Upgrade to Pro — share decks privately, control downloads, hide ads and more …

A Bar-Natan homotopy type - Handle Seminor '21

A Bar-Natan homotopy type - Handle Seminor '21

99e9f426c1e04a86aecb70f65f4d05ef?s=128

Taketo Sano

April 21, 2021
Tweet

Transcript

  1. A Bar - Natan homotopy type (arXiv : 2102 .

    07529 ) 2021 - 04 - 21 Handle Seminor ' 21 #l T . Same
  2. Contents 7 . Overview 2 . Preliminaries 3 . Construction

    of XBN 4 . Homotopy type of XBN 5 . Future Prospects .
  3. Overview I? khovanov homotopy type t.ee/Bar-NatanXInCL ) homotopy type spatial

    ? ftTt: red . ooh . ? IfIT' refinement deformation Khorana homology wm→ Lee homology Rasmussen inv Hiciicl ) < → SCHEI tilt. quot . Bar - Natan homology a categorisation i f Xd : gr. Euler.ch : Jones polynomial JCL)
  4. Main results . - Third . D : link diagram

    . 㱺 XBNCD) : finite CW spectrum se . ITIXBNCD) ) E CINCO) canonically) T reduced cellular t the Bar - Natan cpx of D . - Coch . cpx - Theme . XBGVCD) e Vo Alden St . htpyieq . ( the canonical cells of D . In particular , the stable htpy type of XBNCD) is a link invariant . -
  5. Contents 7 . Introduction 2 . Preliminaries 3 . Construction

    of XBN 4 . Homotopy type of XBN 5 . Future Prospects .
  6. Khovomou homology Ckhovomou . sooo ) link diagram D →

    Hii CD) cbigraded homology) 㱺 D= split nose → abeaos - ( Nif g÷ . . I I¥ { ⑦ 9 * A a ) ( Xv A- ④ A Tm Rlx3kx7@a.x ) Kds { o→ Is c'CD) d- ECD) → o ) = : Cinco)
  7. Deformations of Khorana homology - A - 131×3/1×7 → Hii

    CD) : Khorana homology - A = RANCH- t ) → Hiee CD) : tee homology Good - A = RFXVCXIX ) → HIN CD) : Bon - Natan homology (2005) Fact . When R= , Hie CD;D ) e D2 "" . X' - I = CX- 1)Htt ) . 2E D' ¥0 , For any R , Hin CD; R) E R2" " . X ' - X = Xcx - t ) . I c- R" ¥ ,
  8. Khovanov homotopy type Clipshit - Sarkar , sold > khovamr

    How category EKHCD) 1-framing ( v Cohen- Jones - SLEE . Khovomov spectrum Xkh ) Keable / It : reduced cellular htt't ✓ oooh . opx link diagram D → khovomor complex CKHLD) G- moves 8495cm .
  9. Flow categories A flow category E consists of : D

    lol : Ob Ce ) → 2 . grading function {" "mean - hid:! . :÷: the moduli space from K to y . ( a apt ca - y- D - dim mfd with corners . ~ U µ (x. z ) x NCZ , y) 3) OMG . y) = a > 㱺 y < ° ( Rmd . Here we write do B : x Is y B- z )
  10. e.ge Morse Hou category E = Curse (Mit ) .

    AR a n p b i . r d - oboe) = Grief) = { a. b. c. d } , gr - ind . 2 l l O - Mca , c) = food ) , Mlb , c) = f ? } . MK , d) = for , of } Mca , d) = • → OMG 'd) - Mla . xµk , d ) A T door
  11. Cohen - Jones - Segal constr . Ed P e

    : a flow cat C # oboes < o ) { z : e - Ede ' a large enough Euc . sp. humbly ) ) V : a framing of l with corners . Ms le f : a finite CW cpx . base Pt u { J, } gceobce ) cells . . { * } ( dims , = latte ) ( attaching : da s try x Muggy 㱺 ry collapse \ → X := Ellet - → or a finite CW spectrum . T " . s . \ tryxmcx.gl
  12. Horse moves on flow categories ( by A. Lobb et

    . al . ) E = ( e , 2 , v ) : a framed flow cat kimonos 㱺 e' = cel , i. v ' ) : another framed flow cat sit . 7- f : let → le't htpy equiv . ( D Handle cancellation ② Handle slide { ③ Whitney trick .
  13. ① Handle cancellations . v u ② Handle slides (

    = basis change ) at te g . . to ⾨ i :f÷i
  14. ③ Whitney trick . 11 A b) Mab ) "

    " Mia . g) xucy.to) Mca .sc ) xnxx.yjxncy.to)
  15. Contents 7 . Introduction 2 . Preliminaries 3 . Construction

    of XBN 4 . Homotopy type of XBN 5 . Future Prospects .
  16. (Recall ) I? khovanov homotopy type t.ee/Bar-NatanXInCL ) homotopy type

    spatial ? ftTt: red . ooh . ? IfIT' refinement deformation Khorana homology wm→ Lee homology Rasmussen inv Hii CL ) < → SCHEI tilt. quot . Bar - Natan homology a categorisation i f Xd : gr. Euler.ch : Jones polynomial JCL)
  17. Difficulties in applying the CTS - construction to Lee 1

    Bar - Natan homology ① Relations are more complicated cposets are net abia ! ) { ② Relations involves signs . 1 , merge , I split l X o_O → A → O O ' x x \ b O_O → D - * TID ' Ob x 㱺 DID Lee → " 0×0 (Recall ) A- = REX]/CxZhX - e ) . Kh : Chie) - 10,07 , Lee : Chic) - ( 0,1 ) , BN : chit ) - Cl , o )
  18. Idea . Algebraic For BN homology , when we take

    basis { X. Y } = { X. X - 13 , the operation diagonalizes as : × × merge × × split x x o_O → A → O O Y Y - Y Y y Y O_O → A → 00 Film spatial Construct EBNCD) so that OBCEBNCD) ) = { generators in X. Y } + not Mix }
  19. Construction of EBNCD) Basic relation : × × merge ×

    × split x x O_O → A → O O y y y Y y y O_O → A s 00 link diagram D → poset PBNCD) . e.ge D= ⑦ 2 2 2 2 """ ¥÷÷¥÷÷¥÷:÷÷÷l O O O O
  20. [ Prof . PBNCD) consists of cubic poets . !!::;÷j!!

