LIE ALGEBRAS ASSOCIATED TO FIBER-TYPE ARRANGEMENTS
DANIEL C. COHEN[, FREDERICK R. COHEN", AND MIGUEL XICOTENCATL]
Abstract.Given a hyperplane arrangement in a complex vector space of dim*
*en-
sion `, there is a natural associated arrangement of codimension k subsp*
*aces in
a complex vector space of dimension k`. Topological invariants of the c*
*omple-
ment of this subspace arrangement are related to those of the complement*
* of the
original hyperplane arrangement. In particular, if the hyperplane arrang*
*ement is
fiber-type, then, apart from grading, the Lie algebra obtained from the *
*descend-
ing central series for the fundamental group of the complement of the hy*
*perplane
arrangement is isomorphic to the Lie algebra of primitive elements in th*
*e homol-
ogy of the loop space for the complement of the associated subspace arra*
*ngement.
Furthermore, this last Lie algebra is given by the homotopy groups modul*
*o tor-
sion of the loop space of the complement of the subspace arrangement. Lo*
*oping
further yields an associated Poisson algebra, and generalizations of the*
* "universal
infinitesimal Poisson braid relations."
1. Introduction
Two classical constructions of interest in group theory and topology are:
(i)The Lie algebra arising from the filtration quotients associated to the *
*descend-
ing central series of a discrete group G; and
(ii)The Lie algebra of primitive elements in the singular homology of the lo*
*op
space of a space X, for certain topological spaces X.
The purpose of this article is to illustrate that these two a priori unrelated *
*Lie algebras
are, in fact, isomorphic in certain natural cases. This work is motivated by r*
*ecent
results relating the Lie algebras of (i) and (ii) arising in the context of cla*
*ssical con-
figuration spaces, and resolves a conjecture of the second two authors concerni*
*ng the
generalization of these results to spaces arising from certain hyperplane arran*
*gements.
The main result here asserts that the Lie algebra associated to the fundamen*
*tal
group G of the complement of a fiber-type hyperplane arrangement is, up to regr*
*ading,
isomorphic to the Lie algebra of primitive elements in the homology of the loop*
* space
of the complement of a higher dimensional analogue of the arrangement. The main
theorem is, in fact, stronger. The Samelson product for the loop space gives ri*
*se to a
graded Lie algebra given by the homotopy groups modulo torsion. This Lie algebr*
*a is,
again up to regrading, also isomorphic to the Lie algebra associated to the des*
*cending
central series quotients of G. In addition, after looping further, there are na*
*tural related
Poisson algebras arising from the homology of associated iterated loop spaces.
Given a discrete group G, let Gn be the n-th stage of the descending central*
* series,
defined inductively by G1 = G, and Gn+1 = [Gn; G] for nL 1, and let En0(G) =
Gn=Gn+1 be the n-th associated quotient. Let E*0(G) = n1 En0(G) be the Lie
algebra obtained from the descending central series of G, with Lie algebra stru*
*cture
____________
2000 Mathematics Subject Classification. Primary 52C35, 55P35; Secondary 20F*
*14, 20F36, 20F40.
Key words and phrases. fiber-type arrangement, descending central series, lo*
*op space homology.
[Partially supported by grants LEQSF(1996-99)-RD-A-04 and LEQSF(1999-2002)-R*
*D-A-01 from
the Louisiana Board of Regents, and by grant MDA904-00-1-0038 from the National*
* Security Agency.
"Partially supported by the National Science Foundation.
]Partially supported by the CINVESTAV of the IPN and by the MPI f"ur Mathema*
*tik.
1
2 D. COHEN, F. COHEN, AND M. XICOTENCATL
induced by the commutator map G x G ! G, (x; y) 7! xyx-1y-1. For each positive
integer k, use the ungraded Lie algebra E*0(G) to define a related graded Lie a*
*lgebra
as follows.
Definition 1.1. For a group G, let E*0(G)k be the graded Lie algebra given by
( n
Eq0(G)k = E0(G) if q = 2nk,
0 otherwise,
with Lie bracket structure induced by that of the Lie algebra E*0(G) obtained f*
*rom the
descending central series of G in the obvious manner.
A theorem relating the Lie algebras of (i) and (ii) above is described next.*
* Let Pn be
the Artin pure braid group, the fundamental group of the configuration space F *
*(C; n).
The results on configuration spaces alluded to above, due to Fadell and Hussein*
*i [7]
and Cohen and Gitler [3], may be summarized as follows.
Theorem 1.2. For k 1, the homology of the loop space of the configuration space
F (Ck+1; n) is isomorphic to the universal enveloping algebra of the graded Lie*
* algebra
E*0(Pn)k. Moreover,
(a)The image of the Hurewicz homomorphism
ss*(F (Ck+1; n)) ! H*(F (Ck+1; n); Z)
is isomorphic to E*0(Pn)k; and
(b) The Hurewicz homomorphism induces isomorphisms of graded Lie algebras
ss*(F (Ck+1; n))= Torsion! PrimH*(F (Ck+1; n); Z) ~=E*0(Pn)k;
where Prim o denotes the module of primitive elements, and the Lie algeb*
*ra
structure of the source is induced by the classical Samelson product.
The Lie algebra arising in the above theorem is the "universal Yang-Baxter L*
*ie al-
gebra" L(n), the quotient of the free Lie algebra on a free abelian group of ra*
*nk n2by
relations recorded in (4.1)below, and known variously as the "infinitesimal pur*
*e braid
relations" or the "horizontal four-term relations and framing independence." Fu*
*rther-
more, the homology of an iterated loop space of configuration space, qF (Ck+1; *
*n)
for q > 1, admits the structure of a graded Poisson algebra, see [4]. The assoc*
*iated
relations are called the "universal infinitesimal Poisson braid relations." Th*
*is Pois-
son algebra structure on H*(qF (Ck+1; n)) has recently been used in the context*
* of
algebraic groups by Lehrer and Segal [19].
The universal Yang-Baxter Lie algebra, and infinitesimal pure braid relation*
*s, arise
in a number of contexts. These include the classification of pure braids by Vas*
*siliev in-
variants, see Kohno [18], and the Knizhnik-Zamolodchikov differential equations*
* from
conformal field theory, where the relations appear as integrability conditions *
*on the
associated Gauss-Manin connection, see Varchenko [22]. Moreover, any finite dim*
*en-
sional representation of the Lie algebra L(n) induces a representation of the p*
*ure braid
group Pn on the same vector space, see Kapovich and Millson [15].
An important ingredient in the proof of Theorem 1.2 is a classical result of*
* Fadell and
Neuwirth [8] which shows that configuration spaces admit iterated bundle struct*
*ure.
Similar results are known to hold for certain orbit configuration spaces [24, 2*
*, 6], which
admit analogous bundle structure, and are described in more detail below. All o*
*f these
spaces fit in the following general framework.
For each natural number `, let X` be a functor from Euclidean spaces, with m*
*or-
phisms restricted to endomorphisms, to topological spaces. For a Euclidean spac*
*e E,
let Q`(E) be a discrete subset of E of fixed (possible infinite) cardinality de*
*pending on
`. Assume that there are natural transformations X`(E) ! X`-1(E) which satisfy *
*the
following conditions.
FIBER-TYPE ARRANGEMENTS 3
(1)The space X1(E) = E \ Q1(E) is the complement of a discrete subset of E.
(2)The map X`(E) ! X`-1(E) is a fiber bundle projection, with fiber E \ Q`(*
*E).
(3)Each bundle X`(E) ! X`-1(E) admits a cross-section.
(4)If E ~=C, the fundamental group of X`-1(E) acts trivially on the homolog*
*y of
the fiber E \ Q`(E).
The prototypical examples are given by the configuration spaces X`(E) = F (E; `*
*),
where E = Ck. Further examples are given below.
It seems likely that for many choices of X`, the Lie algebras associated to *
*X`(E) as
E varies are related in a manner analogous to those arising in Theorem 1.2. If *
*E ~=C,
conditions (1) and (2) imply that X`(E) is a K(G; 1) space, where G = ss1(X`(E)*
*) is
the fundamental group of X`(E), as is readily seen from the homotopy sequence o*
*f a
bundle. In this case, condition (3) further implies that the group G admits the*
* structure
of an iterated semidirect product of free groups, and condition (4) restricts t*
*he type
of free group automorphisms arising in this structure. These conditions determ*
*ine
the additive structure of the Lie algebra E*0(G), see [9] and Section 4. For h*
*igher
dimensional E, conditions (1)-(3) imply that the homology of the loop space of *
*X`(E) is
isomorphic to the universal enveloping algebra of the Lie algebra ss*(X`(E))= T*
*orsion,
and determine the additive structure of PrimH*(X`(E); Z), see [6] and Section 5*
*. For
higher dimensional E, these conditions have analogous implications for the homo*
*logy of
an iterated loop space qX`(E) with q > 1, and the Poisson algebra structure adm*
*itted
by this homology, see [4] and Section 6.
