discrete homotopy and homology for graphs€¦ · overview i invariants of dynamic processes:...

Discrete Homotopy and Homology Theory forMetric Spaces

Helene BarceloMathematical Sciences Research Institute

Algebraic and Combinatorial Approachesin Systems Biology — UConn

May 23, 2015

Overview

I Invariants of Dynamic Processes: Aq

(�,�o

)(Atkin, Maurer, Malle, Lovasz 1970’s)

I Discrete Homotopy Theory for Graphs

1(�,�0) ⇠= ⇡1(�q

�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�q�, x0)

I Discrete Homotopy for Cubical Sets

(�,�0)?⇠= ⇡

(X�q�, x0)

I Discrete Homology Theory

I Unexpected Application of Discrete Homotopy Theory

A1(R-Coxeter complex) ⇠= ⇡1�M(W -3-parabolic arr.)

generalizing Brieskorn’s results for C-hyperbolic arrangement

Overview

(�,�o

1(�,�0) ⇠= ⇡1(�q

�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�q�, x0)

(�,�0)?⇠= ⇡

(X�q�, x0)

Overview

(�,�o

1(�,�0) ⇠= ⇡1(�q

�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�q�, x0)

(�,�0)?⇠= ⇡

(X�q�, x0)

Overview

(�,�o

1(�,�0) ⇠= ⇡1(�q

�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�q�, x0)

(�,�0)?⇠= ⇡

(X�q�, x0)

Overview

I Invariants of Dynamic Processes: Aqn(∆, σo)

(Atkin, Maurer, Malle, Lovasz 1970’s)

Aq1(∆, σ0) ∼= π1(Γq

∆, v0)/N(3, 4 cycles) ∼= π1(XΓq∆, x0)

Aqn(∆, σ0)

?∼= πn(XΓq∆, x0)

A1(R-Coxeter complex) ∼= π1

(M(W -3-parabolic arr.)

)generalizing Brieskorn’s results for C-parabolic arrangement

Overview

I Invariants of Dynamic Processes: Aqn(∆, σo)

(Atkin, Maurer, Malle, Lovasz 1970’s)

Aq1(∆, σ0) ∼= π1(Γq

∆, v0)/N(3, 4 cycles) ∼= π1(XΓq∆, x0)

Aqn(∆, σ0)

?∼= πn(XΓq∆, x0)

A1(R-Coxeter complex) ∼= π1

(M(W -3-parabolic arr.)

)generalizing Brieskorn’s results for C-parabolic arrangement

Discrete Homotopy Theory for Graphs

(Babson, B., Kramer, de Longueville, Laubenbacher, Severs, Weaver, White)

Definitions

1. � - graph (� simplicial complex; X metric space)

v0 - distinguished vertex (�0; x0)

- infinite lattice (usual metric)

(�, v0) - set of graph homs f : Zn ! V (�), that is,

if d(~a, ~b ) = 1 in Zn then d(f (~a), f (~b)) = 0 or 1, with

f (~i ) = v0 almost everywhere

3. f , g are discrete homotopic if there exist h 2 An+1(�, v0) and k, ` 2 N

such that for all

~i 2 Zn

~i , k) = f (

~i , ` ) = g(

(�, v0) - set of equivalence classes of maps in An

(�, v0)Note: translation preserves discrete homotopy

Definitions

such that for all

~i 2 Zn

~i , k) = f (

~i , ` ) = g(

Definitions

such that for all

~i 2 Zn

~i , k) = f (

~i , ` ) = g(

Definitions

such that for all

~i 2 Zn

~i , k) = f (

~i , ` ) = g(

Definitions

such that for all

~i 2 Zn

~i , k) = f (

~i , ` ) = g(

Definitions

such that for all

~i 2 Zn

~i , k) = f (

~i , ` ) = g(

Group Structure

I Multiplication: for f , g 2 An

(�, v0) of radius M,N,

f g (~i ) =

(f (~i ) i1 M

g(i1 � (M + N), i2, . . . in) i1 > M

v0 f v0 g v0

[f g ] = [f ] [g ]

Group Structure

f g (~i ) =

(f (~i ) i1 M

g(i1 � (M + N), i2, . . . in) i1 > M

v0 f v0 g v0

[f g ] = [f ] [g ]

Group Structure

f g (~i ) =

(f (~i ) i1 M

g(i1 � (M + N), i2, . . . in) i1 > M

v0 f v0 g v0

[f g ] = [f ] [g ]

Group Structure

f g (~i ) =

(f (~i ) i1 M

g(i1 � (M + N), i2, . . . in) i1 > M

v0 f v0 g v0

[f g ] = [f ] [g ]

