Examples of non- algebraic classes in the Brown-Peterson tower - PowerPoint PPT Presentation

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

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

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

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

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) ↺ +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * CH*(X) Q n Q n 𝝱 ≠ 0 𝝱 if Levine-Morel-Totaro obstruction: If Q n 𝝱 ≠ 0,

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

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

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

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

Voevodsky’ s motivic Milnor operations:

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

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

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

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 ↺ topological realization 2i+2p n -1 H 2i (X;F p ) H (X;F p ) Q n

Obstructions revisited: X smooth complex variety CH i (X;Z/p) ‖ mot Q n 2i,i 2i+2p n -1,i+p n -1 (X;F p ) H (X;F p ) H mot mot ↺ topological realization 2i+2p n -1 H 2i (X;F p ) H (X;F p ) Q n

Obstructions revisited: X smooth complex variety CH i (X;Z/p) ‖ mot Q n 2i,i 2i+2p n -1,i+p n -1 (X;F p ) H (X;F p ) H = 0 mot mot ↺ topological realization 2i+2p n -1 H 2i (X;F p ) H (X;F p ) Q n

Obstructions revisited: X smooth complex variety mot Q n CH i (X;Z/p) ‖ mot Q n 2i,i 2i+2p n -1,i+p n -1 (X;F p ) H (X;F p ) H = 0 mot mot ↺ topological realization 2i+2p n -1 H 2i (X;F p ) H (X;F p ) Q n

Obstructions revisited: X smooth complex variety mot Q n CH i (X;Z/p) ‖ mot Q n 2i,i 2i+2p n -1,i+p n -1 (X;F p ) H (X;F p ) H = 0 mot mot ↺ topological realization 2i+2p n -1 H 2i (X;F p ) H (X;F p ) Q n 𝝱

Obstructions revisited: X smooth complex variety mot Q n CH i (X;Z/p) ‖ mot Q n 2i,i 2i+2p n -1,i+p n -1 (X;F p ) H (X;F p ) H = 0 mot mot ↺ topological realization 2i+2p n -1 H 2i (X;F p ) H (X;F p ) Q n Q n 𝝱 ≠ 0 𝝱 if

Obstructions revisited: X smooth complex variety mot Q n CH i (X;Z/p) ‖ mot Q n 2i,i 2i+2p n -1,i+p n -1 (X;F p ) H (X;F p ) H = 0 mot mot ↺ ✘ topological realization 2i+2p n -1 H 2i (X;F p ) H (X;F p ) Q n Q n 𝝱 ≠ 0 𝝱 if

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

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

Back to our task:

Back to our task: Study Alg 2 *(X) and its complement in E 2 *(X). E top

Back to our task: Study Alg 2 *(X) and its complement in E 2 *(X). E top For example: E=BP ⟨ n ⟩ ?

Back to our task: Study Alg 2 *(X) and its complement in E 2 *(X). E top For example: E=BP ⟨ n ⟩ ? Recall: BP and BP ⟨ n ⟩ exist in the motivic world (e.g. Vezzosi, Hopkins, Hu-Kriz, Ormsby, Hoyois, Ormsby-Østvær).

Back to our task: Study Alg 2 *(X) and its complement in E 2 *(X). E top For example: E=BP ⟨ n ⟩ ? Recall: BP and BP ⟨ n ⟩ exist in the motivic world (e.g. Vezzosi, Hopkins, Hu-Kriz, Ormsby, Hoyois, Ormsby-Østvær). Question: How can we produce non-algebraic elements in BP ⟨ n ⟩ 2 *(X)? top drop the “top” from now on

Back to the cofibre sequence:

Back to the cofibre sequence: Recall the stable cofibre sequence v n |v n | |v n |+1 ∑ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n-1 ⟩ ∑

Back to the cofibre sequence: Recall the stable cofibre sequence v n |v n | |v n |+1 ∑ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n-1 ⟩ ∑ and the induced map q n BP ⟨ n ⟩ * +|v n |+1 (X). BP ⟨ n-1 ⟩ *(X)

Back to the cofibre sequence: Recall the stable cofibre sequence v n |v n | |v n |+1 ∑ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n-1 ⟩ ∑ and the induced map q n BP ⟨ n ⟩ * +|v n |+1 (X). BP ⟨ n-1 ⟩ *(X) For example:

Back to the cofibre sequence: Recall the stable cofibre sequence v n |v n | |v n |+1 ∑ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n-1 ⟩ ∑ and the induced map q n BP ⟨ n ⟩ * +|v n |+1 (X). BP ⟨ n-1 ⟩ *(X) For example: Bockstein homomorphism q 0 H* +1 (X;Z (p) ), n=0: H*(X;F p )

