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

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

Freie Universität Berlin November 16, 2017 Gereon Quick NTNU

Lefschetz’ s theorem: X projective complex surface

Lefschetz’ s theorem:

2-dim. topological cycle

Given 𝜅: 𝛥 ⊂ X. X projective complex surface

Lefschetz’ s theorem:

2-dim. topological cycle

Given 𝜅: 𝛥 ⊂ X. If 𝛥∼C for an alg. curve C, then

homologous

X projective complex surface

Lefschetz’ s theorem:

2-dim. topological cycle

Given 𝜅: 𝛥 ⊂ X. If 𝛥∼C for an alg. curve C, then

homologous form on X

∫ 𝜅*𝛽 = 0

𝛥

(N) X projective complex surface

Lefschetz’ s theorem: 𝛽 is of type (1,1). unless

2-dim. topological cycle

Given 𝜅: 𝛥 ⊂ X. If 𝛥∼C for an alg. curve C, then

homologous form on X

∫ 𝜅*𝛽 = 0

𝛥

(N) X projective complex surface

Lefschetz’ s theorem: 𝛽 is of type (1,1). unless

2-dim. topological cycle

Given 𝜅: 𝛥 ⊂ X. If 𝛥∼C for an alg. curve C, then

homologous form on X

∫ 𝜅*𝛽 = 0

𝛥

(N) Lefschetz: If (N) holds for 𝛥, then [𝛥] is “algebraic”.

there is an algebraic curve C～𝛥

class in homology

X projective complex surface

Higher (co-)dimensions: Let X be a smooth proj. complex algebraic variety and 𝜅:Z⊂X a smooth subvariety of dimension n:

Higher (co-)dimensions: ∫ 𝜅*𝛽 = 0 unless 𝛽 ∈ An,n(X).

Z

Let X be a smooth proj. complex algebraic variety and 𝜅:Z⊂X a smooth subvariety of dimension n:

Higher (co-)dimensions: ∫ 𝜅*𝛽 = 0 unless 𝛽 ∈ An,n(X).

Z

Let X be a smooth proj. complex algebraic variety and 𝜅:Z⊂X a smooth subvariety of dimension n: [Z] ∈ Hp,p(X) ⊂ H2p(X;C)

Poincaré dual Hodge classes of type (p,p) p=dimX-n

This means:

Higher (co-)dimensions: ∫ 𝜅*𝛽 = 0 unless 𝛽 ∈ An,n(X).

Z

Cycle map: Let X be a smooth proj. complex algebraic variety and 𝜅:Z⊂X a smooth subvariety of dimension n: [Z] ∈ Hp,p(X) ⊂ H2p(X;C)

Poincaré dual Hodge classes of type (p,p) p=dimX-n

This means:

Higher (co-)dimensions: ∫ 𝜅*𝛽 = 0 unless 𝛽 ∈ An,n(X).

Z

Cycle map: Let X be a smooth proj. complex algebraic variety and 𝜅:Z⊂X a smooth subvariety of dimension n: Z⊂X Zp(X)

free abelian group on alg. subvarieties

• f codim. p

[Z] ∈ Hp,p(X) ⊂ H2p(X;C)

Poincaré dual Hodge classes of type (p,p) p=dimX-n

This means:

Higher (co-)dimensions: ∫ 𝜅*𝛽 = 0 unless 𝛽 ∈ An,n(X).

Z

Cycle map: Let X be a smooth proj. complex algebraic variety and 𝜅:Z⊂X a smooth subvariety of dimension n: Z⊂X Zp(X)

free abelian group on alg. subvarieties

• f codim. p

⟼ [Zsm]

dual of fund. class (of desingularization)

Hp,p(X) ∩ H2p(X;Z)

clH

[Z] ∈ Hp,p(X) ⊂ H2p(X;C)

Poincaré dual Hodge classes of type (p,p) p=dimX-n

This means:

Higher (co-)dimensions: ∫ 𝜅*𝛽 = 0 unless 𝛽 ∈ An,n(X).