Group Structure

I Identity: e(~i ) = v0 8~i 2 Zn

I Inverses: f �1(~i ) = f (�i1, . . . , in) 8~i 2 Zn

Example (n = 2)

1 d e t i n U0 m E b a r A

�1 s e t a r i

�2 �1 0 1 2 3

f �1 :

1 U n i t e d0 A r a b E m

�1 i r a t e s

�2 �1 0 1 2 3

Group Structure

Example (n = 2)

�1 s e t a r i

�2 �1 0 1 2 3

f �1 :

�1 i r a t e s

�2 �1 0 1 2 3

Group Structure

Example (n = 2)

�1 s e t a r i

�2 �1 0 1 2 3

f �1 :

�1 i r a t e s

�2 �1 0 1 2 3

Group Structure

I Identity: e(~i ) = v0 ∀~i ∈ Zn

I Inverses: f −1(~i ) = f (−i1, . . . , in) ∀~i ∈ Zn

Example (n = 2)

1 r a h n i e R0 n e b u a L d−1 60 r e h c a b

−3 −2 −1 0 1 2 3

f −1 :

1 R e i n h a r0 d L a u b e n−1 b a c h e r 60

−3 −2 −1 0 1 2 3

Group Structure

I Identity: e(~i ) = v0 ∀~i ∈ Zn

I Inverses: f −1(~i ) = f (−i1, . . . , in) ∀~i ∈ Zn

Example (n = 2)

1 r a h n i e R0 n e b u a L d−1 60 r e h c a b

−3 −2 −1 0 1 2 3

f −1 :

1 R e i n h a r0 d L a u b e n−1 b a c h e r 60

−3 −2 −1 0 1 2 3

Group StructureAn

(�, v0) is an abelian group 8 n � 2

f g ⇠g

f⇠ g f

Group StructureAn

(�, v0) is an abelian group 8 n � 2

f g ⇠g

f⇠ g f

Examples

⇣v0 v1 , v0

⌘= 1

, v0⌘= 1

v0 v1 v2 v0

v0 v0 v0 v0

⇣v3 v2

, v0⌘= 1

v0 v1 v2 v3 v0

v0 v0 v1 v0 v0

v0 v0 v0 v0 v0

⇣, v0

⌘⇠= Z

A1(�, v0) ⇠= ⇡1(�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�, v0)

(2-dim cell complex: attach 2-cells to 4, ⇤ of �)

Examples

⇣v0 v1 , v0

⌘= 1

, v0⌘= 1

v0 v1 v2 v0

v0 v0 v0 v0

⇣v3 v2

, v0⌘= 1

v0 v1 v2 v3 v0

v0 v0 v1 v0 v0

v0 v0 v0 v0 v0

⇣, v0

⌘⇠= Z

A1(�, v0) ⇠= ⇡1(�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�, v0)

Examples

⇣v0 v1 , v0

⌘= 1

, v0⌘= 1

v0 v1 v2 v0

v0 v0 v0 v0

⇣v3 v2

, v0⌘= 1

v0 v1 v2 v3 v0

v0 v0 v1 v0 v0

v0 v0 v0 v0 v0

⇣, v0

⌘⇠= Z

A1(�, v0) ⇠= ⇡1(�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�, v0)

Examples

⇣v0 v1 , v0

⌘= 1

, v0⌘= 1

v0 v1 v2 v0

v0 v0 v0 v0

⇣v3 v2

, v0⌘= 1

v0 v1 v2 v3 v0

v0 v0 v1 v0 v0

v0 v0 v0 v0 v0

⇣, v0

⌘⇠= Z

A1(�, v0) ⇠= ⇡1(�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�, v0)

Examples

⇣v0 v1 , v0

⌘= 1

, v0⌘= 1

v0 v1 v2 v0

v0 v0 v0 v0

⇣v3 v2

, v0⌘= 1

v0 v1 v2 v3 v0

v0 v0 v1 v0 v0

v0 v0 v0 v0 v0

⇣, v0

⌘⇠= Z

A1(�, v0) ⇠= ⇡1(�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�, v0)

Examples

⇣v0 v1 , v0

⌘= 1

, v0⌘= 1

v0 v1 v2 v0

v0 v0 v0 v0

⇣v3 v2

, v0⌘= 1

v0 v1 v2 v3 v0

v0 v0 v1 v0 v0

v0 v0 v0 v0 v0

⇣, v0

⌘⇠= Z

A1(�, v0) ⇠= ⇡1(�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�, v0)

Examples

⇣v0 v1 , v0

⌘= 1

, v0⌘= 1

v0 v1 v2 v0

v0 v0 v0 v0

⇣v3 v2

, v0⌘= 1

v0 v1 v2 v3 v0

v0 v0 v1 v0 v0

v0 v0 v0 v0 v0

⇣, v0

⌘⇠= Z

A1(�, v0) ⇠= ⇡1(�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�, v0)

