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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
n
(�,�o
)(Atkin, Maurer, Malle, Lovasz 1970’s)
I Discrete Homotopy Theory for Graphs
Aq
1(�,�0) ⇠= ⇡1(�q
�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�q�, x0)
I Discrete Homotopy for Cubical Sets
Aq
n
(�,�0)?⇠= ⇡
n
(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
I Invariants of Dynamic Processes: Aq
n
(�,�o
)(Atkin, Maurer, Malle, Lovasz 1970’s)
I Discrete Homotopy Theory for Graphs
Aq
1(�,�0) ⇠= ⇡1(�q
�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�q�, x0)
I Discrete Homotopy for Cubical Sets
Aq
n
(�,�0)?⇠= ⇡
n
(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
I Invariants of Dynamic Processes: Aq
n
(�,�o
)(Atkin, Maurer, Malle, Lovasz 1970’s)
I Discrete Homotopy Theory for Graphs
Aq
1(�,�0) ⇠= ⇡1(�q
�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�q�, x0)
I Discrete Homotopy for Cubical Sets
Aq
n
(�,�0)?⇠= ⇡
n
(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
I Invariants of Dynamic Processes: Aq
n
(�,�o
)(Atkin, Maurer, Malle, Lovasz 1970’s)
I Discrete Homotopy Theory for Graphs
Aq
1(�,�0) ⇠= ⇡1(�q
�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�q�, x0)
I Discrete Homotopy for Cubical Sets
Aq
n
(�,�0)?⇠= ⇡
n
(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
I Invariants of Dynamic Processes: Aqn(∆, σo)
(Atkin, Maurer, Malle, Lovasz 1970’s)
I Discrete Homotopy Theory for Graphs
Aq1(∆, σ0) ∼= π1(Γq
∆, v0)/N(3, 4 cycles) ∼= π1(XΓq∆, x0)
I Discrete Homotopy for Cubical Sets
Aqn(∆, σ0)
?∼= πn(XΓq∆, x0)
I Discrete Homology Theory
I Unexpected Application of Discrete Homotopy Theory
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)
I Discrete Homotopy Theory for Graphs
Aq1(∆, σ0) ∼= π1(Γq
∆, v0)/N(3, 4 cycles) ∼= π1(XΓq∆, x0)
I Discrete Homotopy for Cubical Sets
Aqn(∆, σ0)
?∼= πn(XΓq∆, x0)
I Discrete Homology Theory
I Unexpected Application of Discrete Homotopy Theory
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)
Zn
- infinite lattice (usual metric)
2. An
(�, 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
,
h(
~i , k) = f (
~i )
h(
~i , ` ) = g(
~i )
4. A
n
(�, v0) - set of equivalence classes of maps in An
(�, v0)Note: translation preserves discrete homotopy
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)
Zn
- infinite lattice (usual metric)
2. An
(�, 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
,
h(
~i , k) = f (
~i )
h(
~i , ` ) = g(
~i )
4. A
n
(�, v0) - set of equivalence classes of maps in An
(�, v0)Note: translation preserves discrete homotopy
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)
Zn
- infinite lattice (usual metric)
2. An
(�, 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
,
h(
~i , k) = f (
~i )
h(
~i , ` ) = g(
~i )
4. A
n
(�, v0) - set of equivalence classes of maps in An
(�, v0)Note: translation preserves discrete homotopy
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)
Zn
- infinite lattice (usual metric)
2. An
(�, 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
,
h(
~i , k) = f (
~i )
h(
~i , ` ) = g(
~i )
4. A
n
(�, v0) - set of equivalence classes of maps in An
(�, v0)Note: translation preserves discrete homotopy
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)
Zn
- infinite lattice (usual metric)
2. An
(�, 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
,
h(
~i , k) = f (
~i )
h(
~i , ` ) = g(
~i )
4. A
n
(�, v0) - set of equivalence classes of maps in An
(�, v0)Note: translation preserves discrete homotopy
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)
Zn
- infinite lattice (usual metric)
2. An
(�, 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
,
h(
~i , k) = f (
~i )
h(
~i , ` ) = g(
~i )
4. A
n
(�, v0) - set of equivalence classes of maps in An
(�, v0)Note: translation preserves discrete homotopy
Discrete Homotopy Theory for Graphs
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
v0
v0
v0 f v0 g v0
v0
v0
v0
[f g ] = [f ] [g ]
Discrete Homotopy Theory for Graphs
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
v0
v0
v0 f v0 g v0
v0
v0
v0
[f g ] = [f ] [g ]
Discrete Homotopy Theory for Graphs
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
v0
v0
v0 f v0 g v0
v0
v0
v0
[f g ] = [f ] [g ]
Discrete Homotopy Theory for Graphs
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
v0
v0
v0 f v0 g v0
v0
v0
v0
[f g ] = [f ] [g ]
Discrete Homotopy Theory for Graphs
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)
f :
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
Discrete Homotopy Theory for Graphs
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)
f :
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
Discrete Homotopy Theory for Graphs
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)
f :
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
Discrete Homotopy Theory for Graphs
Group Structure
I Identity: e(~i ) = v0 ∀~i ∈ Zn
I Inverses: f −1(~i ) = f (−i1, . . . , in) ∀~i ∈ Zn
Example (n = 2)
f :
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
Discrete Homotopy Theory for Graphs
Group Structure
I Identity: e(~i ) = v0 ∀~i ∈ Zn
I Inverses: f −1(~i ) = f (−i1, . . . , in) ∀~i ∈ Zn
Example (n = 2)
f :
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
Discrete Homotopy Theory for Graphs
Group StructureAn
(�, v0) is an abelian group 8 n � 2
f g ⇠g
f⇠
g
f⇠ g f
Discrete Homotopy Theory for Graphs
Group StructureAn
(�, v0) is an abelian group 8 n � 2
f g ⇠g
f⇠
g
f⇠ g f
Discrete Homotopy Theory for Graphs
Examples
A1
⇣v0 v1 , v0
⌘= 1
A1
⇣v2
v0 v1
, v0⌘= 1
v0 v1 v2 v0
v0 v0 v0 v0
A1
⇣v3 v2
v0 v1
, v0⌘= 1
v0 v1 v2 v3 v0
v0 v0 v1 v0 v0
v0 v0 v0 v0 v0
A1
⇣, v0
⌘⇠= Z
A1(�, v0) ⇠= ⇡1(�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�, v0)
(2-dim cell complex: attach 2-cells to 4, ⇤ of �)
Discrete Homotopy Theory for Graphs
Examples
A1
⇣v0 v1 , v0
⌘= 1
A1
⇣v2
v0 v1
, v0⌘= 1
v0 v1 v2 v0
v0 v0 v0 v0
A1
⇣v3 v2
v0 v1
, v0⌘= 1
v0 v1 v2 v3 v0
v0 v0 v1 v0 v0
v0 v0 v0 v0 v0
A1
⇣, v0
⌘⇠= Z
A1(�, v0) ⇠= ⇡1(�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�, v0)
(2-dim cell complex: attach 2-cells to 4, ⇤ of �)
Discrete Homotopy Theory for Graphs
Examples
A1
⇣v0 v1 , v0
⌘= 1
A1
⇣v2
v0 v1
, v0⌘= 1
v0 v1 v2 v0
v0 v0 v0 v0
A1
⇣v3 v2
v0 v1
, v0⌘= 1
v0 v1 v2 v3 v0
v0 v0 v1 v0 v0
v0 v0 v0 v0 v0
A1
⇣, v0
⌘⇠= Z
A1(�, v0) ⇠= ⇡1(�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�, v0)
(2-dim cell complex: attach 2-cells to 4, ⇤ of �)
Discrete Homotopy Theory for Graphs
Examples
A1
⇣v0 v1 , v0
⌘= 1
A1
⇣v2
v0 v1
, v0⌘= 1
v0 v1 v2 v0
v0 v0 v0 v0
A1
⇣v3 v2
v0 v1
, v0⌘= 1
v0 v1 v2 v3 v0
v0 v0 v1 v0 v0
v0 v0 v0 v0 v0
A1
⇣, v0
⌘⇠= Z
A1(�, v0) ⇠= ⇡1(�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�, v0)
(2-dim cell complex: attach 2-cells to 4, ⇤ of �)
Discrete Homotopy Theory for Graphs
Examples
A1
⇣v0 v1 , v0
⌘= 1
A1
⇣v2
v0 v1
, v0⌘= 1
v0 v1 v2 v0
v0 v0 v0 v0
A1
⇣v3 v2
v0 v1
, v0⌘= 1
v0 v1 v2 v3 v0
v0 v0 v1 v0 v0
v0 v0 v0 v0 v0
A1
⇣, v0
⌘⇠= Z
A1(�, v0) ⇠= ⇡1(�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�, v0)
(2-dim cell complex: attach 2-cells to 4, ⇤ of �)
Discrete Homotopy Theory for Graphs
Examples
A1
⇣v0 v1 , v0
⌘= 1
A1
⇣v2
v0 v1
, v0⌘= 1
v0 v1 v2 v0
v0 v0 v0 v0
A1
⇣v3 v2
v0 v1
, v0⌘= 1
v0 v1 v2 v3 v0
v0 v0 v1 v0 v0
v0 v0 v0 v0 v0
A1
⇣, v0
⌘⇠= Z
A1(�, v0) ⇠= ⇡1(�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�, v0)
(2-dim cell complex: attach 2-cells to 4, ⇤ of �)
Discrete Homotopy Theory for Graphs
Examples
A1
⇣v0 v1 , v0
⌘= 1
A1
⇣v2
v0 v1
, v0⌘= 1
v0 v1 v2 v0
v0 v0 v0 v0
A1
⇣v3 v2
v0 v1
, v0⌘= 1
v0 v1 v2 v3 v0
v0 v0 v1 v0 v0
v0 v0 v0 v0 v0
A1
⇣, v0
⌘⇠= Z
A1(�, v0) ⇠= ⇡1(�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�, v0)
(2-dim cell complex: attach 2-cells to 4, ⇤ of �)
Discrete Homotopy Theory for Graphs
Examples
A1
⇣v0 v1 , v0
⌘= 1
A1
⇣v2
v0 v1
, v0⌘= 1
v0 v1 v2 v0
v0 v0 v0 v0
A1
⇣v3 v2
v0 v1
, v0⌘= 1
v0 v1 v2 v3 v0
v0 v0 v1 v0 v0
v0 v0 v0 v0 v0
A1
⇣, v0
⌘⇠= Z
A1(�, v0) ⇠= ⇡1(�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�, v0)
(2-dim cell complex: attach 2-cells to 4, ⇤ of �)
Discrete Homotopy Theory for Graphs
Examples
A1
⇣v0 v1 , v0
⌘= 1
A1
⇣v2
v0 v1
, v0⌘= 1
v0 v1 v2 v0
v0 v0 v0 v0
A1
⇣v3 v2
v0 v1
, v0⌘= 1
v0 v1 v2 v3 v0
v0 v0 v1 v0 v0
v0 v0 v0 v0 v0
A1
⇣, v0
⌘⇠= Z
A1(�, v0) ⇠= ⇡1(�, v0)/N(3, 4 cycles) ⇠= ⇡1(X�, v0)
(2-dim cell complex: attach 2-cells to 4, ⇤ of �)
Discrete Homotopy Theory
I Aq
n
(�,�0) ⇠= An
(�q�,�0)�q� vertices = all maximal simplices of � of dim� q(�,�0) 2 E (�q�) () dim(� \ �0) � q
I Ar
n
(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
I Aq
n
(�,�0) ⇠= An
(�q�,�0)�q� vertices = all maximal simplices of � of dim� q(�,�0) 2 E (�q�) () dim(� \ �0) � q
I Ar
n
(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 �
i
then
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. . . ?
Is it a Good Analogy to Classical Homotopy?
1. If � is connected, An
(�, v0)independent of v02. Siefert-van Kampen: if
� = �1 [ �2�i
connectedv0 2 �1 \ �2�1 \ �2 connected4, ⇤ lie in one of the �
i
then
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. . . ?
Is it a Good Analogy to Classical Homotopy?
1. If � is connected, An
(�, v0)independent of v02. Siefert-van Kampen: if
� = �1 [ �2�i
connectedv0 2 �1 \ �2�1 \ �2 connected4, ⇤ lie in one of the �
i
then
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. . . ?
Is it a Good Analogy to Classical Homotopy?
1. If � is connected, An
(�, v0)independent of v02. Siefert-van Kampen: if
� = �1 [ �2�i
connectedv0 2 �1 \ �2�1 \ �2 connected4, ⇤ lie in one of the �
i
then
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. . . ?
Is it a Good Analogy to Classical Homotopy?
1. If � is connected, An
(�, v0)independent of v02. Siefert-van Kampen: if
� = �1 [ �2�i
connectedv0 2 �1 \ �2�1 \ �2 connected4, ⇤ lie in one of the �
i
then
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. . . ?
Discrete Homology Theory for Graphs
(B., Capraro, White)
Necessities
1. Discrete n-dim cube Q
n
= {(a1, . . . , an) | ai = 0 or 1}2. Singular n-cube � : Q
n
! � graph homomorphism
3. Ln
(�) := free abelian group generated by all singular n-cubes �
I i th front and back faces of � are singular (n � 1)-cubesI Front: (An
i
�)(a1, . . . , an�1) = �(a1, . . . , ai�1, 0, ai , . . . , an�1)I Back: (Bn
i
�)(b1, . . . , bn�1) = �(b1, . . . , bi�1, 1, bi , . . . , bn�1)I Degenerate singular n-cube: if 9 i s.t. (An
i
�) = (Bn
i
�)I D
n
(�) := free abelian group generated by all degeneratesingular n-cubes
C
n
(�) := Ln
(�)/Dn
(�)
elements of C
n
correspond to n-chains
Discrete Homology Theory for Graphs
(B., Capraro, White)
Necessities
1. Discrete n-dim cube Q
n
= {(a1, . . . , an) | ai = 0 or 1}2. Singular n-cube � : Q
n
! � graph homomorphism
3. Ln
(�) := free abelian group generated by all singular n-cubes �
I i th front and back faces of � are singular (n � 1)-cubesI Front: (An
i
�)(a1, . . . , an�1) = �(a1, . . . , ai�1, 0, ai , . . . , an�1)I Back: (Bn
i
�)(b1, . . . , bn�1) = �(b1, . . . , bi�1, 1, bi , . . . , bn�1)I Degenerate singular n-cube: if 9 i s.t. (An
i
�) = (Bn
i
�)I D
n
(�) := free abelian group generated by all degeneratesingular n-cubes
C
n
(�) := Ln
(�)/Dn
(�)
elements of C
n
correspond to n-chains
Discrete Homology Theory for Graphs
(B., Capraro, White)
Necessities
1. Discrete n-dim cube Q
n
= {(a1, . . . , an) | ai = 0 or 1}
2. Singular n-cube � : Qn
! � graph homomorphism
3. Ln
(�) := free abelian group generated by all singular n-cubes �
I i th front and back faces of � are singular (n � 1)-cubesI Front: (An
i
�)(a1, . . . , an�1) = �(a1, . . . , ai�1, 0, ai , . . . , an�1)I Back: (Bn
i
�)(b1, . . . , bn�1) = �(b1, . . . , bi�1, 1, bi , . . . , bn�1)I Degenerate singular n-cube: if 9 i s.t. (An
i
�) = (Bn
i
�)I D
n
(�) := free abelian group generated by all degeneratesingular n-cubes
C
n
(�) := Ln
(�)/Dn
(�)
elements of C
n
correspond to n-chains
Discrete Homology Theory for Graphs
(B., Capraro, White)
Necessities
1. Discrete n-dim cube Q
n
= {(a1, . . . , an) | ai = 0 or 1}2. Singular n-cube � : Q
n
! � graph homomorphism
3. Ln
(�) := free abelian group generated by all singular n-cubes �
I i th front and back faces of � are singular (n � 1)-cubesI Front: (An
i
�)(a1, . . . , an�1) = �(a1, . . . , ai�1, 0, ai , . . . , an�1)I Back: (Bn
i
�)(b1, . . . , bn�1) = �(b1, . . . , bi�1, 1, bi , . . . , bn�1)I Degenerate singular n-cube: if 9 i s.t. (An
i
�) = (Bn
i
�)I D
n
(�) := free abelian group generated by all degeneratesingular n-cubes
C
n
(�) := Ln
(�)/Dn
(�)
elements of C
n
correspond to n-chains
Discrete Homology Theory for Graphs
(B., Capraro, White)
Necessities
1. Discrete n-dim cube Q
n
= {(a1, . . . , an) | ai = 0 or 1}2. Singular n-cube � : Q
n
! � graph homomorphism
3. Ln
(�) := free abelian group generated by all singular n-cubes �
I i th front and back faces of � are singular (n � 1)-cubesI Front: (An
i
�)(a1, . . . , an�1) = �(a1, . . . , ai�1, 0, ai , . . . , an�1)I Back: (Bn
i
�)(b1, . . . , bn�1) = �(b1, . . . , bi�1, 1, bi , . . . , bn�1)I Degenerate singular n-cube: if 9 i s.t. (An
i
�) = (Bn
i
�)I D
n
(�) := free abelian group generated by all degeneratesingular n-cubes
C
n
(�) := Ln
(�)/Dn
(�)
elements of C
n
correspond to n-chains
Discrete Homology Theory for Graphs
(B., Capraro, White)
Necessities
1. Discrete n-dim cube Q
n
= {(a1, . . . , an) | ai = 0 or 1}2. Singular n-cube � : Q
n
! � graph homomorphism
3. Ln
(�) := free abelian group generated by all singular n-cubes �
I i th front and back faces of � are singular (n � 1)-cubes
I Front: (An
i
�)(a1, . . . , an�1) = �(a1, . . . , ai�1, 0, ai , . . . , an�1)I Back: (Bn
i
�)(b1, . . . , bn�1) = �(b1, . . . , bi�1, 1, bi , . . . , bn�1)I Degenerate singular n-cube: if 9 i s.t. (An
i
�) = (Bn
i
�)I D
n
(�) := free abelian group generated by all degeneratesingular n-cubes
C
n
(�) := Ln
(�)/Dn
(�)
elements of C
n
correspond to n-chains
Discrete Homology Theory for Graphs
(B., Capraro, White)
Necessities
1. Discrete n-dim cube Q
n
= {(a1, . . . , an) | ai = 0 or 1}2. Singular n-cube � : Q
n
! � graph homomorphism
3. Ln
(�) := free abelian group generated by all singular n-cubes �
I i th front and back faces of � are singular (n � 1)-cubesI Front: (An
i
�)(a1, . . . , an�1) = �(a1, . . . , ai�1, 0, ai , . . . , an�1)
I Back: (Bn
i
�)(b1, . . . , bn�1) = �(b1, . . . , bi�1, 1, bi , . . . , bn�1)I Degenerate singular n-cube: if 9 i s.t. (An
i
�) = (Bn
i
�)I D
n
(�) := free abelian group generated by all degeneratesingular n-cubes
C
n
(�) := Ln
(�)/Dn
(�)
elements of C
n
correspond to n-chains
Discrete Homology Theory for Graphs
(B., Capraro, White)
Necessities
1. Discrete n-dim cube Q
n
= {(a1, . . . , an) | ai = 0 or 1}2. Singular n-cube � : Q
n
! � graph homomorphism
3. Ln
(�) := free abelian group generated by all singular n-cubes �
I i th front and back faces of � are singular (n � 1)-cubesI Front: (An
i
�)(a1, . . . , an�1) = �(a1, . . . , ai�1, 0, ai , . . . , an�1)I Back: (Bn
i
�)(b1, . . . , bn�1) = �(b1, . . . , bi�1, 1, bi , . . . , bn�1)
I Degenerate singular n-cube: if 9 i s.t. (An
i
�) = (Bn
i
�)I D
n
(�) := free abelian group generated by all degeneratesingular n-cubes
C
n
(�) := Ln
(�)/Dn
(�)
elements of C
n
correspond to n-chains
Discrete Homology Theory for Graphs
(B., Capraro, White)
Necessities
1. Discrete n-dim cube Q
n
= {(a1, . . . , an) | ai = 0 or 1}2. Singular n-cube � : Q
n
! � graph homomorphism
3. Ln
(�) := free abelian group generated by all singular n-cubes �
I i th front and back faces of � are singular (n � 1)-cubesI Front: (An
i
�)(a1, . . . , an�1) = �(a1, . . . , ai�1, 0, ai , . . . , an�1)I Back: (Bn
i
�)(b1, . . . , bn�1) = �(b1, . . . , bi�1, 1, bi , . . . , bn�1)I Degenerate singular n-cube: if 9 i s.t. (An
i
�) = (Bn
i
�)
I Dn
(�) := free abelian group generated by all degeneratesingular n-cubes
C
n
(�) := Ln
(�)/Dn
(�)
elements of C
n
correspond to n-chains
Discrete Homology Theory for Graphs
(B., Capraro, White)
Necessities
1. Discrete n-dim cube Q
n
= {(a1, . . . , an) | ai = 0 or 1}2. Singular n-cube � : Q
n
! � graph homomorphism
3. Ln
(�) := free abelian group generated by all singular n-cubes �
I i th front and back faces of � are singular (n � 1)-cubesI Front: (An
i
�)(a1, . . . , an�1) = �(a1, . . . , ai�1, 0, ai , . . . , an�1)I Back: (Bn
i
�)(b1, . . . , bn�1) = �(b1, . . . , bi�1, 1, bi , . . . , bn�1)I Degenerate singular n-cube: if 9 i s.t. (An
i
�) = (Bn
i
�)I D
n
(�) := free abelian group generated by all degeneratesingular n-cubes
C
n
(�) := Ln
(�)/Dn
(�)
elements of C
n
correspond to n-chains
Discrete Homology Theory for Graphs
(B., Capraro, White)
Necessities
1. Discrete n-dim cube Q
n
= {(a1, . . . , an) | ai = 0 or 1}2. Singular n-cube � : Q
n
! � graph homomorphism
3. Ln
(�) := free abelian group generated by all singular n-cubes �
I i th front and back faces of � are singular (n � 1)-cubesI Front: (An
i
�)(a1, . . . , an�1) = �(a1, . . . , ai�1, 0, ai , . . . , an�1)I Back: (Bn
i
�)(b1, . . . , bn�1) = �(b1, . . . , bi�1, 1, bi , . . . , bn�1)I Degenerate singular n-cube: if 9 i s.t. (An
i
�) = (Bn
i
�)I D
n
(�) := free abelian group generated by all degeneratesingular n-cubes
C
n
(�) := Ln
(�)/Dn
(�)
elements of C
n
correspond to n-chains
Discrete Homology Theory for Graphs
Necessities
4. Boundary operators @n
for each n � 1
@n
(�) =nX
i=1
(�1)i�An
i
(�)� Bn
i
(�)�
I extend linearly to Ln
(�)I @
n
(Dn
(�)) ✓ Dn�1(�)
I so @n
: Cn
(�) ! Cn�1(�) is well-defined
I @n
� @n+1 = 0
Discrete Homology Theory for Graphs
Necessities
4. Boundary operators @n
for each n � 1
@n
(�) =nX
i=1
(�1)i�An
i
(�)� Bn
i
(�)�
I extend linearly to Ln
(�)
I @n
(Dn
(�)) ✓ Dn�1(�)
I so @n
: Cn
(�) ! Cn�1(�) is well-defined
I @n
� @n+1 = 0
Discrete Homology Theory for Graphs
Necessities
4. Boundary operators @n
for each n � 1
@n
(�) =nX
i=1
(�1)i�An
i
(�)� Bn
i
(�)�
I extend linearly to Ln
(�)I @
n
(Dn
(�)) ✓ Dn�1(�)
I so @n
: Cn
(�) ! Cn�1(�) is well-defined
I @n
� @n+1 = 0
Discrete Homology Theory for Graphs
Necessities
4. Boundary operators @n
for each n � 1
@n
(�) =nX
i=1
(�1)i�An
i
(�)� Bn
i
(�)�
I extend linearly to Ln
(�)I @
n
(Dn
(�)) ✓ Dn�1(�)
I so @n
: Cn
(�) ! Cn�1(�) is well-defined
I @n
� @n+1 = 0
Discrete Homology Theory for Graphs
Necessities
4. Boundary operators @n
for each n � 1
@n
(�) =nX
i=1
(�1)i�An
i
(�)� Bn
i
(�)�
I extend linearly to Ln
(�)I @
n
(Dn
(�)) ✓ Dn�1(�)
I so @n
: Cn
(�) ! Cn�1(�) is well-defined
I @n
� @n+1 = 0
Discrete Homology Theory for Graphs
DefinitionThe discrete homology groups of �:
DHn
(�) = Ker(@n
)/Im(@n+1)
Examples
DHn
(�) = 0 8 n � 1 DHn
(4) = 0 8 n � 1
DHn
(⇤) = 0 8 n � 1 DH1( ) = Z
DefinitionIf �0 ✓ �, then @
n
(Cn
(�0)) ✓ Cn�1(�
0) and there are maps
@n
: Cn
(�, �0) = Cn
(�)/Cn
(�0) ! Cn�1(�, �
0)
The relative homology groups of (�, �0):
DHn
(�, �0) = Ker(@n
)/Im(@n+1)
Discrete Homology Theory for Graphs
DefinitionThe discrete homology groups of �:
DHn
(�) = Ker(@n
)/Im(@n+1)
Examples
DHn
(�) = 0 8 n � 1 DHn
(4) = 0 8 n � 1
DHn
(⇤) = 0 8 n � 1 DH1( ) = Z
DefinitionIf �0 ✓ �, then @
n
(Cn
(�0)) ✓ Cn�1(�
0) and there are maps
@n
: Cn
(�, �0) = Cn
(�)/Cn
(�0) ! Cn�1(�, �
0)
The relative homology groups of (�, �0):
DHn
(�, �0) = Ker(@n
)/Im(@n+1)
Discrete Homology Theory for Graphs
DefinitionThe discrete homology groups of �:
DHn
(�) = Ker(@n
)/Im(@n+1)
Examples
DHn
(�) = 0 8 n � 1
DHn
(4) = 0 8 n � 1
DHn
(⇤) = 0 8 n � 1 DH1( ) = Z
DefinitionIf �0 ✓ �, then @
n
(Cn
(�0)) ✓ Cn�1(�
0) and there are maps
@n
: Cn
(�, �0) = Cn
(�)/Cn
(�0) ! Cn�1(�, �
0)
The relative homology groups of (�, �0):
DHn
(�, �0) = Ker(@n
)/Im(@n+1)
Discrete Homology Theory for Graphs
DefinitionThe discrete homology groups of �:
DHn
(�) = Ker(@n
)/Im(@n+1)
Examples
DHn
(�) = 0 8 n � 1 DHn
(4) = 0 8 n � 1
DHn
(⇤) = 0 8 n � 1 DH1( ) = Z
DefinitionIf �0 ✓ �, then @
n
(Cn
(�0)) ✓ Cn�1(�
0) and there are maps
@n
: Cn
(�, �0) = Cn
(�)/Cn
(�0) ! Cn�1(�, �
0)
The relative homology groups of (�, �0):
DHn
(�, �0) = Ker(@n
)/Im(@n+1)
Discrete Homology Theory for Graphs
DefinitionThe discrete homology groups of �:
DHn
(�) = Ker(@n
)/Im(@n+1)
Examples
DHn
(�) = 0 8 n � 1 DHn
(4) = 0 8 n � 1
DHn
(⇤) = 0 8 n � 1
DH1( ) = Z
DefinitionIf �0 ✓ �, then @
n
(Cn
(�0)) ✓ Cn�1(�
0) and there are maps
@n
: Cn
(�, �0) = Cn
(�)/Cn
(�0) ! Cn�1(�, �
0)
The relative homology groups of (�, �0):
DHn
(�, �0) = Ker(@n
)/Im(@n+1)
Discrete Homology Theory for Graphs
DefinitionThe discrete homology groups of �:
DHn
(�) = Ker(@n
)/Im(@n+1)
Examples
DHn
(�) = 0 8 n � 1 DHn
(4) = 0 8 n � 1
DHn
(⇤) = 0 8 n � 1 DH1( !
) = Z
DefinitionIf �0 ✓ �, then @
n
(Cn
(�0)) ✓ Cn�1(�
0) and there are maps
@n
: Cn
(�, �0) = Cn
(�)/Cn
(�0) ! Cn�1(�, �
0)
The relative homology groups of (�, �0):
DHn
(�, �0) = Ker(@n
)/Im(@n+1)
Discrete Homology Theory for Graphs
DefinitionThe discrete homology groups of �:
DHn
(�) = Ker(@n
)/Im(@n+1)
Examples
DHn
(�) = 0 8 n � 1 DHn
(4) = 0 8 n � 1
DHn
(⇤) = 0 8 n � 1 DH1( !
) = Z
DefinitionIf �0 ✓ �, then @
n
(Cn
(�0)) ✓ Cn�1(�
0) and there are maps
@n
: Cn
(�, �0) = Cn
(�)/Cn
(�0) ! Cn�1(�, �
0)
The relative homology groups of (�, �0):
DHn
(�, �0) = Ker(@n
)/Im(@n+1)
Discrete Homology Theory for Graphs
DefinitionThe discrete homology groups of �:
DHn
(�) = Ker(@n
)/Im(@n+1)
Examples
DHn
(�) = 0 8 n � 1 DHn
(4) = 0 8 n � 1
DHn
(⇤) = 0 8 n � 1 DH1( !
) = Z
DefinitionIf �0 ✓ �, then @
n
(Cn
(�0)) ✓ Cn�1(�
0) and there are maps
@n
: Cn
(�, �0) = Cn
(�)/Cn
(�0) ! Cn�1(�, �
0)
The relative homology groups of (�, �0):
DHn
(�, �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) � =
S�
i
(ii) for each non-degenerate n-cube �, we have � 2 �
i
for some i
(iii) Discrete cover is an n-dim cover for each n
A. Mayer-Vietoris sequence:
· · · @⇤�! DHn
(�1 \ �2)diag��! DH
n
(�1)� DHn
(�2)di↵��! DH
n
(�1 [ �2)@⇤�! DH
n�1(�1 \ �2) · · ·
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) � =
S�
i
(ii) for each non-degenerate n-cube �, we have � 2 �
i
for some i
(iii) Discrete cover is an n-dim cover for each n
A. Mayer-Vietoris sequence:
· · · @⇤�! DHn
(�1 \ �2)diag��! DH
n
(�1)� DHn
(�2)di↵��! DH
n
(�1 [ �2)@⇤�! DH
n�1(�1 \ �2) · · ·
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) � =
S�
i
(ii) for each non-degenerate n-cube �, we have � 2 �
i
for some i
(iii) Discrete cover is an n-dim cover for each n
A. Mayer-Vietoris sequence:
· · · @⇤�! DHn
(�1 \ �2)diag��! DH
n
(�1)� DHn
(�2)di↵��! DH
n
(�1 [ �2)@⇤�! DH
n�1(�1 \ �2) · · ·
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) � =
S�
i
(ii) for each non-degenerate n-cube �, we have � 2 �
i
for some i
(iii) Discrete cover is an n-dim cover for each n
A. Mayer-Vietoris sequence:
· · · @⇤�! DHn
(�1 \ �2)diag��! DH
n
(�1)� DHn
(�2)di↵��! DH
n
(�1 [ �2)@⇤�! DH
n�1(�1 \ �2) · · ·
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) � =
S�
i
(ii) for each non-degenerate n-cube �, we have � 2 �
i
for some i
(iii) Discrete cover is an n-dim cover for each n
A. Mayer-Vietoris sequence:
· · · @⇤�! DHn
(�1 \ �2)diag��! DH
n
(�1)� DHn
(�2)di↵��! DH
n
(�1 [ �2)@⇤�! DH
n�1(�1 \ �2) · · ·
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) � =
S�
i
(ii) for each non-degenerate n-cube �, we have � 2 �
i
for some i
(iii) Discrete cover is an n-dim cover for each n
A. Mayer-Vietoris sequence:
· · · @⇤�! DHn
(�1 \ �2)diag��! DH
n
(�1)� DHn
(�2)di↵��! DH
n
(�1 [ �2)@⇤�! DH
n�1(�1 \ �2) · · ·
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) � =
S�
i
(ii) for each non-degenerate n-cube �, we have � 2 �
i
for some i
(iii) Discrete cover is an n-dim cover for each n
A. Mayer-Vietoris sequence:
· · · @⇤�! DHn
(�1 \ �2)diag��! DH
n
(�1)� DHn
(�2)di↵��! DH
n
(�1 [ �2)@⇤�! DH
n�1(�1 \ �2) · · ·
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) � =
S�
i
(ii) for each non-degenerate n-cube �, we have � 2 �
i
for some i
(iii) Discrete cover is an n-dim cover for each n
A. Mayer-Vietoris sequence:
· · · @⇤�! DHn
(�1 \ �2)diag��! DH
n
(�1)� DHn
(�2)di↵��! DH
n
(�1 [ �2)@⇤�! DH
n�1(�1 \ �2) · · ·
How to Judge if Homology Theory is Good?
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:
DHn
( • , ;) = {0} 8 n � 1
I Long exact sequence:
· · · ! DHn
(�0) ,! DHn
(�) ,! DHn
(�, �0)@⇤�! DH
n�1(�0) · · ·
How to Judge if Homology Theory is Good?
B. Eilenberg-Steenrod axioms:I Homotopy: If
f , 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:
DHn
( • , ;) = {0} 8 n � 1
I Long exact sequence:
· · · ! DHn
(�0) ,! DHn
(�) ,! DHn
(�, �0)@⇤�! DH
n�1(�0) · · ·
How to Judge if Homology Theory is Good?
B. Eilenberg-Steenrod axioms:I Homotopy: If
f , 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 cover
I Dimension:DH
n
( • , ;) = {0} 8 n � 1
I Long exact sequence:
· · · ! DHn
(�0) ,! DHn
(�) ,! DHn
(�, �0)@⇤�! DH
n�1(�0) · · ·
How to Judge if Homology Theory is Good?
B. Eilenberg-Steenrod axioms:I Homotopy: If
f , 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:
DHn
( • , ;) = {0} 8 n � 1
I Long exact sequence:
· · · ! DHn
(�0) ,! DHn
(�) ,! DHn
(�, �0)@⇤�! DH
n�1(�0) · · ·
How to Judge if Homology Theory is Good?
B. Eilenberg-Steenrod axioms:I Homotopy: If
f , 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:
DHn
( • , ;) = {0} 8 n � 1
I Long exact sequence:
· · · ! 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 �
R
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
DHn
(�) =
(G if n = n
0 if n n
I There is a graph Sn such that
DHk
(Sn) =
(Z if k = n
0 if k 6= n
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 �
R
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
DHn
(�) =
(G if n = n
0 if n n
I There is a graph Sn such that
DHk
(Sn) =
(Z if k = n
0 if k 6= n
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 �
R
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
DHn
(�) =
(G if n = n
0 if n n
I There is a graph Sn such that
DHk
(Sn) =
(Z if k = n
0 if k 6= n
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 �
R
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
DHn
(�) =
(G if n = n
0 if n n
I There is a graph Sn such that
DHk
(Sn) =
(Z if k = n
0 if k 6= n
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 �
R
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
DHn
(�) =
(G if n = n
0 if n n
I There is a graph Sn such that
DHk
(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
= zj
, i < j
ARn,3 3-equal subspace arr:�~x 2 Rn
�� xi
= xj
= xk
, 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)
Unexpected Application of Discrete Homotopy Theory
Complex K (⇡, 1) Spaces
Real K (⇡, 1) Spaces
ACn,2 braid arrangement:�~z 2 Cn
�� zi
= zj
, i < j
ARn,3 3-equal subspace arr:�~x 2 Rn
�� xi
= xj
= xk
, 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)
Unexpected Application of Discrete Homotopy Theory
Complex K (⇡, 1) Spaces Real K (⇡, 1) Spaces
ACn,2 braid arrangement:�~z 2 Cn
�� zi
= zj
, i < j
ARn,3 3-equal subspace arr:�~x 2 Rn
�� xi
= xj
= xk
, 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)
Unexpected Application of Discrete Homotopy Theory
Complex K (⇡, 1) Spaces Real K (⇡, 1) Spaces
ACn,2 braid arrangement:�~z 2 Cn
�� zi
= zj
, i < j
ARn,3 3-equal subspace arr:�~x 2 Rn
�� xi
= xj
= xk
, i < j < k
M(ACn,2) is K (⇡, 1)
(Fadell-Neuwirth 1962)
M(ARn,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)
Unexpected Application of Discrete Homotopy Theory
Complex K (⇡, 1) Spaces Real K (⇡, 1) Spaces
ACn,2 braid arrangement:�~z 2 Cn
�� zi
= zj
, i < j
ARn,3 3-equal subspace arr:�~x 2 Rn
�� xi
= xj
= xk
, 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)
Unexpected Application of Discrete Homotopy Theory
Complex K (⇡, 1) Spaces Real K (⇡, 1) Spaces
ACn,2 braid arrangement:�~z 2 Cn
�� zi
= zj
, i < j
ARn,3 3-equal subspace arr:�~x 2 Rn
�� xi
= xj
= xk
, 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)
Unexpected Application of Discrete Homotopy Theory
Complex K (⇡, 1) Spaces Real K (⇡, 1) Spaces
ACn,2 braid arrangement:�~z 2 Cn
�� zi
= zj
, i < j
ARn,3 3-equal subspace arr:�~x 2 Rn
�� xi
= xj
= xk
, 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)
Unexpected Application of Discrete Homotopy Theory
Complex K (⇡, 1) Spaces Real K (⇡, 1) Spaces
ACn,2 braid arrangement:�~z 2 Cn
�� zi
= zj
, i < j
ARn,3 3-equal subspace arr:�~x 2 Rn
�� xi
= xj
= xk
, 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)
Unexpected Application of Discrete Homotopy Theory
Complex K (⇡, 1) Spaces Real K (⇡, 1) Spaces
ACn,2 braid arrangement:�~z 2 Cn
�� zi
= zj
, i < j
ARn,3 3-equal subspace arr:�~x 2 Rn
�� xi
= xj
= xk
, 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)
Unexpected Application of Discrete Homotopy Theory
Complex K (⇡, 1) Spaces Real K (⇡, 1) Spaces
ACn,2 braid arrangement:�~z 2 Cn
�� zi
= zj
, i < j
ARn,3 3-equal subspace arr:�~x 2 Rn
�� xi
= xj
= xk
, 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
Unexpected Application of Discrete Homotopy Theory
Complex K (⇡, 1) Spaces Real K (⇡, 1) Spaces
ACn,2 braid arrangement:�~z 2 Cn
�� zi
= zj
, i < j
ARn,3 3-equal subspace arr:�~x 2 Rn
�� xi
= xj
= xk
, 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
Unexpected Application of Discrete Homotopy Theory
Complex K (⇡, 1) Spaces Real K (⇡, 1) Spaces
ACn,2 braid arrangement:�~z 2 Cn
�� zi
= zj
, i < j
ARn,3 3-equal subspace arr:�~x 2 Rn
�� xi
= xj
= xk
, 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
Unexpected Application of Discrete Homotopy Theory
Complex K (⇡, 1) Spaces Real K (⇡, 1) Spaces
ACn,2 braid arrangement:�~z 2 Cn
�� zi
= zj
, i < j
ARn,3 3-equal subspace arr:�~x 2 Rn
�� xi
= xj
= xk
, 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
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.
(si
sj
)2 = 1 (si
si+1si )
2 = 1
3. Pure braid gp: Ker(�), where � : “Sn
� (i)”! Sn
by �(si
) = si
⇡1(M(ACn,2))
⇠= Ker(�)
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.
(si
sj
)2 = 1 (si
si+1si )
2 = 1
3. Pure braid gp: Ker(�), where � : “Sn
� (i)”! Sn
by �(si
) = si
⇡1(M(ACn,2))
⇠= Ker(�)
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.
(si
sj
)2 = 1 (si
si+1si )
2 = 1
3. Pure braid gp: Ker(�), where � : “Sn
� (i)”! Sn
by �(si
) = si
⇡1(M(ACn,2))
⇠= Ker(�)
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.
(si
sj
)2 = 1 (si
si+1si )
2 = 1
3. Pure braid gp: Ker(�), where � : “Sn
� (i)”! Sn
by �(si
) = si
⇡1(M(ACn,2))
⇠= Ker(�)
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.
(si
sj
)2 = 1 (si
si+1si )
2 = 1
3. Pure braid gp: Ker(�), where � : “Sn
� (i)”! Sn
by �(si
) = si
⇡1(M(ACn,2))
⇠= Ker(�)
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.
(si
sj
)2 = 1 (si
si+1si )
2 = 1
3. Pure braid gp: Ker(�), where � : “Sn
� (i)”! Sn
by �(si
) = si
⇡1(M(ACn,2))
⇠= Ker(�)
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.
(si
sj
)2 = 1 (si
si+1si )
2 = 1
3. Pure braid gp: Ker(�), where � : “Sn
� (i)”! Sn
by �(si
) = si
⇡1(M(ACn,2))
⇠= Ker(�)
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.
(si
sj
)2 = 1 (si
si+1si )
2 = 1
3. Pure braid gp: Ker(�), where � : “Sn
� (i)”! Sn
by �(si
) = si
⇡1(M(ACn,2))
⇠= Ker(�)
Preparation for Proof
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
I Wk
= {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
, sj
i /2 P5. W 0 =“W � {(iii),(iv), . . .}” =) ⇡1(M(W
n,3)) ⇠= Ker(�0)
Preparation for Proof
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
I Wk
= {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
, sj
i /2 P5. W 0 =“W � {(iii),(iv), . . .}” =) ⇡1(M(W
n,3)) ⇠= Ker(�0)
Preparation for Proof
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
I Wk
= {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
, sj
i /2 P5. W 0 =“W � {(iii),(iv), . . .}” =) ⇡1(M(W
n,3)) ⇠= Ker(�0)
Preparation for Proof
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
I Wk
= {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
, sj
i /2 P5. W 0 =“W � {(iii),(iv), . . .}” =) ⇡1(M(W
n,3)) ⇠= Ker(�0)
Preparation for Proof
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
I Wk
= {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
, sj
i /2 P
5. W 0 =“W � {(iii),(iv), . . .}” =) ⇡1(M(Wn,3)) ⇠= Ker(�0)
Preparation for Proof
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
I Wk
= {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
, sj
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
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
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
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)
3= 1
8-cycles (st)
4= 1
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
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)
3= 1
8-cycles (st)
4= 1
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 WI 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)
3= 1
8-cycles (st)
4= 1
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 WI 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)
3= 1
8-cycles (st)
4= 1
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 WI 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)
3= 1
8-cycles (st)
4= 1
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 WI 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)
3= 1
8-cycles (st)
4= 1
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 WI 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)
3= 1
8-cycles (st)
4= 1
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 WI 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)
3= 1
8-cycles (st)
4= 1
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
i
) = 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 )
D. Let W1 := “W � {(ii),(iii),(iv),. . . }” (keep involutions only)and let � : W1 ! W by �(s
i
) = 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 )
D. Let W1 := “W � {(ii),(iii),(iv),. . . }” (keep involutions only)and let � : W1 ! W by �(s
i
) = 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 )
D. Let W1 := “W � {(ii),(iii),(iv),. . . }” (keep involutions only)and let � : W1 ! W by �(s
i
) = 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 )
D. Let W1 := “W � {(ii),(iii),(iv),. . . }” (keep involutions only)and let � : W1 ! W by �(s
i
) = 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 )
D. Let W1 := “W � {(ii),(iii),(iv),. . . }” (keep involutions only)and let � : W1 ! W by �(s
i
) = 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 )
D. Let W1 := “W � {(ii),(iii),(iv),. . . }” (keep involutions only)and let � : W1 ! W by �(s
i
) = 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
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.