Topological K -theory and applications

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Topological K -theory and applications

Geoffrey Powell

CNRS and LAREMA, Angers

Masterclass, Strasbourg, February 2015

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Preliminaries

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Z /2-gradings

Definition (Z/2-grading)

AZ/2-graded vector space is a pair of vector spaces (V₀,V₁).

A morphism of graded vector spaces is a pair of morphisms{f_n:Vn→Wn|n∈Z/2}.

Definition (graded tensor product) M⊗N is defined byˆ

(M⊗N)ˆ ⁰ := (M⁰⊗N⁰)⊕(M¹⊗N¹) (M⊗N)ˆ ¹ := (M⁰⊗N¹)⊕(M¹⊗N⁰).

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Koszul signs

Definition (Transposition)

τ_M,N is the isomorphism ofZ/2-graded vector spaces:

M⊗Nˆ ^τ→^M,N N⊗Mˆ

m⊗n 7→ (−1)^|m||n|n⊗m.

The underlying structure onZ/2-graded vector spaces is encoded by:

Proposition

(V_Z_/2,⊗,ˆ K, τ)is a symmetric monoidal structure onV_Z_/2.

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Flip

Proposition

∃involution

V_Z/2 → V_Z/2 (M₀,M₁) 7→ (M₁,M₀).

Remark

The involution identifies with

−⊗Σˆ K:V_Z/2→ V_Z/2

where(ΣK)₁=Kand(ΣK)₀=0.

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Tensor algebra

Definition (Tensor algebra)

T(V)is the free unital, associative algebra on V : T(V) :=M

n∈N

V^⊗n.

Remark (Universal property)

V_{_} ^ϕ ^//

A

T(V)

∃! ˜ϕ

==

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Graded algebras

Definition (Z/2-graded algebra)

A unitalZ/2-gradedK-algebra is aZ/2-graded vector space A with

η :K → A

µ:A⊗Aˆ → A + associativity and unit axioms.

Proposition

For A,B twoZ/2-graded algebras, A⊗B has the structure of aˆ Z/2-graded algebra with product

(a⊗b)(a⁰⊗b⁰) = (−1)^|b||a⁰^|aa⁰⊗bb⁰.

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Examples: tensor and exterior algebras

Example

T(V)has the structure of aZ/2-graded algebra.

Definition (Exterior algebra)

The Grassmann (or exterior) algebraΛ(V)is theZ/2-graded algebra:

Λ(V) :=T(V)/hx⊗y +y ⊗xi, withZ/2-grading inherited from T(V).

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Examples: C and H

Example

Cis aZ/2-gradedR-algebra, withC0=h1iandC1=hii.

Example (Quaternions) H: underlyingR-vector space:

H=h1,i,j,ki where i²=j²=k²=ijk =−1.

C⇒H(i 7→i and i 7→j) induce C⊗ˆC

∼=

→H.

Here|i|=1=|j|but|k|=0.

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Grothendieck groups of small abelian categories

Definition (Grothendieck group)

The Grothendieck group G₀(A)of a small abelian categoryA is the free abelian group on the set of isomorphism classes of objects (with finite composition series) modulo the relation:

[E] = [M] + [N]if there exists a short exact sequence 0→M →E →N →0.

Proposition

G₀(A)is the free abelian group on the set of isomorphism classes of simple objects ofA.

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Clifford modules and Bott periodicity

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Quadratic spaces

Definition (Quadratic forms)

A quadratic form on V is a map q:V →Ksuch that

1 q(ax) =a²q(x),∀a∈K, x ∈V

2 q(x+y)−q(x)−q(y)is symmetric bilinear in x,y ∈V . Remark (Relationship with bilinear forms)

[char(K)6=2]⇒q(x) =b(x,x)for b(x,y) := 1

2{q(x+y)−q(x)−q(y)}

Definition (Morphisms and orthogonal groups)

A morphism of quadratic spaces(V,q)→(V⁰,q⁰)is aK-linear map f :V →V⁰ such that q⁰(f(v)) =q(v).

Orthogonal group: O(V,q) :=Aut(V,q).

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Clifford algebras

Definition (Clifford algebra)

The Clifford algebra C(V,q)of(V,q)is

C(V,q) :=T(V)/hv^⊗2+q(v)i.

Example

C(V,0) =T(V)/hv^⊗2i ∼= Λ(V).

In particular,dimΛ(V) =2^dim^V. Remark (Anticommutativity)

More generally, if x,y ∈V are orthogonal xy =−yx in C(V,q).

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Universal property of Clifford algebras

∃linearι_(V_,q):V ,→T(V)C(V,q).

