Minimal Stratification for Line Arrangements
Masahiko Yoshinaga (Kyoto U.)
Combinatorial and Geometric aspects of Hyperplane Arrangements
Centro di Ricerca Matematica “Ennio De Giorgi” Pisa, May 25, 2010.
HH
H
HH345 12 45 123 HH
HH HH
HH 1345 2 H
H 4 1235 H H 134
25
14235
H
H
H
H
H
H
H
H
H
H
p
p p p
p p p p p p p p p p p p p p p
p p p
p p p p p p p p p p p p p p p p pH
p p pH
p p p p p p p p p p p p p p pH
pppH
p p p p p p pp
H
H
5
4
3
2
1
OUR OBJECTS
A = {H1 , . . . , Hn }, Hi ⊂ R lines.
n
∪
M = M (A) = C2 \
Hi ⊗ C, complement.
2
i=1
Motivations
How real and combinatorial structure of A are
related to topology of M (A)?
• Topology: π1 (M ), H 1 (M, L).
• Combinatorics: Incidences, chambers.
Contents
§1
§2
§3
§3
A minimal positive presentation for π1 (M ).
Minimal Stratification.
Chamber basis of Orlik-Solomon algebra.
1
Chamber basis and H (M, Lλ ).
Minimal stratification =⇒
⇓
Basis of OS-algebra =⇒
presentation of π1
H 1 (M, Lλ )
1 Minimal positive presentation
1 Minimal positive presentation
F : a generic oriented line near H∞ .
Def. chF (A) := {C : chamber | C ∩ F = ∅}
Q C3
Q
Q
Q
chF (A) = {C1 , C2 , C3 }
C2
Q
Prop. | chF (A)| = b2 (M )
Q
C1 Q
Q
Q
p p p p
p p p p p p p p p p p p p p p p p p p p p p p p p p pQ
p p p p p p ppp F
Q
1 Minimal positive presentation
We also assume H1 ∩ F > · · · > Hn ∩ F ,
and defining equation αi is compatible with F .
Q C3
Q
Q
Q
chF (A) = {C1 , C2 , C3 }
C2
Q
Prop. | chF (A)| = b2 (M )
Q
C1 Q {α1 > 0}
Q
{α1 < 0}Q
p
p p p
p p p p p p p p p p p p p p p p p p p p p p
p
p p p pQ
p p pQ
p p p ppp F
H3
H2
H1
1 Minimal positive presentation
√
Generators: Meridians (with base − −1).
FC ∩ M (A)
γ3
γ2
γ1
b
b
b
@
@
@
@
@s
1 Minimal positive presentation
Relations:
(1) Attach to C ∈ chF (A) a permutation
(i1 , i2 , . . . , in ) of (1, . . . , n) as
i1 < · · · < ik , ik+1 < · · · < in
Going through
right side of C
Going through
left side of C
(2) Associate to C a relation
R(C) : γ1 γ2 . . . γn = γi1 γi2 . . . γin .
Thm. π1 (M ) ∼
= ⟨γ1 , . . . , γn | R(C), C ∈ chF (A)⟩
1 Minimal positive presentation
⟩
12345
γ1 , . . . , γ5 = 14235 = 13425 = 13452
= 34512 = 45123 = 41235
⟨
Example
π1 (M ) ∼
=
HH
H
HH345 12 45 123 HH
HH HH
HH 1345 2 H
H 4 1235 H
H
134
25
14235
H
H
HH
HH H
H
H
H
F
p
p p p
p p p p p p p p p p p p p p p
p p p
p p p p p p p p p p p p p p p p pH
p p pH
p p p p p p p p p p p p p p pH
pppH
p p p p p p pp
H
H
5
4
3
2
1
1 Minimal positive presentation
The correspondence “chamber→relation”
C 7−→ R(C) is natural in the following sense.
Thm. (Y. 2007) ∃! continuous map (up to
homotopy) σC : (D2 , ∂D2 ) → (M, M ∩ FC ) s.t.
(i) σC (D2 ) t C = {pC },
(ii) σC (D2 ) t C ′ = ∅, (C ′ ∈ chF \{C}).
We can read R(C) from σC |∂D2 . We will give
outline of another proof (in §2).
