Algebraic Vs. Topological Vector Bundles Joint with Jean Fasel and Mike Hopkins

Algebraic vs. topological vector bundles joint with Jean Fasel and Mike Hopkins

Aravind Asok (USC)

October 27, 2020

Aravind Asok (USC) Algebraic vs. topological vector bundles 1 / 27 Throughout: k a commutative ring, usually a ﬁeld, frequently C. X is a smooth k-algebraic variety (Zariski top); Smk - category of such objects an if k = C, X is X(C) as a C-manifold (usual top); alg ∼ Vr (X) - =-classes of algebraic vb of rank r; top ∼ an Vr (X) - =-classes of C-vb of rank r on X If k ⊂ C, set: alg top Φr : Vr (X) −→ Vr (X).

Question

Can we characterize the image of Φr?

Notation and the basic question

Aravind Asok (USC) Algebraic vs. topological vector bundles 2 / 27 an if k = C, X is X(C) as a C-manifold (usual top); alg ∼ Vr (X) - =-classes of algebraic vb of rank r; top ∼ an Vr (X) - =-classes of C-vb of rank r on X If k ⊂ C, set: alg top Φr : Vr (X) −→ Vr (X).

Question

Can we characterize the image of Φr?

Notation and the basic question

Throughout: k a commutative ring, usually a ﬁeld, frequently C. X is a smooth k-algebraic variety (Zariski top); Smk - category of such objects

Aravind Asok (USC) Algebraic vs. topological vector bundles 2 / 27 alg ∼ Vr (X) - =-classes of algebraic vb of rank r; top ∼ an Vr (X) - =-classes of C-vb of rank r on X If k ⊂ C, set: alg top Φr : Vr (X) −→ Vr (X).

Question

Can we characterize the image of Φr?

Notation and the basic question

Throughout: k a commutative ring, usually a ﬁeld, frequently C. X is a smooth k-algebraic variety (Zariski top); Smk - category of such objects an if k = C, X is X(C) as a C-manifold (usual top);

Aravind Asok (USC) Algebraic vs. topological vector bundles 2 / 27 top ∼ an Vr (X) - =-classes of C-vb of rank r on X If k ⊂ C, set: alg top Φr : Vr (X) −→ Vr (X).

Question

Can we characterize the image of Φr?

Notation and the basic question

Throughout: k a commutative ring, usually a ﬁeld, frequently C. X is a smooth k-algebraic variety (Zariski top); Smk - category of such objects an if k = C, X is X(C) as a C-manifold (usual top); alg ∼ Vr (X) - =-classes of algebraic vb of rank r;

Aravind Asok (USC) Algebraic vs. topological vector bundles 2 / 27 If k ⊂ C, set: alg top Φr : Vr (X) −→ Vr (X).

Question

Can we characterize the image of Φr?

Notation and the basic question

Aravind Asok (USC) Algebraic vs. topological vector bundles 2 / 27 Question

Can we characterize the image of Φr?

Notation and the basic question

Aravind Asok (USC) Algebraic vs. topological vector bundles 2 / 27 Notation and the basic question

Question

Can we characterize the image of Φr?

Aravind Asok (USC) Algebraic vs. topological vector bundles 2 / 27 Proof. n Case. If X is affine, i.e., X ⊂ C a closed subset; Morse theory (Andreotti–Frankel); in fact, X is a ≤ d-dim’l cell complex Case. If X not affine: ∃ a smooth affine C-variety X˜ and a morphism m π : X˜ → X that is Zariski locally trivial with fibers isomorphic to C (Jouanolou–Thomason); the morphism π is not a vector bundle projection; is topologically trivial.

Warm-up: topology of smooth algebraic varieties

Assume X a smooth C-algebraic variety of dimension d: Theorem The space Xan has the homotopy type of a ﬁnite CW complex.

Aravind Asok (USC) Algebraic vs. topological vector bundles 3 / 27 Case. If X not affine: ∃ a smooth affine C-variety X˜ and a morphism m π : X˜ → X that is Zariski locally trivial with fibers isomorphic to C (Jouanolou–Thomason); the morphism π is not a vector bundle projection; is topologically trivial.

Warm-up: topology of smooth algebraic varieties

Assume X a smooth C-algebraic variety of dimension d: Theorem The space Xan has the homotopy type of a ﬁnite CW complex.

Proof. n Case. If X is afﬁne, i.e., X ⊂ C a closed subset; Morse theory (Andreotti–Frankel); in fact, X is a ≤ d-dim’l cell complex

Aravind Asok (USC) Algebraic vs. topological vector bundles 3 / 27 the morphism π is not a vector bundle projection; is topologically trivial.

Warm-up: topology of smooth algebraic varieties

Assume X a smooth C-algebraic variety of dimension d: Theorem The space Xan has the homotopy type of a ﬁnite CW complex.

Proof. n Case. If X is affine, i.e., X ⊂ C a closed subset; Morse theory (Andreotti–Frankel); in fact, X is a ≤ d-dim’l cell complex Case. If X not affine: ∃ a smooth affine C-variety X˜ and a morphism m π : X˜ → X that is Zariski locally trivial with fibers isomorphic to C (Jouanolou–Thomason);

Aravind Asok (USC) Algebraic vs. topological vector bundles 3 / 27 Warm-up: topology of smooth algebraic varieties

Assume X a smooth C-algebraic variety of dimension d: Theorem The space Xan has the homotopy type of a ﬁnite CW complex.

Aravind Asok (USC) Algebraic vs. topological vector bundles 3 / 27 I = incidence variety of hyperplanes vanishing on a line; π : P](V) → P(V) is induced by projection (ﬁbers are afﬁne spaces)

Definition If X is smooth k-algebraic variety, a Jouanolou device for X is (X˜, π) with π : X˜ → X Zariski locally trivial with affine space fibers, and X˜ smooth affine.

Example

n We call P](V) the standard Jouanolou device of projective space. If X ,→ P is a closed subvariety, then we get a Jouanolou device for X by restricting the n standard Jounolou device for P .

Projective varieties

Example (Projective space)

∨ When X = P(V), set P](V) := P(V) × P(V ) \ I

Aravind Asok (USC) Algebraic vs. topological vector bundles 4 / 27 π : P](V) → P(V) is induced by projection (ﬁbers are afﬁne spaces)

Definition If X is smooth k-algebraic variety, a Jouanolou device for X is (X˜, π) with π : X˜ → X Zariski locally trivial with affine space fibers, and X˜ smooth affine.

Example

n We call P](V) the standard Jouanolou device of projective space. If X ,→ P is a closed subvariety, then we get a Jouanolou device for X by restricting the n standard Jounolou device for P .

Projective varieties

Example (Projective space)

∨ When X = P(V), set P](V) := P(V) × P(V ) \ I I = incidence variety of hyperplanes vanishing on a line;

Aravind Asok (USC) Algebraic vs. topological vector bundles 4 / 27 Definition If X is smooth k-algebraic variety, a Jouanolou device for X is (X˜, π) with π : X˜ → X Zariski locally trivial with affine space fibers, and X˜ smooth affine.

Example

n We call P](V) the standard Jouanolou device of projective space. If X ,→ P is a closed subvariety, then we get a Jouanolou device for X by restricting the n standard Jounolou device for P .

Projective varieties

Example (Projective space)

∨ When X = P(V), set P](V) := P(V) × P(V ) \ I I = incidence variety of hyperplanes vanishing on a line; π : P](V) → P(V) is induced by projection (ﬁbers are afﬁne spaces)

Aravind Asok (USC) Algebraic vs. topological vector bundles 4 / 27 Example

n We call P](V) the standard Jouanolou device of projective space. If X ,→ P is a closed subvariety, then we get a Jouanolou device for X by restricting the n standard Jounolou device for P .

Projective varieties

Example (Projective space)

∨ When X = P(V), set P](V) := P(V) × P(V ) \ I I = incidence variety of hyperplanes vanishing on a line; π : P](V) → P(V) is induced by projection (ﬁbers are afﬁne spaces)

Definition If X is smooth k-algebraic variety, a Jouanolou device for X is (X˜, π) with π : X˜ → X Zariski locally trivial with affine space fibers, and X˜ smooth affine.

Aravind Asok (USC) Algebraic vs. topological vector bundles 4 / 27 n If X ,→ P is a closed subvariety, then we get a Jouanolou device for X by restricting the n standard Jounolou device for P .

Projective varieties

Example (Projective space)

∨ When X = P(V), set P](V) := P(V) × P(V ) \ I I = incidence variety of hyperplanes vanishing on a line; π : P](V) → P(V) is induced by projection (ﬁbers are afﬁne spaces)

Definition If X is smooth k-algebraic variety, a Jouanolou device for X is (X˜, π) with π : X˜ → X Zariski locally trivial with affine space fibers, and X˜ smooth affine.

Example

We call P](V) the standard Jouanolou device of projective space.

Aravind Asok (USC) Algebraic vs. topological vector bundles 4 / 27 Projective varieties

Example (Projective space)

∨ When X = P(V), set P](V) := P(V) × P(V ) \ I I = incidence variety of hyperplanes vanishing on a line; π : P](V) → P(V) is induced by projection (ﬁbers are afﬁne spaces)

Definition If X is smooth k-algebraic variety, a Jouanolou device for X is (X˜, π) with π : X˜ → X Zariski locally trivial with affine space fibers, and X˜ smooth affine.

Example

n We call P](V) the standard Jouanolou device of projective space. If X ,→ P is a closed subvariety, then we get a Jouanolou device for X by restricting the n standard Jounolou device for P .

