A 2-category of dotted cobordisms and a universal odd link homology

XIX Oporto Meeting on Geometry, Topology and PhysicsJuly 20, 2010

Krzysztof PutyraColumbia University, New York

What is covered?Even vs odd link homologies

sketch of the constructions chronological cobordisms

Dotted cobordisms with chronologies chronologies see dots neck-cutting relation and delooping lemma

Chronological Frobenius algebras dotted cobordisms are universal

Cube of resolutions1

2

3110

101

011

100

010

001

000 111vertices

are smoothed diagrams

Observation This is a commutative diagram in a category of 1-mani-folds and cobordisms

edges are cobordis

ms

Khovanov complexEven homology (K, 1999)Apply a graded functor

i.e.

Odd homology (O R S, 2007)Apply a graded pseudo-

functor

i.e.

ModCob2:KhF ModCob2:ORSF

fFORS

XFORS

YFORS

ZFORS gfFORS

gFORS

fFKh

XFKh

YFKh

ZFKh gfFKh

gFKh

Peter Ozsvath

Mikhail Khovano

v

Result: a cube of modules with commutative faces

Result: a cube of modules with both commutative and anticommutative faces

Khovanov complex

0123 CCCC

direct sums create the complex

Theorem Homology groups of the complex C are link invariants.

Peter Ozsvath

Mikhail Khovano

v

Even: signs given explicitely

{+1+3} {+2+3} {+3+3}{+0+3}

Odd: signs given by homological properties

AA

AAAA

AA

3

233

3

000

100

010

001

110

101

011

111

Khovanov complex1

2

3

Dror Bar-NatanTheorem (B-N, 2005) The complex is a link invariant under chain homotopies and relations S/T/4Tu.

edges are cobordisms with

signs Objects: sequences of smoothed diagramsMorphisms: „matrices” of cobordisms

Even vs OddEven homology (B-N, 2005)Complexes for tangles in CobDotted cobordisms:

Neck-cutting relation:

Delooping and Gauss elimination:

Lee theory:

Odd homology (P, 2008)Complexes for tangles in

ChCob

?

??

???

????

= {-1} {+1}

= 1 = 0

= + –

Chronological cobordismsA chronology: a separative Morse function τ.

An isotopy of chronologies: a smooth homotopy H s.th. Ht is a chronology

An arrow: choice of a in/outcoming trajectory of a gradient flow of τ

Pick

one

Fact If τ0 τ1 and dimW = 2, there exist isotopies of M and I that induce an isotopy of these chronologies.

Chronological cobordisms

Theorem (P, 2008) 2ChCob with changes of chronologies is a 2-cate-gory. This category is weakly monoidal with a strict symmetry.

A change of a chronology is a smooth homotopy H. Changes H and H’ are equivalent if H0 H’0 and H1 H’1.

Remark Ht might not be a chronology for some t (so called critical moments).Fact (Cerf, 1970) Every homotopy is equivalent to a homotopy with finitely many critical moments of two types:

type I:

type II:

Chronological cobordismsCritical points cannot be permuted:

Critical points do not vanish:

Arrows cannot be reversed:

Chronological cobordismsA solution in an R-additive extension for changes:

type II: identity

Any coefficients can be replaced by 1’s by scaling:

a b

Chronological cobordismsA solution in an R-additive extension for changes:

type II: identity general type I:MM = MB = BM = BB = X X2 = 1

SS = SD = DS = DD = Y Y2 = 1SM = MD = BS = DB = ZMS = DM = SB = BD = Z-1

Corollary Let bdeg(W) = (#B #M, #D #S). Then

AB = X Y Z

where bdeg(A) = (, ) and bdeg(B) = (, ).

