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Completion of real Johnson-Wilson theory gives fixed points of Morava E -theory Maia Averett Mills College CAT 09 Warsaw, Poland July 2009 Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E

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Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory

Maia Averett

Mills College

CAT ’09 Warsaw, Poland July 2009

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 1 / 18

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Introduction

Two cohomology theories

Fix prime p = 2. Johnson-Wilson theory E(n): Landweber exact theory with E(n)∗ = Z(2)[v1, . . . , vn−1, v±

n ],

|vi| = 2(2i − 1) Morava E-theory En: Landweber exact theory with (En)∗ = W (F2n)[[u1, . . . , un−1]][u±], |ui| = 0, |u| = 2 Related by completion and homotopy fixed points:

  • E(n) = LK(n)E(n),

En(Gal) = E hG

n

  • E(n) ≃ En(Gal)
  • E(n)∗ = (E(n)∗)∧

In =

Z2[[v1, . . . , vn−1]][v±

n ]

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 2 / 18

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Introduction

A natural question

We have E(n) ≃ En(Gal) and... Z/2 acts on E(n) Complex conjugation action Z/2 acts on En(Gal) Action of the subgroup of Morava stabilizer group generated by the formal inverse

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 3 / 18

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Introduction

A natural question

We have E(n) ≃ En(Gal) and... Z/2 acts on E(n) Complex conjugation action Z/2 acts on En(Gal) Action of the subgroup of Morava stabilizer group generated by the formal inverse

Natural question: Are these actions related?

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 3 / 18

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Introduction

A natural question

We have E(n) ≃ En(Gal) and... Z/2 acts on E(n) Complex conjugation action Z/2 acts on En(Gal) Action of the subgroup of Morava stabilizer group generated by the formal inverse

Natural question: Are these actions related? YES

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 3 / 18

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Introduction

A natural question

We have E(n) ≃ En(Gal) and... Z/2 acts on E(n) Complex conjugation action Z/2 acts on En(Gal) Action of the subgroup of Morava stabilizer group generated by the formal inverse

Natural question: Are these actions related? YES

First a little more background...

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 3 / 18

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Background ER(n)

Real theories

Complex conjugation action on E(n) arises in context of Real theories (Z/2-equivariant RO(Z/2)-graded) Atiyah, 1966: Real K-theory KR KR(X) = G

  • cplx v.b. π : E → X
  • E, X Z/2-spaces
  • antilin. on fibers, π equiv
  • Maia Averett (Mills College)

Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 4 / 18

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Background ER(n)

Real theories

Complex conjugation action on E(n) arises in context of Real theories (Z/2-equivariant RO(Z/2)-graded) Atiyah, 1966: Real K-theory KR KR(X) = G

  • cplx v.b. π : E → X
  • E, X Z/2-spaces
  • antilin. on fibers, π equiv
  • Landweber, 1967: Real cobordism MR

Uses Z/2-action of complex conjugation on BU(k).

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 4 / 18

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Background ER(n)

Real theories

Complex conjugation action on E(n) arises in context of Real theories (Z/2-equivariant RO(Z/2)-graded) Atiyah, 1966: Real K-theory KR KR(X) = G

  • cplx v.b. π : E → X
  • E, X Z/2-spaces
  • antilin. on fibers, π equiv
  • Landweber, 1967: Real cobordism MR

Uses Z/2-action of complex conjugation on BU(k). Araki, 1978: Defined BPR using a Quillen idempotent argument

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 4 / 18

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Background ER(n)

Real theories

Complex conjugation action on E(n) arises in context of Real theories (Z/2-equivariant RO(Z/2)-graded) Atiyah, 1966: Real K-theory KR KR(X) = G

  • cplx v.b. π : E → X
  • E, X Z/2-spaces
  • antilin. on fibers, π equiv
  • Landweber, 1967: Real cobordism MR

Uses Z/2-action of complex conjugation on BU(k). Araki, 1978: Defined BPR using a Quillen idempotent argument Hu & Kriz, 2001: Defined KR(n) and ER(n) as MR-modules

