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Extended evaluation maps from knots to the embedding tower
Danica KosanovićMax Planck Institute for
Mathematics, [email protected]
December 2018, Manifolds @Isaac Newton Institute Cambridge
1. Introduce the main players:
2. Summarize a result of work in progresswith Y. Shi and P. Teichner.
3. Explain the geometric trick at the heart of the argument.
Plan
grope cobordism & embedding tower for knots-Flow
DIMENSIONAL HOMOTOPYTOPOLOGY . THEORY
Grope cobordism...
Finite type (Vassiliev) invariants of knots give an ascending filtration:
1.
HYTL ; A) 2 . . . 2
Vent,
2
Ven2 a . . Z Ven2V±o
4 T anyabelian group I knot invariants of typeen } { constants }the spare
otlbonglunotsi. e. those that vanish on singular mots
TL Emb,
( I.ITwith > n double points .
Grope cobordism...
Finite type (Vassiliev) invariants of knots give an ascending filtration:
1.
Dually, there is a tower
where the relation of n-equivalence is defined as:there is a sequence of capped grope cobordisms
of degree at least n connecting K and K’K K’
HYTL ; A) 2 . . . 2
Vent,
2
Ven2 a . . Z Ven2V±o
\ Tonyabelian group I knot invariants of typeen } { constants }the spare
otllonglunotsi. e. those that vanish on singular mots
TL Emb, II.IT
with > n double points .
III. ← tI% ← . . . ←
tI%←tI%F. . .
←TIL
fatalistsIF'¥
run ⇒
Grope cobordism...
Finite type (Vassiliev) invariants of knots give an ascending filtration:
1.
Dually, there is a tower
where the relation of n-equivalence is defined as:
Theorem. [Gusarov, Habiro, Conant-Teichner]Two knots K and K’ share the same invariants of type n-1
if and only if they are n-equivalent.Moreover, are finitely generated abelian groups!
there is a sequence of capped grope cobordisms of degree at least n connecting K and K’K K’
Open problem: What are
these groups?
HYTL ; A) 2 . . . 2
Vent,
2
Ven2 a . . Z Ven2Ho
\ Tonyabelian group I knot invariants of typeen } { constants }the spare
otllonglunotsi. e. those that vanish on singular mots
TL Emb,
( I.ITwith > n double points .
III. ← III. ← . . . ←
tI%←tI%F. . .
←TIL
ofTmItI¥"¥
run⇒
⇐
TIL° In
Fundamental Theorem. [Kontsevich, Conant-Teichner]
2.There is also the descending grope filtration:
Open problem: Is there torsion in
or ?
Open problem: Is it true that
?Tok = g. I
G.I a . . I Gn := I KEITOYLIK } ? . . .
[Ign=f I
Zn -1>Guy
④ Q⇐
± A ④ Q Guy,
AE.
~n
J2m④Id④
Fundamental Theorem. [Kontsevich, Conant-Teichner]
2.There is also the descending grope filtration:
combinatorics offinite type invariants
perturbative quantization of Chern-Simons theory
geometric topology,n-equivalence
where has several equivalent descriptions:
Open problem: Is there torsion in
or ?
Open problem: Is it true that
?Tok = of I
G.I . . . I Gn :=
IKEITOILIK
} ? . . .
guy }
Zn -1>Guyox ④
⇐± A ④ Q Guy
,AE
.~n
J2m0Id④I
Am
Zlarddiagrausofdegn
)4T ,
IT,
SEP % • • • • • • %
degli # cords
1/2 [Bar .
Natal
Ztfawfigraphsofdegn)
•
.
•
⑨.
STU ,IT
,SEP % • ••• • ••• • %112k¥
• a.
ITJaw6i
Treesofdegn) & ¢
% aeaeaeae ¥STU
?IHX
,AS
degp , #Vegt
2.
.2nd
> § ,④my 0 ④ ±G
~ In.io/dQI
Am
Zlowddiagrausifdegn) % • • • • • • %4T ,
IT,
SEPdegli # cords
112• •Ztfawbitteeesofdegn
) XP5TU
?IHX
,As
% •e••to
•%
{
degp.at#VegtItherestofthetree
the rest of the tree therestofthetree the rest of the tree
STUZ• • • ••
§ 00. . . o•00 ¥ § 00. - . o•Be ¥ § 0000 - . .00 ¥ 080800 ¥k n htt k n htt k htt h k htt .
