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Examples of non- algebraic classes in the Brown-Peterson tower Freie Universitt Berlin November 16, 2017 Gereon Quick NTNU Lefschetz s theorem: X projective complex surface Lefschetz s theorem: X projective complex surface 2-dim.

  1. 𝜍 Topological realization: Sm C Man X(C) X ”complex e.g. P 1 P 1 (C)=CP 1 ≃ S 2 manifold of solutions in C” motivic induced map spectrum 𝜍 E a,b (X) E a (X(C)) mot top β€œalgebraic” β€œtopological”

  2. 𝜍 Topological realization: Sm C Man X(C) X ”complex e.g. P 1 P 1 (C)=CP 1 ≃ S 2 manifold of solutions in C” motivic induced map spectrum 𝜍 E a,b (X) E a (X(C)) mot top β€œalgebraic” β€œtopological” Question: How to detect whether classes in E* (X(C)) are algebraic, i.e., top are in the image of 𝜍 ?

  3. 𝜐 motivic Obstruction: Given E HZ. Eilenberg-MacLane Thom map spectrum

  4. 𝜐 motivic Obstruction: Given E HZ. Eilenberg-MacLane Thom map spectrum E 2 * , *(X) mot 𝜐 H 2 * , *(X;Z) mot

  5. 𝜐 motivic Obstruction: Given E HZ. Eilenberg-MacLane Thom map spectrum E 2 *(X) E 2 * , *(X) top mot 𝜐 𝜐 H 2 * , *(X;Z) H 2 *(X;Z) mot singular cohomology

  6. 𝜐 motivic Obstruction: Given E HZ. Eilenberg-MacLane Thom map spectrum 𝜍 E E 2 *(X) E 2 * , *(X) top mot β†Ί 𝜐 𝜐 𝜍 H H 2 * , *(X;Z) H 2 *(X;Z) mot singular cohomology

  7. 𝜐 motivic Obstruction: Given E HZ. Eilenberg-MacLane Thom map spectrum 𝜍 E E 2 *(X) E 2 * , *(X) top mot β†Ί 𝜐 𝜐 =cl H 𝜍 H H 2 * , *(X;Z) H 2 *(X;Z) mot singular cohomology

  8. 𝜐 motivic Obstruction: Given E HZ. Eilenberg-MacLane Thom map spectrum 𝜍 E E 2 *(X) E 2 * , *(X) top mot β†Ί 𝜐 𝜐 =cl H 𝜍 H H 2 * , *(X;Z) Alg 2 *(X) βŠ† H 2 *(X;Z) mot H singular cohomology

  9. 𝜐 motivic Obstruction: Given E HZ. Eilenberg-MacLane Thom map spectrum 𝜍 E Alg 2 *(X) βŠ† E 2 *(X) E 2 * , *(X) top mot E β†Ί 𝜐 𝜐 =cl H 𝜍 H H 2 * , *(X;Z) Alg 2 *(X) βŠ† H 2 *(X;Z) mot H singular cohomology

  10. 𝜐 motivic Obstruction: Given E HZ. Eilenberg-MacLane Thom map spectrum 𝜍 E Alg 2 *(X) βŠ† E 2 *(X) E 2 * , *(X) top mot E must β†Ί 𝜐 𝜐 factor through =cl H 𝜍 H H 2 * , *(X;Z) Alg 2 *(X) βŠ† H 2 *(X;Z) mot H singular cohomology

  11. 𝜐 motivic Obstruction: Given E HZ. Eilenberg-MacLane Thom map spectrum 𝜍 E Alg 2 *(X) βŠ† E 2 *(X) E 2 * , *(X) top mot E must β†Ί 𝜐 𝜐 factor through =cl H 𝜍 H H 2 * , *(X;Z) Alg 2 *(X) βŠ† H 2 *(X;Z) mot H singular cohomology Easier task:

  12. 𝜐 motivic Obstruction: Given E HZ. Eilenberg-MacLane Thom map spectrum 𝜍 E Alg 2 *(X) βŠ† E 2 *(X) E 2 * , *(X) top mot E must β†Ί 𝜐 𝜐 factor through =cl H 𝜍 H H 2 * , *(X;Z) Alg 2 *(X) βŠ† H 2 *(X;Z) mot H singular cohomology Easier task: describe using E 2 *(X(C))\Alg 2 *(X) H 2 *(X;Z)\Alg 2 *(X) top E H

  13. Atiyah-Hirzebruch, Totaro, Levine-Morel: 𝜍 H =cl H H 2 * , *(X;Z) Alg 2 *(X) βŠ† H 2 *(X;Z) mot H

  14. Atiyah-Hirzebruch, Totaro, Levine-Morel: 𝜍 MGL MGL 2 * , *(X) MU 2 *(X) 𝜍 H =cl H H 2 * , *(X;Z) Alg 2 *(X) βŠ† H 2 *(X;Z) mot H

  15. Atiyah-Hirzebruch, Totaro, Levine-Morel: 𝜍 MGL MGL 2 * , *(X) MU 2 *(X) MGL 2 * , *(X) βŠ— L* Z MU 2 *(X) βŠ— L* Z 𝜍 H =cl H H 2 * , *(X;Z) Alg 2 *(X) βŠ† H 2 *(X;Z) mot H

  16. Atiyah-Hirzebruch, Totaro, Levine-Morel: 𝜍 MGL MGL 2 * , *(X) MU 2 *(X) MGL 2 * , *(X) βŠ— L* Z MU 2 *(X) βŠ— L* Z Totaro ⟳ 𝜍 H =cl H H 2 * , *(X;Z) Alg 2 *(X) βŠ† H 2 *(X;Z) mot H

  17. Atiyah-Hirzebruch, Totaro, Levine-Morel: 𝜍 MGL MGL 2 * , *(X) MU 2 *(X) MGL 2 * , *(X) βŠ— L* Z MU 2 *(X) βŠ— L* Z Levine + Totaro Levine-Morel β‰ˆ ⟳ 𝜍 H =cl H H 2 * , *(X;Z) Alg 2 *(X) βŠ† H 2 *(X;Z) mot H

  18. Atiyah-Hirzebruch, Totaro, Levine-Morel: 𝜍 MGL MGL 2 * , *(X) MU 2 *(X) MGL 2 * , *(X) βŠ— L* Z MU 2 *(X) βŠ— L* Z Levine + ≉ in general Totaro Levine-Morel β‰ˆ ⟳ 𝜍 H =cl H H 2 * , *(X;Z) Alg 2 *(X) βŠ† H 2 *(X;Z) mot H

  19. Atiyah-Hirzebruch, Totaro, Levine-Morel: 𝜍 MGL MGL 2 * , *(X) MU 2 *(X) MGL 2 * , *(X) βŠ— L* Z MU 2 *(X) βŠ— L* Z Levine + ≉ in general Totaro Levine-Morel β‰ˆ ⟳ 𝜍 H =cl H H 2 * , *(X;Z) Alg 2 *(X) βŠ† H 2 *(X;Z) mot H β€’ Atiyah-Hirzebruch: cl H is not surjective onto integral Hodge classes.

  20. Atiyah-Hirzebruch, Totaro, Levine-Morel: 𝜍 MGL MGL 2 * , *(X) MU 2 *(X) MGL 2 * , *(X) βŠ— L* Z MU 2 *(X) βŠ— L* Z Levine + ≉ in general Totaro Levine-Morel β‰ˆ ⟳ 𝜍 H =cl H H 2 * , *(X;Z) Alg 2 *(X) βŠ† H 2 *(X;Z) mot H β€’ Atiyah-Hirzebruch: cl H is not surjective onto integral Hodge classes. β€’ Totaro: new classes in kernel of cl H .