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 4 / 18

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Johnson-Wilson theories

Kitchloo and Wilson’s real Johnson-Wilson theory

Real theory E na¨ ıve Z/2-equivariant theory E{e} KR{e} = KU MR{e} = MU ER(n){e} = E(n)

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 5 / 18

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Johnson-Wilson theories

Kitchloo and Wilson’s real Johnson-Wilson theory

Real theory E na¨ ıve Z/2-equivariant theory E{e} KR{e} = KU MR{e} = MU ER(n){e} = E(n) Taking homotopy fixed points gives new theories: KUhZ/2 = KO E(n)hZ/2 = ER(n), Kitchloo and Wilson’s “real Johnson-Wilson”

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 5 / 18

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Johnson-Wilson theories

Kitchloo and Wilson’s real Johnson-Wilson theory

The ER(n) are higher real K-theories. E(1) = KU(2) ER(1) = KO(2) Kitchloo-Wilson: There is a fibration Σλ(n)ER(n)

x(n)

− → ER(n) → E(n) that reduces when n = 1 to the classical fibration ΣKO(2)

η

− → KO(2) → KU(2) Makes computations feasible (Bockstein spectral sequence). λ(n) = 22n+1 − 2n+2 + 1

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 6 / 18

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Johnson-Wilson theories

Bockstein Spectral Sequence

The Bockstein spectral sequence arising from the fibration Σλ(n)ER(n)

x(n)

− → ER(n) → E(n) can be used to compute the coefficients of ER(n).

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 7 / 18

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Johnson-Wilson theories

Bockstein Spectral Sequence

The Bockstein spectral sequence arising from the fibration Σλ(n)ER(n)

x(n)

− → ER(n) → E(n) can be used to compute the coefficients of ER(n). ER(n)∗ = Z(2)[ˆ vk(l) | 0 ≤ k < n, l ∈ Z][x, v±2n+1

n

]/J J is the ideal generated by the relations ˆ v0(0) = 2 x2k+1−1ˆ vk(l) = 0 |x| = λ(n) = 22n+1 − 2n+2 + 1 ER(n)∗ → E(n)∗ sends ˆ vk(l) → vkv−(2k−1)(2n−1)+l2k+1(2n−1)

n

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 7 / 18

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Johnson-Wilson theories

Completed Johnson-Wilson

Bousfield localization gives a completion E(n) → E(n) = LK(n)E(n) such that E(n)∗ → E(n)

∗

is In-adic completion for In = (v0, . . . , vn−1). Z(2)[v1, . . . , vn−1, v±

n ] →

Z2[[v1, . . . , vn−1]][v±

n ]

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 8 / 18

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Johnson-Wilson theories

Completed Johnson-Wilson

Bousfield localization gives a completion E(n) → E(n) = LK(n)E(n) such that E(n)∗ → E(n)

∗

is In-adic completion for In = (v0, . . . , vn−1). Z(2)[v1, . . . , vn−1, v±

n ] →

Z2[[v1, . . . , vn−1]][v±

n ]

Complex conjugation action on E(n) gives action on E(n)

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 8 / 18

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Bridge

Two sides of a picture

Summary so far: Z/2-action on E(n) of complex conjugation gives action on E(n) ER(n) := E(n)hZ/2

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 9 / 18

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Bridge

Two sides of a picture

Summary so far: Z/2-action on E(n) of complex conjugation gives action on E(n) ER(n) := E(n)hZ/2 Other side is Morava E-theory and stabilizer group action...

