Math 676 :Day "

← tea.

All kdtayut bundles are synpluhz & all manifolds are Lagrangian

Tf M"

' 3 a Ll w/ lad comb Ix , , . ,

, xD her a pout pen (meaning forG Hit IU, 4) is a word ohtatij

p& xi : 414 → IR is gim J xi ( 41 air . ,

a D= a :),

thenay I c-

TIMcan be written as

I = If5 ; dxi

fr qe 94 .

Bit th u st a pT

's 614 → 1122 "

1g , 5) - ( x. , .. ,xn

,5, , . in )

then (x , , . ,xn

,}, , . in ) Site load cooks on T

' ylu ).

& th } mhs Tt M into a man : Hd.

Eu betty in local wads.we a define the 2- from

w = I,

dxindci

hhih is girl to - da he ← Esidxi,& here B dyed

.

Morar,its obviously mdgite sine it corresponds

in ah and . pth H the still syyketnfem an1122

"

.

The fans w Balled the canonical sypbhz from anT'sM & a is the tautological fan or Lionville 1- fum .

of case,aeb tochedelhyhindqnhtfile ptlrdoie float cords .

,ht 1h33 the . Rthkohedg . tgh .

we define a & w in a cordite . fee way .

ht ¥pf HIS'

he the new project . Wea ape a pinkie y

a ( p.es) = (dtp ,}D} = 5 ° d this )

Here uhe Tpa, Htm )ditto

TPM

Think 'M ) Kitten

In other wks,

a,µ , ,( D= 5(1dI#}Dv ) fray ve Ip ,}M .

Then,

of case,

w= - da.

Pat : a ichaetntdf : Viner 'lM),

n'a =n

.

Wht desk } men ? Renter tht Its m Mae svth seeks f ¥m^'HIM) = T'sM

,so

T' M

xgnfnBt then

, m Baathup ,so in : silt 'm ) film )

& so µ&( a ) is aI - fm an M

.

pdf : We can just pre th , } pontuie ,so ht PEM ; the god is to shield )

,b) = mph) fray uetpm .

Bt now (u*kDpk) = amp, ldnpv) = dip , up ,( dup D= up ( ditpup , dup ✓ )

=

Mp (d ( ton ) pv )

= up ( d 11,D ,u )

= mph) .

D8

Now,the nifty thing Btht , w/ th . } machinery , aydiftomopkhfmflds to into a canonical syapkdomphsn

flheir cotrt badly as folks :

Tf M ,,Mzae n

. mtbb wltwtobgtl tfms a , .az an Them,,Tthmz & fim ,

→ M ,is a diffo

,Fun He new

lift

f⇐ : HM .

→ Ttmz

( p. , } , ) - ( p. , 52 )

the p .= ftp. ) & 5

,= (dtp ,M£ ( or

, ynwtf , es z=¢dfpD'T'

5,)

One on due th ) D really a ditto ; the only annoying bit is convincing yonvtftht f# & fit'

are Smooth .