adf.in#:YYrIYYisI. y. ' (E) = -. Fu ,. fi. â¢. flakes. in 1 °. Page 3 of 39. Kristen Hendricks notes.pdf. Kristen Hend
Kristen
Hendricks
Floe
Lagrangian
Equivarinnt
&
Homolog >
Floe
Lagrangian
homolog ,
RI
Fyi Hutchings Mcduff
Salamon
-
Pedroza
Plan
:
Lutuwnuks
"
"
A
"
of
to
Sympkotiu
topology
view
of
Lagrangian
Flor
Into
sywpkctiu
homology
Morw
"
,
Into
quite
Review
tko
More
on
Dl
|
topology
-
Lagrangian
Floe homology
Eqnivariant Equivariant
cohomology Floe Las
DZ
|
-
,
homology
D }
|
Applications
-
Morse
Deft at
an
,
A
critical
function pt of •
a
k
fmothdow
homology
"
.
Hessp
A
dpf=O matrix
F :M→1R
,
of
psaeitoiil
(
f)
dams
if
More
is is
non
is
-
Sing
rousing
.
"
{ Morse
}
fetus
ECM ;lR
)
an
Crit ( F)
of
#
tea
={
eigen values
.
indcp )
chat
F
-
in
•
\
is
( NTCF )
pe
,
which
on
fixit
more
,
Hesspf
of
New
.
}
:dpf=O
pen
of
subset
dense
open
an
-
+
'
×
x.
-
,
)z
I
:#
Fix
•
-
.
-
-
xp
.
mtnvgontl
a
adf.in#:YYrIYYisI y
•
flakes
in
1
°
'
(E)
=
-
Fu
fi ,
For
×
>
,
( rit ( FI
e
tilx
,y
let
,
{
=
)
IRSH
vi.
r
:
→
x
To
F
Htautio
a
m↳ Define when
=
and
2.
complex
&4.
IFzk{ writpts :
4.
→
2. (
,
2) 3)
,
,
by
)
translation
.
.
} )
is
,
b
given
.
ind
( y)
being
an
270 This
inrxi
Mlxn
Mt
#
m
#z )
;
>
)=Z(# )y
x
claims 1)
x
Exit"C of
indlx )
Man
(
Y
)=hilxn)/R
,
chain
a
th
on
.
}
flouts
gradient
,
,
)
)
:|
the
made
even
makes
Hulme
)
.
conks
.
( It
:
sense
.
o
.
is
rfo
a
.
)
compact
TO
these
get
win
,
Transvestite
AO
(
metric
In
Morse
or
smooth of
For fctn )
.
:nd( x )
dimn
,
Mlx , , )
=
0
standing
tons
the , -
(y)
ind
This
.
need
up
.
.
.
of
choice
generic
MCs
space
-1
xty
,
is
)
,
the
for
fails
,
.
indGD c-
ind ( x )
if
particular
to
going
standard
:
if Compactness
UB
flouting {8u ; } "
broken
.
fit
which
Given
NC
x. y
sequence
a
)
converges
,
there
to
is a
a
of
subsequence
flouting
possibh
"
:
@#
flouting
Yin III
Let
Illxm ) The
Ex
Ir
(
×
)
,n
Give
"
for
Howlin
( for
xm
)
Howling
)
,
,
is
of
to
y
what
it
to
converge
metric )
(
wl
manifold
a
k
...
,
1
}
nears a
corners
Mou
.
,
precisely far %-)
this
new ×
( 112,1kg
honwmuehnto
is
only need
we
mfds
×
Flodin
broken
Today
to
×
.
formulation
of
generic
a
M(
for
.
15 (
Gluing
ept
precise
a
flows
broken
is
sequence
a
broker
Q
{ possibly
=
for
Kit
,
toy
#
a
,
oktiocn
which
-1
gives
boundary ,
Lenny
:
270 Pfi
.
ind↳=i
Let
.
.
.LI#MCxn)y ]
24=2
indly ) =
III ,#
"
.
=
'
"
'
'
;
.