A brief indication how one may analyze and compare the Lie algebras arising *
*for
various choices of E is given next. First, there is a variant of the classical *
*Freudenthal
suspension, relating reduced suspensions and loop spaces as indicated below, wh*
*ere the
maps are induced by (homology) suspensions.
H2k-2(X`(Ck)) ---- H2k-1(X`(Ck))
??
y
H2k(X`(Ck)) ----! H2k(X`(Ck+1))
If k 2, conditions (1)-(3) above imply that these maps are all (additive) isom*
*or-
phisms. In the case k = 1, these maps yield an additive isomorphism E10(G) =
H1(X`(C)) ~=H2(X`(C2)) where G = ss1(X`(C)). While this comparison does not in
general preserve the structures of these Lie algebras, it does provide a geomet*
*ric way
to compare indecomposable elements in these Lie algebras.
To determine the Lie algebra structure, let S be a sphere of appropriate dim*
*ension
and A : S ! X`(E) a map representing a (reduced) homology generator in minimal
degree. Consider the pullback (E) of the bundle X`+1(E) ! X`(E) along the map A:
E \ Q`(E)----! (E) ----! S
flfl ? ?
fl ?y ?yA
E \ Q`(E)----! X`+1(E) ----! X`(E)
These bundles admit compatible cross-sections by condition (3). There is conseq*
*uently
a morphism of extensions of Lie algebras
0 ----! L(E \ Q`(E))----! L((E)) ----! L(S) ----! 0
?? ? ?
yid ?y ?yA*
0 ----! L(E \ Q`(E))----! L(X`+1(E)) ----! L(X`(E)) ----! 0
4 D. COHEN, F. COHEN, AND M. XICOTENCATL
where L(o) denotes the Lie algebra obtained from the descending central series *
*of the
fundamental group if E ~=C, and the graded Lie algebra of primitive elements in*
* the
homology of the loop space for higher dimensional E. Knowledge of the extension
0 ! L(E \ Q`(E)) ! L((E)) ! L(S) ! 0 and the map A* : L(S) ! L(X`(E))
for all homology generators completely determines the structure of the Lie alge*
*bra
L(X`+1(E)). In favorable situations, one can show that the extensions of Lie al*
*gebras
which arise as E varies are, apart from grading, isomorphic by carefully combin*
*ing
these considerations with the aforementioned comparison of indecomposables.
Several natural families of examples which fit in the framework described ab*
*ove are
given next. These examples either may be or have been studied using (variants o*
*f) the
techniques sketched above. Let M be a manifold, and a group which acts properly
discontinuously on M. The orbit configuration space F (M; `) consists of all `-*
*tuples
of points in M, no two of which lie in the same -orbit.
First, consider orbit configuration spaces of the form F (ExCn; `), where o*
*perates
diagonally of E x Cn, and trivially on Cn. Relevant examples include the follow*
*ing.
(a)A parameterized lattice acting on E = C, so that the orbit space is an *
*elliptic
curve. The orbit configuration spaces associated to the action of the st*
*andard
integral lattice were the subject of [6], where it is shown that the ana*
*logue of
Theorem 1.2 holds for these spaces.
(b)A discrete group acting properly discontinuously on the upper half-plane
E = H, so that the orbit space is a complex curve.
(c)A torsion free subgroup of < Sp(2g; Z) acting properly discontinuously *
*on
Siegel upper half-space E = Hg.
(d)A torsion free subgroup of the mapping class group for genus g surfaces,
acting on Teichmuller space E.
Second, let M = Ck\ {0} and let = Z=pZ act freely on M by rotations. The or*
*bit
configuration spaces F (M; `) were the subject of [24] and [2], the results of *
*which
combine to show that the analogue of Theorem 1.2 also holds for these spaces.
In the instances where conditions (1)-(4) hold, one obtains generalizations *
*of the uni-
versal Yang-Baxter Lie algebra, parameterized by the group . This is the case f*
*or the
family F (M; `) of orbit configuration spaces above, where M = Ck\{0} and = Z=*
*pZ.
As noted by D. Matei, the resulting generalized Yang-Baxter Lie algebra with cy*
*clic
symmetry is of use in constructing Vassiliev invariants of links in the lens sp*
*ace L(p; 1).
The Lie algebras arising from other families of orbit configuration spaces may *
*be of
similar use for other three-manifolds, among other potential applications. The*
* orbit
configuration spaces FZ=pZ(Ck\ {0}; `), and the classical configuration spaces *
*F (Ck; `),
may be realized as complements of finite hyperplane or subspace arrangements. T*
*his
led to speculation in [6] that similar results may hold for fiber-type arrangem*
*ents whose
complements, like configuration spaces, admit iterated bundle structure.
Let A be a hyperplane arrangement in C`, aSfinite collection of codimension *
*one
affine subspaces, with complement M(A) = C`\ H2A H. See Orlik and Terao [20]
as a general reference on arrangements. Given a hyperplane H C`, let Hk be the
codimension k affine subspace of Ck`= (C`)k consisting of all k-tuples of point*
*s in C`,
each of which lies in H. For each positive integer k, the elements of the hyper*
*plane
arrangement A may be used in this way to obtain an arrangementSAk of complex
codimension k subspaces in Ck`, with complement M(Ak) = Ck`\ H2A Hk.
When A is a fiber-type hyperplane arrangement, the behavior of the family of*
* spaces
{X`(Ck) = M(Ak); k 1} is reminiscent of that of the family {F (Ck; n); k 1} of
configuration spaces. Let G = ss1(M(A)) be the fundamental group of the complem*
*ent
of the fiber-type arrangement A in C`, and let E*0(G) be the Lie algebra obtain*
*ed from
the descending central series of G. The main result of this article is as follo*
*ws.
FIBER-TYPE ARRANGEMENTS 5
Theorem 1.3. For k 1, the homology of M(Ak+1), the loop space of the com-
plement of the subspace arrangement Ak+1 in C(k+1)`, is isomorphic to the unive*
*rsal
enveloping algebra of the graded Lie algebra E*0(G)k. Moreover,
(a)The image of the Hurewicz homomorphism
ss*(M(Ak+1)) ! H*(M(Ak+1); Z)
is isomorphic to E*0(G)k; and
(b)The Hurewicz homomorphism induces isomorphisms of graded Lie algebras
ss*(M(Ak+1))= Torsion! PrimH*(M(Ak+1); Z) ~=E*0(G)k;
where the Lie algebra structure of the source is induced by the Samelson*
* product.
The paper is organized as follows.
x2.Given a hyperplane arrangement A C`, there is an associated arrangement
of codimension k subspaces Ak Ck`. The combinatorics and topology of the
subspace arrangement Ak are studied in this section.
x3.The topology of the subspace arrangement Ak, in the instance where the u*
*nder-
lying hyperplane arrangement A is fiber-type, is further studied in this*
* section.
x4.The (known) structure of the Lie algebra E*0(G) associated to the descen*
*ding
central series of the fundamental group G = ss1(M(A)) of the complement *
*of
a fiber-type hyperplane arrangement A is analyzed in this section.
x5.The structure of the Lie algebra of primitive elements in the homology o*
*f the
loop space of the complement of the subspace arrangement Ak is analyzed *
*in
this section, and the isomorphisms of graded Lie algebras asserted in Th*
*eo-
rem 1.3 are established.
x6.The Poisson algebra structure on the homology of an iterated loop space *
*of the
complement of the subspace arrangement Ak is briefly analyzed in this se*
*ction.
2. Redundant Arrangements
Let H be an affine hyperplane in C`, an affine subspace of codimension one. *
*For each
positive integer k, there is an affine subspace Hk of codimension k in Ck`obtai*
*ned from
H in the following manner. Choose coordinates x = (x1; : :;:x`) on C`, and (x1;*
* : :;:xk)
on Ck` = C` x . .x.C`, where for each i, xi =P(xi;1; : :;:xi;`) 2 C`. Then, if*
* the
hyperplane H in C` is given by H = x 2 C` | `j=1ajxj = b , define a codimens*
*ion k
P `
affine subspace Hk in Ck`by Hk = (x1; : :;:xk) 2 Ck`| j=1ajxi;j= b; 1 i k .
Now let A be a hyperplane arrangement in C`, a finite collection of (affine)*
* hy-
perplanes. Via the above process, there is an arrangement Ak = {Hk | H 2 A}
of codimension k affine subspaces in Ck` obtained from A. For evident reasons, *
*call
the subspace arrangement Ak redundant. A brief description of the relationship*
* be-
tween the combinatorics and topology of the hyperplane arrangement A = A1 and t*
*he
redundant subspace arrangement Ak is given in this section.