' it
h
2.
is the realization map:
.2nd
> µ ④ ④my 0 ④ ±G~ Im
Ida
I Jhn - IAm
my
24%49%9
;¥9D* • . • • • • ⇐A. Ift
degli # cords CORDS CROSSING
CHANGES112
• •••ZLJawfiTR.esifdegn) XP me
77STUZIHX,
AS ••••to
••%
••g,•••
. .
'
degt.ES TRIVALENTVERTICES
Itherestofthetree the rest of the tree therestofthetree the rest of the tree
STUZ• • • a•
§ go. . . oo00 ⑤ § OO. - . o•00 ¥ § OOdo - . .00 ¥ % OO0000 ¥k n htt k n htt k htt .
h k htt .
' it
h
A Jacobi tree together with a choice of a root An abstract grope G .3.Claim.
It ⇐ A
ext.
EX2 .
171
3 5* y 6
24g
6 - -
÷.
⇒ .
% ••it•• % • z⇐ '
,
PIs ;4
* ,
•~
1
⇒ I :'
I I
! i3 i Gp
.
3 I
\ i
Gr.
A Jacobi tree together with a choice of a root An abstract grope G .3.
A capped grope cobordism of degree n = an embedding of a grope into together with simple embedding of capsLet = the space of capped grope cobordisms of type
Claim.
Definition.
Two different embeddings of the same abstract grope:
Remark. Define the realization of a tree as the class of any of the knotsobtained from the unknot by a grope cobordism G
I ⇒ A
EX1.
1¥1
3 EX2 .
16245% ••am•• ¥
⇐•
F*3
⇐ >?÷
.
5
,
•
"
;4
⇒
¥'
n
!Gr
.
Ga,
3
!
Is
Emb.CC#tII
A I ':*:*.
.
It.
TA
- I ?
...and the embedding tower for knotsGoodwillie-Weiss embedding calculus applied to the functor Emb ( - , I ) yields a tower of functors T (-) that are polynomial in certain sense. When evaluated at the domain (interval), this gives a tower of spaces:
4.
•13
n
holimTH.TK7
!!
.
a i:
TikW2
I
yg w .⇒
T.TL% I k
EmbfI.IT ImmfI.IT
...and the embedding tower for knotsGoodwillie-Weiss embedding calculus applied to the functor Emb ( - , I ) yields a tower of functors T (-) that are polynomial in certain sense. When evaluated at the domain (interval), this gives a tower of spaces:
4.
For embeddings of codimension > 2, T is homotopy equivalent to the embedding space.
[Goodwillie-Klein-Weiss]
This is not true for the case of classical knots, because is uncountable (not obvious!). However, could still be injective and:
•13
n
holimTK.TKa
7
!: FIL
.
a
""i.
ITHW2
I
yg er .⇒
T.TL% I k
EmbfI.IT ImmfI.IT
...and the embedding tower for knotsGoodwillie-Weiss embedding calculus applied to the functor Emb ( - , I ) yields a tower of functors T (-) that are polynomial in a certain sense. When evaluated at the domain (interval), this gives a tower of spaces:
4.
For embeddings of codimension > 2, T is homotopy equivalent to the embedding space.
[Goodwillie-Klein-Weiss]
This is not true for the case of classical knots, because is uncountable (not obvious!). However, could still be injective and:
Theorem[Budney-Conant-Koytcheff-Sinha]
There is a factorization:
Conjecture.This is an isomorphism, i.e. the n-th evaluation
map is a universal invariant of type <n.
13O
n
hdimTK.TKa
7
!: FIL
.
a
"
": IstoryHard#oTnH:T
t
>
sik
W2 modern,
I r
k en⇒
T.gg% I k
EmbfI.IT ImmfI.IT
Theorem [K.-Shi-Teichner]5.
Given a rooted Jacobi tree let G be the corresponding abstract grope. Then there is a continuous map:
such that the following diagram commutes:
A t*
EmbfaIT
E "-PIK
Emb.CC#ITEVn-PTnK
% Ifa alfa .
ygan Ink
%
Theorem [K.-Shi-Teichner]
Corollary (BCKS Theorem)
5.
