Holonomy Lie algebras and the LCS formula for subarrangements of A_n

0.18 MB·English

by  Paulo Lima-Filho

HOLONOMY LIE ALGEBRAS AND THE LCS FORMULA
FOR SUBARRANGEMENTS OF A
n
9
0 PAULOLIMA-FILHOANDHALSCHENCK1
0
2
Abstract. If X is the complement of a hypersurface in Pn, then Kohno

n showed in [9] that the nilpotent completion of π1(X) is isomorphic to the
a nilpotent completion oftheholonomyLiealgebraofX. WhenX isthe com-
J plementofahyperplanearrangementA,theranksφkofthelowercentralseries

8 quotientsofπ1(X)areknownforisolatedexamples,andfortwospecialclasses:
2 arrangementsforwhichXishypersolvable(inwhichcasethequadraticclosure
ofthecohomologyringisKoszul),oriftheholonomyLiealgebradecomposes

] indegree3asadirectproductoflocalcomponents. Inthispaper,weusethe
T holonomyLiealgebratoobtainaformulaforφk whenAisasubarrangement
A ofAn. Thisextends Kohno’sresult[10]forbraidarrangements,andprovides
aninstance ofanLCSformulaforarrangements whicharenotdecomposable
.
h orhypersolvable.
t
a
m
[
1. Introduction

3
Let X be the complement of an arrangement of projective hypersurfaces. Asso-
v
6 ciated to the fundamental group π1(X) of X is the lower central series: a chain of
5 normal subgroups
2 π (X)=G ≥G ≥G ≥··· ,
1 1 2 3
0
. defined inductively by Gk =[Gk−1,G1]. The gradedvector space (⊕Gi/Gi+1)⊗Q
1 becomes a graded Lie algebra over Q with bracket given by [x,y] =xyx−1y−1. In
0
[9],KohnocombinedresultsofDeligne[4]andMorgan[11]onthe bigradingonthe
9
0 mixed Hodge structure of H1(X,Q) with results of Sullivan [23] on the minimal
: model to show that (⊕G /G )⊗Q is isomorphic to the nilpotent completion of
v i i+1
the holonomy Lie algebra of X.
i
X If X is the complement of an arrangement of hyperplanes A, the holonomy
r Lie algebra h is a purely combinatorial object, defined in terms of the rank two
a X
flats in the intersection lattice L(A). Elements of rank i in L(A) correspond to
maximal sets of hyperplanes {H ,...,H } intersecting in codimension exactly i.
1 k
SlicinganarrangementAwithagenericplaneyieldsanarrangementA′ ⊆P2 with
L(A)≃L(A′)inrank≤2. Inasimilarvein,atheoremofHamm-Lˆe[7]showsthat
thefundamentalgroupofX isunaffectedbytheprocessofslicingdowngenerically
to an arrangement in P2, which shows that the algebra (⊕G /G )⊗Q depends
i i+1
onlyonagenericplanarrestrictionofX. Comparingthesetworeductionssuggests
the possibility of a result like that proved by Kohno. An immediate consequence
of the isomorphism h ≃ (⊕G /G )⊗Q is that φ is the dimension of the kth
X i i+1 k
graded piece of h , where h is graded by bracket depth.
X X
2000 Mathematics Subject Classification. Primary52C35;20F14, 20F40,
Key words and phrases. holonomyLiealgebra,lowercentralseries,hyperplanearrangement.
1PartiallysupportedbyNSFDMS07-07667,NSAMDA904-03-1-0006.
1
2 PAULOLIMA-FILHOANDHALSCHENCK
In [10], Kohno used this framework to prove a beautiful formula for the LCS
ranks of the braid arrangementA :
n
∞ n−1
(1.1) Y(1−tk)φk =P(X,−t)= Y(1−it),
k=1 i=1
where P(X,t)= b ti is the Poincar´epolynomial of the complement of X in Cn.
P i
Definition 1.1. Let G be a (simple) graph on ℓ vertices, with edge-set E, and let
A = {z −z = 0 | (i,j) ∈ E} be the corresponding arrangement in Cℓ; A is
G i j G
called a graphic arrangement. Write U (t) for ∞ 1−tk φk(π1(Cℓ\AG)).
