ñòð. 5 
[1] 1![2] [2] 1![1]
â€¢ â€¢
[2]1
â€¢
â€¢
t![2m] [1]1![I] =1
1 = [1]
â€¢
â€¢
Vk V3 7
V (r=5)
Figure 3
The Hopf algebra structure on A allows one to define Amodule structures on the
duals V* = Homo(V, C) and tensor products V Â® W = V 00 W of Amodules V
and W. In particular, (Af)(v) = f(S(A)v) and A(v ˜ w) = ˜AÂ· (v ˜ w) for A E A,
f E V*, v E V, w E W. Thus the vector spaces in Figure 2 will be Amodules and
the operators will be Alinear.
Kirby and Melvin: Evaluations of the 3manifold invariants
106
The structure of Amodules for k ˜ r and their tensor products Vi 0 Vi for
i + j  1 5 r is parallel to the classical case and is well known:
(10) THEOREM [RT2]. If k 5 r, then the representations V k are irreducible and ::j
self dual. If i + j  1 5 r, then Vi 0 Vi = â‚¬a Vic where k ranges by twos over o˜
1
k
{i+jl, i+j3, i+j5,Â·Â·Â·,lijl+l}. J
If k < r, then
(11) COROLLARY.
= ˜(l)j (k ˜ i) (V2)ilSlA:12 j
vA: ,
J
where the sum is over all 0 ˜ j < ˜.
=Y  W = V, jV = V E9 ... E9 V and
Here we have written U W to mean U Ef)
i times
j times
.
= V 0 . Â· . 0 V. This corollary is the key to our later reduction from arbitrary
yÂ®J
colorings to 2dimensional ones.
The Hop! algebra A has the additional structure of a quasitriangular Hopf alge
bra [D], that is, there exists an invertible element R in A 0 A satisfying
R˜(A)Rl = Li(A) for all A in A
= R 13 R23
(˜0
(12) id)(R)
(idÂ®˜)(R) = R 13 R 12
where ˜(A) = P(˜(AÂ» and peA 0 B) = B 0 A, R 12 = R Â® 1, R23 = 1 0 Rand
R I3 = (P Â® id)(R23 ). R is called a universal Rmatrix. Historically, Rmatrices
have been found for Uk, A and other Hopf algebras by Drinfeld [D], Jimbo [J],
Reshetikhin and Turaev [RT2] and others. We look for an R of the fonn R =
E CnabXn!{a 0 yn Kb, and recursively derive the constants Cnab from the defining
relation R˜(A)Rl = ˜(A). This approach was suggested to us by A. Wa˜serman
who had carried out a similar calculation.
(13) THEOREM. A universal Rmatrix for A is given by
L
=˜ (s  s)n f
R ab+(ba)n+n X n J(a Â® yn J(b
[n]!
4r n,a,b
n < r and 0 :5 a, b < 4r and [n]! = [n][n 1] ... [2][1].
˜
where the sum is over all 0
Â® V k' by
(14) COROLLARY. R acts in the module yk
. _ ˜ (8  s)n . [m + i + n]! + n]!
l
[m
R. j 4ij2n(ij)n(n+l) . .

[m + iJI [m' _ iJ! e.+ n
e. Â® e,  L [nJI t Â® e,n
n
Kirby and Melvin: Evaluations of the 3manifold invariants 107
= 2m + 1, k' = 2m' + 1, and f$ = (P][P 1] ... [n + 1].
where k
In V 2 ˜ V 2 , the Rmatrix is
EXAMPLES:
with respect to the basis el/2 Q:9 el/2' el/2 Q:9 el/2, el/2 Â® el/2, and el/2 Â® el/2,
and
(15)
R=(q)EB(˜ q˜ij)E9(q˜ ˜ l˜ij )EB(˜ q ˜_)E9(q).
(qq)(l+q) (qq)(lq) q
It is now possible to assign operators to the following elementary colored tangles
[RTl]:
˜Rl =Rlp
˜E V
whereE(fÂ®x) =f(x),fe V*,x E
=f(](2 x)
˜ E˜ where E[(i.x Â® f)
˜NwhereN(l)= ˜ ei Â®ei
Figure 4
˜
.˜
Kirby and Melvin: Evaluations of the 3manifold invariants
108 ˜
'˜
.\
l.,\
'.˜
:˜
From these elementary tangle operators, define [RT1] Alinear operators JT,k for 3
A
Â·1
arbitrary oriented, colored, framed tangles T, k. If T is a link L, then we obtain the
".
J
scalar JL,k. The invariance of JL,k under Reidemeister moves on L is well known;
J
the YangBaxter equation (idÂ®R)(RÂ®id)(idÂ®R) = (RÂ®id)(idÂ®R)(RÂ®id) is the
:;
key ingredient, and it follows easily from the defining properties (12) for R. Note
that JL,k is independent of choice of orientation of L.
