/3 α10/3 Browder’s theorem 4/4 α8/5 0 2 4 6 8 10 12 14 16 18 20 Spectral sequences The Adams spectral sequence The Adams-Novikov The box indicates a copy of Z(2). spectral sequence Our strategy

1.28 Black lines indicate α1-multiplication. Red lines indicate differentials. Green lines indicate group extensions.

Here is the v1-periodic part.

1.28 Red lines indicate differentials. Green lines indicate group extensions.

Here is the v1-periodic part.

1.28 Green lines indicate group extensions.

Here is the v1-periodic part.

1.28 A solution to the The Adams-Novikov spectral sequence for p = 2 in low Arf-Kervaire invariant dimensions problem Mike Hill University of Virginia Mike Hopkins Harvard University It is helpful to separate it into two parts having to do with Doug Ravenel University of Rochester v1-periodic and v1-torsion elements. These are related to the image and cokernel of J.

Here is the v1-periodic part.

1.28 2 2 β2/2 = h2 = θ2 and β4/4 = h3 = θ3. Color coding is as before. Blue lines indicate multiplication by ν = h2 = α2,2/2. Green lines indicate group extensions. The ﬁrst differential in this spectral sequence occurs in dimension 26.

Here is the v1-torsion part.

4 ◦

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.29 Color coding is as before. Blue lines indicate multiplication by ν = h2 = α2,2/2. Green lines indicate group extensions. The ﬁrst differential in this spectral sequence occurs in dimension 26.

Here is the v1-torsion part.

4 ◦

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.29 Blue lines indicate multiplication by ν = h2 = α2,2/2. Green lines indicate group extensions. The ﬁrst differential in this spectral sequence occurs in dimension 26.

Here is the v1-torsion part.

4 ◦

η2 Background and β4/4 2 ◦ history β β β β Exotic spheres 2/2 2 β3 4/3 β4/2,2 4 The Pontrjagin-Thom construction 0 The J-homomorphism 0 2 4 6 8 10 12 14 16 18 20 The use of surgery The Hirzebruch signature theorem 2 2 β2/2 = h2 = θ2 and β4/4 = h3 = θ3. The Arf invariant Color coding is as before. Browder’s theorem Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.29 Green lines indicate group extensions. The ﬁrst differential in this spectral sequence occurs in dimension 26.

Here is the v1-torsion part.

4 ◦

η2 Background and β4/4 2 ◦ history β β β β Exotic spheres 2/2 2 β3 4/3 β4/2,2 4 The Pontrjagin-Thom construction 0 The J-homomorphism 0 2 4 6 8 10 12 14 16 18 20 The use of surgery The Hirzebruch signature theorem 2 2 β2/2 = h2 = θ2 and β4/4 = h3 = θ3. The Arf invariant Color coding is as before. Blue lines indicate multiplication by Browder’s theorem Spectral sequences ν = h2 = α . The Adams spectral 2,2/2 sequence The Adams-Novikov spectral sequence

Our strategy

1.29 The ﬁrst differential in this spectral sequence occurs in dimension 26.

Here is the v1-torsion part.

4 ◦

η2 Background and β4/4 2 ◦ history β β β β Exotic spheres 2/2 2 β3 4/3 β4/2,2 4 The Pontrjagin-Thom construction 0 The J-homomorphism 0 2 4 6 8 10 12 14 16 18 20 The use of surgery The Hirzebruch signature theorem 2 2 β2/2 = h2 = θ2 and β4/4 = h3 = θ3. The Arf invariant Color coding is as before. Blue lines indicate multiplication by Browder’s theorem Spectral sequences ν = h2 = α . Green lines indicate group extensions. The Adams spectral 2,2/2 sequence The Adams-Novikov spectral sequence

Our strategy

1.29 A solution to the The Adams-Novikov spectral sequence for p = 2 in low Arf-Kervaire invariant dimensions problem Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

Here is the v1-torsion part.

