The isomorphism conjecture for Artin groups

0.25 MB·English

by  S. K. Roushon

THE ISOMORPHISM CONJECTURE FOR ARTIN GROUPS
5
1 S.K.ROUSHON
0
2
Abstract. WeprovetheFarrell-Jonesfiberedisomorphismconjectureforsev-

v eral classes of Artin groups of finite and affine types. As a consequence, we
o computeexplicitlythesurgeryobstructiongroupsofthefinitetypepureArtin
N groups.

7
1

1. Introduction
]
T
Inthisarticleweareconcernedaboutprovingthefiberedisomorphismconjecture
K
of Farrell and Jones ([11]), for a class of Artin groups of both finite (also called
.
h spherical)andaffine types. The classicalbraidgroupis anexample of a finite type
t Artin group (type A ), and we considered this group in [13] and [28].
a n
m TheisomorphismconjectureisanimportantconjectureinGeometryandTopol-
ogy, and much work has been done in this area in recent times (e.g. [2], [21], [22],
[
[28]). Among other conjectures, the Borel and the Novikov conjectures are conse-

2 quences of the isomorphism conjecture. Although, these two conjectures deal with
v
finitely presented torsion free groups, the isomorphism conjecture is stated for any
6
group. Furthermore, it provides a better understanding of the K- and L-theory of
7
7 the group. It is well-known that the fibered isomorphism conjecture implies van-
0 ishing of the lower K-theory (Wh(−),K˜ (−) and K (−) for i≥1) of any torsion
0 −i
0 free subgroup of the group. In fact, it is still an open conjecture that this should
.
1 be the case for all torsion free groups.
0 Following Farrell-Jonesand Farrell-Hsiang,in all of recentworks on the isomor-
5 phism conjecture, geometric input on the group plays a significant role. Here also,
1
we follow a similar path to prove the conjecture. We prove the following theorem.
:
v
Theorem 1.1. Let Γ be an Artin group of type A , B (= C ), D , F , G ,
i n n n n 4 2
X I (p), A˜ , B˜ , C˜ or D˜ . Then, the fibered isomorphism conjecture wreath product
2 n n n n
r with finite groups is true for any subgroup of Γ. That is, the fibered isomorphism
a
conjecture is true for H ≀G, for any subgroup H of Γ and for any finite group G.
SeeRemark2.1foranextensionofthetheoremtotheArtingroupscorresponding
to the finite complex reflection groups of type G(de,e,r). Also, Theorem 1.1 is
known for right-angled Artin groups, since right-angled Artin groups are CAT(0)
([2]).
As an application, in Theorem 2.4, we extend our earlier computation of the
surgery groups of the classical pure braid groups in [28], to the finite type pure
Artin groups appearing in Theorem 1.1.
2000 Mathematics Subject Classification. Primary: 19D35, 19D50, 19G24, 19J25 Secondary:
57R67,57N37.
Key words and phrases. Artingroups, fibered isomorphismconjecture, Whitehead group, re-
ducedprojectiveclassgroup,surgeryobstructiongroups.
May12,2014.
1
2 S.K.ROUSHON
For the classical braid group case as in [13] and [28], the crucial idea was to see
thatthegeometryofacertainclassof3-manifoldsisinvolvedinthe buildingofthe
group. InthesituationoftheArtingroupsconsideredinthispaper,wefindthatthe
geometryof orbifolds is implicit in the group,and we exploited this information to
provethe conjecture. Moreprecisely,someofthe Artingroupsareunderstoodasa
subgroupoftheorbifoldfundamentalgroupoftheconfigurationspaceofunordered
n-tuples of distinct points, on some 2-dimensional orbifold. In the classical braid
group case, the complex plane played the role of this 2-dimensional orbifold.
The proofs need vanishing of the higher orbifold homotopy groups of the con-
figuration spaces involved. For example, we prove that this configuration space,
which itself is a high dimensional orbifold, has an orbifold covering space which is
a contractible manifold, provided the 2-dimensional orbifold has a similar orbifold
coveringspace(Theorem2.2). Thisresultisnewintheliterature. Inthecase,when
the orbifold is the complex plane, the result is well known ([10]). It is also known
that the finite type Artin groups are fundamental groups of aspherical manifolds
([9]). In the affine type Artin group case, this is still a conjecture.