Z

Cycle map: Let X be a smooth proj. complex algebraic variety and 𝜅:Z⊂X a smooth subvariety of dimension n: Z⊂X Zp(X)

free abelian group on alg. subvarieties

• f codim. p

Hodge’ s question: Is this map surjective? ⟼ [Zsm]

dual of fund. class (of desingularization)

Hp,p(X) ∩ H2p(X;Z)

clH

[Z] ∈ Hp,p(X) ⊂ H2p(X;C)

Poincaré dual Hodge classes of type (p,p) p=dimX-n

This means:

How to do homotopy on Man?

category of complex manifolds

How to do homotopy on Man?

category of complex manifolds

Man Pre

presheaves of sets, i.e., functors: Manop ⟶ Set

How to do homotopy on Man?

category of complex manifolds

M FM

FM: X ⟼ HomMan(X,M)

Man Pre

presheaves of sets, i.e., functors: Manop ⟶ Set

How to do homotopy on Man?

category of complex manifolds

M FM

FM: X ⟼ HomMan(X,M) Presheaves “remember”

HomPre(FM,FM’) = HomMan(M,M’) Man Pre

presheaves of sets, i.e., functors: Manop ⟶ Set

How to do homotopy on Man?

category of complex manifolds

M FM

FM: X ⟼ HomMan(X,M) Presheaves “remember”

HomPre(FM,FM’) = HomMan(M,M’)

• Set

“rigid” presheaves

• f

Man Pre

presheaves of sets, i.e., functors: Manop ⟶ Set

How to do homotopy on Man?

category of complex manifolds

M FM

FM: X ⟼ HomMan(X,M) Presheaves “remember”

HomPre(FM,FM’) = HomMan(M,M’)

• Set

“rigid” presheaves

• f

“allow homotopy” switch to

Set∆

presheaves

• f

Man Pre

presheaves of sets, i.e., functors: Manop ⟶ Set

M FM

FM: X ⟼ discrete simplicial set FM(X)

Man Pre∆

presheaves of simplicial sets functors: Manop ⟶ Set∆

• How to do homotopy on Man?

M FM

FM: X ⟼ discrete simplicial set FM(X)

Man Pre∆

presheaves of simplicial sets functors: Manop ⟶ Set∆

• How to do homotopy on Man?
• Given n≥0, the n-dimensional stalk of F•

= colim F•(Bn(r))

r→0

in Set∆

ball of radius r in n-dim. complex affine space

F•

(n)

M FM

FM: X ⟼ discrete simplicial set FM(X)

Man Pre∆

presheaves of simplicial sets functors: Manop ⟶ Set∆

• How to do homotopy on Man?
• A map F• ⟶ G• is a weak equivalence in Pre∆
• Given n≥0, the n-dimensional stalk of F•

= colim F•(Bn(r))

r→0

in Set∆

ball of radius r in n-dim. complex affine space

F•

(n)

M FM

FM: X ⟼ discrete simplicial set FM(X)

Man Pre∆

presheaves of simplicial sets functors: Manop ⟶ Set∆

• How to do homotopy on Man?
• A map F• ⟶ G• is a weak equivalence in Pre∆
• Given n≥0, the n-dimensional stalk of F•

= colim F•(Bn(r))

r→0

in Set∆

ball of radius r in n-dim. complex affine space

F•

(n)

if F• ⟶ G• is a weak equivalence in Set∆ for all n≥0.

(n) (n)

Homotopy category of Man: Man Pre∆

Homotopy category of Man: Man Pre∆

• homotopy category of

simplicial presheaves on Man

ho

Homotopy category of Man: Man Pre∆

• Given M with an open cover {U𝛽}:

FU ⟶ FM is a weak equivalence.

• ∐U𝛽 ⇉ ∐U𝛽×XU𝛾 ⇶ …

homotopy category of simplicial presheaves on Man

ho

Homotopy category of Man: Man Pre∆

• Can replace Set∆ with Spectra and get a

stable homotopy category hoPreSpectra of Man.

• S1∧- with S1 viewed as a simplicial (constant) presheaf
• Given M with an open cover {U𝛽}:

FU ⟶ FM is a weak equivalence.

• ∐U𝛽 ⇉ ∐U𝛽×XU𝛾 ⇶ …

homotopy category of simplicial presheaves on Man

ho

Homotopy category of SmC: SmC

• smooth complex varieties

Homotopy category of SmC: SmC

• smooth complex varieties

Pre∆

simplicial presheaves on Sm Morel Voevodsky Jardine Joyal Isaksen Dugger …

Homotopy category of SmC:

• Nisnevich covers (replacing open covers)

SmC

• smooth complex varieties

Pre∆

simplicial presheaves on Sm Morel Voevodsky Jardine Joyal Isaksen Dugger …

Homotopy category of SmC:

• Nisnevich covers (replacing open covers)

