Topological K-Theory: some notes

Simone Camosso

e-mail: [email protected], ucio 2112 Edicio U5, secondo piano

Dipartimento di Matematica ed applicazioni

Università di Milano Bicocca

December 21, 2015

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An Introduction to K-Theory

1.1 Semi-groups and Monoids

Denition 1.1.1 A semi-group is an algebraic structure consisting of a nonempty set S and an associative binary operation ·S in S.

Observation 1.1.2 In the denition we have excluded the empty set that is a semi-group with the empty function as the binary operation.

Observation 1.1.3 The operation · : S × S → S is a binary operation in S that for all a, b, c ∈ S satisfy the associative law a · (b · c) = (a · b) · c.

Observation 1.1.4 A semi-group is nite if it contains only a nite number of elements.

Example 1.1.5 The set of positive integers P with the ordinary operation of addi-tion +.

Example 1.1.6 The set of positive integers P with the ordinary operation of mul-tiplication ·.

Example 1.1.7 The set of subsets of S, P(S) with the union ∪ or intersection ∩. Example 1.1.8 The set Z with ordinary + or ·.

Example 1.1.9 The set P with the operation ◦ dened as a ◦ b = a + b + a · b. Denition 1.1.10 A semigroup with an identity is called a monoid.

Example 1.1.11 Every group is a monoid.

Example 1.1.12 The natural numbers N with addition is a commutative monoid. 1

1.2 K-Theory: the idea

The simple idea of Grothendieck [1] is to use the symmetrization as we dene Z from the semi-group N of natural integers. Let S a semi-group, we associate to S the new group G(S) dened as the quotient of S × S by the equivalence relation: (1.1) (a, b) ∼ (c, d) ⇔ ∃e : a + d + e = b + c + e.

The goup structure is given by the addition +G: G(S) × G(S) → G(S)dened

by (a, b) +G(c, d) = (a + c, b + d) with the zero element (0, 0). We dene a map

from S to G(S) sending a to the class of (a, 0) . Note that in general this map is not injective.

Observation 1.2.1 The same construction can be applied to a monoid M and ex-tended to semi-ring as in [2].

Example 1.2.2 (for a ring) Let R a ring with unit and let S = φ(R) be the set of nitely generated projective right R-modules up to isomorphism. Concretely, such a module M is dened as the image of an R-linear map Q : Rn _{→ R}n _{such that}

Q2 _{= Q. We have that S is a semi-group with the operation law M + N = M ⊕ N}

where M is the isomorphism class of the module M. The group G(S) obtained from S by symmetrization is called the K-Theory of R and denoted by K(R).

Observation 1.2.3 K(R) is a functor: a ring homomorphism f : R1 → R2

induces a group map K(R1) → K(R2). On the level of modules, one associates to

an R1-module M1 the R2-module dened by the formula M2 = M1 ⊗R1 R2. If we

write M1 as the image of Q as above, M2 is the image of the projector operator ˜f (Q)

where ˜f is the associated ring homorphism Sn(R1) → Sn(R2).

In particular, there is a unique map Z → R. The cokernel of the natural associated map K(Z) → K(R) is the reduced K-Theory of R and denoted byK(R)e . If R is commutative then K(R) ∼= Z ⊕ eK(R). We can view K(R)e as the non trivial part of the K-Theory of the ring R.

Example 1.2.4 It is easy verify that K(Z) = Z. Example 1.2.5 Considering R = CC(S

1_{), the ring of complex continous functions}

on the circle S1, we have that K(R) = Z. The geometric interpretation is that any

1.3. K-THEORY FOR COMPLEX VECTOR BUNDLE 3 Example 1.2.6 Considering R = CC(S

0_{), the ring of complex continous functions}

on the 0-circle S0 _{= {−1, +1}, we have that K(R) = Z ⊕ Z. The geometric}

inter-pretation is as before.

Example 1.2.7 We have that K(CC(S

2_{)) = Z ⊕ Z, that is equal as in the before}

example for the Bott periodicity.