Examples

⇣v0 v1 , v0

⌘= 1

, v0⌘= 1

v0 v1 v2 v0

v0 v0 v0 v0

⇣v3 v2

, v0⌘= 1

v0 v1 v2 v3 v0

v0 v0 v1 v0 v0

v0 v0 v0 v0 v0

⇣, v0

⌘⇠= Z

A1(�, v0) ⇠= ⇡1(�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�, v0)

Examples

⇣v0 v1 , v0

⌘= 1

, v0⌘= 1

v0 v1 v2 v0

v0 v0 v0 v0

⇣v3 v2

, v0⌘= 1

v0 v1 v2 v3 v0

v0 v0 v1 v0 v0

v0 v0 v0 v0 v0

⇣, v0

⌘⇠= Z

A1(�, v0) ⇠= ⇡1(�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�, v0)

Discrete Homotopy Theory

(�,�0) ⇠= An

(�q�,�0)�q� vertices = all maximal simplices of � of dim� q(�,�0) 2 E (�q�) () dim(� \ �0) � q

(X , x0) r -Lipschitz maps f : Zn ! X (stabilizing in alldirections)

f : X ! Y is r -Lipschitz () d(f (x1), f (x2)) r d(x1, x2)

Discrete Homotopy Theory

(�,�0) ⇠= An

(�q�,�0)�q� vertices = all maximal simplices of � of dim� q(�,�0) 2 E (�q�) () dim(� \ �0) � q

(X , x0) r -Lipschitz maps f : Zn ! X (stabilizing in alldirections)

f : X ! Y is r -Lipschitz () d(f (x1), f (x2)) r d(x1, x2)

Is it a Good Analogy to Classical Homotopy?

1. If � is connected, An

(�, v0)independent of v0

2. Siefert-van Kampen: if

� = �1 [ �2�i

connectedv0 2 �1 \ �2�1 \ �2 connected4, ⇤ lie in one of the �

A1(�, v0) ⇠= A1(�1, v0) ⇤ A1(�2, v0)/N([`] ⇤ [`]�1)

for ` a loop in �1 \ �2

3. Relative discrete homotopy theory and long exact sequences

4. Associated discrete homology theory. . . ?

(�, v0)independent of v02. Siefert-van Kampen: if

� = �1 [ �2�i

A1(�, v0) ⇠= A1(�1, v0) ⇤ A1(�2, v0)/N([`] ⇤ [`]�1)

� = �1 [ �2�i

A1(�, v0) ⇠= A1(�1, v0) ⇤ A1(�2, v0)/N([`] ⇤ [`]�1)

� = �1 [ �2�i

A1(�, v0) ⇠= A1(�1, v0) ⇤ A1(�2, v0)/N([`] ⇤ [`]�1)

� = �1 [ �2�i

A1(�, v0) ⇠= A1(�1, v0) ⇤ A1(�2, v0)/N([`] ⇤ [`]�1)

Discrete Homology Theory for Graphs

(B., Capraro, White)

Necessities

1. Discrete n-dim cube Q

= {(a1, . . . , an) | ai = 0 or 1}2. Singular n-cube � : Q

! � graph homomorphism

(�) := free abelian group generated by all singular n-cubes �

I i th front and back faces of � are singular (n � 1)-cubesI Front: (An

�)(a1, . . . , an�1) = �(a1, . . . , ai�1, 0, ai , . . . , an�1)I Back: (Bn

�)(b1, . . . , bn�1) = �(b1, . . . , bi�1, 1, bi , . . . , bn�1)I Degenerate singular n-cube: if 9 i s.t. (An

�) = (Bn

�)I D

(�) := free abelian group generated by all degeneratesingular n-cubes

(�) := Ln

(�)/Dn

elements of C

correspond to n-chains

Necessities

�) = (Bn

�)I D

(�) := Ln

(�)/Dn

elements of C

Necessities

= {(a1, . . . , an) | ai = 0 or 1}

2. Singular n-cube � : Qn

�) = (Bn

�)I D

(�) := Ln

(�)/Dn

elements of C

Necessities

�) = (Bn

�)I D

(�) := Ln

(�)/Dn

elements of C

Necessities

�) = (Bn

�)I D

(�) := Ln

(�)/Dn

elements of C

Necessities

I i th front and back faces of � are singular (n � 1)-cubes

I Front: (An

�) = (Bn

�)I D

(�) := Ln

(�)/Dn

elements of C

Necessities

�)(a1, . . . , an�1) = �(a1, . . . , ai�1, 0, ai , . . . , an�1)

I Back: (Bn

�) = (Bn

�)I D

(�) := Ln

(�)/Dn

elements of C

Necessities

�)(b1, . . . , bn�1) = �(b1, . . . , bi�1, 1, bi , . . . , bn�1)

