Hyperplane Arrangements and Lefschetz’s
Hyperplane Section Theorem
Masahiko Yoshinaga∗
Groups, Homotopy and Configuration Spaces
5th-11th July, 2005, Tokyo
The purposes
Counting chambers
Recently Dimca-Papadima [1] and
Randell [2] proved that the complement M(A) of a complex hyperplane arrangement A in C` is homotopy equivalent to a minimal CWcomplex, namely, a CW-complex
whose number of k-cells is equal to
its k-th betti number for each k ≥ 0.
The goal here is to provide
Let
F 0 ⊂ F 1 ⊂ F 2 = VR = R2
be a generic flag. And define
½ ¯
¾
¯ C ∩ F k−1 = ∅
chk (A) = C ¯¯
,
C ∩ F k 6= ∅
where C is a chamber.
Fact: ]chk (A) = bk (M(A)).
• a presentation for the fundamental group π1(M(A)),
C5
@
@
@
@
• a minimal chain complex computing local system homology
@
@
RIMS, Kyoto Univ., [email protected]
1
C
C1
C3
F1
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7 @
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@
C0
Ft0
@
obtained from the study of attaching
maps for a real arrangement A.
We concentrate on 2-dimensional
case. See [3] for details.
∗
C6
@
C2
@
@@
ch2 = {C5, C6, C7}
ch1 = {C1, C2, C3}
ch0 = {C0}
π(A, t) = 1 + 3t + 3t2
2
The minimal CW-decomposition
The degree map
chk (A) indexes k-cells.
The degree map is a map
deg : ch2(A)×ch1(A) → {−1, 0, 1}
k = 1.
t
d
C1
d
C2
d
C3
F0
FC1
oo
defined below.
Let C1, . . . , C4 ∈ ch2(A) and C 0 ∈
ch1(A).
,
D
, D
,
D
A
,
A
hh
,hh 1 D
hhh
Dh A ,
D A
A
A
A
A
hhhh
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A
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A
hh2 @
A
h
hhh
@h
A H
@
HH
A
H 3
A
HH
A
H
3
A
A
A
A
Q
A
Q
Q
A
(
4
(
(
(((
A
A
A
Au
u
A
0
1
A
A
A
A
C
+1
C
t
F0
γ1
γ2
γ
0
0
C
−1
C
Cahmber: Ci ←→ 1-cell: γi.
C ∩F
k = 2. Let Q be a defining equation
of A. Let f be a defining equation
of the line F 1. Then
¯ λ¯
¯f ¯
ϕ = ¯¯ ¯¯ : M(A) → R≥0
Q
is a Morse function. C ∈ ch2(A) is
a stable manifold. The corresponding unstable manifold σC is the 2-cell
attaching to M(A) ∩ FC1 .
deg(C1, C 0) = 1
deg(C2, C 0) = deg(C3, C 0) = 0
deg(C4, C 0) = −1.
The degree map will be used to describe the fundamental group and
the boundary map of twisted minimal chain complex.
3
Fundamental group
Local system
ch1(A) = {C1, . . . , Cn}.
Let γ1, . . . , γn be the corresponding
1-cells.
Given C ∈ ch2(A), define a word
R(C) by
Let A = {H1, . . . , Hn}. Choose a
nonzero complex number qi ∈ C∗ for
each i = 1, . . . , n.
We consider the rank one local system L on M(A) such that the local
monodromy around Hi is qi2.
R(C) := γ1e1 γ2e2 · · · γnen γ1−e1 · · · γn−en ,
where ei = deg(C, Ci).
Theorem
The fundamental group π1(M(A))
has the following presentation:
hγ1, . . . , γn |R(C); C ∈ ch2(A)i .
Definition
For two chambers C and C 0,
Sep(C, C 0) := {i|Hi separates C, C 0}.
And define
Y
0
∆(C, C ) :=
qi −
i∈Sep(C,C 0 )
@
@
C6
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p
0
C
@
C 7 @
e
H2
C0
e
@
H1
@
F1
@
@
@
i∈Sep
@
qi−1
C5
@
Ft
Y
γ1
γ2
R(C5) =
R(C6) =
R(C7) =
γ2γ3−1γ2−1γ3
γ1γ3−1γ1−1γ3
γ1γ2−1γ1−1γ2
π1(M(A)) = Z3
@
@
@
γ3
∆(C, C 0) = q1q2 −
1
.
q1 q2
4
Twisted minimal chain complex
A = {H1, . . . , Hn},
ch0(A) = {p},
ch1(A) = {γ1, . . . , γn}
as above. Put
C0 := C[p]
C1 := C[γ1] ⊕ · · · ⊕ C[γn]
M
C2 :=
C[σC ]
C∈ch2
and define ∂ : Ck → Ck−1 as
∂([γi]) = ∆(γi, p)[p]
n
X
∂([σC ]) =
deg(C, γi)∆(C, γi)[γi].
i=1
Theorem
Then (C•, ∂) is a chain complex,
namely, ∂ 2 = 0 and
Hk (C•, ∂) ∼
= Hk (M(A), L).
Reference
[1] A. Dimca, S. Papadima, Hypersurface complements, Milor fibers and
higher homotopy groups of arrangements. Ann. of Math. (2) 158
(2003), 473–507
[2] R. Randell, Morse theory, Milnor fibers
and minimality of hyperplane arrangements. Proc. Amer. Math. Soc. 130
(2002), 2737–2743
[3] M. Yoshinaga, Hyperplane arrangements and Lefschetz’s hyperplane section theorem. (Preprint)