    - Def . The Bar- Natan flow cat . EBNCD) is defined by : • OBCEBNCD) ) = PBNCD) - { . Mca , y ) = CHI - iyl- l ) - dim permntohedron . (0,1 ) Chl ) L T Cl , 1,1 ) • . . • • T • , L ihr v * . i in . • s o • (Oro ) Clio) ( O , 0,0 ) T • (
  21. We have : ← n- dim abettor cat . cubic

    sabot Fi Ei as EBNCD) s Eabecn ) T V F W Pi - PBNCD) > { 0.13" cubic snbposet • • • 0 • Ill • • : • • so ! ÷÷÷Fl - . . . . :* . . . . • • • By Fei ooo • ⑥ ' ' . # Eanbecn) EBNCD)
  22. - Lem . Each Ei can be framed so that

    the associated cellular chain complex satisfies : da = I C-1) skip C- lytic" 'd) pay www my "" F ( std sign assignment , , only when - y . → a - Es
  23. This together gives a framing of EBNCDJ . → XBN

    CD) := Jill EBNCD) ) : the Bon - Natan spectrum f-Thnx . Ethan:p) I. cin.co, of D . :) fry't = ¥y C- Ds" 't > C- Dti" 'd'RE is exactly : × × merge × × split x x o_O → A → O O Y Y - Y Y y y O_O → A → DO Rink . The constr . of XBN is much easier than Xkh .
  24. Contents 7 . Introduction 2 . Preliminaries 3 . Construction

    of XBN 4 . Homotopy type of XBN 5 . Future Prospects .
  25. Recall , HBNCD ) = 22 "' . The basis

    is given explicitly by the canonical cycles . { ACD , o ) / o : orientation on D } e.ge D= ④ ms = : ACD . o ) 3 more for O.O . . Fact 124303 ) are gales, and [ CBNCD) = 25243,037 ④ ( acyclic part )
  26. Similarly , one can define the canonical objects in EBNCD)

    . ¥÷÷¥i÷÷¥÷:÷÷÷l - - Perform handle cancellations ! canonical objects . - PM . By handle cancellations . EBNCD) → to { Xa CD , o ) ) c- disj . union of singletons . -
  27. The canonical cells { ICD , o ) } of

    XBNCD) one the cells corresponding to the canonical objects of EBNCD) . Tims . XBNCD) I Vo Ta CD . o ) In particular , the stable htpy type of XBNCD) / is a link invariant . We immediately recover the structure theorem : HBNCD) THE TTTXBNCD) ) THE titty ra ios ) = ⑦ 2. LACD , 037 .
  28. Contents 7 . Introduction 2 . Preliminaries 3 . Construction

    of XBN 4 . Homotopy type of XBN 5 . Future Prospects .
  29. Rasmussen's S - invariant - A 2- valued knot /

    link ( concordance ) invariant . (- Gives an alternative proof for Milnor conj 4968 ) - Also used to prove the non- sliceness of the Conway lgngot.gg , For a diagram D of a knot K , a • atp put D= ACD) , p = al- D ) . Then sck) - •• AIB > i sck) := If qcatp) + qca- p) } . where q is the quantum grading induced by the quantum filtration on CBNCD) . ( defined using {1. X ) ( Y - X- I is non - homogeneous ! )
  30. Basis change Recall (X. Y ) = Cx , X

    - i ) = Cx , 1) ( f ! ) in A . Then those dual bases correspond as : Cx' , I ' ) = CX's Yt ) (! I) = ( x 't + Yt , - Y' ) . - - A basis change can he realized by cubic handle slides : X④Y → Y④Y X'④Y - . - Y ④ Y X Y - - - Y ?'s ! : of us 'T HIT or X ④ X → Y④X X' ④ X . . . Y ④ X ' x'④ X ' - . - Y X '
  31. - Enp . Perform cubic handle slides on all vertices

    ne 50.13" . This gives a stable htpy equiv : f : XBNCD) Es XB'NCD) ← IX- based and induces : I' TXBNCDDC It Ettxpgvcb)) (X - X n Q z ( x ' - X Y - y ) = Yrs - l ) V L CBNCD) -
  32. =) Case ? O_O IF X' co ) Ya) ÷¥÷.÷.

    . i Case ? splits O g t" so , lo) (2) " " " ¥ : c-i ) Cl ) ④
  33. e.ge L = , Da - U a ⑦ cladybug

    conf . ) " ÷㱺 a :* ÷÷÷÷.÷÷÷ & soy
  34. In general , handle slides produce moduli spaces MGL ,

    y) F ¢ for qcx) s qcy) . In order to introduce a spatial ) filtration on XBN , we need to eliminate all such dcx.gs . - Conj . By an appropriate framing of EBNCD) , all Mex .gs to cqcx) a gig) ) can be eliminated by the Extended ) Whitney tricks . Moreover , we recover : Xian CD) a Fj XBNCD)/Fj - ZXBNCD) . -