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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)
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