The following examples are easily derived from the Rmatrix and the
EXAMPLES:
irreducibility of V k :
[Q] ˜ [k]
D2J
D2J ˜ t{k 21)
t(k 2l),
4
˜
Uk]
˜w
k J
(scalar) operators
tangles
Figure 5
With this definition of JL,k, we have completed the definition of Tr(ML). The
examples in Figure 5 can be used to check the onestrand K move, or conversely,
they may be used to solve for the coefficients of JL,k in the definition of Tr(ML).
When all components of L are colored by the 2dimensional representation, then
JL,2 ˜ JL is just a va˜iant of the Jones polynomial. First note that from (15) R
on V 2 0 V 2 satisfies the characteristic polynomial
tR  tR 1 = (8  8)1. Â·1
..˜
Then, adjusting for framings, JL satisfies the skein relation
= (8 
(16) qJL+  qJL_ S)JLo
(see [L2] for background on skein theory). H VL = V(q) is the version of the Jones
polynomial (for oriented links) satisfying this skein relation and VUnknot = 1, then
it follows that
= JL,2 = [2]t 3LÂ·L VL.
,..,
(17) JL
Kirby and Melvin: Evaluations of the 3manifold invariants 109
Remark: the relation between JL,2 and VL is important because the values of VL
at certain roots of unity have a topological description, as they do for the usual
Jones polynomial VL [LM1], [Lip], [Mur]. In particular, the values at q = e(l/r),
for r = 1, 2, 3, 4 and 6, are as follows:
VL
r VL
2 n 1 (_2)nl
1
detL
2 detL
(18) ( _l)nl
3 1
V2,nl a(V2,)nl
a2
4
y'3d( _i)'" ( _y'3)d( _i)'"
6
where n is the number of components of L, det L is the value at 1 of the (normal
ized) Alexander polynomial of L, a is (_l)Arf(L) when L is proper (so the Arf invari
ant is defined) and 0 otherwise, d is the nullity of Q(mod 3) where Q is the quadratic
form of L (represented by S +5 t for any Seifert matrix 5 of L), and w is the Witt
class of Q(mod3) in W(Z/3Z) = Z/4Z. It is well known that IdetLI = IH1(M)I,
where M is the 2fold branched cover of 53 along L, and d = dimH1 (M;Z/3Z)
(since any matrix representing Q is a presentation matrix for HI (MÂ».
If S is a sublink of L obtained by removing some Icolored compo
PROPOSITION.
nents, then
(19)
Using (11) and (17) we obtain a formula for the general colored framed link
invariant JL,k in terms of J or V for certain cables of L. In particular, a cabling c
of a framed link L is the assignment of nonnegative integers Ci to the L i , and the
associated cable of L, denoted LC, is obtained by replacing each Li with Ci parallel
pushoffs (using the framing).
(20) THEOREM. Using multiindex notation,
Â·(kl i ) hkt2J
h,k = ˜
˜(1), j
J
Â·(k 1j) t
= [] ˜ 3Lk12j.Lk12J
L...J(
2 1)1 â€¢ VL kI2J
. J
J
where the sum is over all 0 ˜ j < ˜.
Lk  12
orientation
{or any  j,
011
Kirby and Melvin: Evaluations of the 3manifold invariants
110
If L = K = framed knot, then
EXAMPLES:
JK,3 = J K 21
JK,. = J](3  2JK
(21)
+ 1.
= JK 4 3JK 2
JK,S 
Tr(ML) = Ec(C)JLc where the sum is over all cables c =
(22) THEOREM. O!L
(crj)
(ct, ... ,cn ), 0 ˜ Ci ˜r = Ej[c + 2j + 1]( 1˜
2, (XL is as in (5), and (c)
0 with c +2j +1 < r.
˜
where the sum is over all j
Remark: a formula like this motivated Lickorish [L3] to give an elementary and
purely combinatorial derivation of essentially the same 3manifold invariant as T r .
The proof reduced to a combinatorial conjecture whose proof has been claimed
by Koh and Smolinsky [KS]. This elegant approach is much shorter and simpler.
However it may be less useful because the above algebra involving A organizes a
great deal of combinatorial information.
For example, using (20) one can give a recursive formula for JH n ,2 = JH n for the
unoriented, ncomponent, Iframed, righthanded Hopf link H n :
=1
JH o
= t 3 [2]
JHl
(23) n/2
+ "'(_1)kl (n 
n2
J H n = t + [2n] 1
k 1) J H n 
k
L.J 2 1e'
k=1
Using deeper properties of A [RT2], one obtains a closed formula:
(24)
It is not clear how to derive such formulae in a combinatorial way from skein theory.
