Algebraic vs topological classes - NTNU · n-dim. cycles (free abelian group of…) modulo boundaries H n(X;Z) = Y continuous map of spaces X Fundamental classes f. Y continuous map

Algebraic vs topological classes

National Mathematicians Meeting in Bergen

September 13, 2018

Gereon QuickNTNU

Algebraic topology in a nutshell


spaces


spacestest spaces continuous maps



•dim 0



•dim 0

dim 1



•dim 0

dim 1

dim 2

dim 3

⋮⋮

dim n ∆n

⋮


spaces

Homology of X:

test spaces continuous maps

•dim 0

dim 1

dim 2

dim 3

⋮⋮

dim n ∆n

⋮


spaces

Homology of X:


•dim 0

dim 1

dim 2

dim 3

⋮⋮

dim n ∆n

⋮

n-dim. cycles

(free abelian group of…)


spaces

Homology of X:


•dim 0

dim 1

dim 2

dim 3

⋮⋮

dim n ∆n

⋮

n-dim. cycles

(free abelian group of…)

modulo boundariesHn(X;Z) =

Y

continuous mapof spaces

X

Fundamental classes

f

Y


∆n

Hn(Y;Z)

X

Fundamental classes

f

Y


∆n

Hn(Y;Z)

X

Fundamental classes

f

induces

Hn(X;Z)fn

Y


∆n

Hn(Y;Z)

X

Fundamental classes

•

Assume: Y=M is a smooth n-manifold.

every point has a neighborhood ≈ Rn

f

induces

Hn(X;Z)fn

Y


∆n

Hn(Y;Z)

X

Fundamental classes

& compact, oriented

•



f

induces

Hn(X;Z)fn

Y


∆n

Hn(Y;Z)

X

Fundamental classes

& compact, oriented

•



f

induces

Hn(X;Z)fn

[M] ∈ Hn(M;Z)“fundamental class of M”

(M,M-p)≈(Rn,Rn-0)

Y


∆n

Hn(Y;Z)

X

Fundamental classes

& compact, oriented

•



f

induces

Hn(X;Z)fn

fn[M] ∈ Hn(X;Z)[M] ∈ Hn(M;Z)“fundamental class of M”

(M,M-p)≈(Rn,Rn-0)

•

Steenrod’s question:

Can every class in Hn(X;Z) be realized as the fundamental class of a smooth n-manifold M → X?

& compact, oriented

•


𝛼 ∈ Hn(X;Z)


& compact, oriented

•


𝛼 ∈ Hn(X;Z)M

X

exist manifold ?

f?fn[M] =


& compact, oriented

•


𝛼 ∈ Hn(X;Z)M

X

exist manifold ?

f?fn[M] =

Example: Every elt in H1(X;Z) is realized as S1 → X.


& compact, oriented

•


𝛼 ∈ Hn(X;Z)M

X

exist manifold ?

f?fn[M] =

Thom’s answer: In general, no!



& compact, oriented

•


𝛼 ∈ Hn(X;Z)M

X

exist manifold ?

f?fn[M] =



manifolds obey some universal geometric operations and 𝛼 may not!

because


& compact, oriented

•


𝛼 ∈ Hn(X;Z)M

X

exist manifold ?

f?fn[M] =



manifolds obey some universal geometric operations and 𝛼 may not!

because


& compact, oriented

With rational coefficients: Yes!

Modify the problem: Step 1

Modify the problem:

• choose a nice X

Step 1

Modify the problem:

• choose a nice X

complex smooth algebraic variety X

set of solutions in some CNof polynomial equations

Step 1

Modify the problem:

Examples:

• choose a nice X



Step 1

Modify the problem:

Examples:

• choose a nice X



• p(x,y,z) = 0 in C3 for p(x,y,z) = xn+yn+zn-1.

Step 1

Modify the problem:

Examples:

• choose a nice X



• p(x,y,z) = 0 in C3 for p(x,y,z) = xn+yn+zn-1.