Aravind Asok (USC) Algebraic vs. topological vector bundles 4 / 27 Basic cohomological invariants: ∗ ∼ top top top H (Grr, Z) = Z[c1 ,..., cr ], deg(ci ) = 2i yields top ∗ top Chern classes: if f : X → Grr represents E, then ci (E) := f ci ; top Qr 2i deﬁnes a function Vr (X) → i=1 H (X; Z) Sub-question: are there restrictions on the possible Chern classes of algebraic vector bundles?

top The set Vr (X)

Representability: Theorem (Pontryagin–Steenrod) top ∼ There is a canonical bijection Vr (X) = [X, Grr].

Aravind Asok (USC) Algebraic vs. topological vector bundles 5 / 27 ∗ ∼ top top top H (Grr, Z) = Z[c1 ,..., cr ], deg(ci ) = 2i yields top ∗ top Chern classes: if f : X → Grr represents E, then ci (E) := f ci ; top Qr 2i deﬁnes a function Vr (X) → i=1 H (X; Z) Sub-question: are there restrictions on the possible Chern classes of algebraic vector bundles?

top The set Vr (X)

Representability: Theorem (Pontryagin–Steenrod) top ∼ There is a canonical bijection Vr (X) = [X, Grr].

Basic cohomological invariants:

Aravind Asok (USC) Algebraic vs. topological vector bundles 5 / 27 top ∗ top Chern classes: if f : X → Grr represents E, then ci (E) := f ci ; top Qr 2i deﬁnes a function Vr (X) → i=1 H (X; Z) Sub-question: are there restrictions on the possible Chern classes of algebraic vector bundles?

top The set Vr (X)

Representability: Theorem (Pontryagin–Steenrod) top ∼ There is a canonical bijection Vr (X) = [X, Grr].

Basic cohomological invariants: ∗ ∼ top top top H (Grr, Z) = Z[c1 ,..., cr ], deg(ci ) = 2i yields

Aravind Asok (USC) Algebraic vs. topological vector bundles 5 / 27 top Qr 2i deﬁnes a function Vr (X) → i=1 H (X; Z) Sub-question: are there restrictions on the possible Chern classes of algebraic vector bundles?

top The set Vr (X)

Representability: Theorem (Pontryagin–Steenrod) top ∼ There is a canonical bijection Vr (X) = [X, Grr].

Basic cohomological invariants: ∗ ∼ top top top H (Grr, Z) = Z[c1 ,..., cr ], deg(ci ) = 2i yields top ∗ top Chern classes: if f : X → Grr represents E, then ci (E) := f ci ;

Aravind Asok (USC) Algebraic vs. topological vector bundles 5 / 27 Sub-question: are there restrictions on the possible Chern classes of algebraic vector bundles?

top The set Vr (X)

Representability: Theorem (Pontryagin–Steenrod) top ∼ There is a canonical bijection Vr (X) = [X, Grr].

Basic cohomological invariants: ∗ ∼ top top top H (Grr, Z) = Z[c1 ,..., cr ], deg(ci ) = 2i yields top ∗ top Chern classes: if f : X → Grr represents E, then ci (E) := f ci ; top Qr 2i deﬁnes a function Vr (X) → i=1 H (X; Z)

Aravind Asok (USC) Algebraic vs. topological vector bundles 5 / 27 top The set Vr (X)

Representability: Theorem (Pontryagin–Steenrod) top ∼ There is a canonical bijection Vr (X) = [X, Grr].

Aravind Asok (USC) Algebraic vs. topological vector bundles 5 / 27 Theorem 2i If we ﬁx a rank r and classes ci ∈ H (X, Z), i = 1,..., r, then the subset of top Vr (X) consisting of bundles with these Chern classes is ﬁnite.

Proof. Qr the map c : Grr → i=1 K(Z, 2i) is a weak equivalence for r = 1, and a Q-weak equivalence for r > 1; the homotopy ﬁber of c has ﬁnite homotopy groups use obstruction theory via the Moore–Postnikov factorization of c

Sub question: are there restrictions on these “further” cohomological invariants for algebraic vector bundles?

top The set Vr (X) ctd.

Further cohomological invariants:

Aravind Asok (USC) Algebraic vs. topological vector bundles 6 / 27 Proof. Qr the map c : Grr → i=1 K(Z, 2i) is a weak equivalence for r = 1, and a Q-weak equivalence for r > 1; the homotopy ﬁber of c has ﬁnite homotopy groups use obstruction theory via the Moore–Postnikov factorization of c

Sub question: are there restrictions on these “further” cohomological invariants for algebraic vector bundles?

top The set Vr (X) ctd.

Further cohomological invariants: Theorem 2i If we ﬁx a rank r and classes ci ∈ H (X, Z), i = 1,..., r, then the subset of top Vr (X) consisting of bundles with these Chern classes is ﬁnite.

Aravind Asok (USC) Algebraic vs. topological vector bundles 6 / 27 the homotopy ﬁber of c has ﬁnite homotopy groups use obstruction theory via the Moore–Postnikov factorization of c

Sub question: are there restrictions on these “further” cohomological invariants for algebraic vector bundles?

top The set Vr (X) ctd.

Further cohomological invariants: Theorem 2i If we ﬁx a rank r and classes ci ∈ H (X, Z), i = 1,..., r, then the subset of top Vr (X) consisting of bundles with these Chern classes is ﬁnite.

Proof. Qr the map c : Grr → i=1 K(Z, 2i) is a weak equivalence for r = 1, and a Q-weak equivalence for r > 1;

Aravind Asok (USC) Algebraic vs. topological vector bundles 6 / 27 use obstruction theory via the Moore–Postnikov factorization of c

Sub question: are there restrictions on these “further” cohomological invariants for algebraic vector bundles?

top The set Vr (X) ctd.

Further cohomological invariants: Theorem 2i If we ﬁx a rank r and classes ci ∈ H (X, Z), i = 1,..., r, then the subset of top Vr (X) consisting of bundles with these Chern classes is ﬁnite.

Proof. Qr the map c : Grr → i=1 K(Z, 2i) is a weak equivalence for r = 1, and a Q-weak equivalence for r > 1; the homotopy ﬁber of c has ﬁnite homotopy groups

Aravind Asok (USC) Algebraic vs. topological vector bundles 6 / 27 Sub question: are there restrictions on these “further” cohomological invariants for algebraic vector bundles?

top The set Vr (X) ctd.

Further cohomological invariants: Theorem 2i If we ﬁx a rank r and classes ci ∈ H (X, Z), i = 1,..., r, then the subset of top Vr (X) consisting of bundles with these Chern classes is ﬁnite.

Aravind Asok (USC) Algebraic vs. topological vector bundles 6 / 27 top The set Vr (X) ctd.

Further cohomological invariants: Theorem 2i If we ﬁx a rank r and classes ci ∈ H (X, Z), i = 1,..., r, then the subset of top Vr (X) consisting of bundles with these Chern classes is ﬁnite.

Sub question: are there restrictions on these “further” cohomological invariants for algebraic vector bundles?

Aravind Asok (USC) Algebraic vs. topological vector bundles 6 / 27 Proposition 1 ∼ alg H (X, GLr) = Vr (X).

Open cover u : U → X, yields C˘(U) → X Bar construction for BGLr Cocycles correspond to morphisms C˘(u) → BGLr Build a homotopy theory Halg(k) for varieties: maps C˘(u) → X are weak equivalences. Proposition

alg Vr (X) = [X, BGLr]Halg(k).

Criticism: BGLr is not equivalent to Grr in Halg(k).

alg The set Vr (X) I: non-abelian sheaf cohomology

Vector bundles are naturally described in terms of sheaf cohomology:

Aravind Asok (USC) Algebraic vs. topological vector bundles 7 / 27 Open cover u : U → X, yields C˘(U) → X Bar construction for BGLr Cocycles correspond to morphisms C˘(u) → BGLr Build a homotopy theory Halg(k) for varieties: maps C˘(u) → X are weak equivalences. Proposition

alg Vr (X) = [X, BGLr]Halg(k).

Criticism: BGLr is not equivalent to Grr in Halg(k).

alg The set Vr (X) I: non-abelian sheaf cohomology

Vector bundles are naturally described in terms of sheaf cohomology: Proposition 1 ∼ alg H (X, GLr) = Vr (X).

Aravind Asok (USC) Algebraic vs. topological vector bundles 7 / 27 Bar construction for BGLr Cocycles correspond to morphisms C˘(u) → BGLr Build a homotopy theory Halg(k) for varieties: maps C˘(u) → X are weak equivalences. Proposition

alg Vr (X) = [X, BGLr]Halg(k).

Criticism: BGLr is not equivalent to Grr in Halg(k).

alg The set Vr (X) I: non-abelian sheaf cohomology

Vector bundles are naturally described in terms of sheaf cohomology: Proposition 1 ∼ alg H (X, GLr) = Vr (X).

Open cover u : U → X, yields C˘(U) → X

Aravind Asok (USC) Algebraic vs. topological vector bundles 7 / 27 Cocycles correspond to morphisms C˘(u) → BGLr Build a homotopy theory Halg(k) for varieties: maps C˘(u) → X are weak equivalences. Proposition

alg Vr (X) = [X, BGLr]Halg(k).

Criticism: BGLr is not equivalent to Grr in Halg(k).

alg The set Vr (X) I: non-abelian sheaf cohomology

Vector bundles are naturally described in terms of sheaf cohomology: Proposition 1 ∼ alg H (X, GLr) = Vr (X).

Open cover u : U → X, yields C˘(U) → X Bar construction for BGLr

Aravind Asok (USC) Algebraic vs. topological vector bundles 7 / 27 Build a homotopy theory Halg(k) for varieties: maps C˘(u) → X are weak equivalences. Proposition

alg Vr (X) = [X, BGLr]Halg(k).

Criticism: BGLr is not equivalent to Grr in Halg(k).

alg The set Vr (X) I: non-abelian sheaf cohomology

Vector bundles are naturally described in terms of sheaf cohomology: Proposition 1 ∼ alg H (X, GLr) = Vr (X).