Here we remark that our results, as well as their proofs, for the Farrell-Jones
fibered isomorphism conjecture are valid for the more general version of the con-
jecture as stated in [2], with coefficients in an additive category.
The rest of the paper is organized as follows. In Section 2 we recall some back-
ground on Artin groups and state our main results. Here, we define a new class of
groups (orbi-braid) based on some intrinsic properties of the orbifold fundamental
groupofthe pure braid space (Definition 2.1)ofa 2-dimensionalorbifold(Theorem
4.2). Section3containssome study ofhomotopytheoryoforbifoldsandconfigura-
tionspaceswhichhelpinprovingTheorem2.2. Theorem2.2providesthenecessary
conditionfortheinductionargumenttoworkintheproofs. Wedescribesomework
from [1] in Section 5, which is the key input for this work. Section 6 contains the
proofs of the main results. The computation of the surgery groups of pure Artin
groups is given in Section 7. Finally, in Section 8 we recall the statement of the
isomorphism conjecture and all the basic facts needed to prove our results.
2. Artin groups and statements of results
Artin groups are an important class of groups, and appear in different areas of
Mathematics.
We are interested in those Artin groups, which appear as extensions of Coxeter
groups by the fundamental groups of hyperplane arrangementcomplements in Cn.
The Coxeter groups are generalization of reflection groups, and is yet another
useful class ofgroups. SeveralCoxetergroupsappear as Weyl groups of simple Lie
algebras. In fact, all the Weyl groups are Coxeter groups.
Next, we give a description of the Coxeter groups in terms of generators and
relations.
Let K = {s ,s ,...,s } be a finite set, and m : K ×K → {1,2,··· ,∞} be a
1 2 k
function with the property that m(s,s)=1, and m(s′,s)=m(s,s′)≥2 for s6=s′.
The Coxeter group associated to the system (K,m) is by definition the following
group.
W ={K | (ss′)m(s,s′) =1, s,s′ ∈S and m(s,s′)<∞}.
(K,m)
THE ISOMORPHISM CONJECTURE FOR ARTIN GROUPS 3
Table of finite Coxeter groups
Coxeter group Coxeter diagram Order of the group
An(n>2) (n+1)!
4 n
Bn(n>3) 2 n!
n−1
Dn (n>5) 2 n!
k 2k
I (k)
2
5 120
H
3
4 1152
F
4
5
H 14400
4
E 51840
6
E 2903040
7
E 696729600
8
Table 1
Table of some afine Coxeter groups
Coxeter group Coxeter diagram
An(n>1)
4
Bn(n>2)
Dn (n>2)
C (n>1) 4 4
n
Table 2
A complete classificationof finite, irreducible Coxeter groups is known (see [8]).
Here, irreducible means the corresponding Coxeter diagram is connected. In this
article, without any loss, by a Coxeter group we will always mean an irreducible
Coxeter group. See Remark 2.5. We reproduce the list of all finite Coxeter groups
in Table 1. These are exactly the finite reflection groups. For a general reference
on this subject we refer the reader to [18].
Also, there are infinite Coxeter groups which are affine reflection groups. See
Table 2, which shows a list of only those we need for this paper. For a complete
list see [18]. In the tables the associated Coxeter diagrams are also shown.
ThesymmetricgroupsS onnlettersandthedihedralgroupI (p)areexamples
n 2
ofCoxetergroups. ThesearetheCoxetergroupoftypeA andI (p)respectivelyin
n 2
4 S.K.ROUSHON
thetable. TheArtingroupassociatedtotheCoxetergroupW is,bydefinition,
(K,m)
A ={K | ss′ss′···=s′ss′s··· ; s,s′ ∈K},
(K,m)
herethenumberoftimesthefactorsinss′ss′··· appearism(s,s′);e.g.,ifm(s,s′)=
3, then the relation is ss′s = s′ss′. A is called the Artin group of type
(K,m)
W . There is an obvious surjective homomorphism A → W . The
(K,m) (K,m) (K,m)
kernel PA (say) of this homomorphism is called the associated pure Artin
(K,m)
group. In the case of type A , the (pure) Artin group is also known as a classical
n
(pure) braid group. When a Coxeter group is a finite or an affine reflection group,
the associated Artin group is called of finite or affine type, respectively.