1.3 K-Theory for Complex Vector Bundle

We can procede using the same procedure with Vect(X)×Vect(X). Where Vect(X) are complex vector bundles over a space X. It is know that on Vect(X) we can add and multiply two vector bundles. We can formally dene an additive inverse or subtraction making Vect(X) a ring. So given two (E1, F1), (E2, F2)) ∈ Vect(X) ×

Vect(X)we dene the relation:

(1.2) (E1, F1) ∼ (E2, F2) ⇔ ∃G : E1⊕ F2⊕ G ∼= E2⊕ F1⊕ G

This relation is an equivalence relation and the quotient denes the group K(X). Each vector bundle E determines an element [E] ∼= [(E, 0)] of K(X) and since (E, 0)+(0, E) ∼ (0, 0), we have −[E] ∼= [(0, E)]. Thus, every element of K(X) is of the form [E] − [F ]. Moreover, there is a complementary bundle F0 _{such that}

F ⊕ F0 ∼= n, the trivial rank n bundle. Consequently

[E] − [F ] = [E ⊕ F0] − [F ⊕ F0] = [E ⊕ F0] − [n],

and so all elements of K(X) are in fact of the form [E]−[n]. It follows that [E] = [F ] in K(X) if and only if E ⊕ n = F ⊕ n for some n. Bundles with this property are called stably isomorphic. From a map f : Y → X one can pull back bundles from X to Y and so induces a map f∗ : K(X) → K(Y ), which only depends on the homotopy class of f.

Observation 1.3.1 Note that in this case K(·) is a contravariant functor from the category of compact topological spaces to the category of abelian groups. In general is not always contravariant, for example in the previous section K(·) was a covariantly functor from the category of rings to the categories of nitely generated projective modules.

Observation 1.3.2 The tensor product between vector bundles on X induces a mul-tiplication in K(X), which makes K(X) a ring.

Also in this case we have a reduced K-Theory. Consider x0 ∈ X, the inclusion

i : {x0} → X induce a map i∗ : K(X) → K(x0) ∼= Z. We dene eK(X) to be the

kernel of i∗_{. The elements of}

e

K(X), if X is connected, are of the form [E]−[rank(E)] and [E] = [F ] in K(x)e if and only if E ⊕ m ∼= F ⊕ n for some m and n. These bundles are called stably equivalent.

Observation 1.3.3 In fact K(X)e is K(X) modulo trivial bundles. The collapsing map c : X → {x0} induces an splitting and

K(X) ∼= eK(X) ⊕ K({x0}) ∼= eK(X) ⊕ Z.

Now we introduce this new notation: K0 for the covariant K-Theory and K0

for the contravariant K-Theory. Then K0_(X) is a ring and K

0(X) is a K0(X)

-module. These are formally analogous to cohomology and homology respectively.

1.4 The classical Bott periodicity Theorem

Theorem 1.4.1 (Periodicity) Let X a compact Hausdor space, then there is an explicit isomorphism between:

(1.3) K(X) ⊗ K(S2) ∼= K(X × S2).

The principal concept used in the proof is the concept of clutching function ϕ for the union of two vector bundle E1, E2 respectively of X1, X2 compact and

Hausdor such that X = X1∪ X2 (see [6]). The vector bundle obtained using this

construction with clutching function is unique (up to isomorphism). Given a vector bundle E over X, we may always view E as obtained by clutching with the identity map 1 on restriction over A = X1∩ X2. To have an idea we can examine the case for

S2. We see S2 _{= S}2

+∪ S−2, upper and lower hemispheres with intersection S1. Now

we consider a vector bundle E (with rank n) over S2 _{and we write S}2

+×Cn= E+and

S2

−× Cn = E−. If f is a clutching function for E we have E = E+∪f E−. As an

example we can ask how is the clutching function for the canonical line bundle H over S2

= CP1. In this case S+2 = {[1 : z−1] : |z−1| ≥ 1} and S−2 = {[z : 1] : |z| ≤ 1}.

We pass from (z, 1) to (1, z−1_{) multiplying by z so f : S}1 _{→ GL}

1(C) is f (z) = z.

We note that if in the same example we change the order of S2

+ = {[z : 1] : |z| ≤ 1}

and S2

1.4. THE CLASSICAL BOTT PERIODICITY THEOREM 5 (E+∪zE−) ⊕ (E+∪z−1 E₋) = H ⊕ H−1 ∼= 1. Here 1 is the one dimensional trivial

bundle. In analogue way we have for Hn _{the clutching function f(z) = z}n_{. Now}

we can use an important fact to prove another important relation useful in the next step. The important fact is the follow: if f : S1 _{→ GL}

n(C) and g : S1 → GLn(C)

are clutching homotopic functions , then

(E+∪f E−) ∼= (E+∪gE−).

