K - theory-

Let Vect Cx) =n

Vectncx) denote the

set of iso classes of finite-dimensionalcomplex vector bundles

.

We have the Whitney sun operations

⑦ : Veetncx) xuecfmcx) → vectntmfx)

This gives Uectcx) the structure of

an abelian monoid-

,

with zero element the

cangue) element of oectocx).

This is a monoid because we dont

necessarily have inverses.

Nonetheless there

is a universal wayto turn an abelian

monoid into an abelian group .

Deff ! Let A be an abelian monoid.

A

group completion is an abelian group KCA)abelian .

with a morphism at a monoid i : A- KCA)

such that for cryabelian

groupB,d

a nap of abelian monads f : A→ B,

there exists a unique abelian group

homomorphism f : KCA)-3 B such that the

diagramA Kcal

! ~

;I !f¥4

B

I

commutes .

Exercise. Give a construction of KCA)

EI: A = So ,I,. - - .

3 under additions,then

KCAIE 21.

Rene : If A is acommutative semi - ring ,

then

KCA) is a commutative ring .This applies to Uectcx) , as this is a

commutative semi -ring ,via the ④ -product

of bundles .

Deff : For a topological space x,

kcx) :-. K( Veatch)

By the previousremark

,this is a commutate're

ring .

Example : what is Klar) ? A vector bundle over

a point is entirely determined by the dimerous

of the fiber,

sie,

Veatch) - 991,2, - . -3

By the example ,KCA) = 21

.

them : An explicit construction of Kcx) shows

that an element of Rtd can be represented

by a formal difference EET - CF] of

ionosphere classes of vector bundles.These

are sometimes called virtual vector bundles.

In this case,the isomorphism KCA) E 21

sends

(CET - EF]) E K (&) → din CE) - dim (F) EZ.

them : Let en denote the trivial bundle

of rank is . If IT : E → X is a

vector bundle over a compact Hausdorff

space ,then there always exists an

embedding ( i.e,a morphism of vector bodies which

is a linear isomorphism on each fiber)

E→ Tn for some n EN.

Then we

can take the orthogonal complement Et of

E with respect to ka ,ie

.

there exists

some NEN such that

E@Et A- Tn.

-

As noted above,an

element at Kfc) can

be represented by a virtual vector bundle

EET -.

By the previous remark ,3- NEN & a Ce sua

that F- ⑦ GET's

.

[ET - [ E ] = [E3 t [a] - ( CFT t Ea])

= [ E@ a] - [ en]

⇒ Every element of ECT) is of the form

[ H ] - [en]

Suppose [ E) = EFI Ek Cx) , An expkct constructor

of KC-),shows that there exists a

bundle G such that

E ⑦G A- FOG

het a'be a bundle such that a a' ee"

Thu,

E⑦ a ⑤ a'

EF a⑦ a'

⇒ E T"I f ⑦ Th

EF

Deff : Two vector bundles " are stably isomorphic-

if there exists m ,n EIN such that

E-⑦ e"t F ⑤ em

het Stan Cx) = Vectcx) /-s

where is is the equivalence relation gives

by stably isomorphic bundles.

Rey : If f : X→ Y is a map of spaces

this induces F. Veetly) → Vet Cx),

I

hence a nap fo : KG)→ kcx).

From what we have seen earlier.

this only

depends on the homotopy class of f.

-

Suppose X is pointed ,so that there is a

map *- X.

We obtain a map

E'

. KCH -3 k¥121 .

The reduced K

theory ECT) : = Kerce).

Ree : Ict) consists of elements EET - CES with

din (E) = din Ce) .

Prof : het x be a pointed compact Hausdorff

space .

Then SBM Cx) is a abelian group,

& EG) ? SBU) .

Root : SB an Ct) is abelian monoid under direct

sum,but it also has riveted

,because I - E '

such that

E- ⑤ E' a en

CE)# CEI - [TIME]

Vectcx) kcx)- E'

r EET - CF]I

,

'

~ I -T

f f ,~

,

'

-e,

-

"-

\⇒ui¥,

'

"

←

a.!¥

's [Eds

Because ICT") =D ,so I will factor through

the map E : KCA→ sBuCx).

I will be swjectue because E is.

To

prove e is injective ,we will construct a

left inverse.

The compose

vectcx) → RGA→ EGO)CET- CET - kid

respects the equivalence relate's -s

,

I

hence indeed a nap j :sBmCx)→ ECT)

.

or

we claim joe -_ id.

If EET - [FT E ECT),

then

n

jukes - CEN -

- EET - Cen] - CES -Cei)

because din E '- din F .

.-

in particular,jeered ⇒ e is injector

& hence an isomorphism .

The fundamental product theorem-

So for we have computed Kca) €21 .

Exercise : K ( 8) a- 21 .

A a- diner-al vector bundle over a sphere Sk

is classified by

[ Sk,

Buen)] I TKBOCN)E- The 0h)

a- Csk-1,ah Ce)]

The nap skit→ Glance) I called the

clutclingfncho# of the bundle.