Open cover u : U → X, yields C˘(U) → X Bar construction for BGLr Cocycles correspond to morphisms C˘(u) → BGLr

Aravind Asok (USC) Algebraic vs. topological vector bundles 7 / 27 Proposition

alg Vr (X) = [X, BGLr]Halg(k).

Criticism: BGLr is not equivalent to Grr in Halg(k).

alg The set Vr (X) I: non-abelian sheaf cohomology

Vector bundles are naturally described in terms of sheaf cohomology: Proposition 1 ∼ alg H (X, GLr) = Vr (X).

Aravind Asok (USC) Algebraic vs. topological vector bundles 7 / 27 Criticism: BGLr is not equivalent to Grr in Halg(k).

alg The set Vr (X) I: non-abelian sheaf cohomology

Vector bundles are naturally described in terms of sheaf cohomology: Proposition 1 ∼ alg H (X, GLr) = Vr (X).

alg Vr (X) = [X, BGLr]Halg(k).

Aravind Asok (USC) Algebraic vs. topological vector bundles 7 / 27 alg The set Vr (X) I: non-abelian sheaf cohomology

Vector bundles are naturally described in terms of sheaf cohomology: Proposition 1 ∼ alg H (X, GLr) = Vr (X).

alg Vr (X) = [X, BGLr]Halg(k).

Criticism: BGLr is not equivalent to Grr in Halg(k).

Aravind Asok (USC) Algebraic vs. topological vector bundles 7 / 27 Deﬁnition

A (contravariant) functor F (valued in some category C) on Smk is homotopy invariant if the pullback map

1 F (U) → F (U × A )

is an isomorphism for all U ∈ Smk.

Proposition alg The functor U 7→ V1 (U) = Pic(U) on Smk is homotopy invariant.

Warning: if we enlarge Smk by including sufﬁciently singular varieties, then the functor U 7→ Pic(U) fails to be homotopy invariant.

alg The set Vr (X) II: homotopy invariance

Homotopy invariance: replace unit interval I by the afﬁne line A1

Aravind Asok (USC) Algebraic vs. topological vector bundles 8 / 27 if the pullback map

1 F (U) → F (U × A )

is an isomorphism for all U ∈ Smk.

Proposition alg The functor U 7→ V1 (U) = Pic(U) on Smk is homotopy invariant.

Warning: if we enlarge Smk by including sufﬁciently singular varieties, then the functor U 7→ Pic(U) fails to be homotopy invariant.

alg The set Vr (X) II: homotopy invariance

Homotopy invariance: replace unit interval I by the afﬁne line A1 Deﬁnition

A (contravariant) functor F (valued in some category C) on Smk is homotopy invariant

Aravind Asok (USC) Algebraic vs. topological vector bundles 8 / 27 Proposition alg The functor U 7→ V1 (U) = Pic(U) on Smk is homotopy invariant.

Warning: if we enlarge Smk by including sufﬁciently singular varieties, then the functor U 7→ Pic(U) fails to be homotopy invariant.

alg The set Vr (X) II: homotopy invariance

Homotopy invariance: replace unit interval I by the afﬁne line A1 Deﬁnition

A (contravariant) functor F (valued in some category C) on Smk is homotopy invariant if the pullback map

1 F (U) → F (U × A )

is an isomorphism for all U ∈ Smk.

Aravind Asok (USC) Algebraic vs. topological vector bundles 8 / 27 Warning: if we enlarge Smk by including sufﬁciently singular varieties, then the functor U 7→ Pic(U) fails to be homotopy invariant.

alg The set Vr (X) II: homotopy invariance

Homotopy invariance: replace unit interval I by the afﬁne line A1 Deﬁnition

A (contravariant) functor F (valued in some category C) on Smk is homotopy invariant if the pullback map

1 F (U) → F (U × A )

is an isomorphism for all U ∈ Smk.

Proposition alg The functor U 7→ V1 (U) = Pic(U) on Smk is homotopy invariant.

Aravind Asok (USC) Algebraic vs. topological vector bundles 8 / 27 alg The set Vr (X) II: homotopy invariance

Homotopy invariance: replace unit interval I by the afﬁne line A1 Deﬁnition

A (contravariant) functor F (valued in some category C) on Smk is homotopy invariant if the pullback map

1 F (U) → F (U × A )

is an isomorphism for all U ∈ Smk.

Proposition alg The functor U 7→ V1 (U) = Pic(U) on Smk is homotopy invariant.

Warning: if we enlarge Smk by including sufﬁciently singular varieties, then the functor U 7→ Pic(U) fails to be homotopy invariant.

Aravind Asok (USC) Algebraic vs. topological vector bundles 8 / 27 n = 1: yes, structure theorem for f.g. modules over a PID n = 2: yes, Seshadri ’58 yes if r > n, Bass ’64 n = 3: yes if k algebraically closed, Murthy–Towber ’74 n = 3, 4, 5: yes, Suslin–Vaserstein ’73/’74

Theorem (Quillen–Suslin ’76) n If k is a PID, then every vector bundle on Ak is trivial.

alg The set Vr (X) II: homotopy invariance (ctd.)

Question (Serre ’55) n If X = Ak, then are all vector bundles trivial?

Aravind Asok (USC) Algebraic vs. topological vector bundles 9 / 27 n = 2: yes, Seshadri ’58 yes if r > n, Bass ’64 n = 3: yes if k algebraically closed, Murthy–Towber ’74 n = 3, 4, 5: yes, Suslin–Vaserstein ’73/’74

Theorem (Quillen–Suslin ’76) n If k is a PID, then every vector bundle on Ak is trivial.

alg The set Vr (X) II: homotopy invariance (ctd.)

Question (Serre ’55) n If X = Ak, then are all vector bundles trivial?

n = 1: yes, structure theorem for f.g. modules over a PID

Aravind Asok (USC) Algebraic vs. topological vector bundles 9 / 27 yes if r > n, Bass ’64 n = 3: yes if k algebraically closed, Murthy–Towber ’74 n = 3, 4, 5: yes, Suslin–Vaserstein ’73/’74

Theorem (Quillen–Suslin ’76) n If k is a PID, then every vector bundle on Ak is trivial.

alg The set Vr (X) II: homotopy invariance (ctd.)

Question (Serre ’55) n If X = Ak, then are all vector bundles trivial?

n = 1: yes, structure theorem for f.g. modules over a PID n = 2: yes, Seshadri ’58

Aravind Asok (USC) Algebraic vs. topological vector bundles 9 / 27 n = 3: yes if k algebraically closed, Murthy–Towber ’74 n = 3, 4, 5: yes, Suslin–Vaserstein ’73/’74

Theorem (Quillen–Suslin ’76) n If k is a PID, then every vector bundle on Ak is trivial.

alg The set Vr (X) II: homotopy invariance (ctd.)

Question (Serre ’55) n If X = Ak, then are all vector bundles trivial?

n = 1: yes, structure theorem for f.g. modules over a PID n = 2: yes, Seshadri ’58 yes if r > n, Bass ’64

Aravind Asok (USC) Algebraic vs. topological vector bundles 9 / 27 n = 3, 4, 5: yes, Suslin–Vaserstein ’73/’74

Theorem (Quillen–Suslin ’76) n If k is a PID, then every vector bundle on Ak is trivial.

alg The set Vr (X) II: homotopy invariance (ctd.)

Question (Serre ’55) n If X = Ak, then are all vector bundles trivial?

n = 1: yes, structure theorem for f.g. modules over a PID n = 2: yes, Seshadri ’58 yes if r > n, Bass ’64 n = 3: yes if k algebraically closed, Murthy–Towber ’74

Aravind Asok (USC) Algebraic vs. topological vector bundles 9 / 27 Theorem (Quillen–Suslin ’76) n If k is a PID, then every vector bundle on Ak is trivial.

alg The set Vr (X) II: homotopy invariance (ctd.)

Question (Serre ’55) n If X = Ak, then are all vector bundles trivial?

n = 1: yes, structure theorem for f.g. modules over a PID n = 2: yes, Seshadri ’58 yes if r > n, Bass ’64 n = 3: yes if k algebraically closed, Murthy–Towber ’74 n = 3, 4, 5: yes, Suslin–Vaserstein ’73/’74

Aravind Asok (USC) Algebraic vs. topological vector bundles 9 / 27 alg The set Vr (X) II: homotopy invariance (ctd.)

Question (Serre ’55) n If X = Ak, then are all vector bundles trivial?

Theorem (Quillen–Suslin ’76) n If k is a PID, then every vector bundle on Ak is trivial.

Aravind Asok (USC) Algebraic vs. topological vector bundles 9 / 27 Quillen’s solution to Serre’s problem actually shows that the Bass-Quillen conjecture holds for k a polynomial ring over a Dedekind domain.

Theorem (Lindel ’81) The Bass–Quillen conjecture is true if k contains a ﬁeld.

Popescu ’89 extended the Lindel’s theorem to some arithmetic situations (e.g., k is regular over a Dedekind domain with perfect residue ﬁelds) Still open in completely generality!

Conjecture (Bass–Quillen ’72) If k is a regular ring of ﬁnite Krull dimension, then for any r ≥ 0

1 Vr(Spec k) −→ Vr(Ak) is a bijection.

Aravind Asok (USC) Algebraic vs. topological vector bundles 10 / 27 Theorem (Lindel ’81) The Bass–Quillen conjecture is true if k contains a ﬁeld.

Popescu ’89 extended the Lindel’s theorem to some arithmetic situations (e.g., k is regular over a Dedekind domain with perfect residue ﬁelds) Still open in completely generality!

Conjecture (Bass–Quillen ’72) If k is a regular ring of ﬁnite Krull dimension, then for any r ≥ 0

1 Vr(Spec k) −→ Vr(Ak) is a bijection.

Quillen’s solution to Serre’s problem actually shows that the Bass-Quillen conjecture holds for k a polynomial ring over a Dedekind domain.