Now,givenafiniteCoxetergroupinGL(n,R),considerthehyperplanearrange-
ment in Rn fixed pointwise by the group. Next, complexify Rn to Cn and consider
the corresponding complexified fixed hyperplanes. We call these complex hyper-
planes in this arrangement,reflecting hyperplanes associatedto the finite reflection
group. The fundamental group of the complement, PA (say), of the arrange-
(K,n)
mentisidentifiedwiththepureArtingroupassociatedtothereflectiongroup([3]).
The reflection group acts freely on this complement. The quotient space has fun-
damental group isomorphic to the Artin group, associated to the reflection group
([3]). That is, we get an exact sequence of the following type.
// // // //
1 PA A W 1.
(K,m) (K,m) (K,m)
We call the space (PA ) PA /W the (pure) Artin space of the
(K,m) (K,m) (K,m)
Artin group of type W .
(K,m)
In this article, this geometric interpretation of the Artin groups is relevant for
us.
Next, our aim is to talk some generalization of this interpretation, replacing C
bysome2-dimensionalorbifold,intheclassicalbraidgroupcase. Thesegeneralized
groups are related to some of the Artin groups. This connection is exploited here
to help extend our earlier result in [13] and [28], to the Artin groups of types A ,
n
B (= C ), D , A˜ , B˜ , C˜ and D˜ . Then, using a different method we treat the
n n n n n n n
cases F , G and I (p).
4 2 2
In [13] and [28], we proved the Farrell-Jones fibered isomorphism conjecture for
H≀F,whereH isanysubgroupofthe classicalbraidgroupandF isafinite group.
In such a situation, we say that the fibered isomorphism conjecture wreath product
with finite groups or FICwF is true for H.
The FICwF version of the conjecture was introduced in [21], and its general
properties were proved in [27] (or see [23]). The importance of this version of
the conjecture was first observed in [13] and [21]. See Section 8, for more on this
subject.
Remark 2.1. Recall that, a complex reflection group is a subgroup of GL(n,C)
generated by complex reflections. A complex reflection is an element of GL(n,C)
whichfixesahyperplaneinCn pointwise. Thereisaclassificationoffinite complex
reflection groups in [32]. There are two classes.
1. G(de,e,r), where d,e,r are positive integers.
2. 34 exceptional groups denoted by G ,...G .
4 37
There are associated Artin groups of the finite complex reflection groups. In
a recent preprint ([[7], Proposition 4.1]) it is shown that A is a subgroup
G(de,e,r)
of A . (In [7], these are called braid groups associated to the reflection groups).
Br
THE ISOMORPHISM CONJECTURE FOR ARTIN GROUPS 5
Therefore,usingLemma8.1andTheorem1.1,FICwFfollowsfortheclassofgroups
A .
G(de,e,r)
Now, we are in a position to state the results we prove in this paper. We begin
with a corollary of the main result. Another corollary (Theorem 1.1) is already
mentioned in the Introduction.
Definition 2.1. Fora topologicalspaceX, wedefine the pure braid space ofX on
n strings by the following.
PB (X)=Xn−{(x ,x ,...,x )∈Xn | x =x for some i,j ∈{1,2,...,n}}.
n 1 2 n i j
ThesymmetricgroupS actsfreelyonthepurebraidspace. Thequotientspace
n
PB (X)/S isdenotedbyB (X),andis calledthe braid spaceofX. WhenX isa
n n n
2-dimensional orbifold, then the orbifold fundamental group πorb(B (X)) is called
1 n
the braid group on the orbifold X. Similarly, πorb(PB (X))iscalledthe pure braid
1 n
group on the orbifold X.
Recall that, an orbifold is called good if it has an orbifold covering space which
is a manifold. See Section 3 for some backgroundon orbifolds.
Definition 2.2. We define a good, connected 2-dimensionalorbifold S to be non-
spherical, if it does not have S2 as an orbifold covering space.
Theorem 2.1. Let S be a non-spherical orbifold. Then, the FICwF is true for any
subgroup of the braid group on the orbifold S.