I
.±tIm#H Ee
5q(#
Nxn )
#
M
of
,
,z
)
)z
broke he
to
#
floulines
,
25 ( ×
=
,z
) 5
T
l
gluing
Hooray
(
0
=
twfd
Pt
2)
need
boundary
1
-
.
?
ix.
¥÷
He # , Ho
If
work
we
Mtn
÷
dependent
Sa ,
f
,g
gen
,
2/-9
one
"
which
)
IF
.
an
of
F
on
by
whereat
to
fotn M
Id ]
;
.
orientations
need "
Morse
Morse
'd
]
Catto
genio by
270
get
"
9m*nµ×{ of of offs
:M×Io,D
112
.
.
Pick
.
→
for
and
tgpnant
or
.
,
on
M×{
1.
tssmllt
÷
f) ÷*"i
Nf
F
those
-1
Define
crit
no
chain
a
§
via
( )
=
x
Deft
:
manifold
A M
-
with
w
is
non
\
.
)
intense
on
¥
is
2
a
at
10
,x
as
.
(1
→
flows
a
)
-
smooth
form
(
closet
's
w
-
'
OF to
.
symplectiv manifold together
.
:
C÷ inog ,
Topology
i
(#from
Fax ,
-
Synpkutic
§
map
.
Into
pts
"
degen
W
Za
.
dim 't
S.l.
:o)
du
(
wlywto
the
⇒
v.
o
)
Equivalently
WM
,
manifolds
Symp .
notion
mfds Area
.
dxndy
,
( × ,n .
On
like
an
( 1124
:
. .
.
xn
.
,
,
form
-
.
+
.
1
a
.
dxnndyn
th
ndisom
.
equipped
:O t
,
,7n)
ITEM
volume
a
2 dimensional
of
Ex
is
)
.
( the
product
cross
area
Thm is
So
C
locally no
Riemannian
Ex
Your
Ii
2
3
local
geometry )
favorite
TTFIRYP Auther
oth
curvature
.
gwfaue
it
.
in
.
ul
gon
0
Ex
( like
inits
)
Synplwtiu
:
5×5
,
.
( M ,w) Any sqmpketomorphi to )
Datsun
natural
for
et
-
( Thun ,Mc Duff lattice
,
,*w those
.
gynplectic
stratus
+
on
form
area
am
Xeajw it
up
onT4 ?
.
to
supp
.
)
×R~
( Dh )
Int
4
not
-
sywphutomophn to Rtl ,
(
If T*U
an
W
×
xn
.
.
;
.
,
w=
,
Not
bk
A
n >
infinite
an -
dx
;)
admits
I
"
,
]
with
many
dimensional
coordinates
.
this
;
local
are
for
impossible
is
no
]¥Oet÷ [w
M→M
There
.
)
yn
;
,
to
Hour
c-
of
symphetomoph.sn f.
.
.
,
here
manifolds
,
,
d(Z%
-
"
S
Y
,
exact
is
compact
6
LT*M,w7
bundles
Cotangent
5
(
f*w=w
gynplectn form EW ,
µ ,w
)
is
s
of
( The ,
act n
diffeomophm
.
infant
sympkctomoehisvsj space
;
then
.
transit
pts ) .
an
,
.nl
,
on
sets of
A
fine
dependent
-
smooth
Hamiltonian
He
maps
Given
He time
M→lR
:
family of
a
,
find
can ,
dependent
.
is
f
v.
dtlc
and
Xµ
with
.
.
WLXN This
looks
Ex
On
:
had
=dH .lv
,v )
.
but
quite
is
11£ ,H↳n)s
dtkxdx
(
dHe× , ,
=
Wart
to
sow
w(
v
-
this -
so
a
v
,
)
,uz
.
+
YVZ
.
,vd=xui+7vz
.
,
but
)
+717
, Xv
.
simple
zbiti
'
)
,
has to be
(
y
-
×
)
.
,
Xµ=G
-
,
x
)
,
#⇐" +1=1
Xp, Ex
:
will
What
be
if
to
parallel
H
:R4
→
1/2
lad
is
sets
prg
.