(25) SYMMETRY PRINCIPLE: Suppose we are given a framed link of n + 1 compo
nents, L U K, L = L 1 U ... U L n , with colors I = 11 U ... U In on Land k on ](. If
we switch the color k to r  k, then
='Y JLU](,luk
JLu/(,Iurk
where'Y = i ra ( _l)˜+ka, a is the framing on ]( and A = Eeven 1.1< . L i == !(. (I +
mod2
I
1)L.
Kirby and Melvin: Evaluations of the 3manifold invariants 111
Use of the Symmetry Principle enables one to cut the number of terms in T r(M L )
from the order of (r  l)n to (˜)n. It also has interesting topological implications.
> 0, then
For r = 5 and L = 1< with framing a
EXAMPLE:
= ˜sinie ( :0) trk]JK'k
Ts(MK)
k=l
+i 5a (_1)4)
+ [2]Jk + 5a
[3]i ( _1)2a JK
= QK(l
(26)
+i +([2] + [3]i )JK) for a even
4
4
= aK(1
= aK(l +i a )(1 + [2]2 t 3a Vlf) since [3] = [2]
= 0 for a == 2 mod 4.
Vk
For a ˜ 2 mod 4, this shows that is an invariant of MK.
Next we discuss the evaluations of Tr(M) when r = 3, 4 and 6. Note that
T2(M) = 1.
For r = 3,
˜ ,sÂ·s
1 u
(27) (M) = L.J ˜
Jnn C
73
v2 S<L
where M = ML' c = e (i) = ˜ and < denotes sublink and we sum over all
sublinks including the empty link (4) Â· 4> = 0). It is not hard to see how the formula
follows from (4) since components with color 1 are dropped (19); it also follows from
the cabling formula (22).
Evidently, Formula 27 depends only on the linking matrix A of L. It is not hard to
/
give an independent proof of the well definedness of (27) by checking its invariance
under blow ups and handle slides' as in the calculus of framed links [Kl]. This means
that T3(M) is an invariant of the stable equivalence class of A (where stabilization
Ineans A E9 (Â±1)). It follows that T3(M) is a homotopy invariant determined by
rank H1 (M; Z) and the linking pairing on Tor H 1 (M; Z), for these determine 'the
stable equivalence class of A.
The cumbersome sum in (27) can be eliminated by using Brown's Z/8Z invariant
/˜ associated with A. View A as giving a Z/4Zvalued quadratic form on a Z/2Z
vector space by reducing D10d4 along the diagonal (to get the form) and reducing
11lOd 2 (to get the inner product on the vector space). A is stably equivalent to a
is
(liagonal matrix and then {3 = nl  n3 mod 8 where ni the number of diagonal
('utries congruent to i (mod4). Observe that
p(M) =
(28) {3 (modS)
(f 
is an invariant of M = A1IJ'
Kirby and Melvin: Evaluations of the 3manifold invariants
112
(29) THEOREM. If all pinvariants of spin structures on M are congruent (mod4),
tben
T3(M) = vI2"t(M) cp(M)
where bt(M) = rankH1 (M;Z/2Z), c = e(i) and p(M) is as above. Otherwise,
T3(M) = O.
If M is a Z/2Zhomology sphere, then
(30) COROLLARY.
= Â±cp(M)
T3(M)
where c = e (l), p,(M) = p,invariant of M and the Â± sign is chosen according to
whetber IH1 (M; Z)I == Â±1 or Â±3 (mod8).
(31) REMARK: 7"3(M) is not always determined by Ht(M; Z) and the pinvariants
of M (although 7"4 is, see below). For example, if M = L(4,1) # L(8,1), then
p(Â±M) = Â±2 so T3(Â±M) = Â±2i, yet M and M have the same homology and
p,invariants.
(32) THEOREM. T4(M 3 ) = Lecll(M,e) where c = e (13 p(M,8) is the p,
6)'
invariant of the spin structure 8 on M and the sum is over all spin structures
onM.
The keys to the proof are these: use the cabling formula (20) to drop Icolored
components, keep 2colored components and double 3colored components; the un
doubled components turn out to be a characteristic sublink and hence to correspond
to a spin structure; at r =4, the Ad invariant (18) comes into play; finally,
cÂ· C +8 Arl(e)
p.(M, 8) =
(33) mod 16
(7 
is a crucial congruence where C is a characteristic sublink corresponding to 8.
The congruence (33) is well known in 4manifold theory [K2], being a generaliza
tion of Rohlin's Theorem. It turns out, motivated by (32) above that we can give
a purely combinatorial proof of (33) without reference to 4manifolds.
At present, we have no general formula for T6( M) in terms of "classical" invariants
of M, although itÂ· is plausible that one exists. Indeed, it is immediate from the
Symmetry Principle (25) and the cabling formula (20) that T6(ML) can be expressed
in tenns of Jones polynomials of cables of L with each component at most doubled.