1 Minimal positive presentation
Example
chF (A) = ∅
F
p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p pp
4
3
2
1
1 Minimal positive presentation
Example
π1 (M ) ∼
= ⟨γ1 , γ2 , γ3 , γ4 | no relations ⟩
chF (A) = ∅
F
p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p pp
4
3
2
1
1 Minimal positive presentation
Example
⟨
π1 (M ) ∼
=
⟩
1234 = 2134
γ1 , γ 2 , γ 3 , γ 4 = 2314 = 2341
234 1
Q
Q 23 14
Q
Q
Q 2 134
Q
Q
Q
F
Q
p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p pQ
p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p pp
Q
4
3
2
1
1 Minimal positive presentation
Example
⟨
π1 (M ) ∼
=
Q
⟩
1234 = 2341
γ1 , γ 2 , γ 3 , γ 4 = 3412 = 4123
234 1 A 34 12
4 123 Q
A
Q
Q A QA
Q
AQ
A Q
Q
Q
A
F
Q
p p p p p p p
p p p p p p p p p p p p p p p p p p p p p p p p p p pAp p p p p p p p p p p p p Q
p p p p p p p p p p p p p p p p p p p pp
Q
A
4
3
2
1
1 Minimal positive presentation
Example
γ1 , . . . , γ6 ⟨
π1 (M ) ∼
=
123456 = 123465
= 234561 = 345612
= 123645 = 236145
= 235614 = 356124 ⟩
= 213456 = 261345
= 612345 = 561234
561234
345612
Z
Z 356124
Z
Z
Z
612345
235614
234561
Z
Z
Z
XXX 236145
261345Z
X
XXX Z
213456 Z
123645X
X
X
X
123465
X
Z
X
F
X
X
Z
p
p p p
p p p
pppppppppppppppppppppppppppppppppppppppppppppppppppppZ
p p p pX
p p pX
p p pp
X
6
5
4
3
2
1
1 Minimal positive presentation
Remark A presentation of group
G = ⟨γ1 , . . . , γn | R1 , . . . , Rb ⟩
is called minimal if n = b1 (G) and b = b2 (G).
• Randell, Falk: Minimal presentation for
π1 (M (A)).
• Dimca, Papadima, Suciu, Randell: minimality
of M (A).
• Generally, the relations have conjugations.
• Eliyahu, Garber, Teicher, “Conjugation-free
geometric presentatioin.”
• Homogeneously presented monoids have
solvable word problem. However our monoids
are rarely embedded in the group.
1 Minimal positive presentation
Corollary. Let
π1 (M (A)) = ⟨γ1 , . . . , γn | R1 , . . . , Rb ⟩
be a minimal presentation. Then there are
n(n−1)
( 2 − b)-words Rb+1 , . . . , R n(n−1) such that
2
Z ∼
= ⟨γ1 , . . . , γn | R1 , . . . , R n(n−1) ⟩.
n
2
Remark. Not all minimally presented
groups
⟩
⟨
have this property, e.g., G = γ1 , γ2 | [γ1 , γ2 ]2 .
1 Minimal positive presentation
(Proof of Cor.)
Perturb generically
R
Z
4123
Z 2341
Z
Z
2134
Z
p p p p
p p p
p p p p p p p p p pZ
p p pZ
ppppppppp
4
3
2 Z1
⟩
⟨ 2413
1234
γi = 2134 = 2341
= 2413 = 4123
1 Minimal positive presentation
q
E 3412
(Proof of Cor.)
E
Perturb generically
E
E
R
E
4123
4123
Z
Z
2413
E
Z 2341
Z 2341 2413 E Z
Z
Z
Zq 2134 E
2134
1243 Z
Z
E
p p p p
p p p
p p p p p p p p p pZ
p p pEZ
ppppppppp
p p p p
p p p
p p p p p p p p p pZ
p p pZ
ppppppppp
4
3
2 Z1
4
3
2 Z1
E
⟩
⟨ ⟨ ⟩
1234
γi = 2134 = 2341
= 2413 = 4123
→
1234
γi = 2134 = 2341 = 1243
= 2413 = 4123 = 3412
= Z4 .
(Q.E.D.)
2 Minimal Stratification
2 Minimal Stratification
b
b
A = {0, 1}
M (A) = C \ {0, 1}
2 Minimal Stratification
c0
r
0
b
6
c
2
≀≀
c0
r
6
c
2
1
b
6
c
1
A = {0, 1}
M (A) = C \ {0, 1}
Homotopy equivalence
6
c
1
π1 (M ) = ⟨c1 , c2 ⟩
2 Minimal Stratification
Instead of looking at homotopy equivalent CW
complex, we consider stratification with
contractible strata.