Example 2.1. Let An be the braid arrangement in Cn, consisting of the hyperplan*
*es
Hi;j= {x 2 Cn | xi= xj}. As is well known, the complement M(An) = F (C; n) is t*
*he
configuration space of n points in C.
For each positive integer k, the associated redundant arrangement Aknconsist*
*s of
subspaces Hki;j= {(x1; : :;:xk) 2 (Cn)k | xp;i= xp;j; 1 p k}. These subspaces
may be realized as Hki;j= {(y1; : :;:yn) 2 (Ck)n | yi = yj}. Thus the compleme*
*nt
M(Akn) = F (Ck; n) is the configuration space of n points in Ck.
6 D. COHEN, F. COHEN, AND M. XICOTENCATL
For an arbitrary hyperplane arrangement A, and for each k, let L(Ak) be the *
*in-
tersection poset of the arrangement Ak, the partially ordered set of non-empty *
*multi-
intersections of elements of Ak. Order L(Ak) by reverse inclusion, and include*
* the
ambient space Ck`in L(Ak) as the minimal element, corresponding to the intersec*
*tion
of no elements of Ak. For the hyperplane arrangement A = A1, it is known that L*
*(A)
is a geometric poset, see [20, Section 2.3]. This need not be the case for an a*
*rbitrary
subspace arrangement. However, for redundant arrangements, the following holds.
Proposition 2.2. If A is a hyperplane arrangement, then L(A) ~=L(Ak) for all k.
Proof.It will be shown that the bijection between A and Ak given by H $ Hk indu*
*ces
an isomorphism of posets L(A) ~=L(Ak). To establish this, it suffices to show t*
*hat to
each codimension r flat X 2 L(A) there correspondsPa codimension kr flat Xk 2 L*
*(Ak).
Write A = {H1; : :;:Hn}, where Hi = x 2 C` | `j=1ai;jxj = bi , and let X =
H1 \ . .\.Hm . The flat X may be realized as the set of solutions of the syste*
*m of
equations Ax = b, where A = (ai;j) is m x ` and b = (b1; : :;:bm ). Then, X has
codimension r in C` if and only if rank[A | b] = rankA = ` - r if and only if
2 3 2 3
A | b A
6 A | b7 6 A 7
rank64 | 75= rank64 75= k(` - r)
A | b A
if and only if Xk = Hk1\ . .\.Hkmhas codimension kr in Ck`.
S
For each k, let M(Ak) = Ck`\ Hk2Ak Hk denote the complement of the (subspac*
*e)
arrangement Ak. In the case k = 1, the cohomology of the hyperplane complement
M(A) = M(A1) is well known. It is isomorphic to the Orlik-Solomon algebra A(A),
see [20, Sections 3.1, 3.2]. A family of algebrasLwhich includes A(A) is define*
*d next.
For each positive integer k, let E2k-1[k] = H2A ZekHbe a free Z-module gen*
*erated
by degree 2kV- 1 elements ekHin one-to-one correspondence with the hyperplanes *
*of A.
Let E[k] = E2k-1[k] be the exterior algebra of E2k-1[k], and denote by I[k] t*
*he ideal
of E[k] generated by the homogeneous elements
Xq
(-1)p-1ekH1^ . .d.ekHp.^.e.kHqif0 codimH1 \ . .\.Hq < q;
p=1
ekH1^ . .^.ekHqifH1 \ . .\.Hq = ;:
Let A[k] = E[k]=I[k]. The Orlik-Solomon algebra is then given by A(A) = A[1].
Proposition 2.2 may be used to determinePthe cohomology of M(Ak) for k > 1 in
terms of that of M(A). Let P (Ak; t) = q0 bq(M(Ak)) . tq be the Poincare poly*
*no-
mial of M(Ak), where bq(X) is the q-th Betti number of X. Results of Goresky and
MacPherson [12], and Yuzvinsky [23], see also Feichtner and Ziegler [11], toget*
*her with
Proposition 2.2, yield the following.
Corollary 2.3. Let A be a hyperplane arrangement in C`.
(1)For each k, the integral (co)homology of M(Ak) is torsion free, and we h*
*ave
P (Ak; t) = P (A; t2k-1).
(2)For each k, the cohomology algebra of M(Ak) isomorphic to the algebra A[*
*k],
H*(M(Ak); Z) ~=A[k].
An explicit basis for the first non-zero (reduced) homology group, H2k-1(M(A*
*k); Z),
of the complement of the subspace arrangement Ak is recorded next. Let L C` be*
* a
complex line that is transverse to the hyperplane arrangement A. Write L = {t.u*
*+v}
where u; v 2 C`are fixed and t 2 C varies. For each hyperplane H of A, the inte*
*rsection
L \ H is a point, say qH = oH . u + v for some oH 2 C. The following is immedia*
*te.
FIBER-TYPE ARRANGEMENTS 7
Lemma 2.4. The subspace Lk = {(t1 . u + v; : :;:tk . u + v) | t1; : :;:tk 2 C} *
*of Ck`
is transverse to the subspace arrangement Ak Ck`. For each subspace Hk of Ak, *
*the
intersection Lk \ Hk is the point (qH ; : :;:qH ) = (oH . u + v; : :;:oH . u + *
*v).
Let S2k-1 be the unit sphere in Ck. For ffl > 0 sufficiently small, the point
0 0 k
(oH + ffl z1) . u + v; : :;:(oH + ffl zk) . u + v 2 L
lies in the intersection Lk \ M(Ak) for all ffl0, 0 < ffl0 ffl. Fix such an ffl*
*, and define a
map ckH: S2k-1 ! Lk \ M(Ak) using the above formula:
(2.1) ckH(z) = ckH(z1; : :;:zk) = (oH + fflz1) . u + v; : :;:(oH + fflzk) .*
* u + v :
Let 2k-1 be the fundamental class of H2k-1(S2k-1; Z), and denote the image of
(ckH)*(2k-1) 2 H2k-1(Lk\ M(Ak); Z) under the map induced by the natural inclusi*
*on
Lk \ M(Ak) ,! M(Ak) by CkH2 H2k-1(M(Ak); Z).
Proposition 2.5. The classes {CkH| H 2 A} form a basis for H2k-1(M(Ak); Z).
Proof.For H 2 A, define pkH: Lk \ M(Ak) ! S2k-1 by
pkH(t1 . u + v; : :;:tk . u + v) = _t_-_oH_._ekt;- o
H . ek
where t = (t1; : :;:tk) and e = (1; : :;:1) are in Ck. It is then readily chec*
*ked that
pkHO ckH= id: S2k-1 ! S2k-1 is the identity map. Furthermore, if H06= H is anot*
*her
hyperplane of A, the composition pkHO ckH0is given by
z + 1_ffl(oH0 - oH ) . e
pkHO ckH0(z) = __________________1;
kz + _ffl(oH0 - oH ) . ek
so is null-homotopic. Consequently, the classes (ckH)*(2k-1) 2 H2k-1(Lk \ M(Ak)*
*; Z)
form a basis. Finally, using stratified Morse theory, one can show that the re*
*lative
homology group Hi(M(Ak); Lk\M(Ak); Z) vanishes for i < 4k-2, see [12, Parts II,*
* III].
It follows that the natural inclusion Lk \ M(Ak) ,! M(Ak) induces an isomorphism
H2k-1(Lk \ M(Ak); Z) -~! H2k-1(M(Ak); Z). So the classes CkH form a basis for
H2k-1(M(Ak); Z) as asserted.
Remark 2.6. The cohomology classes (CkH)* 2 H2k-1(M(Ak); Z) dual to the classes
CkH2 H2k-1(M(Ak); Z) generate the cohomology algebra H*(M(Ak); Z). Let akH2
A[k] denote the image of ekH 2 E[k] under the natural projection. Then the map
H2k-1(M(Ak); Z) ! A2k-1[k], (CkH)* 7! akH, induces an isomorphism of algebras
H*(M(Ak); Z) -~!A[k], see Corollary 2.3.
3. Linearly Fibered Arrangements
In this section, the topology of those redundant arrangements arising from s*
*trictly
linearly fibered and fiber-type hyperplane arrangements is studied further. Rec*
*all the
definition of arrangements of the former type from [9, 20].
Definition 3.1. A hyperplane arrangement A in C`+1is strictly linearly fibered *
*if there
is a choice of coordinates (x; z) = (x1; : :;:x`; z) on C`+1 so that the restri*
*ction, p, of
the projection C`+1 ! C`, (x; z) 7! x, to the complement M(A) is a fiber bundle
projection, with base p(M(A)) = M(B), the complement of an arrangement B in
C`, and fiber the complement of finitely many points in C. Refer to the hyperpl*
*ane
arrangement A as strictly linearly fibered over B.