(If the linking number of each component Li of L with L  L i is odd, then doubled
components may also be eliminated.) Now, since the Jones polynomial of a link at
e (k) is determined by the quadratic form of the link (see 18) it would suffice to
show that the quadratic forms of these cables are invariants of M.
Kirby and Melvin: Evaluations of the 3manifold invariants 113
In particular if M is obtained by surgery on a knot K with framing a, then it
can be shown that
where u is 0 if a = 0, 1 if a > 0 and 1 if a < O. It follows that T6(M) is
determined by a and the Witt class of the quadratic form Q of K. Thus, for odd a,
or a = 0, T6(M) is determined by H1 (M; Z) (with its torsion linking form, needed
to determine the sign of a when a is divisible by 3). For even a, one also needs to
know H1(M; Z) with its torsion linking form (which determines the Witt class of
Q) where M is the canonical 2fold cover of M.
We are especially grateful to N. Yu. Reshetikhin for his lectures and conversa
tions on [RTl] and [RT2], and to Vaughn Jones, Greg Kuperberg and Antony
Wasserman for valuable insights into quantum groups.
REFERENCES
[D] V. G. Drinfel'd, Quantum groups, Proc. Int. Congo Math. 1986 (Amer. Math.
Soc. 1987), 798820.
[FR] R. Fenn and C. Rourke, On Kirby's calculus of links, Topology 18 (1979),
115.
[J] M. Jimbo, A qdifference analogue of U(Q) and the YangBaxter equation,
Letters in Math. Phys. 10 (1985), 6369.
[Kl] R. C. Kirby, A calculus for framed links in S3, Invent. Math. 45 (1978),
3556.
[K2] , "The Topology of 4Manifolds," Lect. Notes in Math., v. 1374,
Springer, N˜w York, 1989.
[KM] R. C. Kirby and P. M. Melvin, Evaluations of new 3manifold invariants,
Not. Amer. Math. Soc., 10 (1989), p. 491, Abstract 89T57254.
[KS] H. K. Ko and L. Smolinsky, A combinatorial matrix in 3manifold theory,
to appear Pacific J. Math..
[Ll] W. B. R. Lickorish, A representation of orientable, combinatorial 3mani
folds, Ann. Math. 76 (1962), 531540.
, Polynomials for links, Bull. London Math. Soc. 20 (1988),
[L2]
558588.
[L3] , Invariants for 3manifolds from the combinatorics of the
Jones polynomial, to appea.r Pacific J. Math.
Kirby and Melvin: Evaluations of the 3manifold invariants
114
[LM1] W. B. R. Lickorish and K. C. Millett, Some evaluations of link polynomials,
Comment. Math. Helv. 61 (1986), 349359.
[Lip] A. S. Lipson, An evaluation of a link polynomial, Math. Proc. Camb. Phil.
Soc. 100 (1986), 361364.
[Mur] H. Murakami, A recursive calculation of the Arf invariant of a link, J.
Math. Soc. Japan 38 (1986), 335338.
[RT1] N. Yu. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants
derived from quantum groups, MSRI preprint (1989).
[RT2] , Invariants of 3manifolds via link polynomials and quan
tum groups, to appear Invent. Math.
[Wa] A. H. Wallace, Modifications and cobounding manifolds, Can. J. Math. 12
(1960), 503528.
[W] Ed Witten, Quantum field theory and the Jones polynomial, Comm. Math.
Phys. 121 (1989), 351399.
Department of Mathematics Department of Mathematics
University of California Bryn Mawr College
Berkeley, CA 94720 Bryn Mawr, PA 19010
Representations of Braid Groups
M.F.ATIYAH
Lecture by
The Mathematical Institute, Oxford
S.K.DONALDSON
Notes by
The Mathematical Institute, Oxford
In this lecture we will review the theory of Heeke algebra representations of braid
1!;roups and invariants of links in 3space, and then describe some of the results
obtained recently by R.J. Lawrence in her Oxford D.Phil. Thesis [3].
(a)Braid group representations, Heeke algebras and link invariants. We
begin by recalling the definition of the braid groups, and their significance for the
theory of links. The braid group on n strands, B n , can be defined as the fundamental
p;roup of the configuration space en of n distinct points in the plane. Thus en is
t.he quotient:
(1)
en = {(Xl, ... ,X n ) E (R 2 )nl xi I Xj for i # j}, and the symmetric group
where
S'n acts on Cn in the obvious way. Elements of the braid group ("braids" ) can be
(lescribed by their graphs in R 2 X [0,1] C R3, as in the diagram.
'I'he action of the braid on its' endpoints defines a homomorphism from B n to Sn,
and this is just the homomorphism corresponding to the Galois covering (1).