I Degenerate singular n-cube: if 9 i s.t. (An

�) = (Bn

�)I D

(�) := Ln

(�)/Dn

elements of C

Necessities

�) = (Bn

(�) := Ln

(�)/Dn

elements of C

Necessities

�) = (Bn

�)I D

(�) := Ln

(�)/Dn

elements of C

Necessities

�) = (Bn

�)I D

(�) := Ln

(�)/Dn

elements of C

Necessities

4. Boundary operators @n

for each n � 1

(�) =nX

(�1)i�An

(�)� Bn

(�)�

I extend linearly to Ln

(�)I @

(�)) ✓ Dn�1(�)

I so @n

(�) ! Cn�1(�) is well-defined

� @n+1 = 0

Necessities

for each n � 1

(�) =nX

(�1)i�An

(�)� Bn

(�)�

(�)) ✓ Dn�1(�)

I so @n

� @n+1 = 0

Necessities

for each n � 1

(�) =nX

(�1)i�An

(�)� Bn

(�)�

(�)I @

(�)) ✓ Dn�1(�)

I so @n

� @n+1 = 0

Necessities

for each n � 1

(�) =nX

(�1)i�An

(�)� Bn

(�)�

(�)I @

(�)) ✓ Dn�1(�)

I so @n

� @n+1 = 0

Necessities

for each n � 1

(�) =nX

(�1)i�An

(�)� Bn

(�)�

(�)I @

(�)) ✓ Dn�1(�)

I so @n

� @n+1 = 0

DefinitionThe discrete homology groups of �:

(�) = Ker(@n

)/Im(@n+1)

Examples

(�) = 0 8 n � 1 DHn

(4) = 0 8 n � 1

(⇤) = 0 8 n � 1 DH1( ) = Z

DefinitionIf �0 ✓ �, then @

(�0)) ✓ Cn�1(�

0) and there are maps

(�, �0) = Cn

(�)/Cn

(�0) ! Cn�1(�, �

The relative homology groups of (�, �0):

(�, �0) = Ker(@n

)/Im(@n+1)

(�) = Ker(@n

)/Im(@n+1)

Examples

(�) = 0 8 n � 1 DHn

(4) = 0 8 n � 1

(⇤) = 0 8 n � 1 DH1( ) = Z

(�0)) ✓ Cn�1(�

(�, �0) = Cn

(�)/Cn

(�0) ! Cn�1(�, �

(�, �0) = Ker(@n

)/Im(@n+1)

(�) = Ker(@n

)/Im(@n+1)

Examples

(�) = 0 8 n � 1

(4) = 0 8 n � 1

(⇤) = 0 8 n � 1 DH1( ) = Z

(�0)) ✓ Cn�1(�

(�, �0) = Cn

(�)/Cn

(�0) ! Cn�1(�, �

(�, �0) = Ker(@n

)/Im(@n+1)

(�) = Ker(@n

)/Im(@n+1)

Examples

(�) = 0 8 n � 1 DHn

(4) = 0 8 n � 1

(⇤) = 0 8 n � 1 DH1( ) = Z

(�0)) ✓ Cn�1(�

(�, �0) = Cn

(�)/Cn

(�0) ! Cn�1(�, �

(�, �0) = Ker(@n

)/Im(@n+1)

(�) = Ker(@n

)/Im(@n+1)

Examples

(�) = 0 8 n � 1 DHn

(4) = 0 8 n � 1

(⇤) = 0 8 n � 1

DH1( ) = Z

(�0)) ✓ Cn�1(�

(�, �0) = Cn

(�)/Cn

(�0) ! Cn�1(�, �

(�, �0) = Ker(@n

)/Im(@n+1)

(�) = Ker(@n

)/Im(@n+1)

Examples

(�) = 0 8 n � 1 DHn

(4) = 0 8 n � 1

(⇤) = 0 8 n � 1 DH1( !

(�0)) ✓ Cn�1(�

(�, �0) = Cn

(�)/Cn

(�0) ! Cn�1(�, �

(�, �0) = Ker(@n

)/Im(@n+1)

(�) = Ker(@n

)/Im(@n+1)

Examples

(�) = 0 8 n � 1 DHn

(4) = 0 8 n � 1

(⇤) = 0 8 n � 1 DH1( !

(�0)) ✓ Cn�1(�

(�, �0) = Cn

(�)/Cn

(�0) ! Cn�1(�, �

(�, �0) = Ker(@n

)/Im(@n+1)

(�) = Ker(@n

)/Im(@n+1)

Examples

(�) = 0 8 n � 1 DHn

(4) = 0 8 n � 1

(⇤) = 0 8 n � 1 DH1( !

(�0)) ✓ Cn�1(�

(�, �0) = Cn

(�)/Cn

(�0) ! Cn�1(�, �

(�, �0) = Ker(@n

)/Im(@n+1)

How to Judge if Homology Theory is Good?

1. Hurewicz Theorem: DH1(�) ⇠= Aab1 (�) n � 2

2. Discrete version of Eilenberg-Steenrod axioms andMayer-Vietoris sequence:

Discrete open cover: n-dim discrete cover of � (or �, X ) is afamily of subgraphs (or subsets) of � such that:

(i) � =

(ii) for each non-degenerate n-cube �, we have � 2 �

for some i

(iii) Discrete cover is an n-dim cover for each n

A. Mayer-Vietoris sequence:

· · · @⇤�! DHn

(�1 \ �2)diag��! DH

(�1)� DHn

(�2)di↵��! DH

(�1 [ �2)@⇤�! DH

n�1(�1 \ �2) · · ·

(i) � =

for some i

· · · @⇤�! DHn

(�1 \ �2)diag��! DH

(�1)� DHn

(�2)di↵��! DH

(�1 [ �2)@⇤�! DH

n�1(�1 \ �2) · · ·

(i) � =

for some i

· · · @⇤�! DHn

(�1 \ �2)diag��! DH

(�1)� DHn

(�2)di↵��! DH

(�1 [ �2)@⇤�! DH

n�1(�1 \ �2) · · ·

(i) � =

for some i

· · · @⇤�! DHn

(�1 \ �2)diag��! DH

(�1)� DHn

(�2)di↵��! DH

(�1 [ �2)@⇤�! DH

n�1(�1 \ �2) · · ·

(i) � =

for some i

· · · @⇤�! DHn

(�1 \ �2)diag��! DH

(�1)� DHn

(�2)di↵��! DH

(�1 [ �2)@⇤�! DH

n�1(�1 \ �2) · · ·

(i) � =

for some i

· · · @⇤�! DHn

(�1 \ �2)diag��! DH

(�1)� DHn

(�2)di↵��! DH

(�1 [ �2)@⇤�! DH

n�1(�1 \ �2) · · ·

(i) � =

for some i

· · · @⇤�! DHn

(�1 \ �2)diag��! DH

(�1)� DHn

(�2)di↵��! DH

(�1 [ �2)@⇤�! DH

n�1(�1 \ �2) · · ·

(i) � =

for some i

· · · @⇤�! DHn

(�1 \ �2)diag��! DH

(�1)� DHn

(�2)di↵��! DH

(�1 [ �2)@⇤�! DH

n�1(�1 \ �2) · · ·

B. Eilenberg-Steenrod axioms:

I Homotopy: Iff , g : (�, �1) ! (�0, �01)

are discrete homotopic maps then their induced maps onhomology are the same

I Excision:DH⇤(�2, �1 \ �2) ⇠= DH⇤(�, �1)

if � = �1 [ �2 is a discrete coverI Dimension:

( • , ;) = {0} 8 n � 1

I Long exact sequence:

· · · ! DHn

(�0) ,! DHn

(�) ,! DHn

(�, �0)@⇤�! DH

n�1(�0) · · ·

B. Eilenberg-Steenrod axioms:I Homotopy: If

f , g : (�, �1) ! (�0, �01)

I Excision:DH⇤(�2, �1 \ �2) ⇠= DH⇤(�, �1)

( • , ;) = {0} 8 n � 1

· · · ! DHn

(�0) ,! DHn

(�) ,! DHn

(�, �0)@⇤�! DH

n�1(�0) · · ·

f , g : (�, �1) ! (�0, �01)

I Excision:DH⇤(�2, �1 \ �2) ⇠= DH⇤(�, �1)

if � = �1 [ �2 is a discrete cover

I Dimension:DH

( • , ;) = {0} 8 n � 1

· · · ! DHn

(�0) ,! DHn

(�) ,! DHn

(�, �0)@⇤�! DH

n�1(�0) · · ·

f , g : (�, �1) ! (�0, �01)

I Excision:DH⇤(�2, �1 \ �2) ⇠= DH⇤(�, �1)

( • , ;) = {0} 8 n � 1

· · · ! DHn

(�0) ,! DHn

(�) ,! DHn

(�, �0)@⇤�! DH

n�1(�0) · · ·

f , g : (�, �1) ! (�0, �01)

I Excision:DH⇤(�2, �1 \ �2) ⇠= DH⇤(�, �1)

( • , ;) = {0} 8 n � 1

· · · ! DHn

(�0) ,! DHn

(�) ,! DHn

(�, �0)@⇤�! DH

n�1(�0) · · ·

How to Judge if Homology Theory is Good?C. Which groups can we obtain?

I For a fine enough rectangulation R of a compact, metrizable,smooth, path-connected manifold M, let �

be the naturalgraph associated to R . Then

⇡1(M) ⇠= A1(�R)

+ (+ suspension)

I For each abelian group G and n 2 N, there is a finiteconnected simple graph � such that

(�) =

(G if n = n

0 if n n

I There is a graph Sn such that

(Sn) =

(Z if k = n

0 if k 6= n

⇡1(M) ⇠= A1(�R)

+ (+ suspension)

(�) =

(G if n = n

0 if n n

(Sn) =

(Z if k = n

0 if k 6= n

⇡1(M) ⇠= A1(�R)

+ (+ suspension)

(�) =

(G if n = n

0 if n n

(Sn) =

(Z if k = n

0 if k 6= n

⇡1(M) ⇠= A1(�R)

+ (+ suspension)

(�) =

(G if n = n

0 if n n

(Sn) =

(Z if k = n

0 if k 6= n

⇡1(M) ⇠= A1(�R)

+ (+ suspension)

(�) =

(G if n = n

0 if n n

(Sn) =

(Z if k = n

0 if k 6= n

Unexpected Application of Discrete Homotopy Theory

Complex K (⇡, 1) Spaces Real K (⇡, 1) Spaces

ACn,2 braid arrangement:�~z 2 Cn

�� zi

, i < j

ARn,3 3-equal subspace arr:�~x 2 Rn

�� xi

, i < j < k

M(ACn,2) is K (⇡, 1)

(Fadell-Neuwirth 1962)M(AR

n,3) is K (⇡, 1)(Khovanov 1996)

⇡1(M(ACn,2))

⇠= pure braid gp.(Fox-Fadell 1963)

⇡1(M(ARn,3))

⇠= pure triplet gp.(Khovanov 1996)

⇡1(M(C-ified refl. arr. type W ))⇠= pure Artin group⇠= Ker(�)(Brieskorn 1971)

⇡1(M(Wn,3)) ⇠= Ker(�0)

where Wn,3 is a 3-parabolic

subgroup of type W(B-Severs-White 2009)

Complex K (⇡, 1) Spaces

Real K (⇡, 1) Spaces

�� zi

, i < j

�� xi

, i < j < k

M(ACn,2) is K (⇡, 1)

n,3) is K (⇡, 1)(Khovanov 1996)

⇡1(M(ACn,2))

⇡1(M(ARn,3))

⇡1(M(Wn,3)) ⇠= Ker(�0)

�� zi

, i < j

�� xi

, i < j < k

M(ACn,2) is K (⇡, 1)

n,3) is K (⇡, 1)(Khovanov 1996)

⇡1(M(ACn,2))

⇡1(M(ARn,3))

⇡1(M(Wn,3)) ⇠= Ker(�0)

�� zi

, i < j

�� xi

, i < j < k

M(ACn,2) is K (⇡, 1)

(Fadell-Neuwirth 1962)

M(ARn,3) is K (⇡, 1)

(Khovanov 1996)

⇡1(M(ACn,2))

⇡1(M(ARn,3))

⇡1(M(Wn,3)) ⇠= Ker(�0)

�� zi

, i < j

�� xi

, i < j < k

M(ACn,2) is K (⇡, 1)

n,3) is K (⇡, 1)(Khovanov 1996)

⇡1(M(ACn,2))

⇡1(M(ARn,3))

⇡1(M(Wn,3)) ⇠= Ker(�0)

�� zi

, i < j

�� xi

, i < j < k

M(ACn,2) is K (⇡, 1)

n,3) is K (⇡, 1)(Khovanov 1996)

⇡1(M(ACn,2))

⇡1(M(ARn,3))

⇡1(M(Wn,3)) ⇠= Ker(�0)

�� zi

, i < j

�� xi

, i < j < k

M(ACn,2) is K (⇡, 1)

n,3) is K (⇡, 1)(Khovanov 1996)

⇡1(M(ACn,2))

⇡1(M(ARn,3))

⇡1(M(Wn,3)) ⇠= Ker(�0)

�� zi

, i < j

�� xi

, i < j < k

M(ACn,2) is K (⇡, 1)

n,3) is K (⇡, 1)(Khovanov 1996)

⇡1(M(ACn,2))

⇡1(M(ARn,3))

⇡1(M(Wn,3)) ⇠= Ker(�0)

�� zi

, i < j

�� xi

, i < j < k

M(ACn,2) is K (⇡, 1)

n,3) is K (⇡, 1)(Khovanov 1996)

⇡1(M(ACn,2))

⇡1(M(ARn,3))

⇡1(M(Wn,3)) ⇠= Ker(�0)

�� zi

, i < j

�� xi

, i < j < k

M(C-ified refl. arr.) is K (⇡, 1)(Deligne 1972)

M(Wn,3) is K (⇡, 1)

(Davis-Janusz.-Scott 2008)

Theorem

An�k+11 (Coxeter complex W ) ⇠= ⇡1(M(W

n,k)) 3 k n

Note: An�k+11

⇠= ⇡1 ⇠= 1 for k > 3

�� zi

, i < j

�� xi

, i < j < k

M(Wn,3) is K (⇡, 1)

Theorem

n,k)) 3 k n

Note: An�k+11

⇠= ⇡1 ⇠= 1 for k > 3

�� zi

, i < j

�� xi

, i < j < k

M(Wn,3) is K (⇡, 1)

Theorem

n,k)) 3 k n

Note: An�k+11

⇠= ⇡1 ⇠= 1 for k > 3

�� zi

, i < j

�� xi

, i < j < k

M(Wn,3) is K (⇡, 1)

Theorem

n,k)) 3 k n

Note: An�k+11

⇠= ⇡1 ⇠= 1 for k > 3

Preparation for Proof

1. Presentation of a Coxeter group (W , S)

(i) s2 = 1 for s 2 S(ii) (st)2 = 1 for s, t such that m(s, t) = 2(iii) (st)3 = 1 for s, t such that m(s, t) = 3

2. Braid group: “Sn

� (i)” i.e.

)2 = 1 (si

si+1si )

3. Pure braid gp: Ker(�), where � : “Sn

� (i)”! Sn

by �(si

) = si

⇡1(M(ACn,2))

⇠= Ker(�)

1. Presentation of a Coxeter group (W , S)

(i) s2 = 1 for s 2 S(ii) (st)2 = 1 for s, t such that m(s, t) = 2(iii) (st)3 = 1 for s, t such that m(s, t) = 3

� (i)” i.e.

)2 = 1 (si

si+1si )

� (i)”! Sn

by �(si

) = si

⇡1(M(ACn,2))

⇠= Ker(�)

1. Presentation of a Coxeter group (W , S)(i) s2 = 1 for s 2 S

(ii) (st)2 = 1 for s, t such that m(s, t) = 2(iii) (st)3 = 1 for s, t such that m(s, t) = 3

� (i)” i.e.

)2 = 1 (si

si+1si )

� (i)”! Sn

by �(si

) = si

⇡1(M(ACn,2))

⇠= Ker(�)

1. Presentation of a Coxeter group (W , S)(i) s2 = 1 for s 2 S(ii) (st)2 = 1 for s, t such that m(s, t) = 2

(iii) (st)3 = 1 for s, t such that m(s, t) = 3...

� (i)” i.e.

)2 = 1 (si

si+1si )

� (i)”! Sn

by �(si

) = si

⇡1(M(ACn,2))

⇠= Ker(�)

1. Presentation of a Coxeter group (W , S)(i) s2 = 1 for s 2 S(ii) (st)2 = 1 for s, t such that m(s, t) = 2(iii) (st)3 = 1 for s, t such that m(s, t) = 3

� (i)” i.e.

)2 = 1 (si

si+1si )

� (i)”! Sn

by �(si

) = si

⇡1(M(ACn,2))

⇠= Ker(�)

� (i)” i.e.

)2 = 1 (si

si+1si )

� (i)”! Sn

by �(si

) = si

⇡1(M(ACn,2))

⇠= Ker(�)

� (i)” i.e.

)2 = 1 (si

si+1si )

� (i)”! Sn

by �(si

) = si

⇡1(M(ACn,2))

⇠= Ker(�)

� (i)” i.e.

)2 = 1 (si

si+1si )

� (i)”! Sn

by �(si

) = si

⇡1(M(ACn,2))

⇠= Ker(�)

4. k-parabolic arrangement (generalization of k-equalarrangement of type W )

I P - a collection of rank k � 1 parabolic subgroups of W closedunder conjugation

= {Fix(G ) | G 2 P}I W 0 - new group with same generators as W and subject to

m0(s, t) =

(1 if hs, ti 2 Pm(s, t) otherwise

Note: if k = 3 we have hs, si /2 P, and if |i � j | � 2 we havehs

i /2 P5. W 0 =“W � {(iii),(iv), . . .}” =) ⇡1(M(W

n,3)) ⇠= Ker(�0)

m0(s, t) =

i /2 P5. W 0 =“W � {(iii),(iv), . . .}” =) ⇡1(M(W

n,3)) ⇠= Ker(�0)

= {Fix(G ) | G 2 P}

I W 0 - new group with same generators as W and subject to

m0(s, t) =

i /2 P5. W 0 =“W � {(iii),(iv), . . .}” =) ⇡1(M(W

n,3)) ⇠= Ker(�0)

m0(s, t) =

i /2 P5. W 0 =“W � {(iii),(iv), . . .}” =) ⇡1(M(W

n,3)) ⇠= Ker(�0)

m0(s, t) =

i /2 P

5. W 0 =“W � {(iii),(iv), . . .}” =) ⇡1(M(Wn,3)) ⇠= Ker(�0)

m0(s, t) =

i /2 P5. W 0 =“W � {(iii),(iv), . . .}” =) ⇡1(M(W

n,3)) ⇠= Ker(�0)

Essence of Proof

A. Bjorner-Ziegler: given a simplicial decomposition � of Sk

�0 a sub-complex of �

+9 regular CW-complex X s.t. ⇡1(X ) ⇠= ⇡1(�/�0)

B. Intersect Sn�1 with hyperplane arrangement W simplicialdecomposition of Sn�1

�0 the 3-parabolic subspace arrangement of type W

+⇡1(�/�0) ⇠= ⇡1(X )

Essence of Proof

+⇡1(�/�0) ⇠= ⇡1(X )

Essence of Proof

+⇡1(�/�0) ⇠= ⇡1(X )

Essence of Proof

What is X?

I The W -permutahedron is the Minkowski sum of unit linesegments ? to hyperplanes of W

I Its 2-skeleton has:

vertices w 2 W

edges (w ,ws), where s is a simple reflection

2-faces are bounded by cycles (w ,ws,wst, . . . ,w(st)

m(s,t))

4-cycles (st)

2= 1 (s and t commute)

6-cycles (st)

8-cycles (st)

I X is the subcomplex of the W -permutahedron gotten byremoving the faces corresponding to �0,

i.e. removing thefaces bounded by 6-cycles, 8-cycles,. . .

Essence of Proof

What is X?I The W -permutahedron is the Minkowski sum of unit line

segments ? to hyperplanes of W

I Its 2-skeleton has:

vertices w 2 W

m(s,t))

4-cycles (st)

6-cycles (st)

8-cycles (st)

Essence of Proof

segments ? to hyperplanes of WI Its 2-skeleton has:

vertices w 2 W

m(s,t))

4-cycles (st)

6-cycles (st)

8-cycles (st)

Essence of Proof

vertices w 2 W

m(s,t))

4-cycles (st)

6-cycles (st)

8-cycles (st)

Essence of Proof

vertices w 2 W

m(s,t))

4-cycles (st)

6-cycles (st)

8-cycles (st)

Essence of Proof

vertices w 2 W

m(s,t))

4-cycles (st)

6-cycles (st)

8-cycles (st)

Essence of Proof

vertices w 2 W

m(s,t))

4-cycles (st)

6-cycles (st)

8-cycles (st)

Essence of Proof

vertices w 2 W

m(s,t))

4-cycles (st)

6-cycles (st)

8-cycles (st)

I X is the subcomplex of the W -permutahedron gotten byremoving the faces corresponding to �0, i.e. removing thefaces bounded by 6-cycles, 8-cycles,. . .

Essence of Proof

C. ⇡1(X ) ⇠= ⇡1(2-skeleton of X )

⇠= ⇡1(1-skeleton of X )/N(3, 4 cycles)⇠= A1(X )

D. Let W1 := “W � {(ii),(iii),(iv),. . . }” (keep involutions only)and let � : W1 ! W by �(s

) = si

Ker(�) ⇠= ⇡1(1-skeleton of X )

= ⇡1(1-skeleton of W -permutahedron)

E. Ker(�)/N ⇠= Ker(�0)

where �0 : “W � {(iii),(iv),. . . }”! W by �(si

) = si

Essence of Proof

C. ⇡1(X ) ⇠= ⇡1(2-skeleton of X )⇠= ⇡1(1-skeleton of X )/N(3, 4 cycles)

⇠= A1(X )

) = si

E. Ker(�)/N ⇠= Ker(�0)

) = si

Essence of Proof

C. ⇡1(X ) ⇠= ⇡1(2-skeleton of X )⇠= ⇡1(1-skeleton of X )/N(3, 4 cycles)⇠= A1(X )

) = si

E. Ker(�)/N ⇠= Ker(�0)

) = si

Essence of Proof

) = si

E. Ker(�)/N ⇠= Ker(�0)

) = si

Essence of Proof

) = si

E. Ker(�)/N ⇠= Ker(�0)

) = si

Essence of Proof

) = si

E. Ker(�)/N ⇠= Ker(�0)

) = si

Essence of Proof

) = si

E. Ker(�)/N ⇠= Ker(�0)

) = si

We have replaced a group (⇡1) defined in terms of the topology ofa space with a group (A1) defined in terms of the combinatorialstructure of the space.

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