SmC

• smooth complex varieties

Pre∆

simplicial presheaves on Sm Morel Voevodsky Jardine Joyal Isaksen Dugger …

• Localize with respect to

Homotopy category of SmC:

• Nisnevich covers (replacing open covers)

FU ⟶ FX for any X and any (hyper)cover U→X

• SmC
• smooth complex varieties

Pre∆

simplicial presheaves on Sm Morel Voevodsky Jardine Joyal Isaksen Dugger …

• Localize with respect to

Homotopy category of SmC:

• Nisnevich covers (replacing open covers)

FU ⟶ FX for any X and any (hyper)cover U→X

• SmC
• smooth complex varieties

ho

homotopy category of

Pre∆

simplicial presheaves on Sm Morel Voevodsky Jardine Joyal Isaksen Dugger …

• Localize with respect to

Homotopy category of SmC:

• Nisnevich covers (replacing open covers)

FU ⟶ FX for any X and any (hyper)cover U→X

• SmC
• smooth complex varieties

ho

homotopy category of

Pre∆

simplicial presheaves on Sm Morel Voevodsky Jardine Joyal Isaksen Dugger …

• Localize with respect to

A1xX⟶X for any X (A1 affine line)

motivic

Homotopy category of SmC:

• stable motivic homotopy category of SmC
• Nisnevich covers (replacing open covers)

FU ⟶ FX for any X and any (hyper)cover U→X

• SmC
• smooth complex varieties

ho

homotopy category of

Pre∆

simplicial presheaves on Sm Morel Voevodsky Jardine Joyal Isaksen Dugger …

• Localize with respect to

A1xX⟶X for any X (A1 affine line)

motivic

Homotopy category of SmC:

• stable motivic homotopy category of SmC
• Nisnevich covers (replacing open covers)

FU ⟶ FX for any X and any (hyper)cover U→X

• SmC
• smooth complex varieties

ho

homotopy category of

• P1∧- the projective line
• S1∧- the “simplicial circle” and (A1-0)∧- the “Tate circle”

Pre∆

simplicial presheaves on Sm Morel Voevodsky Jardine Joyal Isaksen Dugger …

• Localize with respect to

A1xX⟶X for any X (A1 affine line)

motivic

Topological realization:

X(C) X

”complex manifold of solutions in C”

SmC Man 𝜍

Topological realization:

X(C) X

”complex manifold of solutions in C”

SmC Man 𝜍

P1(C)=CP1≃S2 P1 e.g.

Topological realization:

X(C) X

”complex manifold of solutions in C”

SmC Man 𝜍

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

mot top

𝜍

motivic spectrum “topological” “algebraic” induced map P1(C)=CP1≃S2 P1 e.g.

Topological realization:

X(C) X

”complex manifold of solutions in C”

SmC Man 𝜍

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

mot top

𝜍

motivic spectrum “topological” “algebraic” induced map

E* (X(C))

top

are in the image of 𝜍? are algebraic, i.e., Question: How to detect whether classes in

P1(C)=CP1≃S2 P1 e.g.

Obstruction: Given HZ. E 𝜐

motivic Eilenberg-MacLane spectrum Thom map

Obstruction: Given HZ. E 𝜐

motivic Eilenberg-MacLane spectrum Thom map

E2*,*(X)

mot

H2*,*(X;Z)

mot

𝜐

Obstruction: E2*(X)

top

H2*(X;Z)

singular cohomology

𝜐 Given HZ. E 𝜐

motivic Eilenberg-MacLane spectrum Thom map

E2*,*(X)

mot

H2*,*(X;Z)

mot

𝜐

Obstruction: 𝜍E 𝜍H

↺

E2*(X)

top

H2*(X;Z)

singular cohomology

𝜐 Given HZ. E 𝜐

motivic Eilenberg-MacLane spectrum Thom map

E2*,*(X)

mot

H2*,*(X;Z)

mot

𝜐

Obstruction: 𝜍E 𝜍H

↺

=clH E2*(X)

top

H2*(X;Z)

singular cohomology

𝜐 Given HZ. E 𝜐

motivic Eilenberg-MacLane spectrum Thom map

E2*,*(X)

mot

H2*,*(X;Z)

mot

𝜐

Obstruction: 𝜍E 𝜍H

↺

=clH Alg2*(X)⊆

H

E2*(X)

top

H2*(X;Z)

singular cohomology

𝜐 Given HZ. E 𝜐

motivic Eilenberg-MacLane spectrum Thom map

E2*,*(X)