Back to the cofibre sequence: Recall the stable cofibre sequence v n |v n | |v n |+1 ∑ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n-1 ⟩ ∑ and the induced map q n BP ⟨ n ⟩ * +|v n |+1 (X). BP ⟨ n-1 ⟩ *(X) For example: Bockstein homomorphism q 0 H* +1 (X;Z (p) ), n=0: H*(X;F p ) q 1 +2p-1 (X), n=1: H*(X;Z (p) ) BP ⟨ 1 ⟩ * ⋮

A big diagram: H k (X;F p )

A big diagram: H k+1 (X;Z (p) ) q 0 H k (X;F p )

A big diagram: BP ⟨ 1 ⟩ k+1+2p-1 (X) q 1 H k+1 (X;Z (p) ) q 0 H k (X;F p )

A big diagram: +|v 0 |+…+|v n | BP ⟨ n ⟩ k (X) q n ⋮ q 2 BP ⟨ 1 ⟩ k+1+2p-1 (X) q 1 H k+1 (X;Z (p) ) q 0 H k (X;F p )

A big diagram: q n+1 +|v 0 |+…+|v n | BP ⟨ n+1 ⟩ k +|v 0 |+…+|v n+1 | BP ⟨ n ⟩ k (X) (X) q n ⋮ q 2 BP ⟨ 1 ⟩ k+1+2p-1 (X) q 1 H k+1 (X;Z (p) ) q 0 H k (X;F p )

A big diagram: q n+1 +|v 0 |+…+|v n | BP ⟨ n+1 ⟩ k +|v 0 |+…+|v n+1 | BP ⟨ n ⟩ k (X) (X) q n ⋮ BP ⟨ n+1 ⟩ q 2 Thom map BP ⟨ 1 ⟩ k+1+2p-1 (X) HF p q 1 H k+1 (X;Z (p) ) q 0 +|v 0 |+…+|v n+1 | H k (X;F p ) H k (X;F p )

A big diagram: q n+1 +|v 0 |+…+|v n | BP ⟨ n+1 ⟩ k +|v 0 |+…+|v n+1 | BP ⟨ n ⟩ k (X) (X) q n ⋮ BP ⟨ n+1 ⟩ q 2 Thom map BP ⟨ 1 ⟩ k+1+2p-1 (X) HF p q 1 H k+1 (X;Z (p) ) q 0 +|v 0 |+…+|v n+1 | H k (X;F p ) H k (X;F p ) Q n+1 Q n …Q 0

Lifting classes: We get

Lifting classes: We get • a map 𝜒 𝜒 :=q n …q 0 BP ⟨ n ⟩ k +|v 0 |+…+|v n | H k (X;F p ) (X)

Lifting classes: We get • a map 𝜒 𝜒 :=q n …q 0 BP ⟨ n ⟩ k +|v 0 |+…+|v n | H k (X;F p ) (X) 𝜒 ( 𝝱 ) 𝝱

Lifting classes: We get • a map 𝜒 BP k +|v 0 |+…+|v n | (X) • an obstruction 𝜒 :=q n …q 0 BP ⟨ n ⟩ k +|v 0 |+…+|v n | H k (X;F p ) (X) 𝜒 ( 𝝱 ) 𝝱 Q n+1 Q n …Q 0 +|v 0 |+…+|v n+1 | H k (X;F p )

Lifting classes: We get • a map 𝜒 BP k +|v 0 |+…+|v n | (X) • an obstruction 𝜒 :=q n …q 0 BP ⟨ n ⟩ k +|v 0 |+…+|v n | H k (X;F p ) (X) 𝜒 ( 𝝱 ) 𝝱 q n+1 Q n+1 Q n …Q 0 +|v 0 |+…+|v n+1 | BP ⟨ n+1 ⟩ k +|v 0 |+…+|v n+1 | H k (X;F p ) (X)

Lifting classes: We get • a map 𝜒 BP k +|v 0 |+…+|v n | (X) • an obstruction 𝜒 :=q n …q 0 BP ⟨ n ⟩ k +|v 0 |+…+|v n | H k (X;F p ) (X) 𝜒 ( 𝝱 ) 𝝱 q n+1 Q n+1 Q n …Q 0 ≠ 0 +|v 0 |+…+|v n+1 | BP ⟨ n+1 ⟩ k +|v 0 |+…+|v n+1 | H k (X;F p ) (X)

Lifting classes: We get • a map 𝜒 BP k +|v 0 |+…+|v n | (X) • an obstruction 𝜒 :=q n …q 0 BP ⟨ n ⟩ k +|v 0 |+…+|v n | H k (X;F p ) (X) 𝜒 ( 𝝱 ) 𝝱 q n+1 Q n+1 Q n …Q 0 ≠ 0 ≠ 0 +|v 0 |+…+|v n+1 | BP ⟨ n+1 ⟩ k +|v 0 |+…+|v n+1 | H k (X;F p ) (X)