• yn-q(x) = 0 in C2, q without multiple roots, e.g. q(x)=x3+x+1.

Abelian integralsp(x)

√q(x)dx⌠

⌡ n

Step 1

Modify the problem:

• choose a nice X


Step 2

Modify the problem:

• choose a nice X


complex projective space

⊂ CPN

projective

Step 2

Modify the problem:

• choose a nice X


• consider an algebraic subset V ⊂ Xsatisfy additional polynomial equations


⊂ CPN

projective

Step 2

Modify the problem:

• choose a nice X




⊂ CPN

projective

Step 2

[V] ∈ H2n(V;Z)

if V smoothfundamental class of V

Modify the problem:

• choose a nice X




⊂ CPN

projective

Step 2

[V] ∈ H2n(V;Z)

if V smoothfundamental class of V j2n[V] ∈ H2n(X;Z)

j

A new question: X⊂CPN smooth proj. algebraic


Can every class in H2n(X;Z) be realized as the fundamental class of an algebraic subset V⊂X?



desingularization

V

X

Vsm [V]=[Vsm]in H2n(X;Z)



desingularization

V

X


• Example:



desingularization

V

X


• Example:

V ⊂ Xj

curve surface



desingularization

V

X


• Example:z1,z2 local coordinates on X

V = {z2=0} V ⊂ Xj

curve surface



desingularization

V

X


• Example:z1,z2 local coordinates on X

V = {z2=0}

form on X

j*𝛼 V

⌠⌡

V ⊂ Xj

curve surface



desingularization

V

X


• Example:

𝛼 = ∑ g dz1∧dz2

𝛼 = ∑ h dz1∧dz2--

𝛼 = ∑ f dz1∧dz1-

z1,z2 local coordinates on X

V = {z2=0}

form on X

j*𝛼 V

⌠⌡

V ⊂ Xj

curve surface



desingularization

V

X


• Example:


𝛼 = ∑ h dz1∧dz2--

𝛼 = ∑ f dz1∧dz1-


V = {z2=0}

form on X

j*𝛼 V

⌠⌡

V ⊂ Xj

curve surface

= 0 unless



desingularization

V

X


• Example:


𝛼 = ∑ h dz1∧dz2--

𝛼 = ∑ f dz1∧dz1-


V = {z2=0}

form on X

j*𝛼 V

⌠⌡

V ⊂ Xj

curve surface

= 0 unless(Hodge)

Hodge’s question: X⊂CPN smooth proj. algebraic


Can every class in H2n(X;Z) of Hodge type be realized as the fundamental class of an algebraic subset V⊂X?



Atiyah-Hirzebruch: In general, no!




With rational coefficients this is still an open question!





• However, for n = dimX-1, the answer is yes.






given 𝛼 in H2d-2(X;Z)







𝛼PD in H2(X;Z)Poincaré dual








X ∞CP

K(2;Z)

𝛼PD







kCP ⊂𝛼PD in H2(X;Z)Poincaré dual

X ∞CP

K(2;Z)

𝛼PD







kCP ⊂

k-1CP


X ∞CP

K(2;Z)

𝛼PD







kCP ⊂

k-1CP


Vpullback

X ∞CP

K(2;Z)

𝛼PD

Atiyah-Hirzebruch-Totaro obstruction:

algebraic V⊂Xgiven


algebraic V⊂X

fundamental class of Vsm

[V] ∈ H2n(X;Z)H

given


algebraic V⊂X


[V] ∈ H2n(X;Z)H

[V]K ∈ K2n(X)Atiyah-Hirzebruch

K-theory fundamental class

given


algebraic V⊂X


[V] ∈ H2n(X;Z)H

Thom/Quillen: universal fundamental class

[V]MU ∈ MU2n(X)

⋮ ⋮



given


algebraic V⊂X


[V] ∈ H2n(X;Z)H


[V]MU ∈ MU2n(X)