mot

H2*,*(X;Z)

mot

𝜐

Obstruction: 𝜍E 𝜍H

↺

=clH Alg2*(X)⊆

H

Alg2*(X)⊆

E

E2*(X)

top

H2*(X;Z)

singular cohomology

𝜐 Given HZ. E 𝜐

motivic Eilenberg-MacLane spectrum Thom map

E2*,*(X)

mot

H2*,*(X;Z)

mot

𝜐

Obstruction: 𝜍E 𝜍H

↺

=clH Alg2*(X)⊆

H

Alg2*(X)⊆

E must factor through

E2*(X)

top

H2*(X;Z)

singular cohomology

𝜐 Given HZ. E 𝜐

motivic Eilenberg-MacLane spectrum Thom map

E2*,*(X)

mot

H2*,*(X;Z)

mot

𝜐

Obstruction: 𝜍E 𝜍H

↺

=clH Alg2*(X)⊆

H

Alg2*(X)⊆

E must factor through

E2*(X)

top

H2*(X;Z)

singular cohomology

𝜐 Easier task: Given HZ. E 𝜐

motivic Eilenberg-MacLane spectrum Thom map

E2*,*(X)

mot

H2*,*(X;Z)

mot

𝜐

Obstruction: 𝜍E 𝜍H

↺

=clH Alg2*(X)⊆

H

Alg2*(X)⊆

E must factor through

E2*(X)

top

H2*(X;Z)

singular cohomology

𝜐 Easier task: Given HZ. E 𝜐

motivic Eilenberg-MacLane spectrum Thom map

E2*,*(X)

mot

H2*,*(X;Z)

mot

𝜐 E2*(X(C))\Alg2*(X)

E top

describe

H

H2*(X;Z)\Alg2*(X)

using

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

𝜍H=clH

H

Alg2*(X)⊆ H2*,*(X;Z)

mot

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

𝜍H=clH

H

Alg2*(X)⊆ H2*,*(X;Z)

mot

MU2*(X) MGL2*,*(X)

𝜍MGL

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

𝜍H=clH

H

Alg2*(X)⊆ H2*,*(X;Z)

mot

MU2*(X) MGL2*,*(X)

𝜍MGL

MGL2*,*(X)⊗L*Z MU2*(X)⊗L*Z

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

𝜍H=clH

H

Alg2*(X)⊆ H2*,*(X;Z)

mot

MU2*(X) MGL2*,*(X)

𝜍MGL

⟳

Totaro

MGL2*,*(X)⊗L*Z MU2*(X)⊗L*Z

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

𝜍H=clH

H

Alg2*(X)⊆ H2*,*(X;Z)

mot

MU2*(X) MGL2*,*(X)

𝜍MGL

⟳

Totaro

MGL2*,*(X)⊗L*Z MU2*(X)⊗L*Z

Levine + Levine-Morel ≈

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

𝜍H=clH

H

Alg2*(X)⊆ H2*,*(X;Z)

mot

MU2*(X) MGL2*,*(X)

𝜍MGL ≉ in general

⟳

Totaro

MGL2*,*(X)⊗L*Z MU2*(X)⊗L*Z

Levine + Levine-Morel ≈

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

𝜍H=clH

H

Alg2*(X)⊆ H2*,*(X;Z)

mot

MU2*(X) MGL2*,*(X)

𝜍MGL ≉ 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)

𝜍H=clH

H

Alg2*(X)⊆ H2*,*(X;Z)

mot

MU2*(X) MGL2*,*(X)

𝜍MGL ≉ 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 ≈

Fix a prime p. A different perspective:

Fix a prime p.

Brown-Peterson, Quillen

MU(p) splits as a wedge of suspensions of spectra BP with BP = Z(p)[v1,v2,…]. *

|vi|=2(pi-1)

A different perspective:

Fix a prime p.

Brown-Peterson, Quillen

MU(p) splits as a wedge of suspensions of spectra BP with BP = Z(p)[v1,v2,…]. *

|vi|=2(pi-1) quotient map

BP BP/(vn+1,…) =: BP⟨n⟩ with BP⟨n⟩ = Z(p)[v1,…,vn] *

For every n:

A different perspective:

Fix a prime p.