  21. A different perspective: Fix a prime p.

  22. A different perspective: Brown-Peterson, Fix a prime p. Quillen |v i |=2(p i -1) MU (p) splits as a wedge of suspensions of spectra BP with BP = Z (p) [v 1 ,v 2 ,…]. *

  23. A different perspective: Brown-Peterson, Fix a prime p. Quillen |v i |=2(p i -1) MU (p) splits as a wedge of suspensions of spectra BP with BP = Z (p) [v 1 ,v 2 ,…]. * quotient map BP BP/(v n+1 ,…) =: BP ⟨ n ⟩ For every n: with BP ⟨ n ⟩ = Z (p) [v 1 ,…,v n ] *

  24. A different perspective: Brown-Peterson, Fix a prime p. Quillen |v i |=2(p i -1) MU (p) splits as a wedge of suspensions of spectra BP with BP = Z (p) [v 1 ,v 2 ,…]. * quotient map BP BP/(v n+1 ,…) =: BP ⟨ n ⟩ For every n: with BP ⟨ n ⟩ = Z (p) [v 1 ,…,v n ] * The Brown-Peterson tower (Wilson): … … BP BP ⟨ n ⟩ BP ⟨ 1 ⟩ BP ⟨ 0 ⟩ BP ⟨ -1 ⟩ p=2: 2-local HZ (p) HF p connective K-theory

  25. Milnor operations:

  26. Milnor operations: For every n: stable cofibre sequence v n |v n | |v n |+1 βˆ‘ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n-1 ⟩ βˆ‘

  27. Milnor operations: For every n: stable cofibre sequence v n |v n | |v n |+1 βˆ‘ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n-1 ⟩ βˆ‘ with an induced exact sequence (for any space X) +|v n | BP ⟨ n ⟩ * (X) BP ⟨ n ⟩ *(X) q n BP ⟨ n ⟩ * +|v n |+1 (X) BP ⟨ n-1 ⟩ *(X)

  28. Milnor operations: For every n: stable cofibre sequence v n |v n | |v n |+1 βˆ‘ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n-1 ⟩ βˆ‘ with an induced exact sequence (for any space X) +|v n | BP ⟨ n ⟩ * (X) BP ⟨ n ⟩ *(X) q n BP ⟨ n ⟩ * +|v n |+1 (X) BP ⟨ n-1 ⟩ *(X) BP ⟨ n-1 ⟩ Thom map +|v n |+1 (X;F p ) HF p H*(X;F p ) H* Q n

  29. Milnor operations: For every n: stable cofibre sequence v n |v n | |v n |+1 βˆ‘ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n-1 ⟩ βˆ‘ with an induced exact sequence (for any space X) +|v n | BP ⟨ n ⟩ * (X) BP ⟨ n ⟩ *(X) q n BP ⟨ n ⟩ * +|v n |+1 (X) BP ⟨ n-1 ⟩ *(X) BP ⟨ n-1 ⟩ nth Milnor Thom operation: map Q 0 =Bockstein p n-1 p n-1 +|v n |+1 (X;F p ) Q n =P Q n-1 -Q n-1 P HF p H*(X;F p ) H* Q n

  30. The LMT obstruction in action: BP 2 *(X) ⟲ q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) β†Ί +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Q n

  31. The LMT obstruction in action: BP 2 *(X) ⟲ q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) β†Ί +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Q n 𝝱

  32. The LMT obstruction in action: BP 2 *(X) ⟲ q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) β†Ί +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Q n 𝝱 Question: Is 𝝱 algebraic?

  33. The LMT obstruction in action: BP 2 *(X) ⟲ q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) β†Ί +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Z*(X) Q n 𝝱 Question: Is 𝝱 algebraic?

  34. The LMT obstruction in action: BP 2 *(X) LMT ⟲ q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) β†Ί +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Z*(X) Q n 𝝱 Question: Is 𝝱 algebraic?

  35. The LMT obstruction in action: BP 2 *(X) LMT ⟲ 𝝱 n-1 q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) β†Ί +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Z*(X) Q n 𝝱 Question: Is 𝝱 algebraic?