⋮ ⋮



M

X

generators are manifolds

given


algebraic V⊂X


[V] ∈ H2n(X;Z)H

TotaroLevine-Morel


[V]MU ∈ MU2n(X)

⋮ ⋮



M

X


given


algebraic V⊂X


[V] ∈ H2n(X;Z)H

/K∗>0∙K∗(X)

/MU∗>0∙MU∗(X)

TotaroLevine-Morel


[V]MU ∈ MU2n(X)

⋮ ⋮



M

X


given


algebraic V⊂X


[V] ∈ H2n(X;Z)H

/K∗>0∙K∗(X)

/MU∗>0∙MU∗(X)

≉ in general

TotaroLevine-Morel


[V]MU ∈ MU2n(X)

⋮ ⋮



M

X


given


algebraic V⊂X


[V] ∈ H2n(X;Z)H

/K∗>0∙K∗(X)

/MU∗>0∙MU∗(X)

operations on the image

≉ in general

TotaroLevine-Morel


[V]MU ∈ MU2n(X)

⋮ ⋮



M

X


given

Brown-Peterson QuillenWilson⋮

Brown-Peterson spectra BP with BP = Z(p)[v1,v2,…]. * |vi|=2(pi-1)

The Brown-Peterson tower: fix a prime p

evaluationon point

p-local universal theory




evaluationon point

Z(p)[v1,…,vn]

For every n:

Z(p)[v1,v2,…]





evaluationon point “quotient map”

BP BP/(vn+1,…) =: BP⟨n⟩

*= BP⟨n⟩Z(p)[v1,…,vn]

For every n:

Z(p)[v1,v2,…]




The Brown-Peterson tower:

BP⟨n⟩BP … …

HZ(p) HFp

BP⟨0⟩ BP⟨-1⟩

The Brown-Peterson tower:

BP⟨1⟩ p-local connective K-theory

fix a prime p

evaluationon point “quotient map”

BP BP/(vn+1,…) =: BP⟨n⟩

*= BP⟨n⟩Z(p)[v1,…,vn]

For every n:

Z(p)[v1,v2,…]


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:



Milnor operations:



BP⟨n-1⟩*(X) BP⟨n⟩*+|vn|+1 (X)qn

H*(X;Fp) H* +|vn|+1(X;Fp)Qn

BP⟨n-1⟩

HFp

Thommap

For every n:



Milnor operations:



BP⟨n-1⟩*(X) BP⟨n⟩*+|vn|+1 (X)qn

nth Milnor operation:Q0=BocksteinQn=P Qn-1-Qn-1P p

n-1pn-1

H*(X;Fp) H* +|vn|+1(X;Fp)Qn

BP⟨n-1⟩

HFp

Thommap

For every n:



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)⟲


BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X)


qn

↺

BP2*(X)⟲

𝝰


BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X)


qn

↺

BP2*(X)⟲

𝝰Question: Is 𝝰 algebraic?


BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X)


qn

↺

BP2*(X)⟲


algebraic V⊂Xif


BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X)


qn

↺

BP2*(X)⟲


LMT

algebraic V⊂Xif


BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X)


qn

↺

BP2*(X)⟲

𝝰

𝝰n-1

Question: Is 𝝰 algebraic?

LMT

algebraic V⊂Xif


BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X)


qn

↺

BP2*(X)⟲

𝝰

𝝰n-1 qn𝝰n-1


LMT

algebraic V⊂Xif


BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X)


qn

↺

BP2*(X)⟲

𝝰

𝝰n-1 qn𝝰n-1


= 0

ifLMT

algebraic V⊂Xif


BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X)


qn

↺

BP2*(X)⟲

𝝰

𝝰n-1 qn𝝰n-1


= 0

if

𝝰n

thenLMT

algebraic V⊂Xif


BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X)


qn

↺

BP2*(X)⟲

𝝰 Qn𝝰

𝝰n-1 qn𝝰n-1


= 0

if

𝝰n

thenLMT

algebraic V⊂Xif


BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X)


qn

↺

BP2*(X)⟲

𝝰 Qn𝝰

𝝰n-1 qn𝝰n-1 = 0

if

𝝰n

thenLMT

algebraic V⊂Xif


BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X)


qn

↺

BP2*(X)

