neg - university of notre damepmnev/f19/cobordism.pdf · 2019. 11. 20. · fact hqis an iso pf thor...
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n neg Iddithabelian grain
ringstructures x R x In Rien MxN MXNgraded commutative
PMI gz is an Fa algebraMIM dcmxl o.is 10 3
Lee EMT 103G 7 fwm's.t M DW
a neckbunTalk 2I def The i th Pontryagin class of Z is duetoanggato
Picz CD Ca z Q e 4 M Q Ca CzqQ is 2 torsionvanishes
rationally 7h dinM
H't so n Q P pk YC k17EulerclassorientedGrassmanian
Bsoca Bso Cnn1120 IR'sshiftby1ceordnater CBso 7 Shifa Spence
p
BSO I Bso n triobundled
def A characteristic class u is lsteble if wtf E w zFact H Bso stabledeer classes Ex wi Ci piIQ Non example if Eulerclassp Pa T
Def Let ve HTM v TMI C QQ
orientationclasstangentbundle
Let u a charclass The assoc characteristicnumber is wCtm M EmEx Pontryagin s pi pIk 1 can be nonzeroonly if
dem M divisiblebyGThu wall
I rented mfld M U elf all Pontryagin and StiefelWhitney numbers vanish
The Thon Pontryagin s f k KENT o
TheCThon SU o M W as unonated nflds
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thin stability i p Tw p TM
m Hn M0 Hu W
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pent I f PEUT
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PEM fi p Ew Tw Also workswitho 0 Ratyaga SU
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Constructed Let S N a v b Call SphCz _fiberwisept compactificationThe the Tng SphG addedpts at s
H Tz og HEGTrel to thept compactlysupported inat a of director
Gn Let ME IRN w normal bundle 0Then we have a map SN Tv
Neca u
everythingelse as
so have a nap v ta k
SN To Ty classifying bridle over BSON k
2
Feet Ns le this is i variant of e beddingup tohomotopycc cobordism
class
def A prespectney is a sequence ofspace E whaps
En Ente
Exe MSOle Truru E NeeGk t tBSock BSockei
T he C True112
I Tore
def Inc Ex line TurtleCEL
HnCE leg HuwCEH
Pontryagin Thon
Cons 2 gives a map 2 or Tn Nso
In fact this is an 2 4
tf transversality
Proof ofThemthan Hurewicz
LHhcbso If Q Hasso Inconso Q HYDSO HinoEM I
1 I doneTm Em
HYDSO HassoL 2,7
Fact thisannulesFact t is an iso
Pf Then iso inductionFact HQ is an IsoPf Thor Rat Hurwicz X s ply connected
it CX fniteher 1EnT X Q Hi X Q is an iso is 2n
Assumey Pontryagin 4 0 Then WENT _o t wCH BSOW look o 2 Em o t w
look o L or o
EM 1 0
EM is torsion0
Sideelfect Q E thCBSO Ep Pz 3
Fact Eep
Talks Multiplicativegenera
def a multiplicativegenus is a ragmap N
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EI r Ifa M 74cm med2
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y carom y only cm
y DM o med 2
EI If M oriented F non deg symm.bi.lu Son HK or 42givenby Poincare duality
G M signature of thispositiveeigenmly negative ear
this is a multiplicative genes
def let A be a comm graded ringAte It a are I a c A J group
et kn a seq of degree_n honey polynomials in n venashes Xiwhere lXil i
IGiven such a c A white Kca It K.ca Kalai a t
we say kn a mult seq if Kcab Kcal Kcb
Ex ix a t start19ageREQ kn D an multiplicative sequenceded Gwenmultsey kn the K gang fan oriented M
KEM kn pi Tn p CT.m EMT XPontryagin classes ofthetangentbundle
y
Hirzebruch
maltseq FLED
via givenKaymutt KGez Iif z is a power series
the Snc Et In Hi 1 1
It fish Ea a Kakielemsynpolynomials
G her f 2Tiziwrite II Ict z Eze Ezeti ti di di
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If n l writekelt G
Kes is malt seq Prof n Miho Haslett
L gnus is the genus ascec.toftIz
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Remi textet B th Rei
h k k
Ba D Co I Bnctidt 2 ino i o
The Hirzebruch
LIM GEMPreeti Atiyah Sage Co check on PM can use also to
prove rum theoreticpropertiesofRorhyagh numbers
Def the II genus is assoc to
a
Fact A S M spin AEm C It8 II L
Mh spin acmletdef TheToddgeries is R Q
ta cobordism
I Ep Pf Adan SheetalScgAssocto 2
Tea using Chem dares
Fact i V ex var T Ev arithmetic guns