IleH' any braid /3 we can construct a link /3' in the 3manifold 8 1 X R2 by identifying
the top and bottom slices in the graph. The link has an obvious monotonicity
lÂ»l'Operty: the projection from the link to Sl has no critical points. It is easy to see
tha.t if a is another braid then (afja 1 )' is isotopic to fj', and then to show that
I.Ilis construction sets up a 11 correspondence
U Conjugacy cbu..;:
,
Atiyah: Representations of braid groups
116
1
.˜
1
_
o˜
1
The graph of a braid
Thus an invariant of monotone links in Sl x R 2 is just the same thing as a set of
class functions on the braid groups. In particular, for any finite dimensional linear
representation p of B n we obtain 8Jl 51 x R 2 linkinvariant pi through the character
= Tr p((3).
p'(f3')
Pin
53 to a braid (3, using the standard
It is also possible to associate a link
2 3
1
embedding 8 X R C R â€¢ All links in the 3 sphere are obtained in this way and
the isotopy classes of links in 53 can be regarded as obtained from the braids by
imposing an equivalence relation generated by certain "Markov moves ". We get
invariants of links in the 3sphere from representations of the braid groups which
r
ar "consistent" with these moves.
The representations of the braid groups which we will discuss have the feature that
they depend on a continuous parameter q E C. When q = 1 the representations are
just those coming from the symmetric group and in general they factor through an
intermediate object; the Heeke algebra Hn(q). The idea of using Hecke algebra
representations to obtain link invariants is due to Vaughan Jones and we refer to
the extremely readable Annals of Mathematics paper by Jones [2] for a beautiful
account of the more algebraic approach to his theory. Here we will just sun1nlarise
the basic facts and definitions. To define the algebra Hn(q) we recall that there is
a standard system of generators CTi i = 1, ... ,n  1 for the braid group En which
are lifts of transpositions in 5 n , as pictured in the diagram.
The Hecke algebra Hn(q) is the quotient of the group algebra C[B n] :
+q) = 0 ; i = 1, ... , n 
= C[B n]/ < (Ui  1>.
Hn(q) l)(O"i
In fact the braid group can be described concretely as the group generated by
the Uj subject to the relations
Atiyah: Representations of braid groups 117
0 0
: 0
˜
 0
0 0
0 0
The braid (J*2
=
O'i+lO"jO"i+l O'iO"i+lO"j
Ii  jl > 1.
= for
UiUj UjO"i
So Hn(q) is the algebra generated by the aj subject to these relations and the
ar
further conditions (ai  l)(ai + q) = O. If q = 1 we obtain the relations =1
Hatisfied by the transpositions in Sn, and it is easy to check that the algebra Hn(l)
is canonically isomorphic to the group algebra C[Sn]. For any q, a representation of
Hn (q) gives a representation of the group algebra C[En] and hence a representation
of En and from the discussion above we see that when q = 1 these are indeed just
the representations which factor through the symmetric group.
'rhese Hecke algebras arise in many different areas in mathematics, and the param
(˜t,er q can play quite different roles. In algebra we take q to be a prime power and
let F be the field with q elements; then we obtain Hn(q) as the double coset alge
bra of the group G = SL(n, F) with respect to the subgroup B of uppertriangular
luatrices. This is the subalgebra of C[G) generated by the elements
LX
TD=
xED
fen' D E B\G/B. On the other hand in physical applications one should think of q
e ih where h is Planck's constant; the limit q 1 then appears as the classical
+
ItS
lilnit of a quantum mechanical situation.
'I'llere is an intimate relationship between the representations of the Hecke algebra
for general q and the representations of the symmetric group, i.e. of Hn(l). This
n'lationship can be obtained abstractly using the fact that Hn(l)  the group alge
IJra of the symmetric group  is semisimple. We then appeal to a general "rigidity"
I Â»r<>perty: a small deformation of a semisimple algebra does not change the isomor
pl,isln class of the algebra. Hence the Hn(q) are isomorphic, as abstract algebras,
Atiyah: Representations of braid groups
118
to C[Snl for all q sufficiently close to 1  in fact this is true for all values of q except
a finite set E of roots of unity. So for these generic values of q the representations of
Hn(q) can be identified with those of Sn and we obtain a family of representations
Pq,A of B n , with characters Xqt A, indexed by the irreducible representations A of
Sn and a complex number q E C \ E. These are the representations whose char
acters are used to obtain the new link invariants. More precisely, the twovariable
"HOMFLY" polynomial invariant of links can be obtained from a weighted sum of
characters of the form:
X(q,z) = LaA(q,z)Xq,A'
A
for certain rational functions aA of the variables q and z. The earlier Ivariable
Jones polynomial V(q) is obtained from X(q, z) by setting q = z.