Aravind Asok (USC) Algebraic vs. topological vector bundles 10 / 27 Popescu ’89 extended the Lindel’s theorem to some arithmetic situations (e.g., k is regular over a Dedekind domain with perfect residue ﬁelds) Still open in completely generality!

Conjecture (Bass–Quillen ’72) If k is a regular ring of ﬁnite Krull dimension, then for any r ≥ 0

1 Vr(Spec k) −→ Vr(Ak) is a bijection.

Quillen’s solution to Serre’s problem actually shows that the Bass-Quillen conjecture holds for k a polynomial ring over a Dedekind domain.

Theorem (Lindel ’81) The Bass–Quillen conjecture is true if k contains a ﬁeld.

Aravind Asok (USC) Algebraic vs. topological vector bundles 10 / 27 Still open in completely generality!

Conjecture (Bass–Quillen ’72) If k is a regular ring of ﬁnite Krull dimension, then for any r ≥ 0

1 Vr(Spec k) −→ Vr(Ak) is a bijection.

Quillen’s solution to Serre’s problem actually shows that the Bass-Quillen conjecture holds for k a polynomial ring over a Dedekind domain.

Theorem (Lindel ’81) The Bass–Quillen conjecture is true if k contains a ﬁeld.

Popescu ’89 extended the Lindel’s theorem to some arithmetic situations (e.g., k is regular over a Dedekind domain with perfect residue ﬁelds)

Aravind Asok (USC) Algebraic vs. topological vector bundles 10 / 27 Conjecture (Bass–Quillen ’72) If k is a regular ring of ﬁnite Krull dimension, then for any r ≥ 0

1 Vr(Spec k) −→ Vr(Ak) is a bijection.

Quillen’s solution to Serre’s problem actually shows that the Bass-Quillen conjecture holds for k a polynomial ring over a Dedekind domain.

Theorem (Lindel ’81) The Bass–Quillen conjecture is true if k contains a ﬁeld.

Popescu ’89 extended the Lindel’s theorem to some arithmetic situations (e.g., k is regular over a Dedekind domain with perfect residue ﬁelds) Still open in completely generality!

Aravind Asok (USC) Algebraic vs. topological vector bundles 10 / 27 2 E2 ⊂ Q4 deﬁned by x1 = x3 = z + 1 = 0 is isomorphic to A If k = C,X4 = Q4 \ E2 is contractible: in fact, there is an explicit 5 1 morphism A → X4 that is Zariski locally trivial with ﬁbers A Q4 carries an explicit non-trivial rank 2 bundle (the Hopf bundle);

this bundle restricts non-trivially to X4, i.e., contractible varieties may carry non-trivial vector bundles!

Moral: homotopy invariance fails badly for non-afﬁne varieties (even P1)!

alg The set Vr (X) II: homotopy invariance (ctd.)

alg Unfortunately Vr (−) fails to be homotopy invariant for r ≥ 2. Example (A., B. Doran ’08)

Set Q4 = Spec k[x1, x2, x3, x4, z]/hx1x2 − x3x4 = z(z + 1)i

Aravind Asok (USC) Algebraic vs. topological vector bundles 11 / 27 If k = C,X4 = Q4 \ E2 is contractible: in fact, there is an explicit 5 1 morphism A → X4 that is Zariski locally trivial with ﬁbers A Q4 carries an explicit non-trivial rank 2 bundle (the Hopf bundle);

this bundle restricts non-trivially to X4, i.e., contractible varieties may carry non-trivial vector bundles!

Moral: homotopy invariance fails badly for non-afﬁne varieties (even P1)!

alg The set Vr (X) II: homotopy invariance (ctd.)

alg Unfortunately Vr (−) fails to be homotopy invariant for r ≥ 2. Example (A., B. Doran ’08)

Set Q4 = Spec k[x1, x2, x3, x4, z]/hx1x2 − x3x4 = z(z + 1)i 2 E2 ⊂ Q4 deﬁned by x1 = x3 = z + 1 = 0 is isomorphic to A

Aravind Asok (USC) Algebraic vs. topological vector bundles 11 / 27 Q4 carries an explicit non-trivial rank 2 bundle (the Hopf bundle);

this bundle restricts non-trivially to X4, i.e., contractible varieties may carry non-trivial vector bundles!

Moral: homotopy invariance fails badly for non-afﬁne varieties (even P1)!

alg The set Vr (X) II: homotopy invariance (ctd.)

alg Unfortunately Vr (−) fails to be homotopy invariant for r ≥ 2. Example (A., B. Doran ’08)

Set Q4 = Spec k[x1, x2, x3, x4, z]/hx1x2 − x3x4 = z(z + 1)i 2 E2 ⊂ Q4 deﬁned by x1 = x3 = z + 1 = 0 is isomorphic to A If k = C,X4 = Q4 \ E2 is contractible: in fact, there is an explicit 5 1 morphism A → X4 that is Zariski locally trivial with ﬁbers A

Aravind Asok (USC) Algebraic vs. topological vector bundles 11 / 27 this bundle restricts non-trivially to X4, i.e., contractible varieties may carry non-trivial vector bundles!

Moral: homotopy invariance fails badly for non-afﬁne varieties (even P1)!

alg The set Vr (X) II: homotopy invariance (ctd.)

alg Unfortunately Vr (−) fails to be homotopy invariant for r ≥ 2. Example (A., B. Doran ’08)

Aravind Asok (USC) Algebraic vs. topological vector bundles 11 / 27 Moral: homotopy invariance fails badly for non-afﬁne varieties (even P1)!

alg The set Vr (X) II: homotopy invariance (ctd.)

alg Unfortunately Vr (−) fails to be homotopy invariant for r ≥ 2. Example (A., B. Doran ’08)

this bundle restricts non-trivially to X4, i.e., contractible varieties may carry non-trivial vector bundles!

Aravind Asok (USC) Algebraic vs. topological vector bundles 11 / 27 alg The set Vr (X) II: homotopy invariance (ctd.)

alg Unfortunately Vr (−) fails to be homotopy invariant for r ≥ 2. Example (A., B. Doran ’08)

this bundle restricts non-trivially to X4, i.e., contractible varieties may carry non-trivial vector bundles!

Moral: homotopy invariance fails badly for non-afﬁne varieties (even P1)!

Aravind Asok (USC) Algebraic vs. topological vector bundles 11 / 27 • Grr has naturally the structure of a colimit of algebraic varieties • Smk is not “big enough” to do homotopy theory (e.g., Grr is not in this category, cannot form all quotients, etc.) • Spck - “spaces”; simplicial presheaves on Smk • We now force two kinds of maps to be “weak-equivalences”: • Nisnevich local weak equivalences (roughly, u : U → X as Nisnevich covering, build C˘(u) → X, and force C˘(u) → X to be an iso) 1 1 • A -weak equivalences: X × A → X • Halg(k) - invert Nisnevich local weak equivalences and 1 • Hmot(k) - inverting both Nisnevich local and A -weak equivalences (this is the Morel–Voevodsky A1-homotopy category)

The motivic homotopy category

• Naive homotopy (homotopies parameterized by A1) is not an equivalence relation

Aravind Asok (USC) Algebraic vs. topological vector bundles 12 / 27 • Smk is not “big enough” to do homotopy theory (e.g., Grr is not in this category, cannot form all quotients, etc.) • Spck - “spaces”; simplicial presheaves on Smk • We now force two kinds of maps to be “weak-equivalences”: • Nisnevich local weak equivalences (roughly, u : U → X as Nisnevich covering, build C˘(u) → X, and force C˘(u) → X to be an iso) 1 1 • A -weak equivalences: X × A → X • Halg(k) - invert Nisnevich local weak equivalences and 1 • Hmot(k) - inverting both Nisnevich local and A -weak equivalences (this is the Morel–Voevodsky A1-homotopy category)

The motivic homotopy category

• Naive homotopy (homotopies parameterized by A1) is not an equivalence relation • Grr has naturally the structure of a colimit of algebraic varieties

Aravind Asok (USC) Algebraic vs. topological vector bundles 12 / 27 • Spck - “spaces”; simplicial presheaves on Smk • We now force two kinds of maps to be “weak-equivalences”: • Nisnevich local weak equivalences (roughly, u : U → X as Nisnevich covering, build C˘(u) → X, and force C˘(u) → X to be an iso) 1 1 • A -weak equivalences: X × A → X • Halg(k) - invert Nisnevich local weak equivalences and 1 • Hmot(k) - inverting both Nisnevich local and A -weak equivalences (this is the Morel–Voevodsky A1-homotopy category)

The motivic homotopy category

Aravind Asok (USC) Algebraic vs. topological vector bundles 12 / 27 • We now force two kinds of maps to be “weak-equivalences”: • Nisnevich local weak equivalences (roughly, u : U → X as Nisnevich covering, build C˘(u) → X, and force C˘(u) → X to be an iso) 1 1 • A -weak equivalences: X × A → X • Halg(k) - invert Nisnevich local weak equivalences and 1 • Hmot(k) - inverting both Nisnevich local and A -weak equivalences (this is the Morel–Voevodsky A1-homotopy category)

The motivic homotopy category

Aravind Asok (USC) Algebraic vs. topological vector bundles 12 / 27 • Nisnevich local weak equivalences (roughly, u : U → X as Nisnevich covering, build C˘(u) → X, and force C˘(u) → X to be an iso) 1 1 • A -weak equivalences: X × A → X • Halg(k) - invert Nisnevich local weak equivalences and 1 • Hmot(k) - inverting both Nisnevich local and A -weak equivalences (this is the Morel–Voevodsky A1-homotopy category)

The motivic homotopy category

Aravind Asok (USC) Algebraic vs. topological vector bundles 12 / 27 1 1 • A -weak equivalences: X × A → X • Halg(k) - invert Nisnevich local weak equivalences and 1 • Hmot(k) - inverting both Nisnevich local and A -weak equivalences (this is the Morel–Voevodsky A1-homotopy category)