Chronological cobordismsA solution in an R-additive extension for changes:

type II: identity general type I:

exceptional type I:

MM = MB = BM = BB = X X2 = 1SS = SD = DS = DD = Y Y2 = 1SM = MD = BS = DB = ZMS = DM = SB = BD = Z-1

AB = X Y Z bdeg(A) = (, ) bdeg(B) = (, )

1 / XY

X / Y

even oddXYZ 1 -1YXZ 1 -1ZYX 1 -1

Chronological cobordismsA solution in an R-additive extension for changes:

type II: identity general type I:

exceptional type I: 1 / XY or X / Y

Theorem (P, 2010) With the above:• Aut(W) = {1} if #hdls(W) = 0 and #sphr(W) 1• Aut(W) = {1, XY} otherwise

MM = MB = BM = BB = X X2 = 1SS = SD = DS = DD = Y Y2 = 1SM = MD = BS = DB = ZMS = DM = SB = BD = Z-1

AB = X Y Z bdeg(A) = (, ) bdeg(B) = (, )

even oddXYZ 1 -1YXZ 1 -1ZYX 1 -1

Dotted chronological cobordismsMotivation Cutting a neck due to 4Tu:

Add dots formally and assume the usual S/D/N relations:

A chronology takes care of dots, coefficients may be derived from (N):

M M=

= 0(S)

(N) = + –

= 1(D) bdeg( ) = (-1, -1)

M = B = XZS = D = YZ-1

= XY

Z(X+Y) = +

I’m homo-geneous!

I may be 0!

Dotted chronological cobordismsMotivation Cutting a neck due to 4Tu:

Add dots formally and assume the usual S/D/N relations:

A chronology takes care of dots, coefficients may be derived from (N):

= 0(S)

(N) = + –

= 1(D) bdeg( ) = (-1, -1)

M = B = XZ S = D = YZ-1

= XYRemark T and 4Tu can be derived from S/D/N.Notice all coefficients are hidden!

Z(X+Y) = +

I’m homo-geneous!

I may be 0!

Dotted chronological cobordismsTheorem (delooping) The following morphisms are mutually

inverse:

{–1}

{+1}–

Conjecture We can use it for Gauss elimination and a divide-conquer algorithm.

Problem How to keep track on signs during Gauss elimination?

Dotted chronological cobordismsTheorem There are isomorphisms

Mor(, ) [X, Y, Z1, h, t]/(X2-1, Y2-1, (XY – 1)h, (XY – 1)t) =: R

Mor(, ) v+R v-R =: A

given by bdeg(h) = (-1, -1)bdeg(t) = (-2, -2)bdeg(v+) = ( 1, 0)bdeg(v- ) = ( 0, -1)

h

v+ v-

t

= =

left module: right module:

Dotted chronological cobordisms

Algebra/coalgebra structure: given by cobordisms

= XZ=

= XZ=

= Z2

= Operations are right-linear, but not left-linear!

Universality of dotted cobordismsA chronological Frobenius system (R, A) in A is given by a monoidal 2-functor F: 2ChCob A:

R = F()A = F( )

We further assume:• R is graded, A = Rv+ Rv is bigraded• bdeg(v+) = (1, 0) and bdeg(v) = (0, -1)

A base change: (R, A) (R', A') where A' := A R R'A twisting: (R, A) (R, A')

' (w) = (yw)' (w) = (y-1w)

where y A is invertible and deg(y) = (1, 0).

Theorem If (R, A') is a twisting of (R, A) thenC(D; A') C(D; A)

for any diagram D.

Universality of dotted cobordisms

Corollary There is no odd Lee theory:t = 1 X = Y

Corollary There is only one dot in odd theory over a field:X Y XY 1 h = t = 0

Theorem (P, 2010) Any rank 2 chronological Frobenius system with generators in degrees (1, 0) and (0, -1) arrises from (R, A) by a base change and a twisting.

Even vs OddEven homology (B-N, 2005)Complexes for tangles in Cob

Dotted cobordisms:

Neck-cutting relation:

Delooping and Gauss elimination:

Lee theory:

Odd homology (P, 2010)Complexes for tangles in ChCob

Dotted chronological cobordisms- only one dot over a field, if X Y

Neck-cutting with no coefficients

Delooping – yesGauss elimination – sign problem

Lee theory exists only for X = Y= {-1} {+1}

= 1 = 0

= + –

Further remarks Higher rank chronological Frobenius algebras may be

given as multi-graded systems with the number of degrees equal to the rank

For virtual links there still should be only two degrees, and a punctured Mobius band must have a bidegree (–½, –½)

Embedded chronological cobordisms form a (strictly) braided monoidal 2-category; same for the dotted version

The 2-category nChCob of chronological cobordisms of dimension n can be defined in the same way. Each of them is a universal extension of nCob with a strict symmetry in the sense of A.Beliakova and E.Wagner