We will now recall some of the rudiments of the representation theory of symmetric
groups, and the connection with the representations of unitary groups. The irre
ducible representations of Sn are labelled by "Young Diagrams" , or equivalently
by partitions n = PI + ... + Pr, with Pt ;::: ... ˜ Pr > o. For example the trivial
representation corresponds to the "Irow" diagram, or partition n = n, the 1 di
mensional parity representation to the Icolumn diagram, or partition n = 1+Â·Â· Â·+1.
Now let Y be the standard I dimensional representation space of the unitary group
U(l). There are natural commuting actions of U(l) and Sn on the tensor product
vÂ®n = V 0 ... ˜ V, so there is a joint decomposition:
EBAA 0 B˜,
yÂ®n =
˜
where AA is a representation space of U(l) and B A is a represenation space of
Sn. The index A runs over the irreducible representation spaces of Sn, i.e. over the
Young diagrams. So these Young diagrams also label certain representations AA
of the unitary group. It is a fundamental result that the AA are zero except when
the diagram has 1 rows or fewer ( that is, for partitions with at most 1 terms), in
particular the irreducible representations of U(2) are labelled precisely by the 1 and
2row diagrams.
The coefficients aA(q,z) have the property that they vanish when q = z for all
diagrams A with more than 2 rows. Thus the Ivariable polynomial V(q) uses only
the representations of the Hecke algebra associated to the 1 and 2 row diagrams.
These are the diagrams which label the representations of U(2) and this ties in
with the quantumfield theory approach of Witten, in which the Vpolynomial is
obtained in the framework of a gauge theory with structure group U(2)( or rather
SU(2)). The more general X polynomial is obtained from gauge theory using
structure groups SUe1), for all different ˜lues of 1.
119
Atiyah: Representations of braid groups
(b) Geometric constructions of representations.
It is natural to ask for direct geometric constructions of these representations of
the braid group. We now change our point of view slightly: the braid group B n is
the fundamental group of the configuration space Cn, so linear representations of
B n are equivalent to fiat vector bundles over Cn. Thus we seek flat vector bundles
whose monodromy yields the representations Pq,A, and we pay particular attention
to the 2 row diagrams which appear in the Vpolynomial.
Constructions of these fiat bundles are already known in the context of confor
Inal field theory [4], using complex analysis, and these tie in well with Witten's
quantum field theory interpretation of the Jones' invariants. In her thesis, Ruth
Lawrence developed more elementary constructions which used only standard topo
logical notions, specifically homology theory with twisted coefficients. To describe
her construction we begin with a fundamental example which yields the "Burau"
representation and the Alexander polynomial of a link.
en.
Let X = {Xl' ... ' x n } be a point in the configuration space The complement
R \ X retracts on to a wedge of circles, so its' first homology is zn and there
2
iH an ndimensional family of flat complex line bundles over the complement , i.e
representations 71"1 (R2 \ X) + C*. There is a preferred 1 dimensional family of
representations v q , which send each of the standard generators of 1rl to the same
complex number q. These are preserved by the action of the diffeomeorpmsm group
en.
of R2 \ X, and thus extend to families, as X varies in More precisely, let W n
I)c the space
It is not hard to see that H 1 (Wn ) is Z2, generated by a loop in which the point y
t'llcircies one of the points of X and a loop in which one of the points of X encircles
"'Bother. We consider the Iparameter family of representations vq : '1rl(Wn ) + C*
which map the first generator to q and the second generator to 1. These restrict to
l,ll(˜ representations lJ q over the punctured planes R 2 \ X, regarded as the fibres of
the natural map
We let Lq be the flat complex line bundle over W n associated to the representation
1'1/Â·
((t'call now the following general construction. If f : E ˜ B is a fibration and
/\It is a localcoefficient system over. E then for fixed r and for each b E B we
('all obtain a vector space Vb = Hr(fl(b); M). The spaces Vb fit together to
d.Â·fine a vector bundle over B, and this bundle has a natural flat connection, since
Ilolllology is a homotopy invariant. The monodromy of this connection then gives
Il n˜presentation of 1rl (B), the action on the cohomology of the fibres. We apply
l.his in the situation above with the map p and the coefficient system Â£q (or, more
Atiyah: Representations of braid groups
120
precisely, the sheaf of locally constant sections of the flat bundle L,q) taking r = 1.
en,
with fibres Hl(R2' \ X;.c q ). (Note that we
This gives us a flat bundle over
could consider a 2parameter family of representations of 1rl (Wn ), using the extra
generator for H l (Wn ), but this would give no great gain in generality, since it would
just correspond to taking the tensor product with the Idimensional representations
of the braid group.)