2 Minimal Stratification
0
b
1
b
A = {0, 1}
M (A) = C \ {0, 1}
||
Stratifying by contractible sets
S0
0
b
S2
1
b
S1
S1 = (1, ∞)
S2 = (0, 1)
S0 = C \ R≥0
M = S0 ⊔ S1 ⊔ S2
2 Minimal Stratification
0
b
S2
1
b
S1
A = {0, 1}
M (A) = C \ {0, 1}
S2 = (0, 1)
S1 = (1, ∞)
How to recover π1 from stratification?
2 Minimal Stratification
c0
r
0
b
c2
O
S2
1
bO
c1
S1
A = {0, 1}
M (A) = C \ {0, 1}
S2 = (0, 1)
S1 = (1, ∞)
Since strata are contractible,
S1 and S2 determine transversal generators
c1 and c2 uniquely (up to homotopy).
How to recover π1 from stratification?
2 Minimal Stratification
Example. Σ2 : surface of genus g = 2.
Computing π1 (Σ2 ) from stratification.
2 Minimal Stratification
Example. Σ2 : surface of genus g = 2.
Computing π1 (Σ2 ) from stratification.
Σ2
2 Minimal Stratification
Example. Σ2 : surface of genus g = 2.
Computing π1 (Σ2 ) from stratification.
Σ2
C
C: 0-dim stratum.
2 Minimal Stratification
Example. Σ2 : surface of genus g = 2.
Computing π1 (Σ2 ) from stratification.
Σ2
S3
S2
S4
S1
Si : 1-dim strata.
2 Minimal Stratification
Example. Σ2 : surface of genus g = 2.
Computing π1 (Σ2 ) from stratification.
γ1
S3
Σ2
S2
S4
S1
2 Minimal Stratification
Example. Σ2 : surface of genus g = 2.
Computing π1 (Σ2 ) from stratification.
γ2
S3
Σ2
S2
S4
S1
2 Minimal Stratification
Example. Σ2 : surface of genus g = 2.
Computing π1 (Σ2 ) from stratification.
γ3
S3
Σ2
S2
S4
S1
2 Minimal Stratification
Example. Σ2 : surface of genus g = 2.
Computing π1 (Σ2 ) from stratification.
γ4
Σ2
S2
S4
S1
Where is the relations?
2 Minimal Stratification
Example. Σ2 : surface of genus g = 2.
Computing π1 (Σ2 ) from stratification.
γ1 γ2 γ1−1 γ2−1 γ3 γ4 γ3−1 γ4−1
= [γ1 , γ2 ][γ3 , γ4 ] = 1
S3
S2
S4
S1
Look at codim = 2 stratum.
2 Minimal Stratification
To compute π1 (M (A)), we will follow the
Strategy:
⊔
1. Stratify M = λ Sλ by contractible strata.
2. Generator ←→ codim = 1 strata.
3. Relation ←→ codim = 2 strata.
2 Minimal Stratification
Stratify by using the following objects.
S1 := {z ∈ M | α1 (z) ∈ R>0 },
}
{
α2 (z)
S2 := z ∈ M α1 (z) ∈ R<0 ,
@
ppppppppppppppp
@
{
}
αn (z)
@
Sn := z ∈ M αn−1 (z) ∈ R<0 .
@
@
2
p p p p
p p p p p p p p p p p p p p p p p@
ppppppp
What shape? How sit in C ?
