Spaces of embeddings: new directions Danica Kosanovi Zrich, - - PowerPoint PPT Presentation

▶
spaces of embeddings new directions
SMART_READER_LITE
LIVE PREVIEW

Spaces of embeddings: new directions Danica Kosanovi Zrich, - - PowerPoint PPT Presentation

Nov 16, 2023 420 likes •712 views

Spaces of embeddings: new directions Danica Kosanovi Zrich, 10.12.2019 G o a l u n d e r s t a n d s p a c e s o f s m o o t h e m b e d d i n g s ( w i t h W h i t n e y C t o p o l o

slide-1
SLIDE 1

Spaces of embeddings: new directions

Danica Kosanović Zürich, 10.12.2019

slide-2
SLIDE 2

u n d e r s t a n d s p a c e s

  • f

s m

  • t

h e m b e d d i n g s

( w i t h W h i t n e y C t
  • p
  • l
  • g
y )

G

  • a

l

smooth compact manifolds with co
  • qub.(P
. M) "V ' OP , 0M¥
slide-3
SLIDE 3

u n d e r s t a n d s p a c e s

  • f

s m

  • t

h e m b e d d i n g s

( w i t h W h i t n e y C t
  • p
  • l
  • g
y )

G

  • a

l

smooth compact manifolds with

Fix

Then co
  • qubfp
. M) V :P# iii.

flap

= V tap
slide-4
SLIDE 4

u n d e r s t a n d s p a c e s

  • f

s m

  • t

h e m b e d d i n g s

( w i t h W h i t n e y C t
  • p
  • l
  • g
y )

G

  • a

l M

  • t

i v a t i

  • n

Classical knot theory

smooth compact manifolds with

Fix

Then

Huh

?!w.fi#=tnotYisotopy

co
  • qubfp.TN
  • "V
" V :P#

i.⇒

. f- I

Hap

slide-5
SLIDE 5

u n d e r s t a n d s p a c e s

  • f

s m

  • t

h e m b e d d i n g s

( w i t h W h i t n e y C t
  • p
  • l
  • g
y )

G

  • a

l M

  • t

i v a t i

  • n

Classical knot theory Higher-dimensional knots Knot families

Diffeomorphism groups

smooth compact manifolds with

Fix

Then
  • t.EU/5'.Py3l--%aebofI.I4--knotYisotopy
any M

45

'
  • f dimensions
= fin"T%sotopy as "

Eoubfp

, M)
  • Raub.is?D4l=2-knotYisotopy
"( / "
  • Haub , #
"its ") ±

172,2

ftaefliger)

V :P ⇐ M
  • p,o*
  • *.gg#.s....I.y-.z*.gnon.mn.aefDa.y

flop

  • Hap
  • qubdm.my
  • Ditto M)
slide-6
SLIDE 6

My main project concerns the case and the following problem.

For n>0 the evaluation map gives a universal Vassiliev additive knot invariant of type <n.

  • Conjecture. [Budney-Conant-Koytcheff-Sinha '17]

P

  • I
' Wn : Embo (I . I 3) →I
slide-7
SLIDE 7

My main project concerns the case and the following problem.

For n>0 the evaluation map gives a universal Vassiliev additive knot invariant of type <n.

  • Conjecture. [Budney-Conant-Koytcheff-Sinha '17]

The tower of spaces ... ... and evaluation maps come from the embedding calculus of Goodwillie and Weiss '99. This is a powerful technique for studying any space of embeddings . By Goodwillie and Klein '15 if dim(M) - dim(P) > 2 then becomes increasingly better approximation. They predict that for dim(P)=1, dim(M)=3 is surjective on components.

P

= I ' Wn : Embo (I . I 3) →I →I -I → eun
  • n
Embo(P . M)

I

  • wn
slide-8
SLIDE 8

My main project concerns the case and the following problem.

For n>0 the evaluation map gives a universal Vassiliev additive knot invariant of type <n.

Theory of finite type invariants by Vassiliev '90. E.g. all quantum invariants are of finite type. Many questions remain open... Geometrically studied by Gusarov '00, Habiro '00, Conant-Teichner '04. The last uses capped grope cobordisms. This gives a filtration on the monoid of knots:

  • Conjecture. [Budney-Conant-Koytcheff-Sinha '17]

The tower of spaces ... ... and evaluation maps come from the embedding calculus of Goodwillie and Weiss '99. This is a powerful technique for studying any space of embeddings . By Goodwillie and Klein '15 if dim(M) - dim(P) > 2 then becomes increasingly better approximation. They predict that for dim(P)=1, dim(M)=3 is surjective on components.