.
to
first
card ?
look
We
fart
,
afdrffeus
These
flow
the
at
He
Xµ
of
4l|×= Xq(
-1
th ,
.
4€64 )
.
gywbketomurph Classical of
fixed
are
question pts
a
Classically will
What
:
Hamiltonian nekno
find
.
better
the
is
nawbe
lowest
gywphclvmophisw XCM ) ; results
>
.
but
have
?
Day Recall
2 A
:
Flow
Lagrangian Hamiltonian
smooth
An
D±n
n
WIL
if
:
This
-
snbrfd
"
smooth
Htt7→lR
)
Hamiltonian CM
of
9
)
it
family
a
dependent
.
w
,
,w )
.
Lagrangian
is
.
Exercise
the :O
.
0
Example 's
of
M
0
'
is
wlz
it
time
a
dim
-
flow
one
-
(
fetus
of :(
sympketomorphism
time
the
B
homology
The
02
{
plan
Any '
04
s
0
"
)
(4,9*30) in
are
(
(
3
dimensional possible
highest
e-
section
-
g-
E
fare
sywp
any
.
manifold
)
T*X
inside
Graph ( df ) 04 to
5
(
"
}
( Rt ,wo )
orientate
an
1122
E
subnfd
T*X
inside
;
Han
.
isotopic
.
( Nearby which is
Lagrangian is
Ham
closed isotopic
and
conjecture
exact
to the
section
0
MIM
6
•
•
.
a
x { (,x)3
.
{(
iff
×
e-
H[
,
T*X ±
Of
)
has
D=
to
widl
i.e.
,
L
An ,
:
.
,¢k))}
¢
form
sywp the is
is
a
-
wow
diagonal Lagrangian sywpleetowophisn a
.
Note
Dr
:
fixed
=
Pop
pts of
4
M
cowpat
Conjecture
( Arnold
,
1965 )
:
syrpkctiu ¢ , wl gympketomorphism nondegen Then Itixl 4) 7 I manifold
Hamiltonian fixed pts
a
,
(bilhi €s¥ y
.
1
(
on
;
"
( Arnold
,
.
Girvental )
t.in#IiyhEEIEEtIItiiI
:c ;Y .net : "
02
0
tru
if
( Slightly true
tz1M)=0 different
if
"
,
I
,
"
(
M
method )
)
,L
[w]=ac,( M )
0
t
also a
,
>
0
.
Tool
:
Floa
Lagrangian ( 14,44 )
homology HFK ;L
~
}¥ITht
)
,
9 (
fno
ones
chair
-
Lonh
generated by
Kohl Need
the DEI
of
notion
Jx
if
w(
if
I
:T×M→T×m=
so
.
#w(
=
Then
is
w(
W
-
yw
)
structure
Id
we
,
compatible
,J-)
-
-
complex
v€O , and
when
0
>
,
J
almost
an
is '
)
Jv
y
W(
HF
rk
3
,
an
:TM→TM
J
:
Complex
is
say
.
Riemannian
a
metric
g.
Assumptions
(14/4,4)
on •
[
•
Lu
w
.
•
]
.
#
z( M
ept ,
popular
exact
LOXL wa
,
-0
=0
,
of
achieving
Laghs f.
[w]eT2( 17,4 ) .
,
"j M
M→R
this
convex
smooth
?
at
"
a
M (
paper
exact w=
-1
ddef bound !;w -
,
,
Want
to
consider
wklt
JLEI
pseudo Etten
,
holomwphic
-
t
s
:lRxIq
u
M (
x
,y
)
]
uls ,o )
"
Eto
191 )
L
e
,
:#
/
duoj
wlstardw We 'll
define
,
,
tie
]
ACS
CFCLGL
JCE
-
slims
lRx[u
"
.