The motivic homotopy category

• Naive homotopy (homotopies parameterized by A1) is not an equivalence relation • Grr has naturally the structure of a colimit of algebraic varieties • Smk is not “big enough” to do homotopy theory (e.g., Grr is not in this category, cannot form all quotients, etc.) • Spck - “spaces”; simplicial presheaves on Smk • We now force two kinds of maps to be “weak-equivalences”: • Nisnevich local weak equivalences (roughly, u : U → X as Nisnevich covering, build C˘(u) → X, and force C˘(u) → X to be an iso)

Aravind Asok (USC) Algebraic vs. topological vector bundles 12 / 27 • Halg(k) - invert Nisnevich local weak equivalences and 1 • Hmot(k) - inverting both Nisnevich local and A -weak equivalences (this is the Morel–Voevodsky A1-homotopy category)

The motivic homotopy category

Aravind Asok (USC) Algebraic vs. topological vector bundles 12 / 27 and 1 • Hmot(k) - inverting both Nisnevich local and A -weak equivalences (this is the Morel–Voevodsky A1-homotopy category)

The motivic homotopy category

Aravind Asok (USC) Algebraic vs. topological vector bundles 12 / 27 The motivic homotopy category

Aravind Asok (USC) Algebraic vs. topological vector bundles 12 / 27 Example If π : Y → X is Zariski locally trivial with affine space fibers, then π is an A1-weak equivalence (e.g., if π is a vector bundle): 2 1 • SL2 → A \ 0 (project a matrix onto its first column) is an A -weak equivalence; • if X is a smooth k-variety and π : X˜ → X is a Jouanolou device for X, then π is an A1-weak equivalence (thus: every smooth variety has the A1-homotopy type of a smooth affine variety)

Example ∞ • A \ 0 is A1-contractible: the “shift map” is naively homotopic to the ∞ 1 identity =⇒ Gr1 = P → BGL1 is an A -weak equivalence • More generally, Grn → BGLn classifying the tautological vector bundle is an A1-weak equivalence

The motivic homotopy category ctd.

1 Isomorphisms in Hmot(k) are called A -weak equivalences.

Aravind Asok (USC) Algebraic vs. topological vector bundles 13 / 27 2 1 • SL2 → A \ 0 (project a matrix onto its ﬁrst column) is an A -weak equivalence; • if X is a smooth k-variety and π : X˜ → X is a Jouanolou device for X, then π is an A1-weak equivalence (thus: every smooth variety has the A1-homotopy type of a smooth afﬁne variety)

The motivic homotopy category ctd.

Aravind Asok (USC) Algebraic vs. topological vector bundles 13 / 27 • if X is a smooth k-variety and π : X˜ → X is a Jouanolou device for X, then π is an A1-weak equivalence (thus: every smooth variety has the A1-homotopy type of a smooth afﬁne variety)

The motivic homotopy category ctd.

Aravind Asok (USC) Algebraic vs. topological vector bundles 13 / 27 Example ∞ • A \ 0 is A1-contractible: the “shift map” is naively homotopic to the ∞ 1 identity =⇒ Gr1 = P → BGL1 is an A -weak equivalence • More generally, Grn → BGLn classifying the tautological vector bundle is an A1-weak equivalence

The motivic homotopy category ctd.

1 Isomorphisms in Hmot(k) are called A -weak equivalences. Example If π : Y → X is Zariski locally trivial with affine space fibers, then π is an A1-weak equivalence (e.g., if π is a vector bundle): 2 1 • SL2 → A \ 0 (project a matrix onto its first column) is an A -weak equivalence; • if X is a smooth k-variety and π : X˜ → X is a Jouanolou device for X, then π is an A1-weak equivalence (thus: every smooth variety has the A1-homotopy type of a smooth affine variety)

Aravind Asok (USC) Algebraic vs. topological vector bundles 13 / 27 • More generally, Grn → BGLn classifying the tautological vector bundle is an A1-weak equivalence

The motivic homotopy category ctd.

Example ∞ • A \ 0 is A1-contractible: the “shift map” is naively homotopic to the ∞ 1 identity =⇒ Gr1 = P → BGL1 is an A -weak equivalence

Aravind Asok (USC) Algebraic vs. topological vector bundles 13 / 27 The motivic homotopy category ctd.

Aravind Asok (USC) Algebraic vs. topological vector bundles 13 / 27 Morel ’06 if r 6= 2 and k a perfect ﬁeld Schlichting ’15 arbitrary r, k perfect; simpliﬁes part of Morel’s argument A.–M. Hoyois–M. Wendt ’15 (essentially self-contained: in essence, representability is equivalent to the Bass–Quillen conjecture for all smooth k-algebras)

alg The set Vr (X) III: homotopy invariance (ctd.)

alg Cannot expect representability of Vr (X) in Hmot(k) for all smooth X, however: Theorem If k is a ﬁeld or Z, then for any smooth afﬁne k-scheme X,

∼ alg [X, Grr]A1 −→Vr (X)(= [X, Grr]naive).

Aravind Asok (USC) Algebraic vs. topological vector bundles 14 / 27 Schlichting ’15 arbitrary r, k perfect; simpliﬁes part of Morel’s argument A.–M. Hoyois–M. Wendt ’15 (essentially self-contained: in essence, representability is equivalent to the Bass–Quillen conjecture for all smooth k-algebras)

alg The set Vr (X) III: homotopy invariance (ctd.)

alg Cannot expect representability of Vr (X) in Hmot(k) for all smooth X, however: Theorem If k is a ﬁeld or Z, then for any smooth afﬁne k-scheme X,

∼ alg [X, Grr]A1 −→Vr (X)(= [X, Grr]naive). Morel ’06 if r 6= 2 and k a perfect ﬁeld

Aravind Asok (USC) Algebraic vs. topological vector bundles 14 / 27 A.–M. Hoyois–M. Wendt ’15 (essentially self-contained: in essence, representability is equivalent to the Bass–Quillen conjecture for all smooth k-algebras)

alg The set Vr (X) III: homotopy invariance (ctd.)

alg Cannot expect representability of Vr (X) in Hmot(k) for all smooth X, however: Theorem If k is a ﬁeld or Z, then for any smooth afﬁne k-scheme X,

∼ alg [X, Grr]A1 −→Vr (X)(= [X, Grr]naive). Morel ’06 if r 6= 2 and k a perfect ﬁeld Schlichting ’15 arbitrary r, k perfect; simpliﬁes part of Morel’s argument

Aravind Asok (USC) Algebraic vs. topological vector bundles 14 / 27 alg The set Vr (X) III: homotopy invariance (ctd.)

alg Cannot expect representability of Vr (X) in Hmot(k) for all smooth X, however: Theorem If k is a ﬁeld or Z, then for any smooth afﬁne k-scheme X,

∼ alg [X, Grr]A1 −→Vr (X)(= [X, Grr]naive). Morel ’06 if r 6= 2 and k a perfect ﬁeld Schlichting ’15 arbitrary r, k perfect; simpliﬁes part of Morel’s argument A.–M. Hoyois–M. Wendt ’15 (essentially self-contained: in essence, representability is equivalent to the Bass–Quillen conjecture for all smooth k-algebras)

Aravind Asok (USC) Algebraic vs. topological vector bundles 14 / 27 Thus, determining the image of Φr breaks into two stages: alg • characterize the image of Vr (X) −→ [X, Grr]A1 , and • characterize the image of [X, Grr]A1 → [X, Grr]top. Can we give a concrete description of motivic vector bundles? Does this factorization get us anything new?

Comparing algebraic and topological vb

If X a smooth k-variety, then Φr factors:

alg top Vr (X) −→ [X, Grr]A1 −→ [X, Grr] =: Vr (X).

Motivic vector bundles are elements of [X, Grr]A1 .

Aravind Asok (USC) Algebraic vs. topological vector bundles 15 / 27 alg • characterize the image of Vr (X) −→ [X, Grr]A1 , and • characterize the image of [X, Grr]A1 → [X, Grr]top. Can we give a concrete description of motivic vector bundles? Does this factorization get us anything new?

Comparing algebraic and topological vb

If X a smooth k-variety, then Φr factors:

alg top Vr (X) −→ [X, Grr]A1 −→ [X, Grr] =: Vr (X).

Motivic vector bundles are elements of [X, Grr]A1 .

Thus, determining the image of Φr breaks into two stages:

Aravind Asok (USC) Algebraic vs. topological vector bundles 15 / 27 • characterize the image of [X, Grr]A1 → [X, Grr]top. Can we give a concrete description of motivic vector bundles? Does this factorization get us anything new?

Comparing algebraic and topological vb

If X a smooth k-variety, then Φr factors:

alg top Vr (X) −→ [X, Grr]A1 −→ [X, Grr] =: Vr (X).

Motivic vector bundles are elements of [X, Grr]A1 .

Thus, determining the image of Φr breaks into two stages: alg • characterize the image of Vr (X) −→ [X, Grr]A1 , and

Aravind Asok (USC) Algebraic vs. topological vector bundles 15 / 27 Can we give a concrete description of motivic vector bundles? Does this factorization get us anything new?

Comparing algebraic and topological vb

If X a smooth k-variety, then Φr factors:

alg top Vr (X) −→ [X, Grr]A1 −→ [X, Grr] =: Vr (X).

Motivic vector bundles are elements of [X, Grr]A1 .

Thus, determining the image of Φr breaks into two stages: alg • characterize the image of Vr (X) −→ [X, Grr]A1 , and • characterize the image of [X, Grr]A1 → [X, Grr]top.

Aravind Asok (USC) Algebraic vs. topological vector bundles 15 / 27 Does this factorization get us anything new?