G Qk=1(cid:0) (cid:1)
The braid arrangementA is the arrangementassociatedto the graphK . Our
n n
main result is an extension (conjectured in [18]) of Kohno’s theorem:
Theorem 1.2 (Graphic LCS formula). If G is a graph with ℓ vertices, and κ
s
denotes the number of complete subgraphs of G on s+1 vertices, then:
ℓ−1
ℓ−1 X(−1)s−j(cid:0)sj(cid:1)κs
(1.2) UG(t)= Y(1−jt)s=j
j=1
Example 1.3. Suppose G consists of the 1-skeleton of the Egyptian pyramid and
the one skeleton of a tetrahedron, sharing a single triangle, as below:
For this graph, U (t)=(1−2t)4(1−3t). The proof of (1.2) uses a Mayer-Vietoris
G
argument: a graph is determined by the data of the maximal complete subgraphs,
togetherwithgluingdata. Forexample,ifG′denotesthe1-skeletonoftheEgyptian
pyramid, then four triangles are glued along four edges, and Corollary 2.5 shows
thatUG′(t)=((1−t)(1−2t))4/(1−t)4 =(1−2t)4. FortheK4 subgraph,UK4(t)=
(1−t)(1−2t)(1−3t). The graphs G′ and K are glued along a common triangle,
4
hence
(1−2t)4·(1−t)(1−2t)(1−3t)
U (t)= .
G
(1−t)(1−2t)
1.4. Other LCS formulas. To place (1.2) in context, we note that Kohno’s for-
mulahaspreviouslybeenextendedintwodifferentdirections. In[6],FalkandRan-
dell give a generalization to fiber-type arrangements,i.e. those whose complement
canbe factoredas a chainof linear fibrations with fiber F ≃C−{p ,...,p }. As
i 1 ik
shown by Terao in [24], this is equivalent to a lattice-theoretic condition known as
supersolvability. Supersolvability implies that H∗(X) admits a quadratic Gr¨obner
basis,whichimpliesthatH∗(X)isKoszul;in[20],SheltonandYuzvinskyinterpret
the LCSformula (1.1) in terms ofKoszulduality. The requirementthat H∗(X) be
Koszul can be relaxed a bit, as shown in [8], where Jambu and Papadima provide
HOLONOMY LIE ALGEBRAS AND THE LCS FORMULA 3
a version of the LCS formula for hypersolvable arrangements. An arrangement A
is hypersolvable iff there is a supersolvable arrangement B and linear subspace W
with A = B∩W preserving the rank two flats (see [5]). Since an arrangement in
P2 is supersolvable iff there exists x ∈ L (A) such that for each y ∈ L (A) there
2 2
is a line of A connecting x and y, it is clear that supersolvable arrangements have
very constrained geometry.
The second generalization lies at the opposite end of the spectrum. Following
PapadimaandSuciu[13],defineanarrangementtobedecomposableiftheholonomy
Lie algebra behaves in degree 3 like a direct product of holonomy Lie algebras of
pencil arrangements. Arrangements of this type are as simple as possible in many
respects;asshownbyCohen-Suciu[3],thelinearizedAlexanderinvariantisadirect
sum of submodules corresponding to Alexander invariants of rank two subarrange-
ments, with scalarssuitably extended. While the underlying arrangementA is not
in general a product, the LCS ranks behave as if it were. Papadima-Suciu prove
that for decomposable arrangements,
1−µ(p)t
(1.3) UG(t)=(1−t)b1 Y .
1−t
p∈L2(A)
Here µ denotes the Mo¨bius function: for p ∈ L (A), µ(p) is one less than the
2
number of hyperplanes of A which contain p.