Morse
theory
on
)du=0
,t)=×
its
( st )
y
-
j
,
)
)=FzHoh4 2
:
CFCLGL )g
PC
Lo ,
counts
Psegfojphd
,
d paths
Mimics
17
→
{ kf ¢
,
maps
]
L
,
)
starting onlo ending
on
L
,
In an
the
exact
-
ul
exact
-
functional
action
Lagrangian
-
A (
)
y
wehave
easy
=ft*2
+
j
-
folgloj ) )
flow
d2=w
of .=Hw MK
tote
,y )
homolog > class
Want
Transversal 't -
D
For
'
Mblxnl
µ( B)
when
N' 36,7 )
I
=
Maslov
is
a
is
the
index
of
mfl
( µlB?
smooth
a
choice
generic
dimh
OFJGH , -1 ,
additive
algebw
,
-
.
topological )
.
from
Comes
Saro Smale -
TIFT
Fiscx ,n)\Nlxn -
=
,
.
when
.
F%pEFIrB↳)
Compactness
Pseudo
,
hol
discs
)
are
×
&z (g) •* * o
•
t
×→
.
I
HF.CL/4CL))=Zbil4Tz/h,L)=0BkNbhd(L)EMisT*L in
Blcif
T*L
f
,
just
to work
need
.
:L
→
R
is
Morse
,
graph I of )n0
D '
Lagrangian
=
eritpts
Ex
2
:
T*s
'
=Rx5
Lo L
{1%0}
=
=
,
{
}
Casinos
:# ;t***¥÷ high .
↳
¥*
is
on
away supposed to
the
right
be
.
Disc
"
bing.ms.org?spIYe
and
221=0
"
.
Exercise
Calculate
:
CF
of
#
hard
the
Why
else
show
care
you
-
Tqpkotoogigalmpiypntdata
?
( Mr :L ) ,
-
}
Hfggg Topological
invites
'
e.
g
:
HFIF ,
HI ,
Khqmp
...
Part
[
applications
!
More
the
on
index
:€: ×
L
,
|
*
←
%
|
Lo
TL
0
17,44 )
(
e-
Lo
,
a
TMA hetmn \←±¥ *
Uaslov satof How
index
is
Zc
,
(
UMTM )
,u*(TL ) .
U
:
cTlo to twist
much
TM reach
has to
TL
,
J relative
CDZS
v. '
u*TL
'
)
b.
.
over
.
QE 0×6
)
Eqnivariant X
•
•
(X
We
9
X
Z
have
top
2112 )
just space
.
z2=Id
,
)/zyv
EXK
×
( this week ,
theory
=
}
EXK
' Tonnsfrntin
X§nEHz
X→
fixation
a
Xxaa
:t.g.EU Him ( X HE
n
Ex
;
#z )
is module
.
RP
=
over
.ec#;Fd=tFiLoI
lot :L
,
HE,dpH=H*CRP9
:
8
DX
2112
If
fuel
then
,
H9h( X ) =H*CXt4h
03
If
X
→
Y
is
an
equiv
't
)
htteaiy
THE
HE,dY )
then
( Don't
More
need
equiv
(X )
.
.
ariant
)
inverse
.
algebraically
,
•
(
A
*
then (
2*
(
*
C*
i.e. ,
,
,
is
Fz[
an
Take
a
chair
-
(
#D=Ext#µ⇒( C
*
,Fy
module
"
free
resolution
of
(
*
are
E[
2112
]
!
o←C*a#Eui¥¥×o#zEm]¥* Take
px
)
trivial
Unpacking
]
21/2
hon
to
Fz
.
.
:
o→c*±ic*±yc*→
...
]iYnI¥i .
of
the
,
cpx
.
.
,
Get
spectral
a
sequence
(
H*
(
;
*
)
#z
0#z[
:* ( (
Ex
like
This
:
(
(
has the
is
"
equiv
type
htpy
.
iso
fixed
the
he
of
this
follows
wl
the
Uses
fin
⇒
'
considering
differential ifo
is
a
-
T
X×%EZ%→
(
*
132112
( X)
Then
.