@ H1 = {α1 = 0}
H3
H2
2 Minimal Stratification
Consider S = {(z1 , z2 ) ∈ C2 |
z2 `
z2
z1
∈ R>0 }
∗
S∼
C
× R>0
=
`
6
`
`
`
` ` ` ` ` ` ` ` ` `-`
z1
`
(2 + i, −3 + 2i) ∈
/S
`
2
`
r
∼
C
T R2
=
√
`
2
v ∈ Tx R 7→ x + −1v
`
2 Minimal Stratification
Consider S = {(z1 , z2 ) ∈ C2 |
z2 `
z2
z1
∈ R>0 }
∗
S∼
C
× R>0
=
(3, 2 + 2i) ∈
/S
`
6
`
6
(−2i, 2i) ∈
/S `
r (3 + i, 2i) ∈/ S
@
I `
` ` ` ` `@` ` ` `r `-`
z1
`
(2 + i, −3 + 2i) ∈
/S
`
2
`
r
∼
C
T R2
=
√
`
2
v ∈ Tx R 7→ x + −1v
`
2 Minimal Stratification
Consider S = {(z1 , z2 ) ∈ C2 |
z2 `
z2
z1
∈ R>0 }
∗
S∼
C
× R>0
=
` (2 − i,r 4 − 2i) ∈ S
6
3
` 6
(3 + 3i, 2 + 2i) ∈ S
(−2i, 2i) ∈
/S ` r
@
I `
` ` ` ` `@` ` ` `r `-`
z1
`
(2 + i, −3 + 2i) ∈
/S
`
2
`
r
∼
C
T R2
=
√
`
2
v ∈ Tx R 7→ x + −1v
`
2 Minimal Stratification
Consider S = {(z1 , z2 ) ∈ C2 |
z2 `
z2
z1
∈ R>0 }
∗
S∼
C
× R>0
=
`
6
r
3
` 6
` r r
:
@
I ` r
1
` ` ` ` `@`r
` ` `r `-`
z1
:
`
r
9
r3 `
`
r
+
T R2
r
`
2
v ∈ Tx R
`
2
∼
C
=
√
7
→
x + −1v
2 Minimal Stratification
Consider S = {(z1 , z2 ) ∈ C2 |
z2 `
z2
z1
∈ R>0 }
∗
S∼
C
× R>0
=
` 6
>
` r
=
` *
:
` r
(
9
(
(
(` ` `-`
` ` ` `(`(
`
`
(
(
(
z1
(
`
*
`
`
T R2
`
2
v ∈ Tx R
`
2
∼
C
=
√
7
→
x + −1v
2 Minimal Stratification
Example.
We shall outline
the proof with:
C
@
@
@
@
F
@
p p p p p p p p p p p p p p p p p p p p p p p p p pp
@
H2 = {α2 = 0} H1 = {α1 = 0}
(chF (A) = {C})
2 Minimal Stratification
Example.
We shall outline
the proof with:
C
@
@
@
@
SAC @
S
AC
CAS
@
p p
p pp
p@
ppp
p p p CpApS
α2
∈ R<0 } S1 = {α1 ∈ R>0 }
S2 = { α
@
F
@
p p p p p p p p p p p p p p p p p p p p p p p p p pp
@
1
H2 = {α2 = 0} H1 = {α1 = 0}
(chF (A) = {C})
@
@
@
@
@@
@
@
I
@
@
@
@@
@
R
@
@
@@
@
p p p p p p p p p p p@
p@p p
2 Minimal Stratification
Example.
We shall outline
the proof with:
C
@
@
@
@
@
@
@
@@
@
@
I
@
@
@
@@
@
R
@
@
@@
@
p p p p p p p p p p p@
p@p p
α2
∈ R<0 } S1 = {α1 ∈ R>0 }
S2 = { α
1
C@
@S
A
@
@
@
@
@
I
@
@
C
A
S1 ∩ S2 = @
S
@
=C
@
@
@
@
R
@
C
@
A
S
@
@
p p
p pp
p@
p@p p
p p p CpApS
H1 = {α1 = 0}
@
F
@
p p p p p p p p p p p p p p p p p p p p p p p p p pp
@
H2 = {α2 = 0}
@
SAC @
S
AC
CAS
@
p p
p pp
p@
ppp
p p p CpApS
(chF (A) = {C})
2 Minimal Stratification
Lemma.
∪
◦
(i) Si := Si \ C∈chF C, then
∗
3
∼
∼
= (C \ R>0 ) × R>0 = R (Homeo.)
∪
(ii) U := M (A) \ i Si , then U ∼
= R4 .
(iii) Giving a stratification:
◦
Si
M (A) = U ⊔
n
⊔
i=1
◦
Si
⊔
⊔
C∈chF
C.
2 Minimal Stratification
To compute π1 (M (A)), we will follow the
Strategy:
1. Stratify M = λ Sλ by contractible strata.
2. Generator ←→ codim = 1 strata.
3. Relation ←→ codim = 2 strata.
S3
S2
S4
S1
M (A) = U
S1◦ ∪ S2◦ ∪ C
2 Minimal Stratification
J@
B@
@
@
@@
J@
B@
@@
@
@J@
B@
@@
@
JB@
@@
@
B @@ F
J
p p p p p p p
p p p p p pBpJp@
p p p@
p p ppp
B J@
H2
H1
To look at transversal generators,
Si ∩ (F ⊗ C) =?