The complements of hyperplane arrangements of this type are closely related *
*to
configuration spaces, as we now illustrate. ForQeach hyperplane H of A, let fH *
*be a
linear polynomial with H = kerfH . Then Q(A) = H2A fH is a defining polynomial
8 D. COHEN, F. COHEN, AND M. XICOTENCATL
for A. From the definition, if A is strictly linearly fibered over B and |A| = *
*|B| + n,
there is a choice of coordinates for which a defining polynomial for A factors *
*as
(3.1) Q(A) = Q(B) . OE(x; z);
where Q(B) = Q(B)(x) is a defining polynomial for B, and OE(x; z) is a product
OE(x; z) = (z - g1(x))(z - g2(x)) . .(.z - gn(x));
with gj(x) linear. Define g : C` ! Cn by
(3.2) g(x) = g1(x); g2(x); : :;:gn(x) ;
Since OE(x; z) necessarily has distinct roots for any x 2 M(B), the restriction*
* of g to
M(B) takes values in the configuration space F (C; n). The following result was*
* proven
by the first author, see [2, Theorem 1.1.5, Corollary 1.1.6].
Theorem 3.2. Let B be an arrangement of m hyperplanes, and let A be an arrange-
ment of m + n hyperplanes which is strictly linearly fibered over B. Then the *
*bun-
dle p : M(A) ! M(B) is equivalent to the pullback of the bundle of configuration
spaces pn+1 : F (C; n + 1) ! F (C; n) along the map g. Consequently, the bundle
p : M(A) ! M(B) admits a cross-section and has trivial local coefficients in ho*
*mology.
Since it is relevant to the subsequent discussion, a proof is included.
Proof.Denote points in F (C; n + 1) by (y; z), where y = (y1; : :;:yn) 2 F (C; *
*n) and
z 2 C satisfies z 6= yj for each j. Similarly, denote points in M(A) by (x; z),*
* where
x 2 M(B) and OE(x; z) 6= 0. In this notation, we have pn+1(y; z) = y and p(x; z*
*) = x.
Let E = x; (y; z) 2 M(B) x F (C; n + 1) | g(x) = y be the total space of the
pullback of pn+1 : F (C; n + 1) ! F (C; n) along the map g. It is then readily *
*checked
that the map h : M(A) ! E defined by h(x; z) = x; (g(x); z) is an equivalence*
* of
bundles.
Since the bundle pn+1 : F (C; n + 1) ! F (C; n) admits a cross-section, so d*
*oes the
pullback p : M(A) ! M(B). Furthermore, the structure group of the latter is the*
* pure
braid group Pn. Consequently, the action of the fundamental group of the base M*
*(B)
on that of the fiber C\{n points} is by pure braid automorphisms. As such, this*
* action
is by conjugation (see for instance [1] or [13]), hence is trivial in homology.
It is now shown that redundant strictly linearly fibered arrangements admit *
*(linear)
fibrations, just as their codimension one progenitors do.
Theorem 3.3. Let A be a hyperplane arrangement in C`+1 which is strictly linear*
*ly
fibered over B, with projection p : M(A) ! M(B) induced by the map C`+1! C` giv*
*en
by (x1; : :;:x`; z) 7! (x1; : :;:x`). Then for each k, the map Ck(`+1)! Ck` gi*
*ven by
(x1; : :;:x`; z) 7! (x1; : :;:x`) induces a fiber bundle projection pk : M(Ak) *
*! M(Bk).
Furthermore, the bundle pk : M(Ak) ! M(Bk) admits a cross-section.
Proof.By the previous result, the bundle p : M(A) ! M(B) is equivalent to the
pullback of pn+1 : F (C; n+1) ! F (C; n) along the map g of (3.2). An analogous*
* result
for the complements of the subspace arrangements Ak and Bk is established next.*
* For
k 2, view Ck`as (C`)k and Ckn as (Ck)n. Denote points in the configuration spa*
*ce
F (Ck; n + 1) by (y1; : :;:yn; z), where (y1; : :;:yn) 2 F (Ck; n) and z 6= yj *
*for each j.
Define gk : Ck`! Ckn by
i j
(3.3) gk(x1; : :;:xk) = g1(x1); : :;:g1(xk) ; : ::::;:gn(x1); : :;:gn(xk) *
* :
where gi(x1); : :;:gi(xk) 2 Ck for each i. It is readily checked that the res*
*triction of
gk to M(Bk) takes values in the configuration space F (Ck; n). Let ssk : Ek ! M*
*(Bk)
FIBER-TYPE ARRANGEMENTS 9
be the pullback of the bundle pkn+1: F (Ck; n + 1) ! F (Ck; n) along this restr*
*iction,
with total space Ek consisting of all points
k k
(x1; : :;:xk); (y1; : :;:yn; z) 2 M(B ) x F (C ; n + 1)
for which gk(x1; : :;:xk) = pkn+1(y1; : :;:yn; z) = (y1; : :;:yn).
Since the hyperplane arrangement A is strictly linearly fibered over B, the *
*comple-
ment of the subspace arrangement Ak may be realized as
M(Ak) = {(x1; : :;:xk; z) 2 M(Bk) x Ck | z 6= gi(x1); : :;:gi(xk) for1 i *
*n}:
Define hk : M(Ak) ! Ek by hk(x1; : :;:xk; z) = (x1; : :;:xk); (gk(x1; : :;:xk)*
*; z) .
The map hk is a homeomorphism. Moreover, the following diagram commutes.
k
M(Ak) --h--! Ek
? ?
(3.4) ?ypk ?yssk
M(Bk) --id--!M(Bk)
It follows that pk : M(Ak) ! M(Bk) is a bundle which is equivalent to the pullb*
*ack
of the bundle of configuration spaces pkn+1: F (Ck; n + 1) ! F (Ck; n) along th*
*e map
gk : M(Bk) ! F (Ck; n), and therefore has a cross-section.
An analysis of map in homology induced by the map gk : M(Bk) ! F (Ck; n)
defined in (3.3)is given next. For 1 i < j n, define pi;j: F (Ck; n) ! S2k-1 *
*by
pi;j(y1; : :;:yn) = (yj - yi)=kyj - yik. Recall that 2k-1 2 H2k-1(S2k-1; Z) den*
*otes
the fundamental class. The classes p*i;j(2k-1) form a basis for H2k-1(F (Ck; n)*
*), and
generate the cohomology algebra H*(F (Ck; n)), see [4, 5]. Denote the dual clas*
*ses in
H2k-1(F (Ck; n) by Ai;j, 1 i < j n. Note that the classes Ai;jmay be represen*
*ted
by spheres linking the subspaces Hki;j= {yi= yj} in Ckn, as in (2.1).
As in Section 2, let L = {t . u + v} C` be a line transverse to the hyperpl*
*ane
arrangement B, and Lk the corresponding codimension k subspace of Ck`, transver*
*se to
the subspace arrangement Bk. Recall the maps ckH: S2k-1 ! Lk \ M(Bk) from (2.1),
and the resulting basis {CkH| H 2 B} for H2k-1(M(Bk)) exhibited in Proposition *
*2.5.
Proposition 3.4. Let B C` be an arrangement of complex hyperplanes, and let
g : C` ! Cn be an affine transformation whose restriction, g : M(B) ! F (C; n),*
* to
the complement of B takes values in the configuration space F (C; n). Then for *
*every
k 1, the inducedPmap (gk)* : H2k-1(M(Bk); Z) ! H2k-1(F (Ck; n); Z) is given by
(gk)*(CkH) = Ai;jfor each hyperplane H of B, where the sum is over all distin*
*ct i
and j for which g(H) is contained in the hyperplane Hi;j= {yi= yj} in Cn.
Proof.For each hyperplane H of B, let "ckH: S2k-1 ! M(Bk) denote the composition
of ckH: S2k-1 ! Lk \ M(Bk) and the natural inclusion Lk \ M(Bk) ,! M(Bk). It
will be shown that the composition pi;jO gk O "ckH: S2k-1 ! S2k-1 induces the i*
*dentity
in homology if g(H) Hi;j, and induces the trivial homomorphism if g(H) 6 Hi;j,
thereby establishing the result.
For x 2 C`, write g(x) = (g1(x); : :;:gn(x)) as in (3.2). Then gk : Ck` ! C*
*kn is
given by gk(x1; : :;:xk) = (y1; : :;:yn), where yi= (gi(x1); : :;:gi(xk)), see *
*(3.3). Since
the restriction of g to M(B) takes values in F (C; n), the restriction of gk to*
* M(Bk)
takes values in F (Ck; n).