W
dim 't
cpx
H*lxt
"
,
then
is
a
oxtffqo ) '
;Ez)aEEo,o
#z)>,rkH*( xf
Sseq
the
X
if
H*( Xm , E)
→
H*( X ;
rk
from ltt*
that
a
Go ]
⇒
:
of
res
set of
"HFh( X ;Fz )
-0
HHHEIQFZE
Ex
#z )
cohomology :
free
a
,
Fz )
Xg
Borel
t¥dC*
Film ]
Xf
localization
)
IXXEZK
*
RPI
for
generalizes
over
Let
Ss
Sen
⇒
]
o
where
we
3
'
"
#z )
start
' .
smae
.
i.
Xfittdoslspknofxfix
.
Note
fin
:
.
dim
is
qs
5
important
grading
Ex
He
note
Cx
2
has
Also
.
fixed
)
...
,
5
set
doesn't
Sseq
,
ft
=
,×z
)
-
,
!
the
along
split
,
homological
.
Sseq
( owpute
:
try
Let's
( 17,44
•
Ello
•
•
this
We gymp
.
)
)=↳
w/
€
for
Lajn
in
Floe
z(
,
new
"
=L
4)
,[f ,Lj "
,×3
)
I tx
-
,
Xz
-
,
xD
.
gynpkctic
he
( Mf
get mfd
÷
2
invol
,
;
I*w=w
,
"
)
Lagrangian }
-
,
if
connected
,
new
.
QI QZ
How
:
can
we
When lhou
:
HE,n( to ,L
define
relate
we
can
CFCLGL
Want
,
)DE*
T*
,
Extreme ] (
-
) ?
Hfyal ↳ 41
?
HFCE; hat
to
,
chain
a
C Fao
,ln
,
map
.
IE )
x
•
"
"
"
€410
f•¥
Plan
I
Find
;
-
Issue
:
This
;
JCE )
eqnivaviant
I*
chain
a
hard
,
!
e.g.
:
if
map
µ*( B)
homolog
z
,
achieving
transversal 'ty
.
Sometimes A
behdonok
show still
←
prorubli ,
class
µ( B)
impossible of
.
curve
,
:c
.
.
Khovanov then
if Seidel
-
-
condition
Seidel an
Smith )
under
:
no
Such
a
discs
in
( 2010 ) which
JCH
fixed
the
Gave
:
such
a
JZT )
always
can
Set
found
.
IF theoretic
strict exists
be
.
Day 4 As
Equirariant ( 17 ,w,↳,4 )
yesterday
Iz
•
•
can
Maslou
index
( to
make
we
How
Want
Then
Plan I
:
Find
Lnot
I=Id
of Z
-
a
disc from
graded )
to
×
itself
is
.
,
define
HF
it date
A
HEN
-
possible)
2112
Ext
( ↳
L
,
) ?
.
#%gkFC↳4
Jct )
4*9 ?
HFCKY
to
CFCLGH
eqnwariant
always
I
cnn.LY fig
does
t#
:
9
Homology
assumptions
me
when
Floor
Lajn
achieving
,
Fz
)
.
transrasality
0
.
Seidel
.
Smith
Gold
WTM
,
of
Lo
and
Ktkontic
strict
Under
L
and
JCH
F
,
,
.
trans
'HFa( ↳
'
patio
Lfix
F
is
4)
equiv
pointnise
for
.
Chi
cord 's
sane
an
4¥
,
localization
o
In
eq
a
exact ,
,
,
fixing
,
L
assumptions
achieving
Furthermore
↳
exact ,
,
Ham
't
told
,
4)
one
,
.
isotop >
4
'
,
.
o
HFKF
→
"
,4t ) OFEO ,o I "
'
.
Sseq
have
HFCLO
,
4)
too
E
a
'
]
⇒
HFCE.it#90EEo.oJ.Whutassumptio= Mfixem
?
Nlf "x{o}
µh*x{ 9 Normal inside
BE NMF"xIo
,
NCMT ) "
,
,
→
✓
( Ntt "xEo,D
ant
,
NMF
}
|
Maxton ]
normal bundle
"
if
,(NLjmx{ L
the
M*eM
-
triviality
relative
0
})
and
4
( NE "x{ B) one
)
Sseqs
of
lots
table
A
First
net
Khsymp
(k) f
"mP
( k)
this
from
.