P

  • I
' Wn : Embo (I. I3) -I →I -I → eun
  • n

Eoubfp

, M)

I

'HI :=

to Emboli .IM

em . .
slide-9
SLIDE 9
  • Conjecture. [Budney-Conant-Koytcheff-Sinha '17]

For each n>0 the evaluation map gives a universal additive Vassiliev knot invariant of type <n, meaning that the induced map on path components is a monoid homomorphism which factors as and the horizontal map is an isomorphism.

eun : Embo (I . I 3) → I 'ik

!

. ' II n Et TOI
slide-10
SLIDE 10
  • Conjecture. [Budney-Conant-Koytcheff-Sinha '17]

For each n>0 the evaluation map gives a universal additive Vassiliev knot invariant of type <n, meaning that the induced map on path components is a monoid homomorphism which factors as and the horizontal map is an isomorphism.

This was proven in BCKS'17. This is the meaning of universality. This is an equivalence relation that uses gropes. eun : Endo , (I . I 3) → I

/

"

¥

.

Et TOI

←
slide-11
SLIDE 11
  • Conjecture. [Budney-Conant-Koytcheff-Sinha '17]

For each n>0 the evaluation map gives a universal additive Vassiliev knot invariant of type <n, meaning that the induced map on path components is a monoid homomorphism which factors as and the horizontal map is an isomorphism.

Theorem [K.] The horizontal map is surjective.

This was proven in BCKS'17.

This is also one case of the Goodwillie-Klein prediction. It will follow from a more general result we state later.

eun : Embo (I . I 3) → I

7

"¥t

' II n E- Io

T

slide-12
SLIDE 12

Two knots are n-equivalent K ~ K' if there is a finite sequence of capped grope cobordisms of degree n from K to K'. Definition.

h
slide-13
SLIDE 13

Two knots are n-equivalent K ~ K' if there is a finite sequence of capped grope cobordisms of degree n from K to K'. A capped grope cobordism of degree n is a certain 2-complex built

  • ut of embedded tori and disks and shaped after a rooted planar tree

with n labelled leaves. Definition.

h
slide-14
SLIDE 14

Two knots are n-equivalent K ~ K' if there is a finite sequence of capped grope cobordisms of degree n from K to K'. A capped grope cobordism of degree n is a certain 2-complex built

  • ut of embedded tori and disks and shaped after a rooted planar tree

with n labelled leaves. Examples. Definition.

h

i

. ⇐ o . K '

v

l K ' ° ⇐ U
slide-15
SLIDE 15

The knots are both isotopic to the trefoil T. and Hence: U ~ T and U ~ T. Examples continued.

A 2
slide-16
SLIDE 16

The knots are both isotopic to the trefoil T. and Hence: U ~ T and U ~ T. But, one can prove that U ~ T. Moreover, T is the generator of . Examples continued. Theorem [Gusarov '00, Habiro '00, Conant-Teichner '04] is a finitely generated abelian group.

i.

1 2 13

Mh

.

Whn

slide-17
SLIDE 17

The knots are both isotopic to the trefoil T. and Hence: U ~ T and U ~ T. But, one can prove that U ~ T. Moreover, T is the generator of . Examples continued. Theorem [Gusarov '00, Habiro '00, Conant-Teichner '04] is a finitely generated abelian group. However, this group has not been computed for any n > 7.

1 2 13

Kh

. I n "'

Is: in

#

÷ ÷ ÷

'

I

slide-18
SLIDE 18

Theorem [Conant-Teichner '04b] There is a surjective homomorphism of abelian groups .

Here is the abelian group generated by rooted planar binary trees with <n leaves modulo certain relations (related to graph complexes).

Ren

: Atanas I n

a:

slide-19
SLIDE 19

Theorem [Conant-Teichner '04b] There is a surjective homomorphism of abelian groups . Theorem [Kontsevich '93, Bott-Taubes '94, Altschuler-Freidel '97] has a rational inverse (This is a universal additive Vassiliev invariant over Q.)

Here is the abelian group generated by rooted planar binary trees with <n leaves modulo certain relations (related to graph complexes).