Given a rooted Jacobi tree let G be the corresponding abstract grope. Then there is a continuous map:
such that the following diagram commutes:
are connected by a path in and
factors through
A t*
F-mb.la#ITEVn-PTnKEmbfG*ITEVn-PTnK
% Ifa alfa .
ygeun
, IH
Kuk'
⇒ wuk wuk'
Tnyg
⇒ stolen ) TI%n
%
Proof ideas:
Use the ‘punctured knots’ definition of and the following observations:
6.
Embfa#IT
E "
-pink
offor offoryg
eun, TIL
IH
✓iii.¥÷÷:
%I ¥µ
%
Proof ideas:
Use the ‘punctured knots’ definition of and the following observations:
On the complement of J the two knots are the same.1.
6.Ge Embfa
ITE "
-pink
offor offoryg
eun, Tink
TIL
,
O
s
%so
I µµ
Proof ideas:
Use the ‘punctured knots’ definition of and the following observations:
On the complement of J the two knots are the same.
On the complement of the i-th cap-body intersection, for 1 i n, K can be isotoped to U: use the i-th cap to ambient surger the grope into a disk.
1.
2.
6.Ge Embfa*
ITE "
-PIK
offor do 1foryg
eun, IH
IH
-
O
s
iii.i.÷:%
Proof ideas:
Use the ‘punctured knots’ definition of and the following observations:
1.
2.
3.
6.
On the complement of J the two knots are the same.
On the complement of the i-th cap-body intersection, for 1 i n, K can be isotoped to U: use the i-th cap to ambient surger the grope into a disk.
Different surgeries are isotopic when restrictedonto the complement of the union of the corresponding cap-body intersections.
abstractly:
surgery on the blue cap
surgery on the green cap
Ge
Embfaa.ITEh
-Pink
offor do Iforyg
eun, TIL
1TK
⑧-
O
I I
/ I
1 I
I I - .
•
- •
i.I
%I ,
I 1
Definition of T
Let J , J , J , ... be a sequence of closed disjoint subintervals of I.
For S [n]={0,1,...,n} let:
Use restriction maps to get a punctured-cubical diagram:
Finally, define: with the canonical map:
“the S-punctured interval”
“the S-punctured knots”
7.
J
,
n
0 n 2
I I ;U .
E S
es'
EEmbfis.IT-E
holimEE.it?rlnIIop* n
0
II
•
Till := holim Es Aah
> TILOF5E
.
%
Definition of T
Let J , J , J , ... be a sequence of closed disjoint subintervals of I.
For S [n]={0,1,...,n} let:
Use restriction maps to get a punctured-cubical diagram:
Finally, define: with the canonical map:
“the S-punctured interval”
“the S-punctured knots”
More on the proof
7.
We construct a family of disks parametrized by the (n-1)-simplex:
so that is an embedded disk with boundary for every
How will this imply the theorem?
J
so an element of this space is a collection of maps
so to define we need to specify one-parameter family of such collections.
,
n
0 n 2
I I ;U .
E S
es'
E
Embfis.IT-
E
holimEE.it?vlnIIop* "
0
"
•
Till := holim Es Aah
> TIL0FS ELM
.
TY2
h :D'xJ'
IsGhglD'HIMdo
GwangIe
Till :=
holimEs-IM.A.p.to#m(BlPvmt-
I.E.)RIN
JIB- Es
% EVNG
8.
base of induction: deg( ) = 2
const.
thickened embedded
caps
an embedded grope
Set to be constantly equal to for all S 0 (use observation 1).
For any other S [n] use isotopies through S-punctured knots along disks .
idea for
EVNGIHS with a
E 0
lightHEH01 02
012
11 12 2
h :D 'x& I'
r i his ~ >→
G •
→µ
!-
I , IT I I
h.
G I I
-
"
-
.
-→
¥-
,
D40hg Is
'
% I I
Thank you!Danica Kosanović
Max Planck Institute for Mathematics, Bonn
Questions?
'② %
Thank you!Danica Kosanović
Max Planck Institute for Mathematics, Bonn
Questions?
Can we define ‘a space of capped grope cobordisms’?Can we prove it is homotopy equivalent to ?Are the evaluation maps surjective?
÷:- f-