Comparing algebraic and topological vb

If X a smooth k-variety, then Φr factors:

alg top Vr (X) −→ [X, Grr]A1 −→ [X, Grr] =: Vr (X).

Motivic vector bundles are elements of [X, Grr]A1 .

Thus, determining the image of Φr breaks into two stages: alg • characterize the image of Vr (X) −→ [X, Grr]A1 , and • characterize the image of [X, Grr]A1 → [X, Grr]top. Can we give a concrete description of motivic vector bundles?

Aravind Asok (USC) Algebraic vs. topological vector bundles 15 / 27 Comparing algebraic and topological vb

If X a smooth k-variety, then Φr factors:

alg top Vr (X) −→ [X, Grr]A1 −→ [X, Grr] =: Vr (X).

Motivic vector bundles are elements of [X, Grr]A1 .

Aravind Asok (USC) Algebraic vs. topological vector bundles 15 / 27 ∗,∗ ∼ ∗,∗ H (Grr, Z) = H (Spec k, Z)[c1,..., cr], deg(ci) = (2i, i) yields ∗ Chern classes: if f : X → Grr corresponds to E , then ci(E) := f ci; alg Qr 2i,i deﬁnes a function Vr (X) → i=1 H (X; Z) 2i,i 2i top if k = C, cycle class map cl : H (X, Z) → H (X, Z) sends ci to ci i,i i Fundamental difference: H2 (X, Z) more complicated than H2 (X, Z) (e.g., X i afﬁne, H2 (X, Z) vanishes for 2i > dim X; false for motivic cohomology!)

Basic cohomological invariants

If k a ﬁeld, integral cohomology is replaced with motivic cohomology

Aravind Asok (USC) Algebraic vs. topological vector bundles 16 / 27 ∗ Chern classes: if f : X → Grr corresponds to E , then ci(E) := f ci; alg Qr 2i,i deﬁnes a function Vr (X) → i=1 H (X; Z) 2i,i 2i top if k = C, cycle class map cl : H (X, Z) → H (X, Z) sends ci to ci i,i i Fundamental difference: H2 (X, Z) more complicated than H2 (X, Z) (e.g., X i afﬁne, H2 (X, Z) vanishes for 2i > dim X; false for motivic cohomology!)

Basic cohomological invariants

If k a ﬁeld, integral cohomology is replaced with motivic cohomology ∗,∗ ∼ ∗,∗ H (Grr, Z) = H (Spec k, Z)[c1,..., cr], deg(ci) = (2i, i) yields

Aravind Asok (USC) Algebraic vs. topological vector bundles 16 / 27 alg Qr 2i,i deﬁnes a function Vr (X) → i=1 H (X; Z) 2i,i 2i top if k = C, cycle class map cl : H (X, Z) → H (X, Z) sends ci to ci i,i i Fundamental difference: H2 (X, Z) more complicated than H2 (X, Z) (e.g., X i afﬁne, H2 (X, Z) vanishes for 2i > dim X; false for motivic cohomology!)

Basic cohomological invariants

Aravind Asok (USC) Algebraic vs. topological vector bundles 16 / 27 2i,i 2i top if k = C, cycle class map cl : H (X, Z) → H (X, Z) sends ci to ci i,i i Fundamental difference: H2 (X, Z) more complicated than H2 (X, Z) (e.g., X i afﬁne, H2 (X, Z) vanishes for 2i > dim X; false for motivic cohomology!)

Basic cohomological invariants

Aravind Asok (USC) Algebraic vs. topological vector bundles 16 / 27 i,i i Fundamental difference: H2 (X, Z) more complicated than H2 (X, Z) (e.g., X i afﬁne, H2 (X, Z) vanishes for 2i > dim X; false for motivic cohomology!)

Basic cohomological invariants

Aravind Asok (USC) Algebraic vs. topological vector bundles 16 / 27 Basic cohomological invariants

If k a field, integral cohomology is replaced with motivic cohomology ∗,∗ ∼ ∗,∗ H (Grr, Z) = H (Spec k, Z)[c1,..., cr], deg(ci) = (2i, i) yields ∗ Chern classes: if f : X → Grr corresponds to E , then ci(E) := f ci; alg Qr 2i,i defines a function Vr (X) → i=1 H (X; Z) 2i,i 2i top if k = C, cycle class map cl : H (X, Z) → H (X, Z) sends ci to ci i,i i Fundamental difference: H2 (X, Z) more complicated than H2 (X, Z) (e.g., X i affine, H2 (X, Z) vanishes for 2i > dim X; false for motivic cohomology!)

Aravind Asok (USC) Algebraic vs. topological vector bundles 16 / 27 Further cohomological invariants

Just like in topology, further invariants are torsion: Theorem (A., J. Fasel, M. Hopkins) If k a ﬁeld and −1 is a sum of squares in k, then

r Y Grr −→ K(Z(i), 2i) i=1

is a rational A1-weak equivalence.

Aravind Asok (USC) Algebraic vs. topological vector bundles 17 / 27 Conjecture (A., J. Fasel, M. Hopkins)

For “cellular” smooth C-varieties X, [X, Grr]A1 → [X, Grr] is a bijection.

Motivic vector bundles and algebraicity

Theorem (A., J. Fasel, M. Hopkins) Suppose X is a smooth complex afﬁne variety of dimension 4, and Ean → Xan is a rank 2 complex analytic vector bundle with Chern classes top 2i an top an ci ∈ H (X , Z). Assume the Chern classes ci of E are algebraic, i.e., lie in the image of the cycle class map cl. The bundle Ean is algebraizable if 2,1 4,2 and only if we may ﬁnd (c1, c2) ∈ H (X) × H (X) with top top 2 6,3 (cl(c1), cl(c2)) = (c1 , c2 ) such that Sq c2 + c1 ∪ c2 = 0 ∈ H (X, Z/2).

Aravind Asok (USC) Algebraic vs. topological vector bundles 18 / 27 Motivic vector bundles and algebraicity

Conjecture (A., J. Fasel, M. Hopkins)

For “cellular” smooth C-varieties X, [X, Grr]A1 → [X, Grr] is a bijection.

Aravind Asok (USC) Algebraic vs. topological vector bundles 18 / 27 alg Vr (X) / [X, Grr]A1

π∗ =∼ alg ˜ =∼ ˜ Vr (X) / [X, Grr]A1 commutes, i.e., motivic vector bundles represented by actual vector bundles on a Jouanolou device. Thus, to understand image of top horizontal map, sufﬁces to: • characterize the image of π∗, which is the problem of descent for vector bundles.

Motivic vector bundles, concretely

Assume X a smooth k-variety and π : X˜ → X is a Jouanolou device for X;

Aravind Asok (USC) Algebraic vs. topological vector bundles 19 / 27 i.e., motivic vector bundles represented by actual vector bundles on a Jouanolou device. Thus, to understand image of top horizontal map, sufﬁces to: • characterize the image of π∗, which is the problem of descent for vector bundles.

Motivic vector bundles, concretely

Assume X a smooth k-variety and π : X˜ → X is a Jouanolou device for X;

alg Vr (X) / [X, Grr]A1

π∗ =∼ alg ˜ =∼ ˜ Vr (X) / [X, Grr]A1 commutes,

Aravind Asok (USC) Algebraic vs. topological vector bundles 19 / 27 Thus, to understand image of top horizontal map, sufﬁces to: • characterize the image of π∗, which is the problem of descent for vector bundles.

Motivic vector bundles, concretely

Assume X a smooth k-variety and π : X˜ → X is a Jouanolou device for X;

alg Vr (X) / [X, Grr]A1

π∗ =∼ alg ˜ =∼ ˜ Vr (X) / [X, Grr]A1 commutes, i.e., motivic vector bundles represented by actual vector bundles on a Jouanolou device.

Aravind Asok (USC) Algebraic vs. topological vector bundles 19 / 27 which is the problem of descent for vector bundles.

Motivic vector bundles, concretely

Assume X a smooth k-variety and π : X˜ → X is a Jouanolou device for X;

alg Vr (X) / [X, Grr]A1

Aravind Asok (USC) Algebraic vs. topological vector bundles 19 / 27 Motivic vector bundles, concretely

Assume X a smooth k-variety and π : X˜ → X is a Jouanolou device for X;

alg Vr (X) / [X, Grr]A1

Aravind Asok (USC) Algebraic vs. topological vector bundles 19 / 27 ∗ • π fails to be injective (very general phenomenon!), even for X = P1 (e.g., because X˜ is afﬁne, π∗ splits extensions) • Can π∗ be surjective? Long-standing conjectures in algebraic geometry (e.g., the ’77 ∗ n Grauert–Schneider conjecture) imply that π fails to be surjective for P when n ≥ 5. If π∗ does fail to be surjective, can we ﬁnd a counter-example? What happens in low dimensions?

Jouanolou descent I

Suppose π : X˜ → X is a Jouanolou device for X.

Aravind Asok (USC) Algebraic vs. topological vector bundles 20 / 27 • Can π∗ be surjective? Long-standing conjectures in algebraic geometry (e.g., the ’77 ∗ n Grauert–Schneider conjecture) imply that π fails to be surjective for P when n ≥ 5. If π∗ does fail to be surjective, can we ﬁnd a counter-example? What happens in low dimensions?

Jouanolou descent I

Suppose π : X˜ → X is a Jouanolou device for X.

∗ • π fails to be injective (very general phenomenon!), even for X = P1 (e.g., because X˜ is afﬁne, π∗ splits extensions)

Aravind Asok (USC) Algebraic vs. topological vector bundles 20 / 27 Long-standing conjectures in algebraic geometry (e.g., the ’77 ∗ n Grauert–Schneider conjecture) imply that π fails to be surjective for P when n ≥ 5. If π∗ does fail to be surjective, can we ﬁnd a counter-example? What happens in low dimensions?

Jouanolou descent I

Suppose π : X˜ → X is a Jouanolou device for X.

