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
Introduction Two cohomology theories Fix prime p = 2. Johnson-Wilson theory E ( n ): Landweber exact theory with | v i | = 2(2 i − 1) E ( n ) ∗ = Z (2) [ v 1 , . . . , v n − 1 , v ± n ] , Morava E -theory E n : Landweber exact theory with ( E n ) ∗ = W ( F 2 n )[[ u 1 , . . . , u n − 1 ]][ u ± ] , | u i | = 0 , | u | = 2 Related by completion and homotopy fixed points: � E n (Gal) = E hG E ( n ) = L K ( n ) E ( n ) , n � E ( n ) ≃ E n (Gal) � I n = � E ( n ) ∗ = ( E ( n ) ∗ ) ∧ Z 2 [[ v 1 , . . . , v n − 1 ]][ v ± n ] Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E -theory CAT ’09 2 / 18
Introduction A natural question We have � E ( n ) ≃ E n (Gal) and... Z / 2 acts on E n (Gal) Z / 2 acts on � E ( n ) Action of the subgroup of Complex conjugation action 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
Introduction A natural question We have � E ( n ) ≃ E n (Gal) and... Z / 2 acts on E n (Gal) Z / 2 acts on � E ( n ) Action of the subgroup of Complex conjugation action 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
Introduction A natural question We have � E ( n ) ≃ E n (Gal) and... Z / 2 acts on E n (Gal) Z / 2 acts on � E ( n ) Action of the subgroup of Complex conjugation action 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
Introduction A natural question We have � E ( n ) ≃ E n (Gal) and... Z / 2 acts on E n (Gal) Z / 2 acts on � E ( n ) Action of the subgroup of Complex conjugation action 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
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 � � � � E , X Z / 2-spaces � KR ( X ) = G cplx v.b. π : E → X � 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
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 � � � � E , X Z / 2-spaces � KR ( X ) = G cplx v.b. π : E → X � 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
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 � � � � E , X Z / 2-spaces � KR ( X ) = G cplx v.b. π : E → X � 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
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 � � � � E , X Z / 2-spaces � KR ( X ) = G cplx v.b. π : E → X � 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
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
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: KU h Z / 2 = KO E ( n ) h Z / 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
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 x ( n ) Σ λ ( n ) ER ( n ) − → ER ( n ) → E ( n ) that reduces when n = 1 to the classical fibration η Σ KO (2) − → KO (2) → KU (2) Makes computations feasible (Bockstein spectral sequence). λ ( n ) = 2 2 n +1 − 2 n +2 + 1 Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E -theory CAT ’09 6 / 18
Johnson-Wilson theories Bockstein Spectral Sequence The Bockstein spectral sequence arising from the fibration x ( n ) Σ λ ( n ) ER ( 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
Johnson-Wilson theories Bockstein Spectral Sequence The Bockstein spectral sequence arising from the fibration x ( n ) Σ λ ( n ) ER ( n ) − → ER ( n ) → E ( n ) can be used to compute the coefficients of ER ( n ). v k ( l ) | 0 ≤ k < n , l ∈ Z ][ x , v ± 2 n +1 ER ( n ) ∗ = Z (2) [ˆ ] / J n J is the ideal generated by the relations x 2 k +1 − 1 ˆ ˆ v 0 (0) = 2 v k ( l ) = 0 | x | = λ ( n ) = 2 2 n +1 − 2 n +2 + 1 v k ( l ) �→ v k v − (2 k − 1)(2 n − 1)+ l 2 k +1 (2 n − 1) ER ( n ) ∗ → E ( n ) ∗ sends ˆ n Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E -theory CAT ’09 7 / 18
Johnson-Wilson theories Completed Johnson-Wilson Bousfield localization gives a completion E ( n ) → � E ( n ) = L K ( n ) E ( n ) such that ∗ E ( n ) ∗ → � E ( n ) is I n -adic completion for I n = ( v 0 , . . . , v n − 1 ). n ] → � Z (2) [ v 1 , . . . , v n − 1 , v ± Z 2 [[ v 1 , . . . , v n − 1 ]][ v ± n ] Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E -theory CAT ’09 8 / 18
Johnson-Wilson theories Completed Johnson-Wilson Bousfield localization gives a completion E ( n ) → � E ( n ) = L K ( n ) E ( n ) such that ∗ E ( n ) ∗ → � E ( n ) is I n -adic completion for I n = ( v 0 , . . . , v n − 1 ). n ] → � Z (2) [ v 1 , . . . , v n − 1 , v ± Z 2 [[ v 1 , . . . , v n − 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
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 ) h Z / 2 Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E -theory CAT ’09 9 / 18
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 ) h Z / 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
Morava E -theory The Morava stabilizer group Morava E -theory: Landweber exact cohomology theory E n ( E n ) ∗ = W ( F 2 n )[[ u 1 , . . . , u n − 1 ]][ u ± ] Maia Averett (Mills College) Completion of real Johnson-Wilson theory gives fixed points of Morava E -theory CAT ’09 10 / 18
Morava E -theory The Morava stabilizer group Morava E -theory: Landweber exact cohomology theory E n ( E n ) ∗ = W ( F 2 n )[[ u 1 , . . . , u n − 1 ]][ u ± ] F E n is the universal deformation of the Honda formal group law H n 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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