Brown-Peterson, Quillen

MU(p) splits as a wedge of suspensions of spectra BP with BP = Z(p)[v1,v2,…]. *

|vi|=2(pi-1) quotient map

BP BP/(vn+1,…) =: BP⟨n⟩ with BP⟨n⟩ = Z(p)[v1,…,vn] *

For every n:

A different perspective: BP⟨n⟩ BP … …

HZ(p) HFp

BP⟨0⟩ BP⟨-1⟩ The Brown-Peterson tower (Wilson): BP⟨1⟩

p=2: 2-local connective K-theory

Milnor operations:

Milnor operations:

For every n:

BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑

stable cofibre sequence

Milnor operations:

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 For every n:

BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑

stable cofibre sequence

Milnor operations:

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 For every n:

BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑

stable cofibre sequence

Milnor operations:

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 For every n:

BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑

stable cofibre sequence

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?

Z*(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?

Z*(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?

Z*(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?

Z*(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?

Z*(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?

Z*(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?

Z*(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?

Z*(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

Z*(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

Z*(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

Z*(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

Z*(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

Z*(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

Z*(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

Z*(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.

Z*(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

= 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

= 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

= 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

= 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

= 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

= 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

= 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).

will drop the “top” again

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: BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑

stable cofibre sequence

Back to the cofibre sequence: BP⟨n-1⟩*(X) BP⟨n⟩* +|vn|+1(X).

qn and the induced map

Recall: BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑

stable cofibre sequence

Back to the cofibre sequence: BP⟨n-1⟩*(X) BP⟨n⟩* +|vn|+1(X).

qn and the induced map

For example: Recall: BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑

stable cofibre sequence

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

vn |vn|

∑

|vn|+1

∑

stable cofibre sequence

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

vn |vn|

∑

|vn|+1

∑

stable cofibre sequence

A diagram chase: Hk(X;Fp)

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

q0

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

q0 q1

BP⟨1⟩k+1+2p-1(X)

A diagram chase: 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 diagram chase: 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 diagram chase: 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 diagram chase: 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

𝝱 𝜒(𝝱) (X) BPk+|v0|+…+|vn|

Qn+1Qn…Q0

Hk (X;Fp)

+|v0|+…+|vn+1|

• an obstruction

(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

𝝱 𝜒(𝝱) ≠0 (X) BPk+|v0|+…+|vn|

Qn+1Qn…Q0

Hk (X;Fp)

+|v0|+…+|vn+1|

• an obstruction

(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

𝝱 𝜒(𝝱) ≠0 (X) BPk+|v0|+…+|vn|

Qn+1Qn…Q0

Hk (X;Fp)

+|v0|+…+|vn+1|

• an obstruction

(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

𝝱 𝜒(𝝱) ≠0 ✘ (X) BPk+|v0|+…+|vn|

Qn+1Qn…Q0

Hk (X;Fp)

+|v0|+…+|vn+1|

• an obstruction

(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

𝝱 𝜒(𝝱) ≠0 ✘ (X) BPk+|v0|+…+|vn|

Qn+1Qn…Q0

Hk (X;Fp)

+|v0|+…+|vn+1|

• an obstruction

(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

𝝱 𝜒(𝝱) ≠0 ✘ (X) BPk+|v0|+…+|vn|

Qn+1Qn…Q0

Hk (X;Fp)

+|v0|+…+|vn+1|

• an obstruction

(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

𝝱 𝜒(𝝱) ≠0 ✘ (X) BPk+|v0|+…+|vn|

Qn+1Qn…Q0

Hk (X;Fp)

+|v0|+…+|vn+1|

• an obstruction

(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);

|yi|=2 |xi|=1

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:

• Qj(xi)=yi , Qj(yi)=0.

pj

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

|yi|=2 |xi|=1

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.

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

Some final remarks:

Some final remarks:

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

Some final remarks:

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

Some final remarks:

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

･ Kollar: non-torsion classes on hypersurface in P4.

Some final remarks:

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

･ Kollar: non-torsion classes on hypersurface in P4. ･ Voisin: torsion classes based on Kollar’ s example.

Some final remarks:

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

･ Kollar: non-torsion classes on hypersurface in P4. ･ Voisin: torsion classes based on Kollar’ s example.

“not topological”

Some final remarks:

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

･ Kollar: non-torsion classes on hypersurface in P4. ･ Voisin: torsion classes 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

Some final remarks:

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

･ Kollar: non-torsion classes on hypersurface in P4. ･ Voisin: torsion classes 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.

Some final remarks:

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

･ Kollar: non-torsion classes on hypersurface in P4. ･ Voisin: torsion classes 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. non-torsion classes in BP⟨n⟩*(BG)

Thank you!