Remark 1.5. Theorem 6.4 of [18] shows that a graphic arrangement has a coho-
mology ring which is Koszul iff the graph is supersolvable, and in [14], Papadima
and Suciu give necessary and sufficient conditions for a graphic arrangementto be
hypersolvable. ResultsofStanley[21]andTerao[24]showthatA issupersolvable
G
iff G is chordal. As noted in Remark 6.18 of [18], the chromatic polynomial of a
chordal graph is given by
(1.4) χG(t)=tκ0κY0−1(cid:0)1−jt−1(cid:1)Pκs=0−j1(−1)s−j(cid:0)sj(cid:1)κs,
j=1
andinthiscase,(1.2)specializestotheFalk-RandellLCSformula. Attheopposite
end of the spectrum are the decomposable graphic arrangements,classified in [13]:
they are exactly those with κ = 0 for all i ≥ 3. For these arrangements, (1.2)
i
specializes to (1.3). Equations 1.2 and 1.3 are instances of the Resonance LCS
conjecture of [22].
The point of this paper is that for certain special classes of arrangements, it is
possible to find a formula which encompasses both the decomposable and super-
solvable classes, and indeed, everything in between. The main tool involved in the
proof is a careful analysis of the holonomy Lie algebra, which we quickly review.
It is worth noting that in [16], Peeva shows that for the D arrangement, the LCS
3
ranks cannot be obtained as P(N,−t) for any standard graded algebra N, and in
[17], Roos shows that in fact there are arrangements for which
∞
Y(1−ti)φi
i=1
is a transcendental function.
4 PAULOLIMA-FILHOANDHALSCHENCK
1.6. The holonomyLie algebraandformal spaces. LetX bethecomplement
of a complex hypersurface in Pn. Cup product gives a map
H1(X,Q)∧H1(X,Q)−→H2(X,Q);
the dual of this map is the comultiplication map
H (X,Q)−∪→t H (X,Q)∧H (X,Q)−→L(H (X,Q)),
2 1 1 1
where L(H (X,Q)) is the free Lie algebra on H (X,Q). Following Chen [2], define
1 1
the holonomy Lie algebra of X as the quotient L(H (X,Q))/I , where I is the
1 X X
ideal generated by the image of the comultiplication map. In the case of arrange-
ments,Kohnoshowsthatthe imageofthe comultiplicationmapisgeneratedbyall
brackets of the form
k
[xj,Xxi],
i=1
where x is the generator of L(H (X,Q)) corresponding to the i-th hyperplane e
i 1 i
ande ∈{e ,...,e }is a (maximal)setofhyperplanesintersectingincodimension
j 1 k
two; {e ,...,e } corresponds to an element of L (A). This is also an immediate
1 k 2
consequence of the description of the cohomology ring of an arrangement comple-
ment given by Orlik-Solomonin [12].
The proof of Kohno’s main result runs as follows. In [23], Sullivan introduced
the minimalmodelofa differentialgradedalgebra(DGA) andassociatednotionof
formality. A space X is formal if the minimal model of the DGA of Q-polynomial
forms on X is formal. Sullivan shows that for a formal space, (⊕G /G )⊗Q is
i i+1
isomorphic to the associated graded Lie algebra L of a filtration L arising in the
i
construction of the 1-minimal model.
If A is defined by the vanishing of {α1,...,αn}, let R = C[dαα11,...,dααnn].
Brieskorn [1] shows that the map R → H∗(X) sending dαi to its class is an iso-
αi
morphism,soanarrangementcomplementX isformalandSullivan’sresultsapply.
Using work of [4] and [11], Kohno analyzes the bigrading on the Hirsch extensions
appearing in the construction of the 1-minimal model and proves that there is an
isomorphismfromtheholonomyLiealgebratoL. ItfollowsfromPoincar´e-Birkhoff-
Witt that ∞ (1−tk)−φk is the Hilbert series of the universalenvelopingalgebra
Qk=1
of the holonomy Lie algebra.
2. Gluing h from subgraphs
X
In this section, we prove an amalgamation lemma, which will allow us to relate
theholonomyLiealgebrasoftwographicarrangements,identifiedalongacommon
subgraph of a special type.