In the process of proving Theorem 2.1 we need to prove the following theorem.
This result was unknown in the literature, and is of independent interest.
Theorem 2.2. Let S be a non-spherical orbifold. Then, the universal orbifold
covering space of the pure braid space PB (S) of S is a contractible manifold.
n
Proof. See Theorem 4.1 and Corollary 4.1. (cid:3)
Tostateourmaintheoremweneedtodefinethefollowingclassofgroups,which
is a generalization of the class of groups defined in [[22], Definition 1.2.1].
Definition 2.3. A discrete group Γ is called orbi-braid if there exists a finite
filtration of Γ by subgroups; 1 = Γ ⊂ Γ ⊂ ··· ⊂ Γ = Γ such that the following
0 1 n
conditions are satisfied.
1. Γ is normal in Γ for each i.
i
2. For eachi, Γ /Γ is isomorphicto the orbifoldfundamentalgroupπorb(S ),
i+1 i 1 i
of a connected 2-dimensional orbifold S .
i
3. Foreachγ ∈Γofinfiniteorder,andi∈{1,2,...n−2}thefollowingconditions
are satisfied.
(a) There is a finite sheeted, connected orbifold covering space q : S˜ → S , so
i i
that S˜ is a manifold.
i
(b) The conjugation action c of γk, for some k ≥ 1, on Γ /Γ leaves the
γk i+1 i
subgroup q (π (S˜)) < πorb(S ) (q is injective by Lemma 3.2) invariant, where
∗ 1 i 1 i ∗
πorb(S ) is identified with Γ /Γ via a suitable isomorphism.
1 i i+1 i
(c) There is a homeomorphism f :S˜ →S˜, so that the induced homomorphism
i i
f on π (S˜) is equal to c | in Out(π (S˜)).
# 1 i γk π1(S˜i) 1 i
In such a situation we say that the group Γ has rank ≤n.
6 S.K.ROUSHON
Wewillshow,inTheorem4.2,thatthepurebraidgroupsonmost2-dimensional
orbifolds are orbi-braid.
Our main theorem is the following.
Theorem 2.3. Let Γ be a group which contains an orbi-braid subgroup of finite
index. Then, the FICwF is true for any subgroup of Γ.
ThemotivationbehindtheisomorphismconjecturearetheBorelandtheNovikov
conjectures,whichclaimthatanytwoclosedasphericalhomotopyequivalentmani-
foldsarehomeomorphic,andthehomotopyinvarianceofrationalPontryaginclasses
ofasphericalmanifolds,respectively. Thereare twoclassicalexactsequences inK-
and L-theory which summarizes the history of the two conjectures.
A :H (BΓ,K)→K (Z[Γ])
K ∗ ∗
and
A :H (BΓ,L)→L (Z[Γ]).
L ∗ ∗
In the case of torsion free groups,it is conjectured that the above two maps are
isomorphisms. This is also implied by the isomorphism conjecture, which is stated
for any discrete group ([[11], §1.6.1]).
Therefore, we have the following corollary.
Corollary 2.1. For any torsion free subgroup of Γ, where Γ is as in Theorem 1.1,
2.1 or 2.3, the above two assembly maps are isomorphisms.
BelowwementionawellknownconsequenceoftheisomorphismoftheK-theory
assembly map. See [[11], §1.6.1].
Corollary 2.2. The Whitehead group Wh(−), the reduced projective class group
K˜ (−) and the negative K-groups K (−), for i ≥ 1, vanish for any torsion free
0 −i
subgroup of the groups considered in Theorems 1.1, 2.1 and 2.3.
Remark 2.2. It is conjectured that the Whitehead group and K˜ (−) of a torsion
0
free group vanish. Also, W.-c. Hsiang conjectured that K (−), for i ≥ 2, should
−i
vanish for any group ([17]).
Finally, we state a corollary regarding an explicit computation of the surgery
groups L () of the pure Artin groups of finite type. This comes out of the isomor-
∗
phismsofthetwoassemblymapsA andA ,forthegroupsstatedinTheorem1.1.
K L
The calculation was done in [28] for the classical pure braid group case. Together
with the isomorphisms of A and A , the proof basically used the homotopy type
K L
of the suspension of the pure braid space and the knowledge of the surgery groups
of the trivial group.