From (2.1), the map "ckH: S2k-1 ! M(Bk) is given by "ckH(z) = (w1; : :;:wk),*
* where
wj = (oH + fflzj) . u + v, and L \ H is the point qH = oH . u + v. Let ffi= gi(*
*qH ), and
define fii by the equation
gi(wj) = gi((oH + fflzj) . u + v) = gi(qH + fflzj. u) = ffi+ fflfiizj:
10 D. COHEN, F. COHEN, AND M. XICOTENCATL
Then, a calculation yields gk O "ckH(z) = (ff1 . e + fflfi1 . z; : :;:ffn . e +*
* fflfin . z) and
pi;jO gk O "ckH(z) = _ffl(fij-_fii)z_+_(ffj-_ffi)ekffl(fi;
j- fi*
*i)z + (ffj- ffi)ek
where, as before, e = (1; : :;:1). Recall that ffl > 0 was chosen sufficiently *
*small so as to
insure that the point (w01; : :;:w0k), where w0j= (oH +ffl0zj).u+v, lies in Lk\*
*M(Bk) for
all ffl0, 0 < ffl0 ffl. Since gk : M(Bk) ! F (Ck; n), it follows that gk O "ckH*
*(z) 2 F (Ck; n)
for all z 2 S2k-1. In other words, ffl(fij- fii)z + (ffj- ffi)e 6= 0 for all di*
*stinct i and j.
If g(H) 6 Hi;j, then g(qH ) =2Hi;jsince qH = L \ H is generic in H. Thus,
ffi = gi(qH ) 6= gj(qH ) = ffj, and the point (ffie + ffl0fiiz; ffje + ffl0fijz*
*) lies in the
configuration space F (Ck; 2) for all ffl0 ffl, including ffl0= 0. It follows t*
*hat pi;jO gk O ckH
is trivial in homology in this instance.
If, on the other hand, g(H) Hi;j, then ffi = ffj and thus fij - fii is nece*
*ssarily
non-zero. In this instance, pi;jO gk O "ckH(z) = . z, where 2 S1 C* is give*
*n by
= (fij- fii)=|fij- fii|, which clearly induces the identity in homology.
These results extend immediately to fiber-type arrangements, defined next.
Definition 3.5. An arrangement A = A1 of finitely many points in C1 is fiber-ty*
*pe.
An arrangement A = A`of hyperplanes in C`is fiber-type if A is strictly linearl*
*y fibered
over a fiber-type hyperplane arrangement A`-1 in C`-1.
Let A be a fiber-type hyperplane arrangement in C`. Then there is a choice *
*of
coordinatesQ(x1; : :;:x`) on C` so that a defining polynomial for A factors as *
*Q(A) =
` Q dj
j=1Qj(x1; : :;:xj), see (3.1). Write Qj = i=1xj - gi;j(x1; : :;:xj-1) , wh*
*ere dj
is the degree of Qj and each gi;jis linear. The non-negativeQintegers {d1; : :;*
*:d`} are
called the exponents of A. For each j `, the polynomial ji=1Qidefines a fibe*
*r-type
arrangement Aj in Cj with exponents {d1; : :;:dj}, and Aj is strictly linearly *
*fibered
over Aj-1. Furthermore, the map gj = (g1;j; : :;:gdj;j) : Cj-1 ! Cdjgives rise *
*to maps
gkj: M(Akj-1) ! F (Ck; dj) for each k. Theorems 3.2 and 3.3 yield
Theorem 3.6. LetQA be a fiber-type hyperplane arrangement in C` with defining p*
*oly-
nomial Q(A) = `j=1Qj. Then, for each j, 2 j `, and each k 1, the pro-
jection Cj ! Cj-1, (x1; : :;:xj-1; xj) 7! (x1; : :;:xj-1), gives rise to a bund*
*le map
pkj: M(Akj) ! M(Akj-1). This bundle is equivalent to the pullback of the bundle*
* of con-
figuration spaces F (Ck; dj+ 1) ! F (Ck; dj) along the map gkj: M(Akj-1) ! F (C*
*k; dj).
Consequently, the bundle pkj: M(Akj) ! M(Akj-1) admits a cross-section, has tri*
*vial
local coefficients in homology, and the fiber is the complement of dj points in*
* Ck.
Proposition 3.4 also extends to fiber-type arrangements. The specific statem*
*ent is
omitted.
4. The Descending Central Series
In this section, the structure of the Lie algebra E*0(G) associated to the d*
*escending
central series of the fundamental group G of the complement of a fiber-type arr*
*ange-
ment is analyzed. Additively, this structure is given by well known results of *
*Falk and
Randell [9, 10] stated below. Moreover, as shown by Jambu and Papadima [14], th*
*is
Lie algebra is isomorphic to the (integral) holonomy Lie algebra of the arrange*
*ment A
defined by Kohno [16]. An alternate description of E*0(G), which will facilitat*
*e com-
parison with the Lie algebra of primitives in the homology of the loop space of*
* the
complement of the subspace arrangement Ak in Section 5, is given here.
Example 4.1. Let Pn be the Artin pure braid group, the fundamental group of the*
* con-
figuration space F (C; n). The structure of the Lie algebra E*0(Pn) was first d*
*etermined
rationally by Kohno [17]. As observed by many authors, the following descriptio*
*n holds
FIBER-TYPE ARRANGEMENTS 11
over the integers as well. For each n 2, let L[n] be the free Lie algebra gene*
*rated by
elements A1;n+1;L: :;:An;n+1. Then the Lie algebra E*0(Pn) is additively isomor*
*phic to
the direct sum n-1j=1L[j], and the Lie bracket relations in E*0(Pn) are the i*
*nfinitesimal
pure braid relations, given by
[Ai;j+ Ai;k+ Aj;k; Am;k]= 0 for m = i or m = j, and
(4.1)
[Ai;j; Ak;l]= 0 for {i; j} \ {k; l} = ;.
Note that this description realizes the Lie algebra E*0(Pn+1) as the semidir*
*ect prod-
uct of E*0(Pn) by L[n] determined by the Lie homomorphism n : E*0(Pn) ! Der(L[n*
*])
given by n(Ai;j) = ad(Ai;j). From the infinitesimal pure braid relations, one h*
*as
(
(4.2) ad(Ai;j)(Am;n+1) = [Am;n+1; Ai;n+1+ Aj;n+1]if m = i or m = j,
0 otherwise.
This extension of Lie algebras arises topologically from the bundle of configur*
*ation
spaces F (C; n + 1) ! F (C; n).
The additive structure noted above may be obtained from the following result*
* of
Falk and Randell [9, 10].
Theorem 4.2. Let 1 ! H ! G ! K ! 1 be a split extension of groups such that
the conjugation action of K is trivial on H1=H2. Then there is a short exact se*
*quence
of Lie algebras 0 ! E*0(H) ! E*0(G) ! E*0(K) ! 0 which is split as a sequence of
abelian groups. Furthermore, if the descending central series quotients of H an*
*d K are
free abelian, then so are those of G.
Now let A = A` be a fiber-type hyperplane arrangement in C`. The complement *
*of
A` sits atop a tower of fiber bundles
M(A`) -p`-!M(A`-1) p`-1---!.p.2.--!M(A1) = C \ {d1 points};
where the fiber of pj is homeomorphic to the complement of dj points in C. For
simplicity, write B = A`-1 and n = d`. Then A is strictly linearly fibered over*
* B, and
by Theorem 3.2, the bundle p : M(A) ! M(B) is equivalent to the pullback of the
configuration space bundle pn+1 : F (C; n + 1) ! F (C; n) along the map g of (3*
*.2).
Application of the homotopy exact sequence of a bundle (and induction) shows*
* that
M(A) is a K(G; 1) space, where G = G(A) = ss1(M(A)). In light of Theorem 3.2,
there is also a commutative diagram
1 ----! Fn ----! G(A) ----! G(B) ----! 1
? ? ?
(4.3) ?yid ?y ?yg#
1 ----! Fn ----! Pn+1 ----! Pn ----! 1
where g# : G(B) ! Pn is induced by g : M(B) ! F (C; n), and the fundamental gro*
*up
of the fiber C \ {n points} is identified with the free group Fn on n generator*
*s. Since
the underlying bundles admit cross-sections, the rows in the diagram above are *
*split
exact.
Theorem 4.3. Let A be a fiber-type hyperplane arrangement. If the exponents of A
are {d1; : :;:d`}, then E*0(G(A)) ~=L[d1] . . .L[d`] as abelian groups.
Proof.The proof is by induction on `.