:
related to
page
Kh
come
doubt penman
.
°
-
Page
related
to
the
KHSM (
k )
t.MY#IYaEYm**IIfsEHFlo4 HF
(4)
( Seidel using
hypotheses
wlustnitiw prow
different
th
more
,
methods
.
)
.
broadly
Plan
2
:
of
Lots
( Seidel
Version
Smith,
-
( Hendricks
Version
A
variant
noneqni
(
5
w
modeled
Lipchitz
-
's
structures
complex
More
on
,
theory
)
.
Sarka )
-
worth
ufsct
) .
Construction
( 14,44
;
a)
TFCLGL
-
'
,
)
'
feed
Floe Coyotes
' '
the Yi |uµ}¥|# th !
IYI (2*5) honutop
:#
J
#
-
t
a
(
→
,
st
,
:%
that
f.
*
↳ (5)
that
I
⇒
)
are
.
6,62
not
equivalent
Keepgoirsfot '
C
*
G)
e-
,
'
!
Tt
( to ,L )
CFCLGL )
=
,
TF (
to ,L
(
I
Theorem
,
)
is
Then
complexes
-
Fz
free
a
HLS )
Fz
as
,
exists
(
H Fan
[2/12]
which
,
Ham
equivariant
complex
.
well-defined
a
to ,L )
-
is
in
't
under
uflgl
isotope
&
,
a
Sseq
(
HF
÷
(
2
transversal 'ty
exists
,
certain
is
If
Hls )
eg
situations
in
HF
,
.
In of
pwtr
,
th
( Therefore, topological
,
,
every
egnivanint
their
generally ,
input
und
data ) .
's
uohicnn , hob
for
a
eqnir
weak
,
Ham
each
,
JCE )
ETFCLGL aftk
4)
.
.
what
3) page
't
Seidel Smith
if
g
( CFCL ,l )
inv
equivariant
an
,
HE,a( ↳
→
which
of
page
hear
QIEIOI
↳ 4)
Sscqg isotoe,
also
type inits
)
Fatah
as
before
of
the
.
auinits ( Lo
of
]
,
4)
the
an
.
Example 0
Floe
Knot (
0
#
vath
-
homology
Szabo
( 53
.
; Rasmussen )
,K)msH¥K(s3k W
bigram )
recovers
Deft
Alexander polynomial ,
genus ,
KG
A knot preserved
by fun
disjoint
53
212
a
I
* @ K
fibeowness
2- periodic fixing
action
an
if
FL
.
veut
,
it
unknotted
is axis
.
I.
~
is
)
/
→
i•@ .
quotient
knot
Thin
(
Hendricks
§
diagram localization
)
how
,
KY
for
2- periodic
Assoc 'd
to
this
spectral
a
a
sequence
&
map
HFrTlsykTaVawlaw@tFzIoiET-7oiHFkTadp7_tTFkC5.K1oxV0CmlowaFEo.o ' .
( dim V
din
=
W=2
;
]
distinguished
by grading )
Uhy
oris
via
.
Fox
•
cakau
.
n'
a
Surface
vh
minimal •
theory
,
is
Murasogi Edmonds
Go
0
A 3
? info
'
's
"
by (E)
condition
criterion
g
(F)
=
( lit÷ ) DKCE) mod
ZZGLK )
+
'
¥
.
bad ?
this
2
good
classical
Recovers
•
orig
this
is
man
priori
terse
,
depends
Computability
( need
factors th
on
a
! diagram
large
transvestite )
rotor
to
when
2
Thu
HFT Ever , and
IHLS ) 153,E ) page
th
The @
of
Sseq
Von
th car
is
a
spectral
alfiero 's
Sseq be
is
computed
Segura
the ,zLs3
⇒
an
init
(
of
combinatorial }
,
.
E) E
,T )