Theorem 2.4. The surgery groups of finite type pure Artin groups take the follow-
ing form
Z if i=4k,

Li(PA)=ZZN iiff ii==44kk++12,,
2
ZN2 if i=4k+3.
where, N is the number of reflecting hyperplanes associated to the finite Coxeter
group, as given in the following table.
THE ISOMORPHISM CONJECTURE FOR ARTIN GROUPS 7
PA=pure Artin group of type N
A n(n−1)
n−1 2
B (=C ) n2
n n
D n(n−1)
n
F 24
4
I (p) p
2
G 6
2
Table 3
Remark 2.3. Recall here, that there are surgery groups for different kinds of
surgery problems, and they appear in the literature with the notations L∗(−),
i
where ∗ = h,s,h−∞i or hji for j ≤ 0. But, all of them are naturally isomorphic
for groups G, if the Whitehead group Wh(G), the reduced projective class group
K˜ (ZG), and the negative K-groups K (ZG), for i ≥ 1, vanish. This can be
0 −i
checked by the Rothenberg exact sequence ([[31], 4.13]).
···→Lhj+1i(R)→Lhji(R)→Hˆi(Z/2;K˜ (R))→Lhj+1i(R)→Lhji (R)→··· .
i i j i−1 i−1
Where R=Z[G], j ≤1, Wh(G)=K˜ (R), Lh1i =Lh, Lh2i =Ls and Lh−∞i is the
1 i i i i i
hji
limit of L . Therefore, because of Corollary 2.2, we use the simplified notation
i
L (−) in the above corollary.
i
Remark 2.4. The same calculation holds for the other pure Artin groups corre-
spondingtothefinitetypeCoxetergroupsandfinitecomplexreflectiongroups(see
Remark2.1)also,providedweknowthatA andA areisomorphisms. Wefurther
k L
needthefactthattheArtinspacesareaspherical([9]),whichimpliesthattheArtin
groups are torsionfree. The last fact can also be provedgroup theoretically ([14]).
The Artin groups corresponding to the finite complex reflection groups G(de,e,r)
is also torsion free, since A is a subgroup of A ([[7], Proposition 4.1]).
G(de,e,r) Br
We conclude this section with the following useful remark.
Remark 2.5. If we do not assume a Coxeter group C to be irreducible, then
C ≃ C ×···×C , where C is an irreducible Coxeter group for i = 1,2,...,k.
1 k i
Hence, A ≃ A ×···×A . Therefore, by Lemma 8.6, FICwF is true for A ,
C C1 Ck C
if it is true for A for each i.
Ci
3. Some homotopy theory of orbifolds
In this section we give a short background on orbifolds, and then prove some
basic facts. Also, we recall some homotopy theory of orbifolds from [5].
Theconceptoforbifoldwasfirstintroducedin[29],andwascalled‘V-manifold’.
Later,it was revivedin [33], with the new name orbifold, and orbifold fundamental
group(denotedbyπorb(X))ofaconnectedorbifoldwasdefined. Orbifold homotopy
1
groups (denoted by πorb(X)) are defined as the ordinary homotopy groups of the
n
classifying space of the topological groupoid associated to the orbifold. This was
developed in [15]. More recently, in [5], all these were generalized in the more
general category of orbispaces (see [[15], Appendix §6]), their homotopy groups
were defined and the homotopy theory of orbispaceswas developed. See Theorems
4.1.12 and 4.2.7 in [5]. We will be using this homotopy theory for orbispaces, and
apply it in our situation of orbifolds.
8 S.K.ROUSHON
Recall that an orbifold is a topological space, which at every point looks like
the quotient space of an Euclidean space Rn, for some n, by some finite group
action. This finite group is called the local group. The image of the fixed point set
of this finite group action is called the singular set. Points outside the singular set
are called regular points. In case the local group is cyclic (of order k) acting by
rotation about the origin on the Euclidean space, the image of the origin is called
a cone point of order k. If the local group at some point is trivial then it is called
a manifold point. As in the case of manifold, one can also talk of the dimension
of a connected orbifold. One obvious example of an orbifold is the quotient of
a manifold by a finite group. More generally, the quotient of a manifold by a
properly discontinuous faithful action of some discrete group ([[33], Proposition
5.2.6]) is an orbifold. The notion of orbifold covering space, orbifold fiber bundle
etc. were defined in [33]. We refer the reader to this source for the basic materials
and examples.