To identify the representation which is obtained in this way we begin by looking at
the twisted cohomology of R 2 \ X. We can replace this punctured plane by a wedge
of n circles, and use the corresponding cellular cochains :
zn,
C1 =
CO = Z ,
with twisted coboundary map h': Co . Cl given by Â«5 = Â«lq),(lq), ... ,(1
qÂ». For q 1= 1 the Idimensional twisted cohomology has dimension n  1. It is not
hard to identify the n  1 dimensional representation of the braid group which this
leads to. It is obtained from a representation on en by restricting to the subspace
of vectors whose entries sum to o. The standard generator qi of En acts on en by
fixing all the basis vectors el, ... en except for ei, ei+l and acting by the matrix
(1 ˜ 6)q
on the subspace spanned by ei, ei+l. The representation on the vectors whose
entries sum to zero is the reduced Burau representation, and this is in fact the
representation P>..,q obtained from the partition n = (n  1) + 1 . (There is some
choice in sign convention in here: the automorphism Gi t+ (1i of Hn(q) switches
rows and columns in our labelling of representations by Young diagrams.) The
representation is clearly a deformation of the reduced permutation representation
of Sn, which is obtained by taking q = 1. The Alexander polynomial appears in the
following way: if (3 is a braid and tPfj is the matrix given by the Burau representation
then
+ q +... +qnl )˜,8(q),
det(l 1/Jfj(qÂ» = (1
where ˜iJ is the Alexander polynomial of the knot /3 in the 3sphere.
Lawrence extends this idea to obtain other representations of the braid group.
The extension involves iteration of the configuration space construction. Let Cn,m
be the space :
= {({Xl, ... , Xn }, {Yl' ... ' Yrn}) E C n X CmlXi ˜ Yj for any i,j }.
Cn,m
There is an obvious fibration Pn,m : Cn,m . Cn . Notice that On,1 is just the
space W n which we considered before, and Pn,l = p. In general we will obtain
representations of the braid group from the twisted cohomology of the fibres of
Pn,mÂ·
121
Atiyah: Representations of braid groups
For m > 1 the group H 1 (Cn,m) is Z3, with generators represented by loops in
which
(1) one of the points Yj encircles one of the points Xi,
(2) one of the points Yj encircles another,
(3) one of the points of Xi encircles another.
For the same reason as before we may restrict attention to representations which
are trivial on the third generator. Thus we consider a 2parameter family of repre
sentations iiq,a of '1rl(Cn ,m), which map the first generator to a and the second to
q. We let L,q,a be the corresponding flat line bundle over Cn,m, then for each a and
q we have a representation Â¢Jm,q,o: of the braid group B n on the middle cohomology
of the fibre:
Hm(p;;,lm(X) ; Â£q,a).
Lawrence proves that these yield all the representations corresponding to 2row
Young diagrams. More precisely, we have:
THEOREM ,[3].
Suppose 2m ˜ n and let ,\ be the representation of the symmetric group corre
sponding to the partition n = (n  m) + m (a 2row Young diagram). If a = q2
t,he representation <Pm,q,o: of B n contains with multiplicity 1 tbe irreducible Hecke
algebra representation Pq,A'
Ilemarks.
(1) This irreducible piece makes up "most " of the space. For example, when
m = 2 the representation Pq,A has dimension (1/2)n(n3) and the remaining
piece has dimension n.
(2) Lawrence also studies the representations <Pm,q,o: when a is not equal to q2.
For generic a and q they are irreducible and she analyses the degeneration
of the representation when a + q2.
(3) We have seen that the 2row Young diagrams covered by the theorem arise
naturally in the Jones theory as the representations associated to the group
8U(2) which appear in the one variable Vpolynomial. It seems very prob
able that the other representations, corresponding to Young diagrams with
more rows and to the groups SUeZ), will be obtained in the same fashion by
considering configuration spaces of 1 disjoint sets of points.
(4) There are striking general similarities between this approach and an idea
proposed by Hitchin for studying the moduli space of rank 2 holomorphic
bundles over a Riemann surface, which comes to the fore in Witten's in
terpretation of the Jones invariants. (See the reports on the contributions
of Witten and Hitchin in these Proceedings.) Hitchin's idea is to represent
rank 2 bundles on a Riemann surface E as the direct images of line bundles
on a fixed branched cover E + E (cf.[l]). The branched cover is speci
fied by a configuration of points in E, and the general effect is to reduce
Atiyah: Representations of braid groups
122
the nonAbelian theory on E to the Abelian theory (of line bundles) on the
cover.
REFERENCES
1. N.J.Hitchin, Stable bundles and integrable systems, Duke Mathematical Jour. 54 (1987),
91114.
2. V.R.F.Jones, Hecke algebra representations of braid groups and link polynomials, Annals of
Mathematics 126 (1987), 335388.