S1 = {α1 ∈ R>0 },
2
S2 = { α
∈ R<0 }.
α1
2 Minimal Stratification
J@
B@
@
@
@@
J@
B@
@@
@
@J@
B@
@@
@
JB@
@@
@
B @@ F
J
p p p p p p p
p p p p p pBpJp@
p p p@
p p ppp
B J@
H2
H1
To look at transversal generators,
Si ∩ (F ⊗ C) =?
S1 = {α1 ∈ R>0 },
2
S2 = { α
∈ R<0 }.
α1
FC = F ⊗ C
=
b
H2
b
H1
S2 ∩ FC
S1 ∩ FC
2 Minimal Stratification
J@
B@
@
@
@@
J@
B@
@@
@
@J@
B@
@@
@
JB@
@@
@
B @@ F
J
p p p p p p p
p p p p p pBpJp@
p p p@
p p ppp
B J@
H2
To look at transversal generators,
Si ∩ (F ⊗ C) =?
S1 = {α1 ∈ R>0 },
2
S2 = { α
∈ R<0 }.
α1
H1
FC = F ⊗ C
=
b
H2
r
b
H1
γ2
S2 ∩ FC
S1 ∩ FC
γ1 γ 2
The generator corresponding to Si is γi γi+1 . . . γn .
2 Minimal Stratification
Transversal
slice to C.
@
@
I
@ AK γ2
@
@r
A
q
(γ1 γ2 )−1
@ C
@
q Ar
qγ1 γ2 @
@
Aq
−1
γ
@ @
2 A
@
@Relation:
@
look at codim = 2.
@
γ2 (γ1 γ2 )−1 γ2−1 (γ1 γ2 )
@
= γ1−1 γ2−1 γ1 γ2
@
@
=⇒ γ1 γ2 = γ2 γ1 . (Q.E.D.)
3 Chamber basis of OS-algebra
3 Chamber basis of OS-algebra
ωi :=
dαi
(∈
α
2π −1 i
1
√
H 1 (M, Z)).
•
Ω : meromorphic differential forms.
∪
A• := C⟨ωi | i = 1, . . . , n⟩,
0
1
2
=A ⊕A ⊕A .
•
≃
Thm. (Brieskorn) A −→ H • (M, C).
We are connecting differential forms and
chambers.
3 Chamber basis of OS-algebra
Def. Borel-Moore homology H•BM (M, Z) is a
homology of (infinite) locally finite chains.
Since M = M (A) is an oriented 4-manifold, we
have natural isomorphisms:
BM
H4 (M, Z)
H3BM (M, Z)
H2BM (M, Z)
0
∼
= H (M, Z)(∼
= Z),
1
n
∼
∼
H
(M,
Z)(
Z
=
= ),
2
∼
H
(M, Z).
=
3 Chamber basis of OS-algebra
Prop. The isomorphism H3BM ∼
= H 1 is given by:
1
H3BM (M, Z) ∼
H
(M, Z)
=
[S1 ] 7−→ ω1
C
Q
[Si+1 ] 7−→ ωi+1 − ωi
C
Q
Q
C
Q
C
Q
Q
C
αi+1
Q
C
S
i+1 = { αi ∈ R<0 }
Q
p p p p p pp p pp p p p p p p p p Cp p p p p p p pQ
p ppp
Hi+1
Hi
3 Chamber basis of OS-algebra
(Proof.) Use the fact:
ωi+1 − ωi ∈
∼
=
1
H (M )
@
I
dual @
(Cap product or@
R
integration) @
- H3BM (M )∋ [Si+1 ]
dual
(Intersection)
H1 (M )
∫
(ωi+1 − ωi ) =
γj

S
i+1
b
b r- r b
 −1 j = i,
+1 j = i + 1,
0
+1 −1
=

γj
γi+1
γi
0
else.
∴ Si = ωi+1 − ωi . (Q.E.D.)
3 Chamber basis of OS-algebra
≃
The isomorphism H −→ H3BM is extended as:
1
2
H (M )
(ωi+1 − ωi ) ∧ (ωj+1 − ωj )
≃
−→ H2BM (M )
7−→ [Si ∩ Sj ]
We have
BM
H 2 (M ) ∼
H
= 2 (M ) ∼
=
⊕
C∈chF
C · [C].
3 Chamber basis of OS-algebra
Def. Let chF (A) = {C1 , . . . , Cb }.