We denote by I the ideal in L(H (X,Q)) describing the holonomy Lie algebra
G 1
for the graphic arrangement A . In this case, the brackets generating the ideal
G
are as follows: either they have the form [x ,x ] for edges e ,e not contained in a
1 2 1 2
triangleofG,ortheyhavetheform[x ,x +x +x ], i=1,2,3,if {e ,e ,e } isa
i 1 2 3 1 2 3
triangleinG. Notethatinthelattercaseitsufficestousethebrackets[x ,x +x ]
1 2 3
and [x ,x +x ].
2 1 3
To simplify notation, we write h for h and L(G) for the free Lie algebra on
G AG
the edges of G; edges of a graph will be denoted by e , with x the corresponding
j j
generator in L(G) or h . Let ∆ denote set of triangles in a graph G.
G G
HOLONOMY LIE ALGEBRAS AND THE LCS FORMULA 5
Definition 2.1. For a subgraph K of G, the pair (G,K) is triangle-complete (TC)
if whenever {e ,e ,e } ∈ ∆ and {e ,e } ⊆ K with {e ,e } ⊆ {e ,e ,e },
i1 i2 i3 G j1 j2 j1 j2 i1 i2 i3
then {e ,e ,e } ⊆K. So if two edges of a triangle of G lie in K, the third edge
i1 i2 i3
is also in K.
Lemma 2.2. Let b = [x [x ,[x ,[...[x ,x ]]]...] ∈ h be a bracket of depth
n n−1 n−2 1 0 G
n, and suppose that e ∈ G\K for some i ∈ {0,...,n}. If (G,K) is TC, then b
i
can be written as a sum of brackets of depth n
b=X [xjn,[xjn−1,[xjn−2,[...[xj1,xj0]]]...]
with all the e ’s in G\K.
jk
Proof. We induct on the depth of the bracket. Consider [x ,x ], with x ∈ G\K
1 0 1
and x ∈ K. If {e ,e } is not the edge of a triangle in G, then [x ,x ] = 0
0 1 0 1 0
in h , and we can replace x with x . Otherwise, there exists an edge e with
G 0 1 j
{e ,e ,e }∈∆ , so that in h we have [x ,x +x ]=0, hence [x ,x ]=[x ,x ].
1 0 j G G 1 0 j 1 0 j 1
Since (G,K) is TC, the assumption that e 6∈K implies that e 6∈K.
1 j
Let B denote a bracket of the form [x ,[x ,[...[x ,x ]]]...], and consider
n−1 n−2 1 0
[x ,B]. If B has some element in G\K, then by induction we can expand B as a
n
sum of brackets of the desired type. So it suffices in this case to prove the result
for [x ,[x ,B′]] with all entries of [x ,B′] in G\K. But
n n−1 n−1
(2.1) [x ,[x ,B′]]=−[B′,[x ,x ]]−[x ,[B′,x ]],
n n−1 n n−1 n−1 n
andtheresultholdsbyinduction. IfBishasnoelementinG\K,thenx ∈G\K,
n
and again the result follows from the Jacobi identity (2.1) and induction. (cid:3)
Corollary 2.3. If (G,K) is TC, then there are canonical surjections π: h →h
G K
and I →I . Furthermore, the kernel K(π) of π is the subalgebra of h generated
G K G
by {x |e ∈G\K}.
i i
Proof. The map E(G)→L(K), which is the identity on the edges of K and sends
ψ
all other edges to zero, extends to a surjection of free Lie algebras L(G) −→L(K).
Since K ⊂ G, one can consider L(K) as a subalgebra of L(G), and ψ restricts to
the identity on L(K). The induced surjection L(G)→h descends to a surjection
K
π: h →h , since the TC property and Lemma 2.2 guarantee that any generator
G K
of I either is sent to zero, or is unchanged and hence is a generator of I . This
G K
not only gives the surjectivity of ψ : I →I , but also shows that
|IG G K
(2.2) I ∩L(K) = I ,
G K
whenever (G,K) is TC. Using the snake lemma, one concludes that K(π) is the
ideal in h generated by hx | e ∈ G\Ki. We now proceed to show it is indeed
G i i
the subalgebra generated by these elements.