Proposition (Universal property)

For(V,q)and A an associative, unital algebra, f :V →A,

∃˜f :C(V,q)→A morphism of algebras V ^f ^//

ι

A

C(V,q)

˜f

;;

⇔f(v)f(v) =−q(v)1_A in A.

Corollary

1 ϕ: (V,q)→(V⁰,q⁰)induces C(ϕ) :C(V,q)→C(V⁰,q⁰);

2 O(V,q)acts on C(V,q).

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Z /2-grading on C(V , q)

Example

−1∈Z/2⊂O(V,q)inducesα:C(V,q)→^∼⁼ C(V,q).

Corollary (Z/2-grading) C(V,q)is naturallyZ/2-graded:

C⁰(V,q) := image{T^even(V)→C(V,q)}

C¹(V,q) := image{T^odd(V)→C(V,q)}.

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Orthogonal sum and ⊗ ˆ

Notation (Orthogonal sum)

(V,q)⊥(V⁰,q⁰)the orthogonal sum of(V,q),(V⁰,q⁰).

Proposition (Orthogonal sum and⊗)ˆ

∃natural isomorphism ofZ/2-graded algebras C((V,q)⊥(V⁰,q⁰))∼=C(V,q) ˆ⊗C(V⁰,q⁰)

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Introducing C

_n

Notation n∈N+

C_n := C(Rⁿ,q₊) C_n⁰ := C(Rⁿ,q−), where q+is the positive-definite form (resp. q−

negative-definite).

Example C₁∼=C C₁⁰ ∼=R⊕R. Remark

C₁⁰ =h1,ei, e²=1. Basis giving decomposition: ¹₂(1±e)

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Ungraded tensor products. . .

Proposition

1 C(V,q)→^∼⁼ C⁰

(V,q)⊥(R,q₊)

induced by v 7→ve;

2 C_k ∼=C_k+1⁰ ;

3 C_k+2→^∼⁼ C_k⁰ ⊗C₂;

4 C_k+2⁰ →^∼⁼ C_k⊗C₂⁰,

induced by v7→v ⊗e₁e₂for v ∈R^k,{e₁,e₂}an orthonormal basis of(R²,q±).

Corollary

C₄ ∼= C₄⁰ C_k+4 ∼= C_k⊗C4

C_k+4⁰ ∼= C_k⁰ ⊗C₄ C₈ ∼= C₄⊗C₄.

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Quaternions and C

₂

Example (Quaternions and⊗)ˆ C₂∼=H; since

H∼=C⊗ˆC, andC∼=C₁.

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Quaternions and base-change

Notation (Matrix algebras)

Mn(A)the algebra of n×n-matrices with coefficients in A.

Lemma

∃isomorphisms of algebras:

1 C₂⁰ ∼=M₂(R);

2 C⊗_RC∼=C⊕C;

3 C⊗_RH∼=M₂(C)∼=End_C(H);

4 H⊗_RH∼=M₄(R)∼=End_R(H).

C⊗_RH

∼= //

_

End_C_{_}(H)

H⊗_RH _∼₌ ^//End_R(H), vertical arrows induced byC,→HandR,→C.

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Ingredients

1 e₁:=

0 1 1 0

ande₂:=

1 0 0 −1

∈M₂(R).

e₁²=1=e²₂ande₁e₂=−e₂e₁.

2 C⊗_RC∼=C⊕Cby elementary Galois theory.

3 ∃morphism ofR-algebras:

H→End_C(H)

viah7→(x 7→x h). This extends to a morphism of C-algebras:

C⊗_RH

∼=

→End_C(H).

4 Similar.

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Identifying C

_n

and C

_n⁰

Proposition (The ungraded Clifford algebrasCn,C_n⁰)

k C_k C_k⁰

0 R R

1 C R⊕R

2 H M₂(R)

3 H⊕H M₂(C) 4 M₂(H) M₂(H)

5 M₄(C) M₂(H)⊕M₂(H) 6 M₈(R) M₄(H)

7 M₈(R)⊕M₈(R) M₈(C) 8 M₁₆(R) M₁₆(R) and

C_k+8 ∼= C_k ⊗M₁₆(R)∼=M₁₆(C_k) C_k+8⁰ ∼= C_k⁰ ⊗M₁₆(R)∼=M₁₆(C_k⁰).

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Clifford modules

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Clifford modules

Graded and ungraded

Definition (Categories of Clifford modules) For k∈N, define the abelian categories:

1 M_k, the category ofZ/2-graded C_k-modules;

2 N_k, the category of (ungraded) C_k⁰-modules.

Remark (D ´ecalage)

For k>0, the underlying algebra of C_k−1is isomorphic to C_k⁰. Hence

N_k ∼=ungradedC_k−1−modules.