We call
1 ∈ H 0 (M ),
[S1 ] = ω1 , [Si+1 ] = ωi+1 − ωi ∈ H 1 (M ), and
2
[C1 ], . . . , [Cb ] ∈ H (M ),
the chamber basis of H • (M ) = A• .
A = C · 1, A =
0
1
n
⊕
i=1
C · [Si ], A =
2
b
⊕
p=1
C · [Cp ].
3 Chamber basis of OS-algebra
Example. lC3
l
C2
C2
l
l
ω3 ∧ ω2
2
l
H (M )
l
= [C1 ] + [C2 ] C1l
∼
BM
C
1
l
→ H2 (M )
l
p p
p p p p
p p p p p p p p p p p p pl
p ppp
p p
p p p p
p p p p p p p p p pl
p p pl
p ppp
H3
H2
H1
lC3
C2
l
ω3 ∧ ω1
l
= [C2 ] + [C3 ]
l
p p
p p p p
p p p p p p p p p pl
p p pl
p ppp
H3
H1
H3
H2
lC3
l
ω2 ∧ ω1
l
= [C3 ]
l
p p
p p p p
p p p p p p p p p pl
p p pl
p ppp
H2
H1
3 Chamber basis of OS-algebra
Remark. Ko-ki Ito and I recently constructed
explicit basis of H∗BM (M (A), Z) for any
complexified real arrangements in Rℓ .
4 Chamber basis and H (M, Lλ )
1
4 Chamber basis and H (M, Lλ )
1
Applying chamber basis to local system
cohomology group.
A = {H1 , . . . , Hn }, M = M (A), γi , ωi as
above.
Let λ = (λ1 , . . . , λn ) ∈ Cn ,
Def. Lλ : the rank one local system defined by
π1 (M ) ∋ γi 7−→ e2π
Question: H (M, Lλ ) =?
1
√
−1λi
∈ C∗ .
4 Chamber basis and H (M, Lλ )
1
Thm (Esnault-Schechtman-Viehweg). Set
∑n
ωλ = i=1 λi d log αi . If λi are enough small,
then
∗
•
H ∗ (M, Lλ ) ∼
H
(A
, ωλ ∧).
=
Question: In general, can one recover
H 1 (M, Lλ ) from (A• , ωλ ∧) ?
Yes, using chamber basis.
4 Chamber basis and H (M, Lλ )
1
Looking at (A• , ωλ ∧) via chamber basis.
A →A :
0
1
1 7→ ωλ =
n
∑
ηi · [Si ],
i=1
A → A : [Si ] 7→ ωλ ∧ [Si ] =
1
2
b
∑
p=1
p
ρi
· [Cp ].
To obtain H 1 (M, Lλ ), we need “deform” the
p
coefficients ηi , ρi .
Def. ∆(x) := ex/2 − e−x/2 = 2 sinh(x/2).
4 Chamber basis and H (M, Lλ )
1
Thm. Let us define
∇ : A0 → A1 , 1 7→
n
∑
∆(ηi ) · [Si ],
i=1
∇ : A → A , [Si ] 7→
1
2
b
∑
p=1
p
∆(ρi )
· [Cp ],
then (A• , ∇) is a complex for ∀λ ∈ Cn , and
∗
∼
H (A , ∇) = H (M, Lλ ).
∗
•
4 Chamber basis and H (M, Lλ )
1
Thm. Define
∇ : A0 → A1 ,
1 7→
n
∑
∆(ηi ) · [Si ],
i=1
∇ : A1 → A2 ,
[Si ] 7→
b
∑
p=1
∆(ρpi ) · [Cp ],
then (A• , ∇) is a complex, and
H ∗ (A• , ∇) ∼
= H ∗ (M, Lλ ).
Remark. This does not mean “H 1 (M, Lλ ) is
combinatorial”.
5 References
http://www.math.kyoto-u.ac.jp/∼mhyo/index.html
• Y: Hyperplane arrangements and Lefschetz’s hyperplane
section theorem. Kodai Math. J. 30, no. 2 (2007), 157–194.
• Y: Chamber basis of the Orlik-Solomon algebra and Aomoto
complex. Arkiv for Matematik. vol 47 (2009), 393-407.
• Y: Minimality of hyperplane arrangements and basis of local
system cohomology. arXiv:1002.2038
• Ko-ki Ito, Y: Semi-algebraic partition and basis of
Borel-Moore homology of hyperplane arrangements. (draft)