We can apply this simple statement do f(z) = z₀2 0₁ 

, clutching function for (H ⊗H)⊕1 and g(z) = z 0_{0 z}



clutching function for H ⊕H. The homotopy is given by: F (z, t) = z 0 0 1   cos (πt/2) − sin (πt/2) sin (πt/2) cos (πt/2)   z 0 0 1   cos (πt/2) sin (πt/2) − sin (πt/2) cos (πt/2)  .

So we have (H ⊗H)⊕1 ∼= H ⊕ H that we can rewrite as (H −1)2 = 0. We can procede dening the ring homomorphism µ : Z[H]/(H −1)2 _{→ K(S}2₎. Where Z[H]

is the ideal generated by H and the map is given by a + bH 7→ a1 + bH. Note that Z[H]/(H − 1)2 _{= {a + bH : a, b ∈ Z} ∼}

= Z ⊕ Z. It is possible to prove that the map µ is an isomorphism of rings and so, as abelian groups K(S2_{) ∼}

= Z ⊕ Z. The strategy of the proof is to construct an inverse ν of µ such that νµ and µν are the identity map. Before to dene the inverse map we must have other results about clutching functions, for example we want to reduce the function f to a simple function, as polynomial functions. So we consider clutching functions as Laurent polynomials with coecients in GLn(C). After we can procede with an operation of linearization

of this functions as in the following example.

Example 1.4.2 We take p(z) = z2 _{clutching functions on E on S}2_{. The matrix:}

L3(p) =   0 0 1 −z 1 0 0 −z 1  =   0 0 0 −1 0 0 0 −1 0  z +   0 0 1 0 1 0 0 0 1  

is a linear clutching functions for 3E.

We can generalize the example for a general polynomial clutching function A0+ A1z + · · · + Anzn with

Ln+1(p) =        A0 A1 A2 · · · An−1 An −z 1 0 · · · 0 0 0 −z 1 · · · 0 0 ... ... ... ... ... 0 0 0 · · · −z 1        .

Then we have that if p is a polynomial clutching function of degree n and E a vector bundle over S2_{, then}

E+∪Ln+1_(p)E₋ ∼= E ⊕ n1.

Observation 1.4.3 We observe that if p is a polynomial clutching function of degree n and E a vector bundle over S2, then zp is also a polynomial clutching

function of degree n + 1 and E and

E+∪Ln+2_(zp)E₋ ∼= H ⊕ (n + 1)1.

Now using analysis of clutching functions, that is decomposing vector bundles using the complex projector operators R = 1

2πi R S1A(Az + B) −1_{dz, Q =} 1 2πi R S1(Az +

B)−1Adz in this way:

E = Q(E) ⊕ (1 − Q)(E) E = R(E) ⊕ (1 − R)(E),

we denote by p+(z) : Q(E) → R(E)and p− : (1 − Q)(E) → (1 − R)(E)restrictions

that are well dened for the properties of Q, R. If P is a linear clutching function it is possible to write p+ = A+z + B+ and p− = A−z + B− for the decomposition of

P. If fact this general result holds.

Proposition 1.4.4 Let P a linear clutching function and pt _{= p}t

+ + pt− where

pt

+ = A+z + tB+ and pt− = tA−z + B− for t ∈ [0, 1], then pt is an homotopy from

A+z + B− to p and E ∼= kH ⊕ (n − k)1 where rank(A+) = k and n is the order of

A+_.

Example 1.4.5 In the before example we have that E ∼= 2H ⊕ 1. To proceed and complete the proof we need the following lemma.

1.4. THE CLASSICAL BOTT PERIODICITY THEOREM 7 Lemma 1.4.6 Let E a vector bundle over S2 _{then in K(S}2_):

E = kH−n+ (2n + 1)mH−n− 2nH−n

for some integers k, m, n.