In the case ` = 1, A is an arrangement of d = d1 points in C, the fundamenta*
*l group
of the complement is Fd, the free group on d generators, and it is well known t*
*hat
E*0(Fd) is isomorphic to the free Lie algebra L[d], see for instance [21, Chapt*
*er IV].
In general, assume that the fiber-type arrangement A is strictly linearly fi*
*bered over
B and that d` = n as above. Then there is a split, short exact sequence of gro*
*ups
12 D. COHEN, F. COHEN, AND M. XICOTENCATL
1 ! Fn ! G(A) ! G(B) ! 1, and by Theorem 3.2, the action of G(B) on Fn is by
pure braid automorphisms. As such, this action is by conjugation, hence is triv*
*ial on
H*(Fn; Z). By Theorem 4.2, the descending central series quotients of G(A) are *
*free
abelian, and there is a short exact sequence of Lie algebras
(4.4) 0 ! E*0(Fn) ! E*0(G(A)) ! E*0(G(B)) ! 0;
which splits as a sequence of abelian groups. The result follows by induction.
The additive decomposition provided by this result does not, in general, pre*
*serve the
underlying Lie algebra structure. An inductive description of the Lie algebra s*
*tructure
of E*0(G(A)) is given next.
Theorem 4.4. Let A and B be fiber-type hyperplane arrangements with |A| = |B| +*
* n,
and suppose that A is strictly linearly fibered over B. Then the Lie algebra E**
*0(G(A))
is isomorphic to the semidirect product of E*0(G(B)) by L[n] determined by the *
*Lie
homomorphism = n O g* : E*0(G(B)) ! Der(L[n]), where g* : E*0(G(B)) ! E*0(Pn)
is induced by the map g : M(B) ! F (C; n), and n : E*0(Pn) ! Der(L[n]) is given*
* by
n(Ai;j) = ad(Ai;j).
Proof.From the exact sequence of Lie algebras (4.4)noted above, it follows that
E*0(G(A)) is isomorphic to the semidirect product of E*0(G(B)) by L[n] determin*
*ed
by the Lie homomorphism : E*0(G(B)) ! Der(L[n]) given by (b) = adL[n](b) for
b 2 E*0(G(B)). It suffices to show that the homomorphism factors as asserted.
From the diagram (4.3), and the results of Falk and Randell stated in Theore*
*m 4.2,
there is a commutative diagram of Lie algebras with split exact rows
0 ----! L[n] ----! E*0(G(A))----! E*0(G(B))----! 0
?? ? ?
yid ?y ?yg*
0 ----! L[n] ----! E*0(Pn+1)----! E*0(Pn) ----! 0
Via the splittings, view E*0(G(B)) and E*0(Pn) as Lie subalgebras of E*0(G(A)) *
*and
E*0(Pn+1) respectively. Then for a 2 L[n] and b 2 E*0(G(B)), we have [b; a] = [*
*g*(b); a]
in L[n]. Thus adL[n](b) = adL[n](g*(b)) in Der(L[n]) and = n O g*.
This result, together with Proposition 3.4, provides an inductive descriptio*
*n of the
Lie bracket structure of E*0(G(A)). Recall the basis {C1H| H 2 B} for H1(M(B); *
*Z) =
E10(G(B)) exhibited in Proposition 2.5, and recall that the free Lie algebra L[*
*n] is
generated by A1;n+1; : :;:An;n+1.
Corollary 4.5. For generators C1Hof E10(G(B)) and Am;n+1 of L[n], one has
X
(C1H)(Am;n+1) = [Ai;j; Am;n+1]:
g(H)Hi;j
P
Proof.By Proposition 3.4, one has g*(C1H) = Ai;j, where the sum is over all i*
* and
j for which g(H) Hi;j. The result follows.
This corollary can be used to explicitly record the Lie bracket relations in*
* E*0(G(A)),
and to show that these relations are combinatorial, that is, completely determi*
*ned by
the intersection poset L(A). ThePLie algebra E*0(G(A)) is generated by {C1H| H *
*2 A}.
For a flat X 2 L(A), write C1X= XH C1H. The following was proven by Jambu and
Papadima [14], see also [2].
Theorem 4.6. If A is a fiber-type hyperplane arrangement with exponents {d1; : *
*:;:d`},
then the Lie bracket relations in E*0(G(A)) are given by
[C1X; C1H] = 0;
for codimension two flats X 2 L(A) and hyerplanes H 2 A containing X.
FIBER-TYPE ARRANGEMENTS 13
Proof.The proof is by induction on `.
In the case ` = 1, there is nothing to show since G(A) is a free group on d *
*= d1
generators, E*0(G(A)) is isomorphic to the free Lie algebra L[d], and there are*
* no
codimension two flats in L(A).
In general, assume that A is strictly linearly fibered over B andQthat d` = *
*n as
before. Then A has a defining polynomial of the form Q(A) = Q(B) . nj=1(z - gj*
*(x)),
see (3.1). View B as a subarrangement of A = {H | H 2 B} [ {Hj | 1 j n},
where Hj = ker(z - gj(x)). Then the set {C1H| H 2 B} [ {C1Hj| 1 j n} generates
E*0(G(A)), where the generators C1Hjcorrespond to the hyperplanes Hj of A \ B, *
*and
to the generators Aj;n+1of the free Lie algebra L[n] under the additive isomorp*
*hism
E*0(G(A)) ~=E*0(G(B)) L[n].
By Theorem 4.4, E*0(G(A)) is isomorphic to an extension of E*0(G(B)) by L[n].
Consequently, the Lie bracket relations in E*0(G(A)) consist of those of E*0(G(*
*B)), and
those arising from the extension. By induction, the Lie bracket relations in E**
*0(G(B))
are given by [C1X; C1H] = 0 for codimension two flats X contained only in hyper*
*planes
H 2 B A. So it remains to analyze those relations in E*0(G(A)) arising from the
extension. These are given implicitly in Corollary 4.5.
Recall from (4.1)that [Ai;j; Am;n+1] = [Am;n+1; Ai;n+1+ Aj;n+1] if m 2 {i; j*
*}, and
is zero otherwise. Thus the results of Corollary 4.5 may be recorded as
X
(C1H)(Am;n+1) = [C1H; Am;n+1] = [Am;n+1; (ffii;m+ ffij;m)(Ai;n+1+ Aj;n+*
*1)];
g(H)Hi;j
where C1H2 E*0(G(B)) E*0(G(A)) and ffii;mis the Kronecker delta. Note that the
expression on the right lies in L[n]. Under the above identifications, these r*
*elations
take the form
X
(4.5) [C1H; C1Hm] = [C1Hm; (ffii;m+ ffij;m)(C1Hi+ C1Hj)]
g(H)Hi;j
Now one can check that g(H) Hi;jif and only if H \ Hi\ Hj is a codimension two
flat in L(A) if and only if Hi\ Hj H. Using this observation, the relation (4.*
*5)may
be expressed as X
[C1H; C1Hm] = [C1Hm; C1Hj]:
Hm \HjH
A calculation then shows that this relation is equivalent to [C1X; C1Hm] = 0, w*
*here X is
the codimension two flat in L(A) contained in H and Hm . Since this relation ho*
*lds for
all Hm 2 A \ B for which X H \ Hm , it follows that [C1X; C1H] = 0 as well.
5.Homology of the Loop Space
The structure of the Lie algebra of primitive elements in the homology of th*
*e loop
space of the complement of a redundant subspace arrangement associated to a fib*
*er-
type hyperplane arrangement is analyzed in this section. In analogy with the p*
*re-
vious section, begin by recalling this structure for the classical configuratio*
*n spaces
F (Ck+1; n) for k 1.
Example 5.1. The integral homology of the loop space F (Ck+1; n) was calculated
by Fadell and Husseini [7]. The structure of the Lie algebra PrimH*(F (Ck+1; n)*
*; Z)
was subsequently determined by Cohen and Gitler [3]. For brevity, denote this *
*Lie
algebra by L(n)k. The structure of L(n)k may be described as follows.
For each n 2, let L[n]k denote the free Lie algebra generated by elements B*
*i;n+1,
1L i n, of degree 2k. Then L(n)k is additively isomorphic to the direct sum
n-1
j=1 L[j]k, and the Lie bracket relations in L(n)k are given by the infinitesi*
*mal pure
14 D. COHEN, F. COHEN, AND M. XICOTENCATL
braid relations on the Bi;j, see (4.1). Thus, there is an isomorphism of grade*
*d Lie
algebras L(n)k ~=E*0(Pn)k, see Definition 1.1, Theorem 1.2, and Example 4.1.