4 ◦

η2 Background and β4/4 2 ◦ history β β β β Exotic spheres 2/2 2 β3 4/3 β4/2,2 4 The Pontrjagin-Thom construction 0 The J-homomorphism 0 2 4 6 8 10 12 14 16 18 20 The use of surgery The Hirzebruch signature theorem 2 2 β2/2 = h2 = θ2 and β4/4 = h3 = θ3. The Arf invariant Color coding is as before. Blue lines indicate multiplication by Browder’s theorem Spectral sequences ν = h2 = α . Green lines indicate group extensions. The Adams spectral 2,2/2 sequence The ﬁrst differential in this spectral sequence occurs in The Adams-Novikov spectral sequence

dimension 26. Our strategy

1.29 Fortunately our proof does not involve these complexities.

The Arf-Kervaire invariant question translates to the following: In the Adams-Novikov spectral sequence, is the 2,2j+1 element θj = β2j−1/2j−1 ∈ E2 a permanent cycle? It cannot be the target of a differential because its ﬁltration is too low. We will show that it is the source of a nontrivial differential for j ≥ 7.

A solution to the The Arf-Kervaire invariant in the Adams-Novikov spectral Arf-Kervaire invariant sequence problem Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

These spectral sequences are very complicated and have been studied very closely.

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

1.30 The Arf-Kervaire invariant question translates to the following: In the Adams-Novikov spectral sequence, is the 2,2j+1 element θj = β2j−1/2j−1 ∈ E2 a permanent cycle? It cannot be the target of a differential because its ﬁltration is too low. We will show that it is the source of a nontrivial differential for j ≥ 7.

These spectral sequences are very complicated and have been studied very closely. Fortunately our proof does not involve these complexities.

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

1.30 In the Adams-Novikov spectral sequence, is the 2,2j+1 element θj = β2j−1/2j−1 ∈ E2 a permanent cycle? It cannot be the target of a differential because its ﬁltration is too low. We will show that it is the source of a nontrivial differential for j ≥ 7.

These spectral sequences are very complicated and have been studied very closely. Fortunately our proof does not involve these complexities.

Background and The Arf-Kervaire invariant question translates to the following: 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

1.30 It cannot be the target of a differential because its ﬁltration is too low. We will show that it is the source of a nontrivial differential for j ≥ 7.

These spectral sequences are very complicated and have been studied very closely. Fortunately our proof does not involve these complexities.

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.30 We will show that it is the source of a nontrivial differential for j ≥ 7.

These spectral sequences are very complicated and have been studied very closely. Fortunately our proof does not involve these complexities.

Background and The Arf-Kervaire invariant question translates to the following: history Exotic spheres In the Adams-Novikov spectral sequence, is the The Pontrjagin-Thom 2,2j+1 construction element θj = β j−1/ j−1 ∈ E a permanent cycle? The J-homomorphism 2 2 2 The use of surgery The Hirzebruch signature theorem It cannot be the target of a differential because its ﬁltration is The Arf invariant too low. Browder’s theorem Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.30 A solution to the The Arf-Kervaire invariant in the Adams-Novikov spectral Arf-Kervaire invariant sequence problem Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester

These spectral sequences are very complicated and have been studied very closely. Fortunately our proof does not involve these complexities.

Background and The Arf-Kervaire invariant question translates to the following: history Exotic spheres In the Adams-Novikov spectral sequence, is the The Pontrjagin-Thom 2,2j+1 construction element θj = β j−1/ j−1 ∈ E a permanent cycle? The J-homomorphism 2 2 2 The use of surgery The Hirzebruch signature theorem It cannot be the target of a differential because its ﬁltration is The Arf invariant too low. We will show that it is the source of a nontrivial Browder’s theorem Spectral sequences differential for j ≥ 7. The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.30 (i) It has an Adams-Novikov spectral sequence in which the image of each θj is nontrivial. (ii) It is 256-periodic, meaning Σ256M =∼ M.

(iii) π−2(M) = 0.

(ii) and (iii) imply that π254(M) = 0.

If θ7 exists, (i) implies it has a nontrivial image in this group, so it cannot exist.

The argument for θj for larger j is similar, since j+1 |θj | = 2 − 2 ≡ −2 mod 256 for j ≥ 7.

A solution to the Our strategy Arf-Kervaire invariant problem

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

1.31 (ii) It is 256-periodic, meaning Σ256M =∼ M.

(iii) π−2(M) = 0.

(ii) and (iii) imply that π254(M) = 0.

If θ7 exists, (i) implies it has a nontrivial image in this group, so it cannot exist.

The argument for θj for larger j is similar, since j+1 |θj | = 2 − 2 ≡ −2 mod 256 for j ≥ 7.

A solution to the Our strategy Arf-Kervaire invariant problem

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

1.31 (iii) π−2(M) = 0.

(ii) and (iii) imply that π254(M) = 0.

If θ7 exists, (i) implies it has a nontrivial image in this group, so it cannot exist.

The argument for θj for larger j is similar, since j+1 |θj | = 2 − 2 ≡ −2 mod 256 for j ≥ 7.

A solution to the Our strategy Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester We will produce a map S0 → M, where M is a nonconnective spectrum with the following properties. (i) It has an Adams-Novikov spectral sequence in which the image of each θj is nontrivial. (ii) It is 256-periodic, meaning Σ256M =∼ M. 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

1.31 (ii) and (iii) imply that π254(M) = 0.

If θ7 exists, (i) implies it has a nontrivial image in this group, so it cannot exist.

The argument for θj for larger j is similar, since j+1 |θj | = 2 − 2 ≡ −2 mod 256 for j ≥ 7.

A solution to the Our strategy Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester We will produce a map S0 → M, where M is a nonconnective spectrum with the following properties. (i) It has an Adams-Novikov spectral sequence in which the image of each θj is nontrivial. (ii) It is 256-periodic, meaning Σ256M =∼ M. Background and (iii) π−2(M) = 0. 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

1.31 If θ7 exists, (i) implies it has a nontrivial image in this group, so it cannot exist.

The argument for θj for larger j is similar, since j+1 |θj | = 2 − 2 ≡ −2 mod 256 for j ≥ 7.

A solution to the Our strategy Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester We will produce a map S0 → M, where M is a nonconnective spectrum with the following properties. (i) It has an Adams-Novikov spectral sequence in which the image of each θj is nontrivial. (ii) It is 256-periodic, meaning Σ256M =∼ M. Background and (iii) π−2(M) = 0. history Exotic spheres (ii) and (iii) imply that π (M) = 0. The Pontrjagin-Thom 254 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

1.31 The argument for θj for larger j is similar, since j+1 |θj | = 2 − 2 ≡ −2 mod 256 for j ≥ 7.

A solution to the Our strategy Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester We will produce a map S0 → M, where M is a nonconnective spectrum with the following properties. (i) It has an Adams-Novikov spectral sequence in which the image of each θj is nontrivial. (ii) It is 256-periodic, meaning Σ256M =∼ M. Background and (iii) π−2(M) = 0. history Exotic spheres (ii) and (iii) imply that π (M) = 0. The Pontrjagin-Thom 254 construction The J-homomorphism The use of surgery If θ7 exists, (i) implies it has a nontrivial image in this group, so The Hirzebruch signature theorem it cannot exist. The Arf invariant Browder’s theorem

Spectral sequences The Adams spectral sequence The Adams-Novikov spectral sequence

Our strategy

1.31 A solution to the Our strategy Arf-Kervaire invariant problem

Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester We will produce a map S0 → M, where M is a nonconnective spectrum with the following properties. (i) It has an Adams-Novikov spectral sequence in which the image of each θj is nontrivial. (ii) It is 256-periodic, meaning Σ256M =∼ M. Background and (iii) π−2(M) = 0. history Exotic spheres (ii) and (iii) imply that π (M) = 0. The Pontrjagin-Thom 254 construction The J-homomorphism The use of surgery If θ7 exists, (i) implies it has a nontrivial image in this group, so The Hirzebruch signature theorem it cannot exist. The Arf invariant Browder’s theorem

Spectral sequences The argument for θj for larger j is similar, since The Adams spectral j+1 sequence |θj | = 2 − 2 ≡ −2 mod 256 for j ≥ 7. The Adams-Novikov spectral sequence

Our strategy

1.31