∗ • π fails to be injective (very general phenomenon!), even for X = P1 (e.g., because X˜ is afﬁne, π∗ splits extensions) • Can π∗ be surjective?

Aravind Asok (USC) Algebraic vs. topological vector bundles 20 / 27 If π∗ does fail to be surjective, can we ﬁnd a counter-example? What happens in low dimensions?

Jouanolou descent I

Suppose π : X˜ → X is a Jouanolou device for X.

∗ • π fails to be injective (very general phenomenon!), even for X = P1 (e.g., because X˜ is afﬁne, π∗ splits extensions) • Can π∗ be surjective? Long-standing conjectures in algebraic geometry (e.g., the ’77 ∗ n Grauert–Schneider conjecture) imply that π fails to be surjective for P when n ≥ 5.

Aravind Asok (USC) Algebraic vs. topological vector bundles 20 / 27 What happens in low dimensions?

Jouanolou descent I

Suppose π : X˜ → X is a Jouanolou device for X.

Aravind Asok (USC) Algebraic vs. topological vector bundles 20 / 27 Jouanolou descent I

Suppose π : X˜ → X is a Jouanolou device for X.

Aravind Asok (USC) Algebraic vs. topological vector bundles 20 / 27 Proof. • Idea: Describe the target of π∗ and construct enough vector bundles on X.

• Use obstruction theory to do describe [X, Grr]A1 in cohomological terms; • Use the Hartshorne–Serre correspondence (between codimension 2 lci schemes and rank 2 vector bundles) to construct the required vb on X.

Jouanolou descent II

Theorem (A., J. Fasel, M. Hopkins) If X is a smooth projective variety of dimension ≤ 2 over C, and (X˜, π) is a ∗ alg alg Jouanolou device for X, then π : Vr (X) → Vr (X˜) is surjective.

Aravind Asok (USC) Algebraic vs. topological vector bundles 21 / 27 • Use obstruction theory to do describe [X, Grr]A1 in cohomological terms; • Use the Hartshorne–Serre correspondence (between codimension 2 lci schemes and rank 2 vector bundles) to construct the required vb on X.

Jouanolou descent II

Theorem (A., J. Fasel, M. Hopkins) If X is a smooth projective variety of dimension ≤ 2 over C, and (X˜, π) is a ∗ alg alg Jouanolou device for X, then π : Vr (X) → Vr (X˜) is surjective.

Proof. • Idea: Describe the target of π∗ and construct enough vector bundles on X.

Aravind Asok (USC) Algebraic vs. topological vector bundles 21 / 27 • Use the Hartshorne–Serre correspondence (between codimension 2 lci schemes and rank 2 vector bundles) to construct the required vb on X.

Jouanolou descent II

Theorem (A., J. Fasel, M. Hopkins) If X is a smooth projective variety of dimension ≤ 2 over C, and (X˜, π) is a ∗ alg alg Jouanolou device for X, then π : Vr (X) → Vr (X˜) is surjective.

Proof. • Idea: Describe the target of π∗ and construct enough vector bundles on X.

• Use obstruction theory to do describe [X, Grr]A1 in cohomological terms;

Aravind Asok (USC) Algebraic vs. topological vector bundles 21 / 27 Jouanolou descent II

Theorem (A., J. Fasel, M. Hopkins) If X is a smooth projective variety of dimension ≤ 2 over C, and (X˜, π) is a ∗ alg alg Jouanolou device for X, then π : Vr (X) → Vr (X˜) is surjective.

Proof. • Idea: Describe the target of π∗ and construct enough vector bundles on X.

Aravind Asok (USC) Algebraic vs. topological vector bundles 21 / 27 If (X , x) is a pointed space, we may deﬁne A1-homotopy sheaves A1 πi (X , x). A1-Postnikov tower: given a pointed A1-connected space, we can build X inductively out of Eilenberg-Mac Lane spaces K(π, n); these have exactly 1 non-trivial A1-homotopy sheaf in degree n

We can inductively describe the set of maps [U, X ]A1 using sheaf cohomology with coefﬁcients in A1-homotopy sheaves

Obstruction theory in A1-homotopy

Aravind Asok (USC) Algebraic vs. topological vector bundles 22 / 27 A1-Postnikov tower: given a pointed A1-connected space, we can build X inductively out of Eilenberg-Mac Lane spaces K(π, n); these have exactly 1 non-trivial A1-homotopy sheaf in degree n

We can inductively describe the set of maps [U, X ]A1 using sheaf cohomology with coefﬁcients in A1-homotopy sheaves

Obstruction theory in A1-homotopy

Aravind Asok (USC) Algebraic vs. topological vector bundles 22 / 27 We can inductively describe the set of maps [U, X ]A1 using sheaf cohomology with coefﬁcients in A1-homotopy sheaves

Obstruction theory in A1-homotopy

Classical homotopy theory gives techniques for providing a “cohomological” description of homotopy classes: one factors a space into homotopically simple spaces (Eilenberg–Mac Lane spaces). F. Morel developed these ideas in algebraic geometry. If (X , x) is a pointed space, we may deﬁne A1-homotopy sheaves A1 πi (X , x). A1-Postnikov tower: given a pointed A1-connected space, we can build X inductively out of Eilenberg-Mac Lane spaces K(π, n); these have exactly 1 non-trivial A1-homotopy sheaf in degree n

Aravind Asok (USC) Algebraic vs. topological vector bundles 22 / 27 Obstruction theory in A1-homotopy

We can inductively describe the set of maps [U, X ]A1 using sheaf cohomology with coefﬁcients in A1-homotopy sheaves

Aravind Asok (USC) Algebraic vs. topological vector bundles 22 / 27 Example

A1 For any integer n ≥ 1, π0 (SLn) = 1 Turns out it sufﬁces to check this on sections over ﬁelds

For any ﬁeld F, any matrix in SLn(F) may be factored as a product of elementary (shearing) matrices Any elementary shearing matrix is A1-homotopic: if a ∈ F then use 1 at 0 1

1 Any matrix in SLn(F) is naively A -homotopic to the identity.

Intuition: a space should be A1-connected if points can be connected by chains of afﬁne lines

Aravind Asok (USC) Algebraic vs. topological vector bundles 23 / 27 Turns out it sufﬁces to check this on sections over ﬁelds

For any ﬁeld F, any matrix in SLn(F) may be factored as a product of elementary (shearing) matrices Any elementary shearing matrix is A1-homotopic: if a ∈ F then use 1 at 0 1

1 Any matrix in SLn(F) is naively A -homotopic to the identity.

Intuition: a space should be A1-connected if points can be connected by chains of afﬁne lines Example

A1 For any integer n ≥ 1, π0 (SLn) = 1

Aravind Asok (USC) Algebraic vs. topological vector bundles 23 / 27 For any ﬁeld F, any matrix in SLn(F) may be factored as a product of elementary (shearing) matrices Any elementary shearing matrix is A1-homotopic: if a ∈ F then use 1 at 0 1

1 Any matrix in SLn(F) is naively A -homotopic to the identity.

Intuition: a space should be A1-connected if points can be connected by chains of afﬁne lines Example

A1 For any integer n ≥ 1, π0 (SLn) = 1 Turns out it sufﬁces to check this on sections over ﬁelds

Aravind Asok (USC) Algebraic vs. topological vector bundles 23 / 27 Any elementary shearing matrix is A1-homotopic: if a ∈ F then use 1 at 0 1

1 Any matrix in SLn(F) is naively A -homotopic to the identity.

Intuition: a space should be A1-connected if points can be connected by chains of afﬁne lines Example

A1 For any integer n ≥ 1, π0 (SLn) = 1 Turns out it sufﬁces to check this on sections over ﬁelds

For any ﬁeld F, any matrix in SLn(F) may be factored as a product of elementary (shearing) matrices

Aravind Asok (USC) Algebraic vs. topological vector bundles 23 / 27 1 Any matrix in SLn(F) is naively A -homotopic to the identity.

Intuition: a space should be A1-connected if points can be connected by chains of afﬁne lines Example

A1 For any integer n ≥ 1, π0 (SLn) = 1 Turns out it sufﬁces to check this on sections over ﬁelds

For any ﬁeld F, any matrix in SLn(F) may be factored as a product of elementary (shearing) matrices Any elementary shearing matrix is A1-homotopic: if a ∈ F then use 1 at 0 1

Aravind Asok (USC) Algebraic vs. topological vector bundles 23 / 27 Intuition: a space should be A1-connected if points can be connected by chains of afﬁne lines Example

A1 For any integer n ≥ 1, π0 (SLn) = 1 Turns out it sufﬁces to check this on sections over ﬁelds

For any ﬁeld F, any matrix in SLn(F) may be factored as a product of elementary (shearing) matrices Any elementary shearing matrix is A1-homotopic: if a ∈ F then use 1 at 0 1

1 Any matrix in SLn(F) is naively A -homotopic to the identity.

Aravind Asok (USC) Algebraic vs. topological vector bundles 23 / 27 A1 GL1 is discrete, i.e., π0 (GL1) = GL1: there are no non-constant 1 algebraic maps A → GL1 ∞ the map A \ 0 → BGL1 is a principal GL1-bundle and this yields an A1-ﬁber sequence ∞ since A \ 0 is A1-contractible, the result follows from the long exact sequence in homotopy

Example

A1 A1 For any n ≥ 2, π1 (BSLn) = 1; one identiﬁes π1 (BSLn) = π0(SLn) using a ﬁber sequence.

Example

For any n ≥ 2, the map BGLn → BGL1 coming from det : GLn → GL1 A1 induces an isomorphism π1 (BGLn) = GL1.