Take 12=2,we want to study bundles

oooo 5 EEP?

We have the caramel bundle H,

{ (e,v) le Eep? vee }- EPIESZ

( gu)- e

This bus clutching function f : s'→ Glace)

= EX

I given by flz) = Z .

Under the equivalence

[ s ',alencar] a Vectis ( SD

Whitney sum of bundles ⇐ Bloch sum of

matures

Tensed product ⇐ Kronecker of

matrices

mo . .÷÷÷÷:3Therefore H H has clutching functions

f : s'

→ GLZCQ)

z-i-sf.IT

#④ H has clutching function

ng : s'→ alzce)

2- 1-3 z2

CH④H) ⑤ Ts has clutching function

g: s

'

- Glace)

2-1- ( Ef g)

claim : f e g gue Ionoptic bundles,

ie

As rate 2 bundles our S2

H ① HE CH H) Tz

Indeed a homotopy is given by

s 't [off→ alack)( Z

,t)

Ie: :X:S::i: c : :n:÷÷:÷÷÷

.

Rene : In k (5) we therefore have

2h = tf t I

= H2 - 2h + I

0=CH - 1)2

We therefore have a well defined group homomorphism

pl : ICH]

IF→ KCSZ)

H- CHI1- Ce

,]

This ( Fundamental product theorem)

tf X is compact Hausdorff,

then the ing

homomorphism

Kcx) ④ 21Gt]CHIP→ KC > +

S2)

x ④ y- Tite)④T5(ply))

where it,

- xx S2 → x

Iz : xx S2 → S2

is an isomorphism .

Prod : very different to what we have seen

is class so far,but if you

want to see

the proof , you can find a nice expositionsin Hatcher 's book on K-theory a vector

bundles.

Cori : Taking X to be a point

KC S2) E 21Gt] ICH - 15

As an abelian group KCSZIE 2h04,but it has

an interesting multiplication .

E

Hf

u

A-→ B-0g

C- ngETI g-IT

-

Bott periodicity-

On Tuesday we introduced a factor

K : h Top- cringe

which is a"

cohomology theory"

for

Uectc- spaces ,

Lennie : het IT : E→ x be a vector

bundle over a compact Hausdorff space ,

& let A C X closed subset such

Ela→ A is trivial.Then there

exists an open neighborhood USA

such that Elo→ U is trivial .

Props : If X is a compact Handoff

space ,A a closed subset ACX

contain a basepont & EA,

then the

based maps

( A. se) ( x.A) E- CHA, AIA)

gives a sequence

rica) K Cx) Ecxla)

which is exact at EG)

Proof : imlq) e ke Cio) ( ioq=o)-

Note that isq = ( qoi5,

but goiu the constant mop to the basepoert , so

ioq - o , a in Cgs) C-Kocis).

Kocis) E imcq)-

Recall I I sBunCx)= Vectcx) 1ns .

Let IT : E→ X be abundle such

that i- = Ela → A is stably trivial .

By adding a trivial bundle if necessary

we can just assume that EIA→ A,

I choose a trivializations

h ? Ela ~→ A X Kh.

het Elh be the quotient of E

under the relations h- ' Gav) - h-1cy.no) for

x. y C- A,

& we get a induced projection

Eth- XIA

Clair : Eth → XIA is a bundle,ie

-

locally trivial. To see this

,observe that

over the set ( x# SCALA) a- XSA

this bundle I identified with

Elma→ XIA

Using the previous lemma it follows we

car fond or open neighborhood USA says

that

Elo- u

is trivial.

Restricted to OIA CNA

we can identify via h

( Eth) Toya a- Cola) x ¢ "

so this bundle is locally trivial -

Finally ,

the square

quotientE- Eth

t. ¥

.x-

q

is a pad- back

,d so E A- 9- (Eth) .

Therefore her cimcq) as regained . D

-

Let us remind ourselves of some constructors

from earlier To the semester .

Det : het f : X-Y be a map ,then

the mapping cyclades in the pushout

x y

iii.→ u! '"

Moreover f factors as

f : X Mf -537

The napping cone of f Cf : - Mft

There is a collapse map

c: Cf- 71ft)

Ree : If X,Y are compact Hausdorff,then

so are Mf & Cf.

hemmed : het x be a compact Hausdorff

space,

A CX acontractible closed

subspace and DEA a basepart . Then

the collapse map x→ xla induces

en isomorphism or I.

Proot : If IT : E- X is a vector bundle,

then the esthetics Ela → A must

be trivial,as A is contractible GUI was

an exercise earlier in the semester).

Choose a trivializationh : EIA ~→ A x en

,

we can then construct a bundle

Eth → XIA

by the construction in the previous proportionwhich satisfies a (Eth) E- E

.

In other

words Icc) is surjective .

We want to show it is injective .We must

show that Eth is independent of the choice

of h. , up to iuomorphym .

Let ho,

h,

be two trivializations, they

differ by postonposition with

hoth ,'- Axe

"

→ Axe"

la,

u)- ca, glasco))

for a map g: A-→ Glace) .