  36. The LMT obstruction in action: BP 2 *(X) LMT ⟲ 𝝱 n-1 q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) β†Ί +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Z*(X) Q n 𝝱 Question: Is 𝝱 algebraic?

  37. The LMT obstruction in action: BP 2 *(X) LMT ⟲ q n 𝝱 n-1 𝝱 n-1 q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) β†Ί +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Z*(X) Q n 𝝱 Question: Is 𝝱 algebraic?

  38. The LMT obstruction in action: BP 2 *(X) LMT if ⟲ = 0 q n 𝝱 n-1 𝝱 n-1 q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) β†Ί +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Z*(X) Q n 𝝱 Question: Is 𝝱 algebraic?

  39. The LMT obstruction in action: BP 2 *(X) LMT then if ⟲ = 0 q n 𝝱 n-1 𝝱 n 𝝱 n-1 q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) β†Ί +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Z*(X) Q n 𝝱 Question: Is 𝝱 algebraic?

  40. The LMT obstruction in action: BP 2 *(X) LMT then if ⟲ = 0 q n 𝝱 n-1 𝝱 n 𝝱 n-1 q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) β†Ί +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Z*(X) Q n Q n 𝝱 𝝱 Question: Is 𝝱 algebraic?

  41. The LMT obstruction in action: BP 2 *(X) LMT then if ⟲ = 0 q n 𝝱 n-1 𝝱 n 𝝱 n-1 q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) β†Ί +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Z*(X) Q n Q n 𝝱 𝝱

  42. The LMT obstruction in action: BP 2 *(X) LMT then if ⟲ = 0 q n 𝝱 n-1 𝝱 n 𝝱 n-1 q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) β†Ί +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Z*(X) Q n Q n 𝝱 𝝱 Levine-Morel-Totaro obstruction:

  43. The LMT obstruction in action: BP 2 *(X) LMT then if ⟲ = 0 q n 𝝱 n-1 𝝱 n 𝝱 n-1 q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) β†Ί +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Z*(X) Q n β‰  0 Q n 𝝱 𝝱 if Levine-Morel-Totaro obstruction: If Q n 𝝱 β‰  0,

  44. The LMT obstruction in action: BP 2 *(X) LMT then ⟲ q n 𝝱 n-1 𝝱 n 𝝱 n-1 q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) β†Ί +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Z*(X) Q n β‰  0 Q n 𝝱 𝝱 if Levine-Morel-Totaro obstruction: If Q n 𝝱 β‰  0,

  45. The LMT obstruction in action: BP 2 *(X) LMT then ⟲ q n 𝝱 n-1 β‰  0 𝝱 n 𝝱 n-1 q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) β†Ί +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Z*(X) Q n β‰  0 Q n 𝝱 𝝱 if Levine-Morel-Totaro obstruction: If Q n 𝝱 β‰  0,

  46. The LMT obstruction in action: BP 2 *(X) LMT then ⟲ ✘ q n 𝝱 n-1 β‰  0 𝝱 n 𝝱 n-1 q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) β†Ί +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Z*(X) Q n β‰  0 Q n 𝝱 𝝱 if Levine-Morel-Totaro obstruction: If Q n 𝝱 β‰  0,

  47. The LMT obstruction in action: BP 2 *(X) LMT then ⟲ ✘ ✘ q n 𝝱 n-1 β‰  0 𝝱 n 𝝱 n-1 q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) β†Ί +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Z*(X) Q n β‰  0 Q n 𝝱 𝝱 if Levine-Morel-Totaro obstruction: If Q n 𝝱 β‰  0,

  48. The LMT obstruction in action: BP 2 *(X) LMT then ⟲ ✘ ✘ q n 𝝱 n-1 β‰  0 𝝱 n 𝝱 n-1 q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) β†Ί +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Z*(X) Q n β‰  0 Q n 𝝱 𝝱 if Levine-Morel-Totaro obstruction: If Q n 𝝱 β‰  0, then 𝝱 is not algebraic.