Furthermore, as is the case for the descending central series of the pure br*
*aid
group, the Lie algebra L(n + 1)k is isomorphic to the semidirect product of L(n*
*)k
by L[n]k determined by the Lie homomorphism kn: L(n)k ! Der(L[n]k) given by
kn(Bi;j) = ad(Bi;j). From the infinitesmial pure braid relations, there is a fo*
*rmula for
ad(Bi;j) analogous to that given in (4.2). As before, this extension of Lie alg*
*ebras arises
topologically from the bundle of configuration spaces F (Ck+1; n + 1) ! F (Ck+1*
*; n).
Now let A be a fiber-type hyperplane arrangement in C`with exponents {d1; : *
*:;:d`}.
Then, for each k, there is a tower of fiber bundles
k pk`-1 pk
M(Ak`) -p`--!M(Ak`-1) ---! . .-.2--!M(Ak1) = Ck \ {d1 points};
where the fiber of pkjis homeomorphic to the complement of dj points in Ck, see
Theorem 3.6. Furthermore, each of the fiber bundles pkj: M(Akj) ! M(Akj-1) invo*
*lving
the complements of the redundant subspace arrangements Akj Cjk admits a cross-
section, and, as indicated above, M(Ak1) is the complement of d1 points in Ck. *
* By
work of the second two authors [6, Theorem 1], the following holds.
Theorem 5.2. Let A be a fiber-type hyperplane arrangement in C` with exponents
{d1; : :;:d`}. Then, for each k 1,
Q `
(a)There is a homotopy equivalence M(Ak+1) ! j=1(Ck+1 \ {dj points});
(b) The integral homologyNof M(Ak+1) is torsion free, and is isomorphic to t*
*he
tensor product `j=1H*((Ck+1 \ {dj points}); Z) as a coalgebra;
(c)The module of primitives in the integral homology of M(Ak+1) is isomorph*
*ic
to ss*(M(Ak+1)) modulo torsion as a Lie algebra.
Remark 5.3. The homotopy groups of a loop space admit a bilinear pairing, which
satisfies the axioms for a graded Lie algebra in case there is no 2 or 3 torsio*
*n in the
homotopy groups. The graded analogue of the symmetry law can fail in case 2-tor*
*sion is
present, while the graded analogue of the Jacobi identity can fail if 3-torsion*
* is present.
Thus, forming the quotient of the homotopy groups by the torsion gives a graded
module which satisfies the axioms for a graded Lie algebra. Analogous propertie*
*s of
iterated loop spaces yield a graded Poisson algebra, see Section 6 below.
Proof of Theorem 5.2.Given a fibration F -i!E -! B with a section oe, there is a
homotopy equivalence E ' B x F given by the composite:
B x F -oexi----!E x E --!E;
where is the loop space multiplication and such that the inclusions of B and F*
* into
E are maps of H-spaces. Moreover, if the spaces involved have torsion free homo*
*logy
then H*(E) ~=H*(B) H*(F ). By a theorem of Milnor and Moore, one obtains
(5.1) PrimH*(E) ~=PrimH*(B) PrimH*(F )
upon passing to the Lie algebra of primitives. This result is a topological ana*
*logue of
Theorem 4.2 as the underlying Lie algebra structure is "twisted."
Now apply these considerations to the fiber bundleWpk+1j: M(Ak+1j) ! M(Ak+1j*
*-1).
The fiber in this case is F = Ck+1 \ {dj points} ' djS2k+1. Assertion (a) fol*
*lows
by induction, and then (b) by the K"unneth theorem. By the Bott-Samelson Theore*
*m,
H*(F ) is isomorphic to T [dj]k, a tensor algebra on dj generators of degree 2k.
Thus the module of primitive elements is generated as a Lie algebra by the p*
*rimitive
elements in degree 2k which are in the image of the Hurewicz map. Since the Hur*
*ewicz
FIBER-TYPE ARRANGEMENTS 15
map takes values in the the module of primitive elements, that module is genera*
*ted as
a Lie algebra by those spherical classes given by the homology classes of degre*
*e 2k.
Next notice that the homology groups here are torsion free. Hence the Hurew*
*icz
map factors through ss*(M(Ak+1))= Torsion. Furthermore, the homotopy groups of a
loop space modulo torsion give a graded Lie algebra where the Lie bracket is in*
*duced
by the classical Samelson product, and the Hurewicz map is a morphism of graded*
* Lie
algebras. Thus the induced map ss*(M(Ak+1))= Torsion! PrimH*(M(Ak+1); Z)
is an epimorphism of Lie algebras.
Since all spaces are simply connected, and are of finite type, the homotopy *
*groups
modulo torsion are finitely generated free abelian groups in any fixed degree. *
*By a clas-
sical theorem of Milnor and Moore concerning rational homotopy groups, the indu*
*ced
map ss*(M(Ak+1))= Torsion! PrimH*(M(Ak+1); Z) is also a monomorphism. The
result follows.
By (5.1)above, there is an isomorphism of graded abelian groups
Prim H*(M(Ak+1j)) ~=PrimH*(M(Ak+1j-1)) L[dj]k:
Proceeding inductively, this implies that the Lie algebra Prim H*(M(Ak+1); Z) is
isomorphic to L[d1]k . . .L[d`]k as a graded abelian group, where {d1; : :;:d`*
*} are
the exponents of A. Thus the Lie algebras Prim H*(M(Ak+1); Z) and E*0(G(A))k
are additively isomorphic, see Theorem 4.3. To show that they are are isomorphi*
*c as
Lie algebras, thereby completing the proof of Theorem 1.3, it remains to show t*
*hat
the Lie bracket structure of PrimH*(M(Ak+1); Z) coincides with that of E*0(G(A)*
*)k.
This analysis parallels the determination of the Lie algebra structure of E*0(G*
*(A)) in
Section 4.
The fiber-type hyperplane arrangement A = A` is strictly linearly fibered ov*
*er
A`-1, and |A| = |A`-1| + d`. As before, write B = A`-1 and n = d`. Recall the
map gk+1 : M(Bk+1) ! F (Ck+1; n) from (3.3). Recall also that the Lie algebra
Prim H*(F (Ck+1; n); Z) is denoted by L(n)k. Analogously, denote the Lie algeb*
*ra
Prim H*(M(Ak+1); Z) by L(A)k.
Theorem 5.4. Let A and B be fiber-type hyperplane arrangements with A strictly
linearly fibered over B and |A| = |B| + n. Then the Lie algebra L(A)k is isomor*
*phic
to the semidirect product of L(B)k by the free Lie algebra L[n]k determined by *
*the
Lie homomorphism k = knO flk*: L(B)k ! Der(L[n]k), where flk*: L(B)k ! L(n)k
is the map in loop space homology induced by gk+1 : M(Bk+1) ! F (Ck+1; n), and
kn: L(n)k ! Der(L[n]k) is given by kn(Bi;j) = ad(Bi;j).
Proof.The realization of the bundle pk+1 : M(Ak+1) ! M(Bk+1) as the pullback of
the bundle of configuration spaces pk+1n+1: F (Ck+1; n + 1) ! F (Ck+1; n) along*
* the map
gk+1 : M(Bk+1) ! F (Ck+1; n) from Theorem 3.3 yields a commutative diagram of
Hopf algebras
H*((Ck+1 \ {n points}))----! H*(M(Ak+1)) ----! H*(M(Bk+1))
?? ? ?
yid ?y ?yflk*
H*((Ck+1 \ {n points}))----! H*(F (Ck+1; n + 1))----! H*(F (Ck+1; n))
with exact rows, and, on the level of primitives, a commutative diagram of Lie *
*algebras
0 ----! L[n]k----! L(A)k ----! L(B)k ----! 0
?? ? ?
y id ?y ?yflk*
0 ----! L[n]k----! L(n + 1)k----! L(n)k ----! 0
16 D. COHEN, F. COHEN, AND M. XICOTENCATL
where flk*: L(B)k ! L(n)k is induced by gk+1 : M(Bk+1) ! F (Ck+1; n). Since the
underlying bundles admit cross-sections, the rows in the above diagrams are spl*
*it exact.
Via these splittings, view L(B)k and L(n)k as Lie subalgebras of L(A)k and L(n *
*+ 1)k
respectively.
From the above considerations, it follows that the Lie algebra L(A)k is isom*
*orphic
to the semidirect product of L(B)k by L[n]k determined by the Lie homomorphism
k : L(B)k ! Der(L[n]k) given by k(b) = adL[n]k(b) for b 2 L(B)k. Moreover, for
a 2 L[n]k, we have [b; a] = [flk*(b); a] in L[n]k. Thus adL[n]k(b) = adL[n]k(f*
*lk*(b)) in
Der(L[n]k) and k = knO flk*.