Lifting classes: We get • a map 𝜒 BP k +|v 0 |+…+|v n | (X) • an obstruction ✘ 𝜒 :=q n …q 0 BP ⟨ n ⟩ k +|v 0 |+…+|v n | H k (X;F p ) (X) 𝜒 ( 𝝱 ) 𝝱 q n+1 Q n+1 Q n …Q 0 ≠ 0 ≠ 0 +|v 0 |+…+|v n+1 | BP ⟨ n+1 ⟩ k +|v 0 |+…+|v n+1 | H k (X;F p ) (X)

Lifting classes: We get • a map 𝜒 BP k +|v 0 |+…+|v n | (X) • an obstruction ✘ 𝜒 :=q n …q 0 BP ⟨ n ⟩ k +|v 0 |+…+|v n | H k (X;F p ) (X) 𝜒 ( 𝝱 ) 𝝱 q n+1 Q n+1 Q n …Q 0 ≠ 0 ≠ 0 +|v 0 |+…+|v n+1 | BP ⟨ n+1 ⟩ k +|v 0 |+…+|v n+1 | H k (X;F p ) (X) • If Q n+1 ..Q 0 ( 𝝱 ) ≠ 0, then 𝜒 ( 𝝱 ) is not algebraic.

Lifting classes: We get • a map 𝜒 BP k +|v 0 |+…+|v n | (X) • an obstruction ✘ 𝜒 :=q n …q 0 BP ⟨ n ⟩ k +|v 0 |+…+|v n | H k (X;F p ) (X) 𝜒 ( 𝝱 ) 𝝱 q n+1 Q n+1 Q n …Q 0 ≠ 0 ≠ 0 +|v 0 |+…+|v n+1 | BP ⟨ n+1 ⟩ k +|v 0 |+…+|v n+1 | H k (X;F p ) (X) • If Q n+1 ..Q 0 ( 𝝱 ) ≠ 0, then 𝜒 ( 𝝱 ) is not algebraic. But we also pay a price…

Lifting classes: We get • a map 𝜒 BP k +|v 0 |+…+|v n | (X) • an obstruction ✘ 𝜒 :=q n …q 0 BP ⟨ n ⟩ k +|v 0 |+…+|v n | H k (X;F p ) (X) 𝜒 ( 𝝱 ) 𝝱 the degree q n+1 Q n+1 Q n …Q 0 increases ≠ 0 ≠ 0 +|v 0 |+…+|v n+1 | BP ⟨ n+1 ⟩ k +|v 0 |+…+|v n+1 | H k (X;F p ) (X) • If Q n+1 ..Q 0 ( 𝝱 ) ≠ 0, then 𝜒 ( 𝝱 ) is not algebraic. But we also pay a price…

Wilson’ s unstable splitting: The price is as little as possible:

Wilson’ s unstable splitting: The price is as little as possible: Theorem (Wilson): For any finite complex X, the map BP i (X) BP ⟨ n ⟩ i (X) is surjective if i ≤ 2(p n +…+p+1).

Wilson’ s unstable splitting: The price is as little as possible: Theorem (Wilson): For any finite complex X, the map BP i (X) BP ⟨ n ⟩ i (X) is surjective if i ≤ 2(p n +…+p+1). Recall |v n |=2p n -1, hence |v 0 |+…+|v n |=2(p n +…+1)-n-1.

Wilson’ s unstable splitting: The price is as little as possible: Theorem (Wilson): For any finite complex X, the map BP i (X) BP ⟨ n ⟩ i (X) is surjective if i ≤ 2(p n +…+p+1). Recall |v n |=2p n -1, hence |v 0 |+…+|v n |=2(p n +…+1)-n-1. BP k +|v 0 |+…+|v n | (X) 𝜒 =q n …q 0 BP ⟨ n ⟩ k +|v 0 |+…+|v n | H k (X;F p ) (X)

Wilson’ s unstable splitting: The price is as little as possible: Theorem (Wilson): For any finite complex X, the map BP i (X) BP ⟨ n ⟩ i (X) is surjective if i ≤ 2(p n +…+p+1). Recall |v n |=2p n -1, hence |v 0 |+…+|v n |=2(p n +…+1)-n-1. BP k +|v 0 |+…+|v n | (X) need to pick k ≥ n+3 ✘ 𝜒 =q n …q 0 BP ⟨ n ⟩ k +|v 0 |+…+|v n | H k (X;F p ) (X)

Examples of non-algebraic classes: