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

Examples of non- algebraic classes in the Brown-Peterson tower

Institut Mittag-Leffler April 20, 2017 Gereon Quick NTNU

Algebraic classes: Let E be a motivic spectrum over SmC.

smooth complex schemes

Algebraic classes: Topological realization: Etop E

in classical stable homotopy category

Let E be a motivic spectrum over SmC.

smooth complex schemes

Algebraic classes: Topological realization: Etop E

in classical stable homotopy category

Ea (X(C)) Ea,b (X)

mot top

𝛓 Let E be a motivic spectrum over SmC.

smooth complex schemes

Algebraic classes: Topological realization: Etop E

in classical stable homotopy category

Ea (X(C)) Ea,b (X)

mot top

𝛓 Question: Which elements in E2*(X(C))

top

i.e., are in the image of 𝛓? are algebraic, Let E be a motivic spectrum over SmC.

smooth complex schemes

Thom map: E2*(X(C)) E2*,*(X)

mot top

𝛓

Thom map: The map 𝛓 is natural in X and E. E2*(X(C)) E2*,*(X)

mot top

𝛓

Thom map: The map 𝛓 is natural in X and E. E2*(X(C)) E2*,*(X)

mot top

𝛓 HZ E Let us assume we have a map of motivic spectra

motivic Eilenberg-MacLane spectrum

𝞄

Thom map: The map 𝛓 is natural in X and E. E2*(X(C)) E2*,*(X)

mot top

𝛓 HZ E Let us assume we have a map of motivic spectra

motivic Eilenberg-MacLane spectrum

𝞄 E2*(X(C)) E2*,*(X)

mot top

𝛓E H2*(X(C);Z) H2*,*(X;Z)

mot top

𝛓H

singular cohomology

↺

Obstruction: E2*,*(X)

mot

E2*(X)

top

𝛓E H2*(X;Z) H2*,*(X;Z)

mot

𝛓H

↺

𝞄

Obstruction: E2*,*(X)

mot

E2*(X)

top

𝛓E H2*(X;Z) H2*,*(X;Z)

mot

𝛓H

↺

CH*(X) ≈ =clH 𝞄

Obstruction: E2*,*(X)

mot

E2*(X)

top

𝛓E H2*(X;Z) H2*,*(X;Z)

mot

𝛓H

↺

CH*(X) ≈ =clH

algebraic classes in singular cohomology

Alg2*(X)⊆

H

𝞄

Obstruction: E2*,*(X)

mot

E2*(X)

top

𝛓E H2*(X;Z) H2*,*(X;Z)

mot

𝛓H

↺

CH*(X) ≈ =clH

algebraic classes in singular cohomology

Alg2*(X)⊆

H

Alg2*(X)⊆

E

𝞄

Obstruction: E2*,*(X)

mot

E2*(X)

top

𝛓E H2*(X;Z) H2*,*(X;Z)

mot

𝛓H

↺

CH*(X) ≈ =clH

algebraic classes in singular cohomology

Alg2*(X)⊆

H

Alg2*(X)⊆

E must factor through

𝞄

Obstruction: E2*,*(X)

mot

E2*(X)

top

𝛓E H2*(X;Z) H2*,*(X;Z)

mot

𝛓H

↺

CH*(X) ≈ =clH

algebraic classes in singular cohomology

Alg2*(X)⊆

H

Alg2*(X)⊆

E must factor through

𝞄 New tasks: Given HZ. E 𝞄

Obstruction: E2*,*(X)

mot

E2*(X)

top

𝛓E H2*(X;Z) H2*,*(X;Z)

mot

𝛓H

↺

CH*(X) ≈ =clH

algebraic classes in singular cohomology

Alg2*(X)⊆

H

Alg2*(X)⊆

E must factor through

𝞄 New tasks: Given HZ. E 𝞄

• Alg2*(X) in terms of Alg2*(X)?

E H

Describe

Obstruction: E2*,*(X)

mot

E2*(X)

top

𝛓E H2*(X;Z) H2*,*(X;Z)

mot

𝛓H

↺

CH*(X) ≈ =clH

algebraic classes in singular cohomology

Alg2*(X)⊆

H

Alg2*(X)⊆

E must factor through

𝞄 New tasks: Given HZ. E 𝞄

• Alg2*(X) in terms of Alg2*(X)?

E H

Describe

• E2*(X(C))\Alg2*(X) using H2*(X;Z)\Alg2*(X)?

E H top

Atiyah-Hirzebruch, Totaro, Levine-Morel: H2*(X;Z) CH*(X) clH

H

Alg2*(X)⊆

Atiyah-Hirzebruch, Totaro, Levine-Morel: H2*(X;Z) CH*(X) clH

H

Alg2*(X)⊆ MU2*(X) MGL2*,*(X) 𝛓

Atiyah-Hirzebruch, Totaro, Levine-Morel: H2*(X;Z) CH*(X) clH

H

Alg2*(X)⊆ MU2*(X) MGL2*,*(X) 𝛓 MGL2*,*(X)⊗L*Z MU2*(X)⊗L*Z

Atiyah-Hirzebruch, Totaro, Levine-Morel: H2*(X;Z) CH*(X) clH

H

Alg2*(X)⊆ MU2*(X) MGL2*,*(X) 𝛓

⟳

Totaro

MGL2*,*(X)⊗L*Z MU2*(X)⊗L*Z

Atiyah-Hirzebruch, Totaro, Levine-Morel: H2*(X;Z) CH*(X) clH

H

Alg2*(X)⊆ MU2*(X) MGL2*,*(X) 𝛓

⟳

Totaro

MGL2*,*(X)⊗L*Z MU2*(X)⊗L*Z

Levine + Levine-Morel ≈

Atiyah-Hirzebruch, Totaro, Levine-Morel: H2*(X;Z) CH*(X) clH

H

Alg2*(X)⊆ MU2*(X) MGL2*,*(X) 𝛓

≉ in general

⟳

Totaro

MGL2*,*(X)⊗L*Z MU2*(X)⊗L*Z

Levine + Levine-Morel ≈

Atiyah-Hirzebruch, Totaro, Levine-Morel: H2*(X;Z) CH*(X) clH

H

Alg2*(X)⊆ MU2*(X) MGL2*,*(X) 𝛓

≉ in general

⟳

Totaro

MGL2*,*(X)⊗L*Z MU2*(X)⊗L*Z

• Atiyah-Hirzebruch: clH is not surjective
• nto integral Hodge classes.

Levine + Levine-Morel ≈

Atiyah-Hirzebruch, Totaro, Levine-Morel: H2*(X;Z) CH*(X) clH

H

Alg2*(X)⊆ MU2*(X) MGL2*,*(X) 𝛓

≉ in general

⟳

Totaro

MGL2*,*(X)⊗L*Z MU2*(X)⊗L*Z

• Atiyah-Hirzebruch: clH is not surjective
• nto integral Hodge classes.
• Totaro: new classes in kernel of clH.

Levine + Levine-Morel ≈

A different perspective: Fix a prime p.

A different perspective: MU(p) splits as a wedge of suspensions of spectra BP with BP = Z(p)[v1,v2,…], |vi|=2(pi-1). * Recall (Brown-Peterson, Quillen): Fix a prime p.

A different perspective: MU(p) splits as a wedge of suspensions of spectra BP with BP = Z(p)[v1,v2,…], |vi|=2(pi-1). * Recall (Brown-Peterson, Quillen): Fix a prime p. For every n, there is a quotient map BP BP/(vn+1,…) =: BP⟨n⟩ with BP⟨n⟩ = Z(p)[v1,…,vn] *

A different perspective: MU(p) splits as a wedge of suspensions of spectra BP with BP = Z(p)[v1,v2,…], |vi|=2(pi-1). * Recall (Brown-Peterson, Quillen): Fix a prime p. For every n, there is a quotient map BP BP/(vn+1,…) =: BP⟨n⟩ with BP⟨n⟩ = Z(p)[v1,…,vn] * BP⟨n⟩ BP … … HZ(p) HFp BP⟨0⟩ BP⟨-1⟩

‖ ‖

The Brown-Peterson tower (Wilson):

Milnor operations: For every n, there is a stable cofibre sequence BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑

Milnor operations: For every n, there is a stable cofibre sequence BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑ with an induced exact sequence (for any space X) BP⟨n⟩* (X)

+|vn|

BP⟨n⟩*(X) BP⟨n-1⟩*(X) BP⟨n⟩*+|vn|+1 (X)

qn

Milnor operations: For every n, there is a stable cofibre sequence BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑ with an induced exact sequence (for any space X) BP⟨n⟩* (X)

+|vn|

BP⟨n⟩*(X) BP⟨n-1⟩*(X) BP⟨n⟩*+|vn|+1 (X)

qn

H*(X;Fp) H*

+|vn|+1(X;Fp) Qn BP⟨n-1⟩ HFp Thom map

Milnor operations: For every n, there is a stable cofibre sequence BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑ with an induced exact sequence (for any space X) BP⟨n⟩* (X)

+|vn|

BP⟨n⟩*(X) BP⟨n-1⟩*(X) BP⟨n⟩*+|vn|+1 (X)

qn nth Milnor

• peration:

Q0=Bockstein Qn=P Qn-1-Qn-1P

pn-1 pn-1

H*(X;Fp) H*

+|vn|+1(X;Fp) Qn BP⟨n-1⟩ HFp Thom map

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱 Question: Is 𝝱 algebraic?

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱 Question: Is 𝝱 algebraic?

CH*(X)

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱 Question: Is 𝝱 algebraic?

CH*(X)

LMT

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱 𝝱n-1 Question: Is 𝝱 algebraic?

CH*(X)

LMT

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱 𝝱n-1 Question: Is 𝝱 algebraic?

CH*(X)

LMT

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱 𝝱n-1 qn𝝱n-1 Question: Is 𝝱 algebraic?

CH*(X)

LMT

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱 𝝱n-1 qn𝝱n-1 Question: Is 𝝱 algebraic?

CH*(X)

= 0

if LMT

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱 𝝱n-1 qn𝝱n-1 Question: Is 𝝱 algebraic?

CH*(X)

= 0

if

𝝱n

then LMT

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱 Qn𝝱 𝝱n-1 qn𝝱n-1 Question: Is 𝝱 algebraic?

CH*(X)

= 0

if

𝝱n

then LMT

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱 Qn𝝱 𝝱n-1 qn𝝱n-1

CH*(X)

= 0

if

𝝱n

then LMT

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X) Levine-Morel-Totaro obstruction:

⟲

𝝱 Qn𝝱 𝝱n-1 qn𝝱n-1

CH*(X)

= 0

if

𝝱n

then LMT

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X) Levine-Morel-Totaro obstruction:

⟲

𝝱 Qn𝝱 𝝱n-1 qn𝝱n-1

CH*(X)

= 0

if

𝝱n

then LMT

If Qn𝝱 ≠ 0, ≠ 0

if

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X) Levine-Morel-Totaro obstruction:

⟲

𝝱 Qn𝝱 𝝱n-1 qn𝝱n-1

CH*(X)

𝝱n

then LMT

If Qn𝝱 ≠ 0, ≠ 0

if

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X) Levine-Morel-Totaro obstruction:

⟲

𝝱 Qn𝝱 𝝱n-1 ≠ 0 qn𝝱n-1

CH*(X)

𝝱n

then LMT

If Qn𝝱 ≠ 0, ≠ 0

if

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X) Levine-Morel-Totaro obstruction:

⟲

𝝱 Qn𝝱 𝝱n-1 ≠ 0 qn𝝱n-1

CH*(X)

𝝱n

then LMT

✘ If Qn𝝱 ≠ 0, ≠ 0

if

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X) Levine-Morel-Totaro obstruction:

⟲

𝝱 ✘ Qn𝝱 𝝱n-1 ≠ 0 qn𝝱n-1

CH*(X)

𝝱n

then LMT

✘ If Qn𝝱 ≠ 0, ≠ 0

if

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X) Levine-Morel-Totaro obstruction:

⟲

𝝱 ✘ Qn𝝱 𝝱n-1 ≠ 0 qn𝝱n-1 then 𝝱 is not algebraic.

CH*(X)

𝝱n

then LMT

✘ If Qn𝝱 ≠ 0, ≠ 0

if

Voevodsky’ s motivic Milnor operations:

Voevodsky’ s motivic Milnor operations: There are motivic operations Qn ∈ 𝓑 2pn-1,pn-1

mod p-motivic Steenrod algebra

mot

Voevodsky’ s motivic Milnor operations: There are motivic operations Qn ∈ 𝓑 2pn-1,pn-1

mod p-motivic Steenrod algebra

mot

H H

i+2pn-1,j+pn-1 (X;Fp)

(X;Fp)

i,j mot mot mod p-motivic cohomology

For a smooth complex variety X: Qn

mot

Voevodsky’ s motivic Milnor operations: H (X;Fp)

2i,i mot

= CHi(X;Z/p) H (X;Fp)

i,j mot

= 0 if i>2j. and Recall: There are motivic operations Qn ∈ 𝓑 2pn-1,pn-1

mod p-motivic Steenrod algebra

mot

H H

i+2pn-1,j+pn-1 (X;Fp)

(X;Fp)

i,j mot mot mod p-motivic cohomology

For a smooth complex variety X: Qn

mot

Obstructions revisited: H H

2i+2pn-1,i+pn-1(X;Fp) Qn

(X;Fp)

2i,i mot mot

H H2i(X;Fp)

2i+2pn-1

X smooth complex variety

(X;Fp)

↺

Qn

topological realization

mot

Obstructions revisited: H H

2i+2pn-1,i+pn-1(X;Fp) Qn

(X;Fp)

2i,i mot mot

H H2i(X;Fp)

2i+2pn-1

X smooth complex variety

(X;Fp)

↺

Qn

CHi(X;Z/p)

‖ topological realization

mot

Obstructions revisited: H H

2i+2pn-1,i+pn-1(X;Fp) Qn

(X;Fp)

2i,i mot mot

H H2i(X;Fp)

2i+2pn-1

X smooth complex variety

(X;Fp)

↺

Qn

CHi(X;Z/p)

‖

= 0

topological realization

mot

Obstructions revisited: H H

2i+2pn-1,i+pn-1(X;Fp) Qn

(X;Fp)

2i,i mot mot

H H2i(X;Fp)

2i+2pn-1

X smooth complex variety

(X;Fp)

↺

Qn

CHi(X;Z/p)

‖

= 0

topological realization

mot

Qn

mot

Obstructions revisited: H H

2i+2pn-1,i+pn-1(X;Fp) Qn

(X;Fp)

2i,i mot mot

H H2i(X;Fp)

2i+2pn-1

X smooth complex variety

(X;Fp)

↺

Qn

CHi(X;Z/p)

‖

= 0 𝝱

topological realization

mot

Qn

mot

Obstructions revisited: H H

2i+2pn-1,i+pn-1(X;Fp) Qn

(X;Fp)

2i,i mot mot

H H2i(X;Fp)

2i+2pn-1

X smooth complex variety

(X;Fp)

↺

Qn

CHi(X;Z/p)

‖

= 0 𝝱

topological realization

mot

Qn

mot

Qn𝝱 ≠ 0

if

Obstructions revisited: H H

2i+2pn-1,i+pn-1(X;Fp) Qn

(X;Fp)

2i,i mot mot

H H2i(X;Fp)

2i+2pn-1

X smooth complex variety

(X;Fp)

↺

Qn

CHi(X;Z/p)

‖

= 0 𝝱 ✘

topological realization

mot

Qn

mot

Qn𝝱 ≠ 0

if

Obstructions revisited: H H

2i+2pn-1,i+pn-1(X;Fp) Qn

(X;Fp)

2i,i mot mot

H H2i(X;Fp)

2i+2pn-1

X smooth complex variety

(X;Fp)

↺

Qn

CHi(X;Z/p)

‖

= 0 𝝱 ✘ Observation: The LMT-obstruction is particular to smooth varieties and bidegrees (2i,i).

topological realization

mot

Qn

mot

Qn𝝱 ≠ 0

if

Obstructions revisited: H H

2i+2pn-1,i+pn-1(X;Fp) Qn

(X;Fp)

2i,i mot mot

H H2i(X;Fp)

2i+2pn-1

X smooth complex variety

(X;Fp)

↺

Qn

CHi(X;Z/p)

‖

= 0 𝝱 ✘ Observation: The LMT-obstruction is particular to smooth varieties and bidegrees (2i,i).

topological realization

mot

Qn

mot

Example: Qn𝛋≠0 for 𝛋 the fundamental class of a suitable Eilenberg-MacLane space, though 𝛋 is algebraic.

Qn𝝱 ≠ 0

if

Back to our task:

Back to our task: Study Alg2*(X) and its complement in E2*(X).

E top

Back to our task: Study Alg2*(X) and its complement in E2*(X).

E top

For example: E=BP⟨n⟩?

Back to our task: Study Alg2*(X) and its complement in E2*(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 Alg2*(X) and its complement in E2*(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).

drop the “top” from now on

Question: How can we produce non-algebraic elements in BP⟨n⟩2*(X)?

top

Back to the cofibre sequence:

Back to the cofibre sequence: Recall the stable cofibre sequence BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑

Back to the cofibre sequence: Recall the stable cofibre sequence BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑ BP⟨n-1⟩*(X) BP⟨n⟩* +|vn|+1(X).

qn

and the induced map

Back to the cofibre sequence: Recall the stable cofibre sequence BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑ BP⟨n-1⟩*(X) BP⟨n⟩* +|vn|+1(X).

qn

and the induced map For example:

Back to the cofibre sequence: Recall the stable cofibre sequence BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑ BP⟨n-1⟩*(X) BP⟨n⟩* +|vn|+1(X).

qn

and the induced map n=0: H*(X;Fp) H*+1 (X;Z(p)),

q0 Bockstein homomorphism

For example:

Back to the cofibre sequence: Recall the stable cofibre sequence BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑ BP⟨n-1⟩*(X) BP⟨n⟩* +|vn|+1(X).

qn

and the induced map n=0: H*(X;Fp) H*+1 (X;Z(p)),

q0 Bockstein homomorphism

For example: n=1: H*(X;Z(p))

+2p-1(X), q1

BP⟨1⟩* ⋮

A big diagram: Hk(X;Fp)

A big diagram: Hk(X;Fp) Hk+1(X;Z(p))

q0

A big diagram: Hk(X;Fp) Hk+1(X;Z(p))

q0 q1

BP⟨1⟩k+1+2p-1(X)

A big diagram: Hk(X;Fp) Hk+1(X;Z(p))

q0 q1

BP⟨1⟩k+1+2p-1(X) (X)

q2

⋮ BP⟨n⟩k

qn +|v0|+…+|vn|

A big diagram: Hk(X;Fp) Hk+1(X;Z(p))

q0 q1

BP⟨1⟩k+1+2p-1(X) (X)

q2

⋮ BP⟨n⟩k

qn +|v0|+…+|vn|

(X) BP⟨n+1⟩k+|v0|+…+|vn+1|

qn+1

A big diagram: Hk(X;Fp) Hk+1(X;Z(p))

q0 q1

BP⟨1⟩k+1+2p-1(X) (X)

q2

⋮ BP⟨n⟩k

qn +|v0|+…+|vn|

(X) BP⟨n+1⟩k+|v0|+…+|vn+1|

qn+1 BP⟨n+1⟩

Hk (X;Fp)

+|v0|+…+|vn+1| HFp Thom map

A big diagram: Hk(X;Fp) Hk+1(X;Z(p))

q0 q1

BP⟨1⟩k+1+2p-1(X) (X)

q2

⋮ BP⟨n⟩k

qn +|v0|+…+|vn|

(X) BP⟨n+1⟩k+|v0|+…+|vn+1|

qn+1 BP⟨n+1⟩

Hk (X;Fp)

+|v0|+…+|vn+1| HFp Thom map Qn+1Qn…Q0

Lifting classes: We get

Lifting classes: We get Hk(X;Fp) (X) BP⟨n⟩k+|v0|+…+|vn|

𝜒:=qn…q0

• a map 𝜒

Lifting classes: We get Hk(X;Fp) (X) BP⟨n⟩k+|v0|+…+|vn|

𝜒:=qn…q0

• a map 𝜒

𝝱 𝜒(𝝱)

Lifting classes: We get Hk(X;Fp) (X) BP⟨n⟩k+|v0|+…+|vn|

𝜒:=qn…q0

• a map 𝜒

(X) BPk+|v0|+…+|vn|

Qn+1Qn…Q0

Hk (X;Fp)

+|v0|+…+|vn+1|

• an obstruction

𝝱 𝜒(𝝱)

Lifting classes: We get Hk(X;Fp) (X) BP⟨n⟩k+|v0|+…+|vn|

𝜒:=qn…q0

• a map 𝜒

(X) BPk+|v0|+…+|vn|

Qn+1Qn…Q0

Hk (X;Fp)

+|v0|+…+|vn+1|

• an obstruction

𝝱 𝜒(𝝱) (X) BP⟨n+1⟩k+|v0|+…+|vn+1|

qn+1

Lifting classes: We get Hk(X;Fp) (X) BP⟨n⟩k+|v0|+…+|vn|

𝜒:=qn…q0

• a map 𝜒

(X) BPk+|v0|+…+|vn|

Qn+1Qn…Q0

Hk (X;Fp)

+|v0|+…+|vn+1|

• an obstruction

𝝱 𝜒(𝝱) ≠0 (X) BP⟨n+1⟩k+|v0|+…+|vn+1|

qn+1

Lifting classes: We get Hk(X;Fp) (X) BP⟨n⟩k+|v0|+…+|vn|

𝜒:=qn…q0

• a map 𝜒

(X) BPk+|v0|+…+|vn|

Qn+1Qn…Q0

Hk (X;Fp)

+|v0|+…+|vn+1|

• an obstruction

𝝱 𝜒(𝝱) ≠0 (X) BP⟨n+1⟩k+|v0|+…+|vn+1|

qn+1

≠0

Lifting classes: We get Hk(X;Fp) (X) BP⟨n⟩k+|v0|+…+|vn|

𝜒:=qn…q0

• a map 𝜒

(X) BPk+|v0|+…+|vn|

Qn+1Qn…Q0

Hk (X;Fp)

+|v0|+…+|vn+1|

• an obstruction

𝝱 𝜒(𝝱) ≠0 ✘ (X) BP⟨n+1⟩k+|v0|+…+|vn+1|

qn+1

≠0

Lifting classes: We get Hk(X;Fp) (X) BP⟨n⟩k+|v0|+…+|vn|

𝜒:=qn…q0

• a map 𝜒

(X) BPk+|v0|+…+|vn|

Qn+1Qn…Q0

Hk (X;Fp)

+|v0|+…+|vn+1|

• an obstruction

𝝱 𝜒(𝝱) ≠0 ✘ (X) BP⟨n+1⟩k+|v0|+…+|vn+1|

qn+1

• If Qn+1..Q0(𝝱)≠0, then 𝜒(𝝱) is not algebraic.

≠0

Lifting classes: We get Hk(X;Fp) (X) BP⟨n⟩k+|v0|+…+|vn|

𝜒:=qn…q0

• a map 𝜒

(X) BPk+|v0|+…+|vn|

Qn+1Qn…Q0

Hk (X;Fp)

+|v0|+…+|vn+1|

• an obstruction

𝝱 𝜒(𝝱) ≠0 ✘ (X) BP⟨n+1⟩k+|v0|+…+|vn+1|

qn+1

But we also pay a price…

• If Qn+1..Q0(𝝱)≠0, then 𝜒(𝝱) is not algebraic.

≠0

Lifting classes: We get Hk(X;Fp) (X) BP⟨n⟩k+|v0|+…+|vn|

𝜒:=qn…q0

• a map 𝜒

(X) BPk+|v0|+…+|vn|

Qn+1Qn…Q0

Hk (X;Fp)

+|v0|+…+|vn+1|

• an obstruction

𝝱 𝜒(𝝱) ≠0 ✘ (X) BP⟨n+1⟩k+|v0|+…+|vn+1|

qn+1

But we also pay a price…

the degree increases

• If Qn+1..Q0(𝝱)≠0, then 𝜒(𝝱) is not algebraic.

≠0

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 BPi(X) BP⟨n⟩i(X) is surjective if i≤2(pn+…+p+1).

Wilson’ s unstable splitting: The price is as little as possible: Theorem (Wilson): For any finite complex X, the map BPi(X) BP⟨n⟩i(X) is surjective if i≤2(pn+…+p+1). Recall |vn|=2pn-1, hence |v0|+…+|vn|=2(pn+…+1)-n-1.

Wilson’ s unstable splitting: The price is as little as possible: Theorem (Wilson): For any finite complex X, the map BPi(X) BP⟨n⟩i(X) is surjective if i≤2(pn+…+p+1). Recall |vn|=2pn-1, hence |v0|+…+|vn|=2(pn+…+1)-n-1. Hk(X;Fp) (X) BP⟨n⟩k+|v0|+…+|vn|

𝜒=qn…q0

(X) BPk+|v0|+…+|vn|

Wilson’ s unstable splitting: The price is as little as possible: Theorem (Wilson): For any finite complex X, the map BPi(X) BP⟨n⟩i(X) is surjective if i≤2(pn+…+p+1). Recall |vn|=2pn-1, hence |v0|+…+|vn|=2(pn+…+1)-n-1. Hk(X;Fp) (X) BP⟨n⟩k+|v0|+…+|vn|

𝜒=qn…q0

(X) BPk+|v0|+…+|vn|

need to pick k≥n+3

✘

Examples of non-algebraic classes:

Examples of non-algebraic classes: Theorem (Q.): For every n, there is a smooth proj. complex variety X and a class in BP⟨n⟩2*(X). BP⟨n⟩2*,*

mot(X)

which is not in the image of the map

2(pn+…+1)+2 (X)

BP⟨n⟩

• top. realization

Examples of non-algebraic classes: Theorem (Q.): For every n, there is a smooth proj. complex variety X and a class in BP⟨n⟩2*(X). BP⟨n⟩2*,*

mot(X)

which is not in the image of the map

2(pn+…+1)+2 (X)

BP⟨n⟩

• top. realization

Proof:

Examples of non-algebraic classes: Theorem (Q.): For every n, there is a smooth proj. complex variety X and a class in BP⟨n⟩2*(X). BP⟨n⟩2*,*

mot(X)

which is not in the image of the map

2(pn+…+1)+2 (X)

BP⟨n⟩

• top. realization

Proof: Let Gk:=∏k(Z/p) (following Atiyah-Hirzebruch).

Examples of non-algebraic classes: Theorem (Q.): For every n, there is a smooth proj. complex variety X and a class in BP⟨n⟩2*(X). BP⟨n⟩2*,*

mot(X)

which is not in the image of the map

2(pn+…+1)+2 (X)

BP⟨n⟩

• top. realization

Proof: Let Gk:=∏k(Z/p) (following Atiyah-Hirzebruch). We know:

Examples of non-algebraic classes: Theorem (Q.): For every n, there is a smooth proj. complex variety X and a class in BP⟨n⟩2*(X). BP⟨n⟩2*,*

mot(X)

which is not in the image of the map

2(pn+…+1)+2 (X)

BP⟨n⟩

• top. realization

Proof: Let Gk:=∏k(Z/p) (following Atiyah-Hirzebruch). We know: • H*(BGk;Fp)= Fp[y1,…,yk]⊗𝚳(x1,…,xk);

Examples of non-algebraic classes: Theorem (Q.): For every n, there is a smooth proj. complex variety X and a class in BP⟨n⟩2*(X). BP⟨n⟩2*,*

mot(X)

which is not in the image of the map

2(pn+…+1)+2 (X)

BP⟨n⟩

• top. realization

Proof: Let Gk:=∏k(Z/p) (following Atiyah-Hirzebruch). We know: • H*(BGk;Fp)= Fp[y1,…,yk]⊗𝚳(x1,…,xk);

• |xi|=1, |yi|=2, |xi|2=yi;

Examples of non-algebraic classes: Theorem (Q.): For every n, there is a smooth proj. complex variety X and a class in BP⟨n⟩2*(X). BP⟨n⟩2*,*

mot(X)

which is not in the image of the map

2(pn+…+1)+2 (X)

BP⟨n⟩

• top. realization

Proof: Let Gk:=∏k(Z/p) (following Atiyah-Hirzebruch). We know: • H*(BGk;Fp)= Fp[y1,…,yk]⊗𝚳(x1,…,xk);

• |xi|=1, |yi|=2, |xi|2=yi;
• Qj(xi)=yi , Qj(yi)=0.

pj

Proof continued: H*(BGk;Fp)= Fp[y1,…,yk]⊗𝚳(x1,…,xk);

Qj(xi)=yi , Qj(yi)=0.

pj

Proof continued: Choose: k=n+3 and 𝝱:=x1…xn+3 in Hn+3(BGk;Fp).

H*(BGk;Fp)= Fp[y1,…,yk]⊗𝚳(x1,…,xk); Qj(xi)=yi , Qj(yi)=0.

pj

Proof continued: Check: Qn+1…Q0(𝝱)≠0. Choose: k=n+3 and 𝝱:=x1…xn+3 in Hn+3(BGk;Fp).

H*(BGk;Fp)= Fp[y1,…,yk]⊗𝚳(x1,…,xk); Qj(xi)=yi , Qj(yi)=0.

pj

Proof continued: Check: Qn+1…Q0(𝝱)≠0. Choose: k=n+3 and 𝝱:=x1…xn+3 in Hn+3(BGk;Fp).

H*(BGk;Fp)= Fp[y1,…,yk]⊗𝚳(x1,…,xk); Qj(xi)=yi , Qj(yi)=0.

pj

is not in the image of the map

2(pn+…+1)+2 (BGn+3)

BP⟨n⟩ Hence x:=qn…q0(𝝱) in BP⟨n⟩ BP (BGn+3)

2(pn+…+1)+2

(BGn+3).

2(pn+…+1)+2

Proof continued: Check: Qn+1…Q0(𝝱)≠0. Choose: k=n+3 and 𝝱:=x1…xn+3 in Hn+3(BGk;Fp).

H*(BGk;Fp)= Fp[y1,…,yk]⊗𝚳(x1,…,xk); Qj(xi)=yi , Qj(yi)=0.

pj

is not in the image of the map

2(pn+…+1)+2 (BGn+3)

BP⟨n⟩ Hence x:=qn…q0(𝝱) in BP⟨n⟩ BP (BGn+3)

2(pn+…+1)+2

(BGn+3).

2(pn+…+1)+2

Finally, set X = Godeaux-Serre variety associated to the group Gn+3 and pullback x via X BGn+3 × CP∞.

a 2(pn+1+…+1)+1- connected map

□

Let’ s check the numbers:

Let’ s check the numbers: The minimal complex dimension of X is 2n+3-1 for p=2. (It’ s huge.)

Let’ s check the numbers: The minimal complex dimension of X is 2n+3-1 for p=2. (It’ s huge.) BP⟨1⟩ is 2-local connective complex K-theory ku. For n=1 and p=2:

Let’ s check the numbers: There is a smooth proj. variety X of dimension 15

• ver C with a non-algebraic class in BP⟨1⟩8(X).

The minimal complex dimension of X is 2n+3-1 for p=2. (It’ s huge.) BP⟨1⟩ is 2-local connective complex K-theory ku. For n=1 and p=2:

Remarks:

Remarks:

• For n=0: the example of Atiyah and Hirzebruch.

Remarks:

• For n=0: the example of Atiyah and Hirzebruch.
• Types of non-algebraic classes in H*(X;Z):

Remarks:

• For n=0: the example of Atiyah and Hirzebruch.
• Types of non-algebraic classes in H*(X;Z):
• Kollar: non-torsion class on hypersurface in P4.

Remarks:

• For n=0: the example of Atiyah and Hirzebruch.
• Types of non-algebraic classes in H*(X;Z):
• Kollar: non-torsion class on hypersurface in P4.
• Voisin: torsion class based on Kollar’

s example.

Remarks:

• For n=0: the example of Atiyah and Hirzebruch.
• Types of non-algebraic classes in H*(X;Z):
• Kollar: non-torsion class on hypersurface in P4.
• Voisin: torsion class based on Kollar’

s example.

“not topological”

Remarks:

• For n=0: the example of Atiyah and Hirzebruch.
• Types of non-algebraic classes in H*(X;Z):
• Kollar: non-torsion class on hypersurface in P4.
• Voisin: torsion class based on Kollar’

s example.

“not topological”

• Yagita, Pirutka-Yagita, Kameko, and others:

non-torsion class in H*(BG;Z) for alg. group G with (Z/p)3⊂G

Remarks:

• For n=0: the example of Atiyah and Hirzebruch.
• Types of non-algebraic classes in H*(X;Z):
• Kollar: non-torsion class on hypersurface in P4.
• Voisin: torsion class based on Kollar’

s example.

“not topological”

• Yagita, Pirutka-Yagita, Kameko, and others:

non-torsion class in H*(BG;Z) for alg. group G with (Z/p)3⊂G

• Antieau: class in H*(BG;Z) for alg. group G, represent. theory
• n which the Qi’

s vanish, but a higher differential in the AH-spectral sequence is nontrivial.

What next?

BP⟨n⟩*(X) H*(X;Fp) 𝞄 non-algebraic non-algebraic. So far: What next?

BP⟨n⟩*(X) H*(X;Fp) 𝞄 non-algebraic non-algebraic. So far: What about non-algebraic algebraic? What next?

Start with n=1: BP⟨1⟩2k,k(X) H2k(X;Fp) 𝞄

mot

BP⟨1⟩2k(X) H2k,k(X;Fp)

mot

𝞄mot 𝛓

Start with n=1: BP⟨1⟩2k,k(X) H2k(X;Fp) 𝞄

mot

BP⟨1⟩2k(X) H2k,k(X;Fp)

mot

𝞄mot 𝝱 𝝱H

algebraic

𝛓

Start with n=1: BP⟨1⟩2k,k(X) H2k(X;Fp) 𝞄

mot

BP⟨1⟩2k(X) H2k,k(X;Fp)

mot

𝞄mot 𝝱 𝝱H

algebraic

𝝲H 𝛓

Start with n=1: BP⟨1⟩2k,k(X) H2k(X;Fp) 𝞄

mot

BP⟨1⟩2k(X) H2k,k(X;Fp)

mot

𝞄mot 𝝱 𝝱H

algebraic

𝝲H

Levine-Morel, Levine-Tripathi: surjective

𝛓

Start with n=1: BP⟨1⟩2k,k(X) H2k(X;Fp) 𝞄

mot

BP⟨1⟩2k(X) H2k,k(X;Fp)

mot

𝞄mot 𝝱 𝝱H

algebraic

𝝲H 𝝲

Levine-Morel, Levine-Tripathi: surjective

𝛓

Start with n=1: BP⟨1⟩2k,k(X) H2k(X;Fp) 𝞄

mot

BP⟨1⟩2k(X) H2k,k(X;Fp)

mot

𝞄mot 𝝱 𝝱H

algebraic

𝝲H 𝝲

?

Levine-Morel, Levine-Tripathi: surjective

𝛓

Start with n=1: BP⟨1⟩2k,k(X) H2k(X;Fp) 𝞄 𝛓(𝝲)-𝝱 ∈ Ker(𝞄)

mot

BP⟨1⟩2k(X) H2k,k(X;Fp)

mot

𝞄mot 𝝱 𝝱H

algebraic

𝝲H 𝝲

?

Levine-Morel, Levine-Tripathi: surjective

𝛓

Start with n=1: BP⟨1⟩2k,k(X) H2k(X;Fp) 𝞄 𝛓(𝝲)-𝝱 ∈ Ker(𝞄)

mot

BP⟨1⟩2k(X) H2k,k(X;Fp)

mot

𝞄mot 𝝱 𝝱H

algebraic

𝝲H 𝝲

?

Levine-Morel, Levine-Tripathi: surjective

𝛓 = v1.BP⟨1⟩2k+2(X)

Larry Smith (true for ku)

For n≥2: BP⟨n⟩2k,k(X) H2k(X;Fp) 𝞄 𝛓(𝝲)-𝝱 ∈ Ker(𝞄)

mot

BP⟨n⟩2k(X) H2k,k(X;Fp)

mot

𝞄mot 𝝱 𝝱H

algebraic

𝝲H 𝝲

?

𝛓

For n≥2: BP⟨n⟩2k,k(X) H2k(X;Fp) 𝞄 𝛓(𝝲)-𝝱 ∈ Ker(𝞄)

mot

BP⟨n⟩2k(X) H2k,k(X;Fp)

mot

𝞄mot 𝝱 𝝱H

algebraic

𝝲H 𝝲

?

𝛓 ⊇ BP⟨n⟩*<0.BP⟨n⟩2k+*(X) ≠ in general

Some new tasks:

Some new tasks:

• Find a smooth proj. complex X and elements in

BP⟨n⟩2k(X) which are

Some new tasks:

• Find a smooth proj. complex X and elements in

BP⟨n⟩2k(X) which are

• in Ker(𝞄) (i.e. map to 0 in H2k(X;Fp)),

Some new tasks:

• Find a smooth proj. complex X and elements in

BP⟨n⟩2k(X) which are

• in Ker(𝞄) (i.e. map to 0 in H2k(X;Fp)),
• but not in BP⟨n⟩*<0.BP⟨n⟩2k+*(X),

Some new tasks:

• Find a smooth proj. complex X and elements in

BP⟨n⟩2k(X) which are

• in Ker(𝞄) (i.e. map to 0 in H2k(X;Fp)),
• but not in BP⟨n⟩*<0.BP⟨n⟩2k+*(X),
• not algebraic.

Some new tasks:

• Find a smooth proj. complex X and elements in

BP⟨n⟩2k(X) which are

• in Ker(𝞄) (i.e. map to 0 in H2k(X;Fp)),
• but not in BP⟨n⟩*<0.BP⟨n⟩2k+*(X),
• not algebraic.
• Find such examples with an interesting

geometric interpretation.

probably not quite as spectacular as the example of Asok-Fasel-Hopkins of non-algebraizable vector bundles…

Thank you!