//
I I
G K
(cid:15)(cid:15) (cid:15)(cid:15)
hx |e ∈G\Ki // L(G) //L(K)
i i
δ
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
K(π) //h // h
G K
6 PAULOLIMA-FILHOANDHALSCHENCK
By Lemma 2.2 any bracket γ ∈ K(π) involving some x with e ∈ G\K may
i i
be written as a sum of brackets with all entries coming from G\K, so it suffices
to show that if α 6= 0 ∈ h is a sum of brackets involving only x ’s with e ∈ K
G i i
then α 6∈ K(π). Indeed, an element α of this form can be lifted to an element
α ∈ L(K) ⊂ L(G), and hence π(α) = 0 if and only if α ∈ I = I ∩L(K) ⊂ I .
K G G
Tbhis gives α=0. b (cid:3)
Let G and G be two graphs, having a common subgraph K. We construct a
1 2
new graph G=G G by identifying G and G along the image of K.
1SK 2 1 2
Theorem 2.4. If G = G G and (G ,K) and (G,G ) have property TC for
1SK 2 i i
i∈{1,2}, then:
a: The diagram
h π2 // h
G G2
π1 φ2
(cid:15)(cid:15) (cid:15)(cid:15)
h // h
G1 φ1 K
is a pull-back diagram of Lie algebras.
b: The diagram
h i2 //h
K G2
i1 j2
(cid:15)(cid:15) (cid:15)(cid:15)
h //h
G1 j1 G
is a push-out diagram of Lie algebras.
Proof. To prove assertion a, it suffices to show that
(2.3) 0−→h −π−1−×−π→2 h ×h −φ−1−−−φ→2 h −→0.
G G1 G2 K
is an exact sequence of vector spaces. It follows from Corollary 2.3 that φ −φ
1 2
is surjective. On the other hand, Lemma 2.2 shows that any element α ∈ h ,
i Gi
i = 1,2, can be written as α = g +k , where g (respectively k ) is a sum of
i i i i i
brackets involving only generators associated to edges in G −K (respectively K).
i
In particular, if (α ,α ) ∈ Ker(φ −φ ) then k = k and one immediately sees
1 2 1 2 1 2
that(α ,α )isintheimageofπ ×π . Theexactnessofthesequenceinthemiddle
1 2 1 2
is now clear. Consider the commutative diagram:
//
K(π ) K(φ )
1 2
(cid:15)(cid:15) (cid:15)(cid:15)
K(π ) //h π2 //h
2 G G2
π1 φ2
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
K(φ ) //h φ1 // h
1 G1 K
HOLONOMY LIE ALGEBRAS AND THE LCS FORMULA 7
Since both (G ,K) and (G,G ) are TC, it follows from Lemma 2.2 that any
i i
bracketinK(φ )maybe writtenas abracketinvolvingonly the edgesinG \K =
2 2
G\G , and any bracket in K(φ ) may be written as a bracket involving only the
1 1
edges in G \K = G\G . Thus, the maps K(π ) → K(φ ) and K(π ) → K(φ )
1 2 1 2 2 1
are surjective.
By Corollary 2.3, K(π ) is the subalgebra of h generated by the variables as-
i G
sociated to edges in G\G , i = 1,2 and, similarly, K(φ ) is the subalgebra of h
i i G
generated by the variables associated to edges in G \K, i=1,2.
i
It follows that the projection L(G)→h induces an isomorphism
G
L(G\G )/I ∩L(G\G )≃K(π ).
1 G 1 1
The surjection L(G)→L(G ) restricts to an isomorphism
2
L(G\G )−≃→L(G \K)
1 2
and under this isomorphism one has
K(π ) ≃ L(G\G )/I ∩L(G\G )
1 1 G 1
≃ L(G \K)/I ∩L(G )∩L(G \K)
2 G 2 2
= L(G \K)/I ∩L(G \K)
2 G2 2
≃ K(φ ),
2
where the equality comes from (2.2), since (G,G ) is TC. It follows that the upper
2
horizontal arrow in the diagram is an isomorphism, and so is the leftmost vertical
arrow. In particular K(π )∩K(π ) = 0, thus showing that the kernel of π ×π
1 2 1 2
is trivial, and this concludes the proof that the sequence (2.3) is exact, and (a)
follows.