This result, together with Proposition 3.4, provides an inductive descriptio*
*n of the
Lie bracket structure of L(A)k. The space M(Bk+1) is 2k-connected, and the coho*
*mol-
ogy algebra H*(M(Bk+1); Z) is generated by classes ak+1Hin one-to-one correspon*
*dence
with the hyperplanes H 2 B, see Corollary 2.3. These classes are of degree 2k +*
* 1, and
are dual to the elements of the basis {Ck+1H| H 2 B} for H2k+1(M(Bk+1); Z) exhi*
*bited
in Proposition 2.5. See also Remark 2.6.
The above observations imply that homology suspension induces an isomorphism
oe* : H2k(M(Bk+1); Z) ! H2k+1(M(Bk+1); Z):
Let fikH2 H2k(M(Bk+1); Z) be the unique class satisfying oe*(fikH) = Ck+1H. Re*
*call
that the free Lie algebra L[n]k is generated by B1;n+1; : :;:Bn;n+1.
Corollary 5.5. For generators fikHof L(B)k and Bm;n+1 of L[n]k, one has
X
k(fikH)(Bm;n+1) = [Bi;j; Bm;n+1]:
g(H)Hi;j
P
Proof.By Proposition 3.4, one has gk+1*(Ck+1H) = Ai;j, where the sum is over *
*all i
and j for which g(H) Hi;j. Since the homology suspension oe* is an isomorphis*
*mP
and flk*is the map in loop space homology induced by gk+1, one has flk*(fikH) =*
* Bi;j,
where the sum is over all i and j for which g(H) Hi;j. The result follows.
To complete the proof of Theorem 1.3, assume inductively that the Lie algebr*
*as
E*0(G(B))k and L(B)k are isomorphic. By Theorem 4.4, the Lie algebra E*0(G(A))
is the extension of E*0(G(B)) by the free Lie algebra L[n] (generated in degree*
* one)
determined by the Lie homomorphism = n O g*. Thus E*0(G(A))k may be realized
as the extension of E*0(G(B))k by the free Lie algebra L[n]k (generated in degr*
*ee 2k)
determined by as specified in Definition 1.1. Similarly, by Theorem 5.4, the *
*Lie
algebra L(A)k is the extension of L(B)k by the free Lie algebra L[n]k determine*
*d by
the Lie homomorphism k = knO flk*. A comparison of the results of Corollary 4.5
and Corollary 5.5 reveals that these extensions coincide. Therefore, the Lie al*
*gebras
E*0(G(A))k and L(A)k are isomorphic.
Alternatively, Corollary 5.5 may be used to explicitly determine the Lie bra*
*cket
structure in L(A)k. As the argument is completely analogous to that which estab*
*lished
Theorem 4.6, the result stated below without proof.PThe Lie algebra L(A)k is ge*
*nerated
by {fikH| H 2 A}. For a flat X 2 L(A), write fikX= XH fikH.
Theorem 5.6. Let A be a fiber-type hyperplane arrangement. Then, for each k 1,
the Lie bracket relations in L(A)k are given by
[fikX; fikH] = 0;
for codimension two flats X 2 L(A) and hyerplanes H 2 A containing X.
FIBER-TYPE ARRANGEMENTS 17
6.Homology of Iterated Loop Spaces
In this final section, the Poisson algebra structure on the homology of an i*
*terated
loop space of the complement of a redundant subspace arrangement associated to a
fiber-type hyperplane arrangement is briefly analyzed.
For q > 1, the homology of an q-fold loop space, qX, admits the structure of*
* a
graded Poisson algebra. Namely, there is a bilinear map given by the Browder op*
*eration
q-1 : Hi(qX) Hj(qX) ! Hi+j+q-1(qX)
which satisfies properties listed in [4, pages 215-217]. In particular, this pa*
*iring satis-
fies the axioms of a (graded) Poisson algebra, and is compatible with the White*
*head
product structure for the classical Hurewicz homomorphism.
In the case where X = X`(Ck+1), k 1, satisfies conditions (1)-(3) from the
Introduction, these structures are analogues of classical constructions in homo*
*topy
theory. First, note that the single suspension X`(Ck+1) is homotopy equivalent*
* to
a bouquet of spheres. Thus there is an induced map oe2 : 2X`(Ck+1) ! X`(Ck+2)
which induces an isomorphism on the first non-trivial homology group. The adjo*
*int
E2 : X`(Ck+1) ! 2X`(Ck+2) also induces an isomorphism on the first non-trivial
homology group. This last map is an analogue of the classical Freudenthal doub*
*le
suspension where the spaces X`(E) are replaced by single odd dimensional sphere*
*s.
Looping E2 is given by (E2) : X`(Ck+1) ! 3X`(Ck+2).
Theorem 6.1. Let A be a fiber-type hyperplane arrangement in C` with exponents
{d1; : :;:d`}. Then, for each k 1,
(a)The multiplicative map (E2) : M(Ak+1) ! 3M(Ak+2) induces an isomor-
phism on H2k(-; Z), and is zero in degrees greater than 2k.
(b) If q > 1, the homology of qM(Ak+1), with any field coefficients, is a gr*
*aded
Poisson algebra with Poisson bracket given by the Browder operation for *
*the
homology of a q-fold loop space.
Q ` W
(c)If q > 1, then qM(Ak+1) is homotopy equivalent to j=1q( djS2k+1).
(d) If 1 < q < 2k + 1, the homology of qM(Ak+1), with coefficients in a fiel*
*d F of
characteristic zero, is generated as a Poisson algebra by elements fiH o*
*f degree
2k + 1 - q for H 2 A. The Poisson bracket is given by the Browder operat*
*ion
q-1, and satisfies the relations
q-1[fiX ; fiH ];
for codimensionPtwo flats X 2 L(A) and hyerplanes H 2 A containing X,
where fiX = XH fiH .
Sketch of Proof.Part (a) follows from the fact that the homology of 3M(Ak+2) is
abelian while the homology of M(Ak+1) is generated by Lie brackets of weight at
least 2 in homological degrees greater than 2k.
Part (b) follows from the remarks at the beginning of this section.
Part (c) follows at once from the fact that the result holds in case q = 1, *
*which was
established in Theorem 5.2.
In case q = 1, the Browder operation q-1 is precisely the natural Lie bracket
in the homology of a 1-fold loop space, M(Ak+1). These Lie bracket relations a*
*re
recorded in Theorem 5.6. As shown in [4, pages 215-217], a further property of *
*the
operation q-1 is that oe*q-1(x; y) = q-2(oe*x; oe*y), where oe* denotes the hom*
*ology
suspension. Thus by induction on q, the asserted Poisson bracket relations are *
*satisfied
modulo elements in the kernel of the suspension. Furthermore, q-1(x; y) is prim*
*itive
in case the classes x and y are primitive.
18 D. COHEN, F. COHEN, AND M. XICOTENCATL
In characteristic zero, and in case q is greater than 1, the homology suspen*
*sion
induces an isomorphism on the module of primitives. Thus the asserted Poisson b*
*racket
relations are satisfied.
Remark 6.2. Let A = An be the braid arrangement in Cn. As noted in Example 2.1,*
* one
then has M(Ak+1n) = F (Ck+1; n) for all k. For the braid arrangement, the codim*
*ension
two flats in L(An) (the partition lattice) are of the forms
Hi;j\ Hi;k\ Hj;kfor1 i < j < k n; and Hi;j\ Hk;lfor{i; j} \ {k; l} = ;:
Thus by Theorem 6.1, for 1 < q < 2k + 1, the homology of qF (Ck+1; n), with
coefficients in a field F of characteristic zero, is generated as a Poisson alg*
*ebra by
elements Bi;j= fiHi;jof degree 2k + 1 - q for 1 i < j n. Moreover, the Poisson
bracket relations are given by the universal infinitesimal Poisson braid relati*
*ons:
q-1[Bi;j+ Bi;k+ Bj;k; Bm;k]= 0 for m = i or m = j, and
q-1[Bi;j; Bk;l]= 0 for {i; j} \ {k; l} = ;.
As shown in [3], these are precisely the infinitesimal pure braid relations in *
*case q = 1,
see also Examples 4.1 and 5.1.
It seems likely that, via the natural universal mapping property, one could *
*define the
"universal infinitesimal Poisson braid algebra," and that the homology of qF (C*
*k+1; n)
with coefficients in a field F of characteristic zero is that algebra over F.
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Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803
E-mail address: cohen@math.lsu.edu
URL: http://www.math.lsu.edu/"cohen
Department of Mathematics, University of Rochester, Rochester, NY 14627
E-mail address: cohf@math.rochester.edu
URL: http://www.math.rochester.edu/u/cohf
Depto. de Mathematicas, Cinvestav del IPN, Mexico City
Max-Plank-Institut f"ur Mathematik, P.O. Box 7280, D-53072 Bonn, Germany
E-mail address: xico@mpim-bonn.mpg.de