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 9 / 18

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Morava E-theory

The Morava stabilizer group

Morava E-theory: Landweber exact cohomology theory En (En)∗ = W (F2n)[[u1, . . . , un−1]][u±]

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 10 / 18

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Morava E-theory

The Morava stabilizer group

Morava E-theory: Landweber exact cohomology theory En (En)∗ = W (F2n)[[u1, . . . , un−1]][u±] FEn is the universal deformation of the Honda formal group law Hn

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 10 / 18

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Morava E-theory

The Morava stabilizer group

Morava E-theory: Landweber exact cohomology theory En (En)∗ = W (F2n)[[u1, . . . , un−1]][u±] FEn is the universal deformation of the Honda formal group law Hn Morava stabilizer group: Sn := Aut(Hn)

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 10 / 18

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Morava E-theory

The Morava stabilizer group

Morava E-theory: Landweber exact cohomology theory En (En)∗ = W (F2n)[[u1, . . . , un−1]][u±] FEn is the universal deformation of the Honda formal group law Hn Morava stabilizer group: Sn := Aut(Hn) Note: Sn has a subgroup of order two generated by the formal inverse i(x).

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 10 / 18

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Morava E-theory

The Morava stabilizer group

Morava E-theory: Landweber exact cohomology theory En (En)∗ = W (F2n)[[u1, . . . , un−1]][u±] FEn is the universal deformation of the Honda formal group law Hn Morava stabilizer group: Sn := Aut(Hn) Note: Sn has a subgroup of order two generated by the formal inverse i(x). Extended Morava stabilizer group: Gn := Gal(F2n/F2) ⋉ Sn

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 10 / 18

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Morava E-theory

Hopkins-Miller-Goerss

Lubin-Tate theory ⇒ Gn acts on (En)∗

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 11 / 18

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Morava E-theory

Hopkins-Miller-Goerss

Lubin-Tate theory ⇒ Gn acts on (En)∗ Hopkins-Miller-Goerss ⇒ Gn acts on En by E∞-maps

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 11 / 18

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Morava E-theory

Hopkins-Miller-Goerss

Lubin-Tate theory ⇒ Gn acts on (En)∗ Hopkins-Miller-Goerss ⇒ Gn acts on En by E∞-maps Get interesting E∞-ring spectra by E hK

n

for K ⊆ Gn e.g. E hGn

n

= LK(n)S

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 11 / 18

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Morava E-theory

Hopkins-Miller-Goerss

Lubin-Tate theory ⇒ Gn acts on (En)∗ Hopkins-Miller-Goerss ⇒ Gn acts on En by E∞-maps Get interesting E∞-ring spectra by E hK

n

for K ⊆ Gn e.g. E hGn

n

= LK(n)S Define En(Gal) := E hK

n

for K = Gal(F2n/F2) ⋉ F×

2n.

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 11 / 18

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Morava E-theory

Hopkins-Miller-Goerss

Lubin-Tate theory ⇒ Gn acts on (En)∗ Hopkins-Miller-Goerss ⇒ Gn acts on En by E∞-maps Get interesting E∞-ring spectra by E hK

n

for K ⊆ Gn e.g. E hGn

n

= LK(n)S Define En(Gal) := E hK

n

for K = Gal(F2n/F2) ⋉ F×

2n.

En(Gal)∗ = Z2[[v1, . . . , vn−1]][v±

n ]

Order 2 subgroup generated by i(x) acts on En(Gal)

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 11 / 18

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Morava E-theory

A natural question

We have E(n) ≃ En(Gal) and... Z/2 acts on E(n) Complex conjugation action Z/2 acts on En(Gal) Action of the subgroup of Morava stabilizer group generated by the formal inverse

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 12 / 18

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Morava E-theory

A natural question

We have E(n) ≃ En(Gal) and... Z/2 acts on E(n) Complex conjugation action Z/2 acts on En(Gal) Action of the subgroup of Morava stabilizer group generated by the formal inverse

Natural question: Are these actions related?

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 12 / 18

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Morava E-theory

A natural question

We have E(n) ≃ En(Gal) and... Z/2 acts on E(n) Complex conjugation action Z/2 acts on En(Gal) Action of the subgroup of Morava stabilizer group generated by the formal inverse

Natural question: Are these actions related? YES

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 12 / 18

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Answering that question Conclusions

Answer: Yes

Theorem (A.)

There is an equivalence

  • E(n)

hZ/2

≃ En(Gal)hZ/2 and the natural map ER(n) = E(n)hZ/2 → E(n)

hZ/2

induces an algebraic completion on coefficients.

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 13 / 18

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Answering that question Conclusions

Consequences

Corollary

After completion, ER(n) is an E∞-ring spectrum.

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 14 / 18

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Answering that question Conclusions

Consequences

Corollary

After completion, ER(n) is an E∞-ring spectrum.

Corollary

(En(Gal)hZ/2)∗ = Z2[[ˆ vk(l) | 0 ≤ k < n, l ∈ Z]][x, v±2n+1

n

]/J J is the ideal generated by the relations ˆ v0(0) = 2 x2k+1−1ˆ vk(l) = 0 and for k ≤ m, ˆ vm(l)ˆ vk(2m−ks) = ˆ vm(l + s)ˆ vk(0) |x| = λ(n) = 22n+1 − 2n+2 + 1 |v2n+1

n

| = 2n+2(2n − 1)2 |ˆ vk(l)| = 2(2k − 1) + l2k+2(2n − 1)2 − 2(2k − 1)(2n − 1)2

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 14 / 18

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The proof

A bit about the proof

We’d like equivariant map ϕ : E(n) → En(Gal) that is also an equivalence. But we only have a homotopy equivariant one.

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 15 / 18

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The proof

A bit about the proof

We’d like equivariant map ϕ : E(n) → En(Gal) that is also an equivalence. But we only have a homotopy equivariant one. Try replacing E(n) by E(n) ∧ F( E(n), En(Gal))ϕ ϕ homotopy equivariant ⇒ conjugation action on F( E(n), En(Gal))ϕ ev : E(n) ∧ F( E(n), En(Gal))ϕ → En(Gal) is honestly equivariant

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 15 / 18

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The proof

A bit about the proof

We’d like equivariant map ϕ : E(n) → En(Gal) that is also an equivalence. But we only have a homotopy equivariant one. Try replacing E(n) by E(n) ∧ F( E(n), En(Gal))ϕ ϕ homotopy equivariant ⇒ conjugation action on F( E(n), En(Gal))ϕ ev : E(n) ∧ F( E(n), En(Gal))ϕ → En(Gal) is honestly equivariant If F( E(n), En(Gal))ϕ ≃ pt, then E(n) ∧ F( E(n), En(Gal))ϕ ≃ E(n). Need appropriate category so that F( E(n), En(Gal))ϕ ≃ pt. Try S-algebra maps. Problem: not known if Z/2-action on E(n) is a S-algebra map.

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 15 / 18

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The proof

A bit about the proof

Instead use FS−alg(v−1

n

  • MU, En(Gal)). New problem: not contractible.

Dirty trick: create S-algebra T so that FT−alg(v−1

n

  • MU, En(Gal)) is

homotopy discrete.

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 16 / 18

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The proof

A bit about the proof

Instead use FS−alg(v−1

n

  • MU, En(Gal)). New problem: not contractible.

Dirty trick: create S-algebra T so that FT−alg(v−1

n

  • MU, En(Gal)) is

homotopy discrete.

T = free commutative S-algebra on a bunch of spheres π∗(v −1

n

  • MU) = π∗(

E(n) ∧ T) Compute BKSS for FT−alg(−, En(Gal)) A map E(n) ∧ T → v −1

n

  • MU gives a map of spectral sequences that is

an iso on E2 Since that for E(n) ∧ T collapses, so does that for v −1

n

  • MU

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 16 / 18

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The proof

A bit about the proof

Now v−1

n

  • MU ∧ FT−alg(v−1

n

  • MU, En(Gal))ν → En(Gal)

is equivariant. After taking homotopy fixed points, obtain a factorization v−1

n

  • MU

hg Z/2

En(Gal)hg

Z/2

  • E(n)

hg Z/2

✲ ❄ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ✸

≃

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 17 / 18

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Fin!

Thank you!

Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E-theory CAT ’09 18 / 18