Example

A1 π1 (BGL1) = GL1

Aravind Asok (USC) Algebraic vs. topological vector bundles 24 / 27 ∞ the map A \ 0 → BGL1 is a principal GL1-bundle and this yields an A1-ﬁber sequence ∞ since A \ 0 is A1-contractible, the result follows from the long exact sequence in homotopy

Example

A1 A1 For any n ≥ 2, π1 (BSLn) = 1; one identiﬁes π1 (BSLn) = π0(SLn) using a ﬁber sequence.

Example

For any n ≥ 2, the map BGLn → BGL1 coming from det : GLn → GL1 A1 induces an isomorphism π1 (BGLn) = GL1.

Example

A1 π1 (BGL1) = GL1 A1 GL1 is discrete, i.e., π0 (GL1) = GL1: there are no non-constant 1 algebraic maps A → GL1

Aravind Asok (USC) Algebraic vs. topological vector bundles 24 / 27 ∞ since A \ 0 is A1-contractible, the result follows from the long exact sequence in homotopy

Example

A1 A1 For any n ≥ 2, π1 (BSLn) = 1; one identiﬁes π1 (BSLn) = π0(SLn) using a ﬁber sequence.

Example

For any n ≥ 2, the map BGLn → BGL1 coming from det : GLn → GL1 A1 induces an isomorphism π1 (BGLn) = GL1.

Example

Aravind Asok (USC) Algebraic vs. topological vector bundles 24 / 27 Example

A1 A1 For any n ≥ 2, π1 (BSLn) = 1; one identiﬁes π1 (BSLn) = π0(SLn) using a ﬁber sequence.

Example

For any n ≥ 2, the map BGLn → BGL1 coming from det : GLn → GL1 A1 induces an isomorphism π1 (BGLn) = GL1.

Example

A1 π1 (BGL1) = GL1 A1 GL1 is discrete, i.e., π0 (GL1) = GL1: there are no non-constant 1 algebraic maps A → GL1 ∞ the map A \ 0 → BGL1 is a principal GL1-bundle and this yields an A1-ﬁber sequence ∞ since A \ 0 is A1-contractible, the result follows from the long exact sequence in homotopy

Aravind Asok (USC) Algebraic vs. topological vector bundles 24 / 27 Example

For any n ≥ 2, the map BGLn → BGL1 coming from det : GLn → GL1 A1 induces an isomorphism π1 (BGLn) = GL1.

Example

A1 A1 For any n ≥ 2, π1 (BSLn) = 1; one identiﬁes π1 (BSLn) = π0(SLn) using a ﬁber sequence.

Aravind Asok (USC) Algebraic vs. topological vector bundles 24 / 27 Example

Example

A1 A1 For any n ≥ 2, π1 (BSLn) = 1; one identiﬁes π1 (BSLn) = π0(SLn) using a ﬁber sequence.

Example

For any n ≥ 2, the map BGLn → BGL1 coming from det : GLn → GL1 A1 induces an isomorphism π1 (BGLn) = GL1.

Aravind Asok (USC) Algebraic vs. topological vector bundles 24 / 27 M K2 is the second Milnor K-theory sheaf 1 A the map BSLn → BGL∞ induces an isomorphism on π2 (−) for n ≥ 3 and the latter represents Quillen’s algebraic K-theory 1 M A K2 = π1 (SLn), n ≥ 3 and can be thought of as “non-trivial relations among elementary matrices” (classic presentation of Milnor K2) MW K2 is the second Milnor–Witt K-theory sheaf 1 A SL2 = Sp2 and the map BSL2 → BSp∞ is an isomorphism on π2 (−) the latter represents symplectic K-theory and includes information about symplectic forms over our base MW M the map BSp∞ → BGL∞ yields a map K2 → K2 ; this map is an epimorphism of sheaves and its kernel may be described via the “fundamental ideal” in the Witt ring (A. Suslin)

Example (F. Morel) There are isomorphisms

( MW 1 K if n = 2 πA (BSL )−→∼ 2 2 n M K2 if n ≥ 3.

Aravind Asok (USC) Algebraic vs. topological vector bundles 25 / 27 1 A the map BSLn → BGL∞ induces an isomorphism on π2 (−) for n ≥ 3 and the latter represents Quillen’s algebraic K-theory 1 M A K2 = π1 (SLn), n ≥ 3 and can be thought of as “non-trivial relations among elementary matrices” (classic presentation of Milnor K2) MW K2 is the second Milnor–Witt K-theory sheaf 1 A SL2 = Sp2 and the map BSL2 → BSp∞ is an isomorphism on π2 (−) the latter represents symplectic K-theory and includes information about symplectic forms over our base MW M the map BSp∞ → BGL∞ yields a map K2 → K2 ; this map is an epimorphism of sheaves and its kernel may be described via the “fundamental ideal” in the Witt ring (A. Suslin)

Example (F. Morel) There are isomorphisms

( MW 1 K if n = 2 πA (BSL )−→∼ 2 2 n M K2 if n ≥ 3.

M K2 is the second Milnor K-theory sheaf

Aravind Asok (USC) Algebraic vs. topological vector bundles 25 / 27 1 M A K2 = π1 (SLn), n ≥ 3 and can be thought of as “non-trivial relations among elementary matrices” (classic presentation of Milnor K2) MW K2 is the second Milnor–Witt K-theory sheaf 1 A SL2 = Sp2 and the map BSL2 → BSp∞ is an isomorphism on π2 (−) the latter represents symplectic K-theory and includes information about symplectic forms over our base MW M the map BSp∞ → BGL∞ yields a map K2 → K2 ; this map is an epimorphism of sheaves and its kernel may be described via the “fundamental ideal” in the Witt ring (A. Suslin)

Example (F. Morel) There are isomorphisms

( MW 1 K if n = 2 πA (BSL )−→∼ 2 2 n M K2 if n ≥ 3.

M K2 is the second Milnor K-theory sheaf 1 A the map BSLn → BGL∞ induces an isomorphism on π2 (−) for n ≥ 3 and the latter represents Quillen’s algebraic K-theory

Aravind Asok (USC) Algebraic vs. topological vector bundles 25 / 27 MW K2 is the second Milnor–Witt K-theory sheaf 1 A SL2 = Sp2 and the map BSL2 → BSp∞ is an isomorphism on π2 (−) the latter represents symplectic K-theory and includes information about symplectic forms over our base MW M the map BSp∞ → BGL∞ yields a map K2 → K2 ; this map is an epimorphism of sheaves and its kernel may be described via the “fundamental ideal” in the Witt ring (A. Suslin)

Example (F. Morel) There are isomorphisms

( MW 1 K if n = 2 πA (BSL )−→∼ 2 2 n M K2 if n ≥ 3.

Aravind Asok (USC) Algebraic vs. topological vector bundles 25 / 27 1 A SL2 = Sp2 and the map BSL2 → BSp∞ is an isomorphism on π2 (−) the latter represents symplectic K-theory and includes information about symplectic forms over our base MW M the map BSp∞ → BGL∞ yields a map K2 → K2 ; this map is an epimorphism of sheaves and its kernel may be described via the “fundamental ideal” in the Witt ring (A. Suslin)

Example (F. Morel) There are isomorphisms

( MW 1 K if n = 2 πA (BSL )−→∼ 2 2 n M K2 if n ≥ 3.

Aravind Asok (USC) Algebraic vs. topological vector bundles 25 / 27 the latter represents symplectic K-theory and includes information about symplectic forms over our base MW M the map BSp∞ → BGL∞ yields a map K2 → K2 ; this map is an epimorphism of sheaves and its kernel may be described via the “fundamental ideal” in the Witt ring (A. Suslin)

Example (F. Morel) There are isomorphisms

( MW 1 K if n = 2 πA (BSL )−→∼ 2 2 n M K2 if n ≥ 3.

Aravind Asok (USC) Algebraic vs. topological vector bundles 25 / 27 MW M the map BSp∞ → BGL∞ yields a map K2 → K2 ; this map is an epimorphism of sheaves and its kernel may be described via the “fundamental ideal” in the Witt ring (A. Suslin)

Example (F. Morel) There are isomorphisms

( MW 1 K if n = 2 πA (BSL )−→∼ 2 2 n M K2 if n ≥ 3.

M K2 is the second Milnor K-theory sheaf 1 A the map BSLn → BGL∞ induces an isomorphism on π2 (−) for n ≥ 3 and the latter represents Quillen’s algebraic K-theory 1 M A K2 = π1 (SLn), n ≥ 3 and can be thought of as “non-trivial relations among elementary matrices” (classic presentation of Milnor K2) MW K2 is the second Milnor–Witt K-theory sheaf 1 A SL2 = Sp2 and the map BSL2 → BSp∞ is an isomorphism on π2 (−) the latter represents symplectic K-theory and includes information about symplectic forms over our base

Aravind Asok (USC) Algebraic vs. topological vector bundles 25 / 27 Example (F. Morel) There are isomorphisms

( MW 1 K if n = 2 πA (BSL )−→∼ 2 2 n M K2 if n ≥ 3.

Aravind Asok (USC) Algebraic vs. topological vector bundles 25 / 27 Motivic vector bundles on algebraic surfaces

Theorem If k is algebraically closed, and X˜ is the Jouanolou device of a smooth projective surface X, then for r ≥ 2 the map

alg ˜ ˜ 4,2 ˜ (c1, c2): Vr (X) −→ Pic(X) × H (X, Z)

is an isomorphism.

Proof. Obstruction theory! Case of trivial determinant: there is a canonical “Euler class” map MW BSL2 −→ K(K2 , 2); ˜ 2 ˜ MW 2 ˜ M ∼ 4,2 ˜ if X is as in the statement, then H (X, K2 ) → H (X, K2 ) = H (X, Z) is 2 ˜ MW ˜ an isomorphism; any class in H (X, K2 ) lifts uniquely to [X, BSL2]A1 .

Aravind Asok (USC) Algebraic vs. topological vector bundles 26 / 27 Thank you!

Aravind Asok (USC) Algebraic vs. topological vector bundles 27 / 27