  49. Voevodsky’ s motivic Milnor operations:

  50. Voevodsky’ s motivic Milnor operations: There are motivic operations mod p-motivic Q n ∈ 𝓑 2p n -1,p n -1 mot Steenrod algebra

  51. Voevodsky’ s motivic Milnor operations: There are motivic operations mod p-motivic Q n ∈ 𝓑 2p n -1,p n -1 mot Steenrod algebra For a smooth complex variety X: mot Q n i+2p n -1,j+p n -1 (X;F p ) i,j H (X;F p ) H mot mot mod p-motivic cohomology

  52. Voevodsky’ s motivic Milnor operations: There are motivic operations mod p-motivic Q n ∈ 𝓑 2p n -1,p n -1 mot Steenrod algebra For a smooth complex variety X: mot Q n i+2p n -1,j+p n -1 (X;F p ) i,j H (X;F p ) H mot mot mod p-motivic cohomology 2i,i Recall: = CH i (X;Z/p) H (X;F p ) and mot i,j H (X;F p ) = 0 if i>2j. mot

  53. Obstructions revisited: X smooth complex variety mot Q n 2i,i 2i+2p n -1,i+p n -1 (X;F p ) H (X;F p ) H mot mot β†Ί topological realization 2i+2p n -1 H 2i (X;F p ) H (X;F p ) Q n

  54. Obstructions revisited: X smooth complex variety mot Q n 2i,i 2i+2p n -1,i+p n -1 (X;F p ) H (X;F p ) H = 0 mot mot β†Ί topological realization 2i+2p n -1 H 2i (X;F p ) H (X;F p ) Q n

  55. Obstructions revisited: X smooth complex variety mot Q n mot Q n 2i,i 2i+2p n -1,i+p n -1 (X;F p ) H (X;F p ) H = 0 mot mot β†Ί topological realization 2i+2p n -1 H 2i (X;F p ) H (X;F p ) Q n

  56. Obstructions revisited: X smooth complex variety mot Q n mot Q n 2i,i 2i+2p n -1,i+p n -1 (X;F p ) H (X;F p ) H = 0 mot mot β†Ί topological realization 2i+2p n -1 H 2i (X;F p ) H (X;F p ) Q n 𝝱

  57. Obstructions revisited: X smooth complex variety mot Q n mot Q n 2i,i 2i+2p n -1,i+p n -1 (X;F p ) H (X;F p ) H = 0 mot mot β†Ί topological realization 2i+2p n -1 H 2i (X;F p ) H (X;F p ) Q n Q n 𝝱 β‰  0 𝝱 if

  58. Obstructions revisited: X smooth complex variety mot Q n mot Q n 2i,i 2i+2p n -1,i+p n -1 (X;F p ) H (X;F p ) H = 0 mot mot β†Ί ✘ topological realization 2i+2p n -1 H 2i (X;F p ) H (X;F p ) Q n Q n 𝝱 β‰  0 𝝱 if

  59. Obstructions revisited: X smooth complex variety mot Q n mot Q n 2i,i 2i+2p n -1,i+p n -1 (X;F p ) H (X;F p ) H = 0 mot mot β†Ί ✘ topological realization 2i+2p n -1 H 2i (X;F p ) H (X;F p ) Q n Q n 𝝱 β‰  0 𝝱 if Observation: The LMT-obstruction is particular to smooth varieties and bidegrees (2i,i).

  60. Obstructions revisited: X smooth complex variety mot Q n mot Q n 2i,i 2i+2p n -1,i+p n -1 (X;F p ) H (X;F p ) H = 0 mot mot β†Ί ✘ topological realization 2i+2p n -1 H 2i (X;F p ) H (X;F p ) Q n Q n 𝝱 β‰  0 𝝱 if Observation: The LMT-obstruction is particular to smooth varieties and bidegrees (2i,i). Example: Q n 𝛋 β‰  0 for 𝛋 the fundamental class of a suitable Eilenberg-MacLane space, though 𝛋 is algebraic.

  61. Back to our task:

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