As A

is contractible this map g is homotopi to

the constant trap to the identity matrix .

From such a homotopy we produce a homotopy H

of trurhzatus over A,

a hence a Xriulizatcj

ol Elaexco , IT → Accord

which is ho at one end d h,at the

other . Using the same constantin as in

the previous kung,we construct

⇐ xco.IT ) ht → CHAI -16,1T

which is Elko at one end a Eth,

at the other.

You can check that this

implies that Elko E. Eth , .

⇐ : If f :X → Y is the inclusion of a

closed subspace into a compact Hausdorff

space and * EX a basepont , then the

map c : Cf → TIX induces a isomorphism

on K.

Poot : As f is the inclusion of a closed

subspace,the core on X

( Cx) := xxco.IT¥03

I a closed subspace of Cf,& the map

c is exactly gives by collapsing the

core to a point .

Because ccx) is

contractible we we apply the previous

lemma . D

X '- a compact Hausdorff space

A- CX a closed subspace

* EA a basepont

f : A-X be the inclusion map

A# x Cf#Cj

t.tt '

↳XIA G- IX

Moreover Cftx EEA.

rica) ri G)← Ecxtt)SH

ECCE) # feat

⇒ we get a long exact sequence Ceecept

possibly at the way left)

Ei

ICA) EG) EICH← EGAD ← Elena← . . -

This sequence is exact everywhere exact possibly

at the very left .

Recall that is cohomologyIit ' (ex) a- File)

so we can define

E- icy) : = Elix) for is, o

Ther thy sequence ca be written

E. Ca) EEN EIGHTIETH← E-267 ← EIGHT

C- , -.

-

the external product or reduced K-theory-

we have an external product

kct) ④KG) -3 Kcxxy)A ④ B- ITE (A) ④ Tty (B)

where it : xx Y- X

Ty : xx 't- Y

For based spaces the catenin product is

less useful than the ssmasb product

XAY = Xx Y-

xvy- Xxl 01%3×7

xvy

We want to show that there-

is a added

external product

E. G) ④ Ely)→ riocxny )

To do this we consider the"

excet"

sequence

riofxuy) Eocxxy)EE°CxiD←E4xu'DC- . . .

Lemmy : the inclusions in : x- xvy

iy : y→ xvy

induce an isomorphism

i.⑦ op : E- icxuy)- E- iGd⑤EiG)

proof : Exercise Great week)

is : Eicxxy)- E- icxuy)=EiCxI⑤EiG)

is split surjective , split by TF ⑦IT -5 ,the

projector maps ontoeach factor .

In particular

riocxxy)=I°(xny)⑦R°Cx)⑦ETx)

hold -

- tucked → recon)

If xtkocx) , ytR°( Y) then we have

iTxH④Ty4y) EEOC xxx)

This class vanishes when restated to Sto ) XY

& xx Syd,therefore TECH④ TFG) actually

lies is the summand riocxry).

this defines

the reduced external product

- * - : E°Cx7④E°Cy)→ Rocky).

BottpeiodicutyyThere is a map

c : Ex→ s'nx

given by collapsing 6,17×5×03 C EX.

This is a contractible closed subspace , d so

by the lemma we saw earlier

c- : E. (s'm) EYED.

Recall the fundamental product theorem :

K°( 5) = ICHI ICH - IT

In redued K-theory

E. ( 55=215*13 .

We may thetoe form a map Cthe Bott map)

B : riocx)- TEC a- ECG)

X 1- ( H - IT X

theorem ? The Bout map is a isomorphism for all

compact Hausdorff spaces X.

Root : By the fundamental product theorem

* (5) ④ KG)→ kCs2xx)

is an isomorphism .

Recall that

KCXP a- Ito ECT) ,I so we obtain an isomorphic

Eo ⑤ to ⑦ (Etsy ④ KOCH) Telstra

Comparing this with the decomposer of teocsex)

we obtained earlier :

knock,x)=E%hx)⑦E°Cx) ECT)

shows the the external product map

race)④I°Cx)→E°C5nx)

i ar isomorphism . Under the identification

£062) EZSH - 13 ,this is precisely the

BIA map .

D.

Therefore there is a isomorphism

p : riocx) → E-zcx)

or, replacing X by six , on isomorphism

p: E - icx)- E - itza)

for all i > co .So we only have two groups

ETA & E- Cx).

Deff :

Eigg .

.= grew) ie 't even

E- 7-Cx) i EX odd

Ex)

Coe : Ii ( su) I {21 i= 0 nod 2

O iz I nod 2

Ii ( suit) goi=o nod 2

21 i' I nod 2

Poot : By BoA periodicity we can redee to

5 I s'

,which we have already clones

them : Using Bott periodicity we can extend the

hattexauf sequence we constructed earlier into an

exact se queue continuing ndefrtely to the left.

But by period -city we can write this as a

following six-term exact sequence

ITAI TEG) I Ecua)

tf fBooE- 'Center-1Gt E

-E)