To prove assertion b, first observe that (2.2) implies that the natural maps
i : h → h and j : h → h are injections, for r = 1,2. Now, consider the
r K Gr r Gr G
sequence
0−→h −i1−×−i→2 h ×h −j−1−−−j→2 h −→0.
K G1 G2 G
A similar argument to the one used in previous assertionshows that this sequence
is exact and that the diagram is in fact a push-out of Lie algebras. (cid:3)
Corollary 2.5. With assumptions above,
U (t)·U (t)
U (t)= G1 G2 .
G
U (t)
K
3. Main Theorem
To prove the main theorem, we induct on the number of vertices v(G) of G, the
basic case being trivial. If G is a graph on {v ,...,v }, then let G denote the
1 n 1
subgraph of G consisting of all edges of G not containing v , and G′ denote the
n 2
subgraphof G consisting of all edges of G containing v . Note that v(G )=n−1.
n 1
Lemma 3.1. Let K consist of all edges in G which connect vertices {a ,a } such
1 1 2
that {a ,v },{a ,v } ∈ G′, and define G = G′ ∪K. Then (G ,K) and (G,G )
1 n 2 n 2 2 2 i i
are TC.
8 PAULOLIMA-FILHOANDHALSCHENCK
Proof. If a triangle with vertices {a ,a ,a } of G has two edges {a ,a },{a ,a }
1 2 3 1 2 1 3
in K, then {a ,v }∈G′ and hence {a ,a }∈K. So (G,K) is TC, which implies
i n 2 2 3
(G ,K) is TC. Clearly (G,G ) is TC, since any triangle of G with two edges in G
i 1 1
does notcontainthe vertexv , sohas the third edge inG . Finally, if a triangle of
n 1
G has two edges {a ,a },{a ,a } in G , then there are three possibilities.
1 2 1 3 2
(1) If v =a , then {a ,a }∈K ⊆G .
n 1 2 3 2
(2) If v =a , then {a ,a }∈K, hence {v ,a }∈G .
n 2 1 3 n 3 2
(3) If v 6∈{a ,a ,a }, then {v ,a },{v ,a }∈G , so {a ,a }∈K ⊆G .
n 1 2 3 n 2 n 3 2 2 3 2
(cid:3)
IfG≃K thenthe formulaholdsby[10](this canalsobe proveddirectly,using
n
[6], [21] and [24]), so we may assume G 6≃ K . Using the notation of Lemma 3.1
n
choose v so that v(G ) = v(G′) < v(G). By induction, Lemma 3.1 and Corol-
n 2 2
lary 2.5, we find that U (t)=
G
v(G1)−1 v(G2)−1
X (−1)s−j`sj´κs(G1) X (−1)s−j`sj´κs(G2)
v(G1)−1(1−jt) s=j · v(G2)−1(1−jt) s=j
Qj=1 Qj=1
v(K)−1
X (−1)s−j`sj´κs(K)
v(K)−1(1−jt) s=j
Qj=1
Observing that for any graph H, one has κ (H) = 0 whenever s ≥ v(H), we can
s
write this formula as
2X(−1)s−j(cid:0)sj(cid:1){κs(G1)+κs(G2)−κs(K)}3
6 7
Y (1−jt)4s≥j 5.
j≥1
Partition the complete subgraphs of G into two sets: those which contain v , and
n
those which do not. A complete subgraph lying in G and not containing v
2 n
corresponds to a complete subgraph lying in K, and so is counted in both G
1
and G . This concludes the proof of (1.2).
2
Remark 3.2. The ideas here may be extended to the class of Lie algebras where
an analog of Theorem 2.4 holds, and will appear elsewhere shortly.
References
[1] E.Brieskorn,Sur les groupes de tresses,S´eminaireBourbaki,1971/72, LNM317,Springer-
Verlag,(1973), 21–44.