Levine-Morel-Totaro obstruction:

⟲

𝝰 Qn𝝰


if

𝝰n

thenLMT

algebraic V⊂Xif


BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X)


qn

↺

BP2*(X)


⟲

𝝰 Qn𝝰


if

𝝰n

thenLMT

If Qn𝝰 ≠ 0,

≠ 0if

algebraic V⊂Xif


BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X)


qn

↺

BP2*(X)


⟲

𝝰 Qn𝝰

𝝰n-1 qn𝝰n-1𝝰n

thenLMT

If Qn𝝰 ≠ 0,

≠ 0if

algebraic V⊂Xif


BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X)


qn

↺

BP2*(X)


⟲

𝝰 Qn𝝰

𝝰n-1 ≠ 0qn𝝰n-1𝝰n

thenLMT

If Qn𝝰 ≠ 0,

≠ 0if

algebraic V⊂Xif


BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X)


qn

↺

BP2*(X)


⟲

𝝰 Qn𝝰


thenLMT

✘

If Qn𝝰 ≠ 0,

≠ 0if

algebraic V⊂Xif


BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X)


qn

↺

BP2*(X)


⟲

𝝰

✘

Qn𝝰


thenLMT

✘

If Qn𝝰 ≠ 0,

≠ 0if

algebraic V⊂Xif


BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X)


qn

↺

BP2*(X)


⟲

𝝰

✘

Qn𝝰


thenLMT

✘

If Qn𝝰 ≠ 0,

≠ 0if

algebraic V⊂Xif

then 𝝰 is not algebraic.

✘

Generalize the question:

Generalize the question:

so far

Generalize the question: given

𝛼 ∈ H2*(X;Z)so far


𝛼 ∈ H2*(X;Z)V ⊂ X

is there an algebraic?

[V] =so far


𝛼 ∈ H2*(X;Z)V ⊂ X


[V] =so far

now


𝛼 ∈ H2*(X;Z)V ⊂ X


[V] =

BP⟨n⟩s (X)

so far

now


𝛼 ∈ H2*(X;Z)V ⊂ X


[V] =

BP⟨n⟩s (X)motBP⟨n⟩s,r (X)

motivic/algebraic spectrum

so far

now


𝛼 ∈ H2*(X;Z)V ⊂ X


[V] =VoevodskyMorelVezzosi Hopkins Hu-KrizOrmsby Hoyois Ormsby-Østvær⋮



so far

now


𝛼 ∈ H2*(X;Z)V ⊂ X





so far

SmC ManC

topological realization

top. real.induced hom.

now


𝛼 ∈ H2*(X;Z)V ⊂ X





“topological”“algebraic”

so far

SmC ManC



now


𝛼 ∈ H2*(X;Z)V ⊂ X





“topological”“algebraic”

so far

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

Question:

SmC ManC



now

We can lift…

Hk(X;Fp)

We can lift…

Hk(X;Fp)

Hk+1(X;Z(p))q0

We can lift…

Hk(X;Fp)

Hk+1(X;Z(p))q0

q1

BP⟨1⟩k+1+2p-1(X)

We can lift…

Hk(X;Fp)

Hk+1(X;Z(p))q0

q1

BP⟨1⟩k+1+2p-1(X)

(X)

q2

⋮

BP⟨n⟩kqn

+|Q0|+…+|Qn|

We can lift…

Hk(X;Fp)

Hk+1(X;Z(p))q0

q1

BP⟨1⟩k+1+2p-1(X)