The proof is very simple because every clutching function on E over S2 is

homotopic to z−n_{p, where p is a polynomial clutching function of degree ≤ 2n.}

We have:

E = E+∪z−n_pE₋ = E₊∪_z−nE₋⊗ E₊∪_pE₋= H−n⊗ [E₊∪_L2n+1_(p)E₋− 2n1] =

= H−n⊗ [kH + ((2n + 1)m − k)1 − 2n1] = kH1−n+ ((2n + 1)m − k)H−n− 2nH−n, where rank(L2n+1_(p)+₎and m is the order of the matrix p(z). This say that K(S2₎

is generated by 1 and H. By using H2 _{= 2H − 1in K(S}2_{)we get H}3 _{= 2H}2_{− H =}

2(2H − 1) − H = 4H − 2 − H = 3H − 2and H4 _{= 4H − 3so ...H}n _{= nH − (n − 1)}

for n ≥ 0 by induction on n. We can dene ν(Hn_{) = nH − (n − 1)} and ν(1) = 1.

We extend linearly to be a ring homomorphism. Finally is simple to prove that νµ and µν are the identity map and we have proved the Bott Theorem.

Example 1.4.7 If X = Sn we have e K(S2n+1_{) = 0} and e K(S2n_{) = Z or for} odd-dimensional spheres K(S2n+1 ) = Z and even-dimensional K(S2n) = Z ⊕ Z.

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Calculations of K-groups

We rst note some facts. If X is a contractible Hausdor space then K0_{(X) ∼}

= Z (because homotopic invariant se [4]). Let x ∈ X and 0 ≤ t ≤ 1 we consider the homotopy map φt(x) = (1 − t)xbetween the identity map on [0, 1] and the map 0.

Thus K0_{([0, 1]) ∼}

= Z. Furthermore we consider the following split theorem:

Theorem 2.0.8 (split) Suppose A is a closed subspace of a locally compact Haus-dor space X and let i : A → X the inclusion, then there exist an exact sequence:

K(X \ A) → K(X) → K(A),

furthermore if j : X → A is a map continuous at innity and has the property that ji is the identity on A, then there exists a split exact sequence:

0 → K(X \ A) → K(X) i

∗

→ K(A) → 0, whence K(X) is isomorphic to K(A) ⊕ K(X \ A).

2.1 K-Theory of R

K0_(Rn) ∼= K0({p} × Rn) ∼= 

Z n even 0 n odd because the n sphere is the one point compactication of Rn_.

2.2 K-Theory of of the gure eight

Denition 2.2.1 The wedge of pointed spaces (X, x0) and (Y, y0) is the pointed

space:

(2.1) X ∨ Y = (X × {y0}) ∪ ({x0} × Y ) ⊆ X × Y,

endowed with the subspace topology and with (x0, y0) a basepoint.

The gure eight is the space S1 _{∨ S}1_{. We consider the map that identify two}

copies of S1 to a single one S1. The topological space (S1_{∨ S}1_{) \ S}1is homeomorphic

to R. Then

K0(S1∨ S1) ∼= K0(S1) ⊕ K0_{(R) ∼}= Z.

2.3 K-Theory of of the Torus

For an inclusion map i from S1 _{× {1} ∼}

= S1 into S1 × S1 _{and j the projection of}

S1× S1 _{onto S}1_{× {1}. Then ji is the identity. The complement of S}1_{× {1} in the}

torus is homeomorphic to S1_{× R, then}

K0(S1× S1_{) ∼}

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The Grothendieck-Riemann-Roch

Theorem

3.1 The Chern and Todd characters

Denition 3.1.1 (Chow ring) Let X a non singular projective algebraic variety, the i-Chow ring CHi_{(X) of X is the free abelian group generated by irreducible}

closed subvarieties of X of codimension i modulo rational equivalence. The general Chow ring CH(X) of Xis the direct sum LdimX

i=0 CH

i_(X).

Example 3.1.2 CH0_{(X) = n · [X] or CH}1_{(X) = P ic(X).}

The multiplicative structure is induced by the intersection of subvarieties [X0_{] ·}

[X00] = [X0∩ X00_{] ∈ CH}i+j_{(X). If E is a vector bundle of rank k we assign its Chern}

classes

(3.1) ci(E) ∈ CHi(X) i = 0, . . . , k

Example 3.1.3 c0(E) = [X], c1(E) = c1(∧kE), where c1(∧kE) is the chern class

of line bundles dened as the corrisponding divisor class.

Given a proper map π : X → Y , there is a map π∗ : CH(X) → CH(Y ). If X0

is a subvariety of dimension k then the dimension of π(X0_{) is ≤ k, if is strictly less}

then k we set π∗([X0]) = 0and if is k we set π∗([X0]) = nX0·[π(X0)]( is the closure).

