neg - university of notre damepmnev/f19/cobordism.pdf · 2019. 11. 20. · fact hqis an iso pf thor...

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Matt Tako cobordism ept smooth J Mn N sixth Cd at wn MINM.N are abordant if 7 Watt can ask for additional structures M N orientedly use dart if 7 W at 2W MIN T T X or or In n nfds be.d.su r donate n neg Iddith abelian grain ring structures x R x In Rien M xN MXN graded commutative PMI gz is an Fa algebra MIM dcmxl o.is 10 3 Lee EMT 103 G 7 fwm's.t M DW a neck bun Talk 2 I def The i th Pontryagin class of Z is due to ang gato Picz CD Ca z Q e 4 M Q Ca CzqQ is 2 torsion vanishes rationally 7 h din M H't so n Q P pk YC k 17 Eulerclass oriented Grassmanian B soca Bso Cnn 1120 IR's shift by 1 ceordnate r C Bso 7 Shifa Spence

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Page 1: neg - University of Notre Damepmnev/f19/cobordism.pdf · 2019. 11. 20. · Fact HQis an Iso Pf Thor Rat Hurwicz X s plyconnected it CX fniteher 1En T X Q Hi X Q is an iso is 2n Assumey

Matt Tako cobordismept smoothJ Mn N sixth Cdat wnMINM.Nare abordant if 7Watt

can ask for additionalstructuresM N orientedly use dart if 7 W at 2W MINT T Xor or

In n nfdsbe.d.su

rdonate

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

Page 2: neg - University of Notre Damepmnev/f19/cobordism.pdf · 2019. 11. 20. · Fact HQis an Iso Pf Thor Rat Hurwicz X s plyconnected it CX fniteher 1En T X Q Hi X Q is an iso is 2n Assumey

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

If Essene MT103 M 8WConedirection TWI TM E

bycollared nude in

thin stability i p Tw p TM

m Hn M0 Hu W

O u MTSi tin M H t w m

pent I f PEUT

HYw HYMIE tint Cen nexact

PEM fi p Ew Tw Also workswitho 0 Ratyaga SU

c auroriented

Page 3: neg - University of Notre Damepmnev/f19/cobordism.pdf · 2019. 11. 20. · Fact HQis an Iso Pf Thor Rat Hurwicz X s plyconnected it CX fniteher 1En T X Q Hi X Q is an iso is 2n Assumey

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

Page 4: neg - University of Notre Damepmnev/f19/cobordism.pdf · 2019. 11. 20. · Fact HQis an Iso Pf Thor Rat Hurwicz X s plyconnected it CX fniteher 1En T X Q Hi X Q is an iso is 2n Assumey

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

Page 5: neg - University of Notre Damepmnev/f19/cobordism.pdf · 2019. 11. 20. · Fact HQis an Iso Pf Thor Rat Hurwicz X s plyconnected it CX fniteher 1En T X Q Hi X Q is an iso is 2n Assumey

Talks Multiplicativegenera

def a multiplicativegenus is a ragmap N

Ree i for thistalk SLG150 A Qso a bet so Q

EI r Ifa M 74cm med2

y mum y M ex n

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

Page 6: neg - University of Notre Damepmnev/f19/cobordism.pdf · 2019. 11. 20. · Fact HQis an Iso Pf Thor Rat Hurwicz X s plyconnected it CX fniteher 1En T X Q Hi X Q is an iso is 2n Assumey

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

Txsymmetric

If n l writekelt G

Kes is malt seq Prof n Miho Haslett

L gnus is the genus ascec.toftIz

2ZkB e

2K

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

Page 7: neg - University of Notre Damepmnev/f19/cobordism.pdf · 2019. 11. 20. · Fact HQis an Iso Pf Thor Rat Hurwicz X s plyconnected it CX fniteher 1En T X Q Hi X Q is an iso is 2n Assumey

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