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Graded to ungraded

Lemma

For k>0and M aZ/2-graded C_k-module,

1 M⁰and M¹are C_k⁰-modules;

2 multiplication by v ∈R^k\{0}induces an isomorphism of C_k⁰-modules M⁰→^∼⁼ M¹.

Proposition (Equivalence graded/ ungraded) For k>0,

(−)⁰:M_k → N_k M 7→ M⁰ induces an equivalence of categoriesM_k ∼=N_k.

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Morita equivalence

Algebraic Bott periodicity

Lemma

C_k₊₈∼=C_k ⊗M₁₆(R)∼=M₁₆(C_k).

Proposition (Morita equivalence)

For A an associative algebra and0<n∈N, the categories of modules over the rings A and M_n(A)are equivalent as abelian categories.

Corollary (Algebraic Bott Periodicity)

For k>0,N_k andN_k+8are equivalent as abelian categories.

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Grothendieck groups of Clifford modules

Notation (Grothendieck groups) M_k :=G₀(M_k)

N_k :=G₀(N_k).

Corollary (Algebraic Bott periodicity forG0) For k>0,

1 (−)⁰:M_k →N_k induces M_k →^∼⁼ N_k;

2 M_k is equivalent to the Grothendieck group of the category of ungraded C_k−1-modules;

3 M_k ∼=M_k+8.

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Restriction of Clifford modules

Proposition (Restriction) R^k ,→R^k ⊥R∼=R^k+1induces

C_k ,→C_k+1

and the restriction functorM_k+1→M_k induces M_k+1→M_k.

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Identifying C

_k

→ C

_k₊₁

Lemma (Identifying restrictions)

C₀ ^//C₁ ^//C₂ ^//C₃ ^//C₄ ^//C₅ ^//C₆ ^//C₇ ^//C₈

R ^//C ^//H

diag//H^⊕2 ^//M₂H ^//M₄C ^//M₈R

diag//M₈R^⊕2 ^//M₁₆R

1 H⊕H,→M₂(H)is the inclusion as diagonal matrices

2 M₂(H),→M₄(C)is induced byH,→M₂(C)via C⊗_RH∼=M₂(C)

3 M₄(C),→M₈(R)is induced byC,→M₂(R).

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Sketch proof

Inclusions of Clifford algebras

The inclusionsC_k ,→C_k₊₁andC_k⁰ ,→C_k+1⁰ identify for 0≤k ≤1 as:

C₀ ^//C₁ ^//C₂ C₀⁰ ^//C₁⁰ ^//C₂⁰

R ^//C ^//H R ^//R⊕R ^//M₂(R) whereR,→R⊕Ris the diagonal embedding and

R⊕R,→M₂(R)the inclusion of diagonal matrices.

Extends using⊗(ungraded).

Need bothC_k andC_k⁰.

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Simple algebras

Remark

M_n(F), forF∈ {R,C,H}aresimple algebras.

In particular∃unique isomorphism class of irreducible modules, the natural representationFⁿ.

Hencecan identify M_k and M_k →M_k−1.

M_k is considered (fork >0) as the Grothendieck group of the category ofungradedC_k−1-modules.

Notation (Cokernel) For0<k ∈N,

A_k :=Coker{M_k+1→M_k}.

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The groups A

_k

Theorem

k 1 2 3 4 5 6 7 8

C_k C H H⊕H M₂H M₄C M₈R M₈R⊕M₈R M₁₆R

M_k Z Z Z Z⊕Z Z Z Z Z⊕Z

A_k Z/2 Z/2 0 Z 0 0 0 Z

For k>8, M_k ∼=Mk−8and A_k ∼=A_k−8. Remark

By convention, M₀=Z=A₀.

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Multiplicative structures

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Using ⊗ ˆ

Recall thatC_k+l →^∼⁼ C_k⊗Cˆ _l. Lemma

For k,l ∈N,⊗ˆ induces

⊗ˆ :M_k×M_l →M_k+l.

hence

M_k ⊗M_l →M_k_+l.

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The algebra L A

_k

Theorem (An Atiyah-Bott-Shapiro theorem)

1 L

k∈NM_k has the structure of anN-graded associative algebra.

2 This passes toL

k∈NA_k and this structure is (graded) commutative.

3

M

k∈N

A_k ∼=Z[η,a,b]/(2η, η³, ηa,a²−4b) where|η|=1,|a|=4and|b|=8.

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Closing remarks

Remark

1 Thisalgebraicpicture is intimately related with K -theory forR-vector bundles. This will be explained tomorrow. . .

2 There is asimplerpicture overC. These are related by Galois descent.

3 The above foreshadows the relationship between KU and KO. . .