Ren

: Atanas I n

at

.

Ran

  • 2. n
: I n ④ IQ

Atan ④ Q

I
slide-20
SLIDE 20

Open Problems.

  • Are there any torsion elements in ?
  • Are there any torsion elements in ?
  • Is it true that ?

Theorem [Conant-Teichner '04b] There is a surjective homomorphism of abelian groups . Theorem [Kontsevich '93, Bott-Taubes '94, Altschuler-Freidel '97] has a rational inverse (This is a universal additive Vassiliev invariant over Q.)

Here is the abelian group generated by rooted planar binary trees with <n leaves modulo certain relations (related to graph complexes).

Ren

: Atanas I n

a:

Ren

  • 2. n
: I n ④ IQ

Atan ④ Q

I t n

a:

Ron

: At Eos I n
slide-21
SLIDE 21

Theorem [K.] The following diagram commutes:

d

d is given by higher differentials in the Bousfield-Kan spectral sequence.

t

Ann

Ray y

B. B. Am

ikyn

Io T

slide-22
SLIDE 22

Theorem [K.] The following diagram commutes:

d

In particular, this proves one half of the Conjecture: is surjective.

d is given by higher differentials in the Bousfield-Kan spectral sequence.

t

Ann

Ray \

B.B. A.

ikyn

TOI

etn

slide-23
SLIDE 23

Theorem [K.] The following diagram commutes:

  • Corollaries. [K]
  • 1. If d is injective over some coefficient group A (i.e. the spectral sequence over A

collapses on the second page along the diagonal), then is a universal additive Vassiliev invariant over A.

  • 2. Since is rationally injective, d is as well. Hence the Conjecture is true over Q.
  • 3. Boavida-Weiss and BCKS group structures on agree.

d

In particular, this proves one half of the Conjecture: is surjective.

d is given by higher differentials in the Bousfield-Kan spectral sequence.

t

Ann

Ray \

b. B. Am

ikyn

TOI

ein

etn

Ren

l

Io T

slide-24
SLIDE 24

A large part of [K] is devoted to developing tools for proving that the two roles of trees

  • one in the finite type theory and other in the embedding calculus - are compatible.
These results can be used further for the remaining open cases of Goodwillie-Klein, as well as for studying analogues of the Dax invariant. We study the punctured knots model for for any M with dim(M) > 2. In particular we describe geometrically the fibres of the maps (the surjectivity is also new). We provide certain explicit homotopy equivalences for total fibres of cubes which have left homotopy inverses.

I

Embo(I . M)

I

am I . .
slide-25
SLIDE 25

A large part of [K] is devoted to developing tools for proving that the two roles of trees

  • one in the finite type theory and other in the embedding calculus - are compatible.
These results can be used further for the remaining open cases of Goodwillie-Klein, as well as for studying analogues of the Dax invariant.

Another important ingredient of the proof is the following theorem from our joint work.

  • Theorem. [K.-Shi-Teichner]

For a capped grope cobordism from K to K' there is a path in T from ev (K) to ev (K'). Moreover, this gives a continuous map from the space of gropes on K' to the fibre of

  • ver ev (K').
We study the punctured knots model for for any M with dim(M) > 2. In particular we describe geometrically the fibres of the maps (the surjectivity is also new). We provide certain explicit homotopy equivalences for total fibres of cubes which have left homotopy inverses.

I

Embo(I. M)

I

  • oe I . .
n n n Wn: Eeubo (I. I 3) →I n
slide-26
SLIDE 26

The crucial geometric manoeuvre for [KST]: The symmetric surgery on a capped torus

slide-27
SLIDE 27

The crucial geometric manoeuvre for [KST]: The symmetric surgery on a capped torus

slide-28
SLIDE 28

References. Current and future projects.

  • [with Shi and Teichner] Restate the injectivity part of the conjecture in terms of grope spaces.
Build in geometric IHX relations.
  • Extend the result about surjectivity of the evaluation map to any 3-manifold with boundary M.
  • Knots in the d-dimensional space: homotopy classes which are integral lifts of rational classes
  • btained from certain graph homology.
  • [with Teichner] More generally, Borromean link families: homotopy classes in spaces of links
detected rationally in corresponding graph complexes.
  • [with Teichner] Dax invariant detects non-isotopic homotopic disks with dual spheres?