(X)

q2

⋮

BP⟨n⟩kqn

+|Q0|+…+|Qn| (X)BP⟨n+1⟩k+|Q0|+…+|Qn+1|qn+1

We can lift…

Hk(X;Fp)

Hk+1(X;Z(p))q0

q1

BP⟨1⟩k+1+2p-1(X)

(X)

q2

⋮

BP⟨n⟩kqn

+|Q0|+…+|Qn| (X)BP⟨n+1⟩k+|Q0|+…+|Qn+1|qn+1

BP⟨n+1⟩

Hk (X;Fp)+|Q0|+…+|Qn+1|

HFp

Thommap

We can lift…

Hk(X;Fp)

Hk+1(X;Z(p))q0

q1

BP⟨1⟩k+1+2p-1(X)

(X)

q2

⋮

BP⟨n⟩kqn

+|Q0|+…+|Qn| (X)BP⟨n+1⟩k+|Q0|+…+|Qn+1|qn+1

BP⟨n+1⟩

Hk (X;Fp)+|Q0|+…+|Qn+1|

HFp

Thommap

Qn+1Qn…Q0

We can lift…

Hk(X;Fp)

Hk+1(X;Z(p))q0

q1

BP⟨1⟩k+1+2p-1(X)

(X)

q2

⋮

BP⟨n⟩kqn

+|Q0|+…+|Qn| (X)BP⟨n+1⟩k+|Q0|+…+|Qn+1|qn+1

BP⟨n+1⟩

Hk (X;Fp)+|Q0|+…+|Qn+1|

HFp

Thommap

Qn+1Qn…Q0

but we pay a price…

Examples of non-algebraic classes:


Theorem (Q.): For every n, there is a smooth projective complex algebraic 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





2(pn+…+1)+2 (X)BP⟨n⟩

top. realization

Proof:





2(pn+…+1)+2 (X)BP⟨n⟩

top. realization

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





2(pn+…+1)+2 (X)BP⟨n⟩

top. realization


We know:





2(pn+…+1)+2 (X)BP⟨n⟩

top. realization


We know: • H*(BGk;Fp) = Fp[y1,…,yk]⊗𝚲(x1,…,xk);|yi|=2 |xi|=1





2(pn+…+1)+2 (X)BP⟨n⟩

top. realization


We know:

• Qj(xi)=yi , Qj(yi)=0.pj

• H*(BGk;Fp) = Fp[y1,…,yk]⊗𝚲(x1,…,xk);|yi|=2 |xi|=1

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

pj

Proof continued:

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

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

pj

Proof continued:

Check: Qn+1…Q0(𝝰)≠0.



pj

Proof continued:




pj

is not in the image of the map

2(pn+…+1)+2 (BGn+3)BP⟨n⟩Hence x:=qn…q0(𝝰) in

BP⟨n⟩BP (BGn+3)2(pn+…+1)+2 (BGn+3).2(pn+…+1)+2

Proof continued:




pj

is not in the image of the map

2(pn+…+1)+2 (BGn+3)BP⟨n⟩Hence x:=qn…q0(𝝰) 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

□

Some final remarks:

Some final remarks:

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

Some final remarks:


• Other types of non-alg. classes in H*(X;Z):

Some final remarks:


• Other types of non-alg. 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:





･ Benoist-Ottem: torsion classes of degree 4 on 3-folds.

Some final remarks:





･ 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:







･ Antieau: class in H*(BG;Z) for alg. group G, represent. theoryon which the Qi’s vanish, but a higher differential in the AH-spectral sequence is nontrivial.


Some final remarks:







･ Antieau: class in H*(BG;Z) for alg. group G, represent. theoryon 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!

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Algebraic vs topological classes - NTNU · n-dim. cycles (free abelian group of…) modulo boundaries H n(X;Z) = Y continuous map of spaces X Fundamental classes f. Y continuous map