In general, an orbifold need not have a manifold as an orbifold covering space,
but, if this is the case then the orbifold is called good. One can show that a good
compact orbifold has a finite sheeted orbifold covering space, which is a manifold.
In the case of closed 2-dimensional orbifolds, only the sphere with one cone point
and the sphere with two cone points of different orders are not good orbifolds. See
the figure below. Also, see [[33], Theorem 5.5.3].
m
m
m > n
n
Figure 1: Bad orbifolds
A boundary of the underlying space of an orbifold has two types, one which
we call manifold boundary and the other orbifold boundary. These are respectively
definedaspointsontheboundary(oftheunderlyingspace)withlocalgrouptrivial
or non-trivial.
We begin with the following two lemmas. The proof of the first lemma follows
from the classification of 2-manifolds.
Lemma 3.1. Let S be a non-spherical orbifold. Then, the universal orbifold cov-
ering space of S is homeomorphic to a submanifold of R2.
Remark 3.1. We make here the obvious remark, that if S is non-spherical then
S−{finitely many points} is also non-spherical.
The following second lemma follows from standard covering space theory for
orbifolds. We will be using this lemma frequently throughout the paper without
referring to it.
Lemma 3.2. Let q : S˜ →S be a finite sheeted orbifold covering map between two
connected orbifolds. Then, the induced map q :πorb(S˜)→πorb(S) is an injection,
∗ 1 1
and the image q (πorb(S˜)) is a finite index subgroup of πorb(S).
∗ 1 1
Proof. See [[6], Corollary 2.4.5]. (cid:3)
THE ISOMORPHISM CONJECTURE FOR ARTIN GROUPS 9
Recallthat,the quotientmapfromanorbifoldby afinite groupactionisalways
an orbifold covering map, and the quotient space is called a global quotient.
Hence, we havethe followingconsequence. We need this corollaryto deduce the
FICwF for πorb(B (S)), after proving the FICwF for πorb(PB (S))
1 n 1 n
Corollary 3.1. Let S be a connected 2-dimensional orbifold. Then, q is injective
∗
and q (πorb(PB (S))) is a finite index subgroup of πorb(B (S)). Here, q denotes
∗ 1 n 1 n
the global quotient map
q :PB (S)→PB (S)/S =B (S).
n n n n
Now,recallthataconnectedspaceiscalledaspherical ifallitshomotopygroups,
except the fundamental group, vanish. The following is a parallel concept in the
category of orbifold (or orbispaces)
Definition 3.1. We call a connected orbifold (or orbispace) X orbi-aspherical if
πorb(X)=0 for all i≥2.
i
Consequently, a good orbifold is orbi-asphericalif it has a contractible manifold
universal covering space.
Next, we give a necessary condition on the orbifold fundamental group of a 2-
dimensionalorbifoldto ensure that the orbifoldis good. This resultwill be needed
todeducetheFICwFfortheorbifoldfundamentalgroupsof2-dimensionalorbifolds.
Also, we will need the lemma to begin an induction step to prove Theorem 2.3.
Lemma 3.3. Let S be a 2-dimensional orbifold with finitely generated, infinite
orbifold fundamental group. Then, S is good.
Proof. LetS bea2-dimensionalorbifoldasinthestatement. Recallthat,thereare
only three kinds of singularities in a 2-dimensional orbifold: cone points, reflector
lines (or mirror) and corner reflectors, as there are only three kinds of finite sub-
groupofO(2). See [[33],Proposition5.4.2]. After goingtoafinitesheetedcovering
we can make sure that it has only cone points. So, assume that S has only cone
points.
First,weconsiderthecasewhenSisnoncompact. Sincetheorbifoldfundamental
group is finitely generated we can find a compact suborbifold S′ ⊂ S with circle
boundarycomponentssothattheinteriorofS′ishomeomorphictoSasanorbifold.
Clearly, all these circle boundary components consists of manifold points of S.