It is clear that π∗ : CHdimX(X) → CHdimX−dimY(Y ). Note that we have another

map π! : K(X) → K(Y )from the K-groups dened as π!([E]) =

P

i(−1)i[Riπ∗(E)]

(the fact that is well dened is not trivial!).

We have two importants maps Ch, T odd : K(X) → CH(X) ⊗ Q dened in terms of chern classes.

Denition 3.1.4 (Chern character) Ch(E) = rankE + c1(E) +

1 2(c1(E) 2_{− 2c} 2(E)) + . . . = k Y i=1 eαi_,

where αi are the formal roots of c(E) = 1+c1(E) + . . . + ck(E)and c(E) = Qki=1(1 +

αi)(note: k is the rank of E). Ch(E) is a ring homomorphism that is: Ch(E ⊗F ) =

Ch(E) · Ch(F ) and Ch(E ⊕ F ) = Ch(E) + Ch(F ). Denition 3.1.5 (Todd character)

T odd(E) = k Y i=1 αi 1 − eαi = 1 + 1 2c1(E) + 1 12(c1(E) 2 + c2(E)) + . . . ,

and T odd(E ⊗ F ) = T odd(E) · T odd(F ).

Theorem 3.1.6 (Grothendieck-Riemann-Roch) Let X, Y non singular projec-tive varieties and π : X → Y a proper algebraic map, then for a locally free coherent sheaf E on X we have:

(3.2) Ch(π!(E)) · T odd(Y ) = π∗(Ch(E) · T odd(X)).

3.2 Examples

Example 3.2.1 We consider a family of nonsingular curves on a nonsingular base variety π : X → B and L a line bundle.

Ch(π!(L)) = π∗  c1(L) + 1 2c1(L) 2 + . . .  ·  1 + 1 2c1(ω ∨ ) + 1 12c1(ω ∨ )2+ . . .  ,

in general we have c1(E∨) = −c1(E) then c1(ω∨) = −c1(ω), c1(ω∨)2 = c1(ω)2.

Mul-tiplying rank(π!L)+c1(π!L)+. . . = π∗  1 + c1(L) − 1 2c1(ω) + 1 2c1(L) 2 + 1 12c1(ω) 2₋ 1 2c1(L)c1(ω) + . . .  ,

3.2. EXAMPLES 13 c1 π∗(L) − R1π∗L
= π∗  1 2c1(L) 2 + 1 12c1(ω) 2₋ 1 2c1(L)c1(ω)  , if we take L = ω, we have R1_π ∗ω = OB and c1(R1π∗ω) = c1(OB) = 0 so c1(π∗ω) = c1(∧gπ∗ω) = λ1 = 1 12π∗ c1(ω) 2
_. If we take L = ωn then λn= c1(π∗ωn) = π∗  1 2c1(ω n₎2₊ 1 12c1(ω) 2 ₋1 2c1(ω n_)c 1(ω) + . . .  , = 1 12(6n 2_{− 6n + 1)π} ∗(c1(ω)2),

comparing λ1 and λn we obtain:

λn= (6n2− 6n + 1)λ1,

and in terms of line bundle we obtain the Mumford isomorphism: λn ∼= λ(6n

2_−6n+1)

1 .

Example 3.2.2 Considering the constant map π : X → p we have Ri_π

∗E(p) =

Hi_{(X, E)}. We have CH(p) = Z[p] and π

∗ : CHn→ CH0(p). If [D] ∈ CHn(X)then

D = P

p∈Xnp[p] where deg(D) = Pp∈Xnp and we nd the

Hirzebruch-Riemann-Roch Theorem

Bibliography

[1] Borel A., Serre J.-P., Le théorème de Riemann-Roch (d'après Grothendieck), Bull. Soc. Math. France 86, 97-136 (1958).

[2] Varvara Karpova, Complex Topological K-Theory, Semester Project, EPFL Spring 2009.

[3] N.Jacobson, Lectures in Abstract Algebra I. Basic Concepts , Springer-Verlag, New York Heidelberg Berlin.

[4] Efton Park, Complex Topological K-Theory, Cambridge Studies in ad-vanced mathematics.

[5] Martin Schlichenmaier, An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces, Springer (Second Edition).

[6] David Manuel Murrugarra Tomairo, Bott Periodicity, Master Thesis, Faculty of the Virginia Polytechnic Institute.