[2] K.T.Chen,Iteratedintegralsofdifferentialformsandloopspace cohomology,Ann.ofMath.
97(1973), 217-246.
[3] D.Cohen,A.Suciu,Alexanderinvariantsofcomplexhyperplanearrangements,Trans.Amer.
Math.Soc.351(1999), 4043–4067.
[4] P.Deligne,Th´eorie de Hodge II,Inst.Hautes E´tudesSci.Publ.Math.40(1971), 5-58.
[5] G.Denham, A.Suciu, On the homotopy Lie algebra of an arrangement, MichiganMath. J.
54(2006), 319–340.
[6] M.Falk,R.Randell,The lower central seriesof a fiber-type arrangement,Invent. Math.82
(1985), 77–88.
[7] H. Hamm, D. T. Lˆe, Un th´eor`eme de Zariski du type de Lefschetz, Ann. Sci. E´cole Norm.
Sup.6(1973), 317–366.
[8] M.Jambu,S.Papadima,Ageneralizationoffiber-typearrangementsandanewdeformation
method,Topology 37(1998), 1135–1164.
HOLONOMY LIE ALGEBRAS AND THE LCS FORMULA 9
[9] T. Kohno, On the holonomy Lie algebra and the nilpotent completion of the fundamental
group of the complement of hypersurfaces, NagoyaMathJ.92(1983), 21–37.
[10] T.Kohno,S´eriedePoincar´e-Koszul associ´ee auxgroupes detressespures,Invent.Math.82
(1985), 57–75.
[11] J. Morgan, The algebraic topology on smooth algebraic varieties, Inst. Hautes E´tudes Sci.
Publ.Math.48(1978), 137-204.
[12] P. Orlik, L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent.
Math.56(1980), 167–189.
[13] S.Papadima,A.Suciu,WhendoestheassociatedgradedLiealgebraofanarrangementgroup
decompose?, CommentariiMathematici Helvetici,81(2006), 859–875.
[14] S.Papadima, A. Suciu, Higher homotopy groups of complements of complex hyperplane ar-
rangementsAdv.Math.165(2002), 71–100.
[15] S.Papadima,S.Yuzvinsky,Onrational K[π,1]spaces and Koszulalgebras,J.PureApplied
Algebra144(1999), 157–167.
[16] I. Peeva, Hyperplane arrangements and linear strands in resolutions, Trans. Amer. Math.
Soc.355(2003), 609–618.
[17] J.ERoos,Thehomotopy Liealgebraofacomplexhyperplanearrangementisnotnecessarily
finitely generated,ExperimentalMathematics 17(2008), 129–143.
[18] H.Schenck, A.Suciu,Lowercentral seriesand free resolutions of hyperplane arrangements,
Trans.Amer.Math.Soc.354(2002), 3409–3433.
[19] H.Schenck, A.Suciu,Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-
Gelfand correspondence, Trans.Amer.Math.Soc.,358,(2006), 2269-2289.
[20] B.Shelton,S.Yuzvinsky,Koszulalgebrasfromgraphsandhyperplane arrangements,J.Lon-
donMath.Soc.56(1997), 477–490.
[21] R.P.Stanley, Supersolvable lattices,AlgebraUniversalis2(1972), 197–217.
[22] A. Suciu, Fundamental groups of line arrangements: Enumerative aspects, in: Advances in
algebraicgeometry motivated by physics, ContemporaryMath., vol. 276, Amer.Math. Soc,
Providence,RI,2001, pp.43–79.
[23] D.Sullivan,Infinitesimalcomputations intopology,Inst.HautesE´tudesSci.Publ.Math.47
(1977), 269–331.
[24] H.Terao,Modular elementsoflatticesandtopological fibration,Adv.Math.62(1986),135–
154.
Departmentof Mathematics,Texas A&MUniversity, CollegeStation,TX77843
E-mail address: [email protected]
URL:http://www.math.tamu.edu/~paulo.lima-filho
Departmentof Mathematics,University ofIllinois, Urbana,IL 61801
E-mail address: [email protected]
URL:http://www.math.uiuc.edu/~schenck