Now, let DS′ be the double of S′. Then DS′ has infinite orbifold fundamental
group. Hence, from the classification of 2-dimensional orbifolds using geometry
(see [[33], Theorem 5.5.3]), it follows that DS′ is a good orbifold. Since, DS′ has
evennumber ofcone points, and in the case of two cone points they have the same
orders. See Figure 1. Clearly, then S′, and hence S, also has an orbifold covering
space, which is a manifold.
Next, if S is compact then the same argument as in the previous paragraph,
shows that S is good.
This completes the proof of the Lemma. (cid:3)
Wenowrecallacoupleofhomotopytheoreticresultsfororbifoldsfrom[5]. These
results were proven in the general category of orbispaces, but here we reformulate
them for orbifolds.
10 S.K.ROUSHON
Lemma 3.4. ([[5], Theorem 4.1.4] or [[6], Corollary 2.4.5]) Let q : M → N
be an orbifold covering map between two connected orbifolds M and N. Then,
q :πorb(M)→πorb(N) is an isomorphism for all n≥2.
∗ n n
Lemma3.5. ([[5],Theorem 4.2.5]) Letf :M →N beanorbifold fibrationbetween
twoconnectedorbifolds M andN, withfiberF over someregularpoint ofN. Then,
there is a long exact sequence of orbifold homotopy groups.
··· //πorb (N) // πorb(F) //πorb(M) // πorb(N) //···
k+1 k k k
··· //πorb(N) // πorb(F) //πorb(M) // πorb(N).
1 0 0 0
Animmediate corollaryisthe following. Thisresultis paralleltothe 2-manifold
situation, where we know that any 2-manifold of genus ≥1 is aspherical.
Lemma 3.6. A non-spherical orbifold is orbi-aspherical.
Proof. The proof is immediate from Lemmas 3.1 and 3.4. (cid:3)
In this paper wewill avoidmentioning basepoints in the notations ofhomotopy
groups and fundamental groups, as we will be considering connected orbifolds (or
manifolds) only. But, we always take a regular point as a base point.
4. Topology of orbifold braid spaces
Inthissectionweestablishsomeresultsonthetopologyofthepurebraidspaces
ofanorbifold. Then,weprovethatthepurebraidgrouponanon-sphericalorbifold
is orbi-braid.
Theorem4.1. LetS beanon-sphericalorbifold. Then, PB (S)isorbi-aspherical.
n
Proof. Recall that PB (S)=Sn−∪n H , where
n i<j;i,j=1 ij
H ={(x ,x ,...,x )∈Sn | x =x }.
ij 1 2 n i j
Considertheprojectionp:PB (S)→PB (S)tothelast(n−1)coordinates. pis
n n−1
alocallytrivialorbifoldfibrationwithgenericfiberS′ =S−{(n−1)regular points}.
We can now prove the theorem by induction on n. For n = 1, PB (S) =
n
S. Hence, the induction starts by Lemma 3.6. So, we assume the statement for
dimension n−1. Now, applying Lemma 3.5 to the fibration p, and the induction
hypothesis we complete the proof of the theorem. We only need to note here that,
S′ is non-spherical by Remark 3.1 and hence, Lemma 3.6 is applicable on S′. (cid:3)
Corollary 4.1. Let S be a non-spherical orbifold. Then, the universal orbifold
covering space of PB (S) is a contractible manifold.
n
Proof. By hypothesis, S has a finite sheeted orbifoldcoveringspace S˜ (say),where
S˜ is a manifold. Therefore, S˜n −(qn)−1(∪n H ) is a manifold, where qn :
i<j;i,j=1 ij
S˜n → Sn is the n-times product of the covering map q : S˜ → S. Hence, we have
an orbifold covering map
S˜n−(qn)−1(∪n H )→PB (S),
i<j;i,j=1 ij n
whosetotalspaceisamanifold. Next,applyingTheorem4.1andLemma3.4weget
that all the orbifold homotopy groups of the universal cover of PB (S) is trivial.
n
Now,since(byuniqueness)theuniversalcoverisamanifold,itsordinaryhomotopy
groups are isomorphic to the orbifold homotopy groups. Therefore, the ordinary
homotopygroupsarealsotrivial, andhence the universalcoveris contractible. (cid:3)