3. R.J.Lawrence, "Homology Representations of Braid Groups," D.Phil. Thesis, Oxford, 1989.
4. A.Tsuchiya and Y.Kanie, Vertex operators in conforma field theory on pI and monodromy
representations of braid groups, Advanced Studies in Pure Maths. 16 (1988), 297372.
PART 3
THREEDIMENSIONAL MANIFOLDS
125
Introduction
At present the study of 3dimensional manifolds is dominated by Thurston's con
jecture that a compact 3maniold can be decomposed into a finite family of sum
mands, which are either bounded by spheres (and can be closed), or bounded by
tori, each summand admitting one of eight geometric structures. And although this
conjecture is far from being proved for example if the universal covering manifold
]{;[3 is diffeomeorphic to S3 , is M 3 elliptic? it does seem to provide the framework
in which to pose interesting questions. In his lecture H. Rubinstein asks to what
extent the rigidity proved by Mostow for hyperbolic manifolds extends to other
geometries. Starting with an irreducible manifold M3 with infinite fundamental
group, his method is to construct a nonpositively curved metric with singularities,
using a restricted type of polyhedral decomposition. This exists, for example, if M3
is a covering of S3 branched along the figure of eight knot with all degrees greater
than or equal to 3. In this situation it is possible to prove that a homotopy equiv
alence f : M + M' (irreducible) can be replaced by a homeomorphism. Note that
we only need to assume that one manifold M admits a suitable polyhedral metric,
and that we do not restrict attention to Haken manifolds. This result shows that
in some sense hyperbolic 3manifolds are generic. Their automorphisms are the
subject of the article by Thomas, in which two proofs are given of the result that an
arbitrary finite group G can be realised as a subgroup of the (finite) isometry group
l(M) of some hyperbolic 3manifold M of finite volume. The motivation of this
paper is the famous result of A. Hurwitz that the order of the group of conformal
a.utomorphisms of a closed Riemann surface of genus 9 ˜ 2 is bounded above by
84(g  1).
'L'he remaining paper in this section, by I{irby and Taylor, studies Pin structures
OIl lowdimensional manifolds. For example they show that, if Pin+ denotes the
usual central extension of Z/2 by O(n) classified by W2, then n.fin+ is cyclic of order
1(3 and generated by the class of RP4. Furthermore the fake projective space of
(˜a.ppell and Shaneson represents Â±9, the !{ummer surface Â±8, and these relations
n: pin
= 0, any oriented 3manifold M 3
Hurvive in the topological category. Since
with a chosen spin structure q bounds a. spin manifold W 4 the signature of which,
l'C\duced modulo 16, gives the classical JLinvariant of the pair (M3, q). I{irby and
'I'aylor now introduce a function f3 : H2 (M; Z/2) + Z/8, related to the symmetric,
trilinear, intersection map T, which up to a factor of 2 detects the difference between
t.lt(˜ Jlinvariants associated to different spin structures. This enables them to give an
f'xtcnsion and geometric interpretation of Turaev's work on T. They also consider
f.1l(˜ bordism of characteristic pairs (A1 4 , F2), where F2 is an embedded surface dual
wi,
to 102 + showing that the group n˜ is isomorphic to Z/8 $ Z/4 (f) Z/2. And in
til( ˜ same way that the Jlinvariant can be defined for pairs (M 3 , q), rather than just
fc ... Z/2homology spheres, they extend Robertello's description of the Arf invariant
uf a knot to a suitable class of links L embedded in M 3 with fixed structure q. It
i˜l clear that this paper is once again the sta.rting point for new developments 
'e)1' example, does the definition of the Casson invariant extend in the same way as
Introduction
126
that of p., and what classes in !lfin+ are represented by the exotic projective spaces
constructed by Fintushel and Stem ?
An Introduction to Polyhedral Metrics of
NonPositive Curvature on 3Manifolds
I. R. AITCHISON
University of Melbourne
J. H. RUBINSTEIN
Institute for Advanced Study and
University of Melbourne
Â§O INTRODUCTION
Polyhedral differential geometry has been an active area of resarch for a long time.
In general relativity it is often called Regge calculus. In the 1960's, work was done
by T. Banchoff and D. Stone. More recently, beautiful results have been obtained
by J. Cheeger [9], M. Gromov [13] and M. Gromov and W. Thurston [14].
The Geometrization Programme of Thurston (see [33], [34] and the survey article
of P. Scott [29]) seeks to classify all closed 3manifolds by dividing them canonically
into pieces, which admit locally symmetric Riemannian metrics called geometries.
There are eight geometries 53, 52 X R, R3, Nil, Solv, PSL(2, R), H2 x R and H3.
Our aim is to introduce polyhedral metrics which are applicable to the geometries
R3, H2 X R and H3 â€¢ In particular, these metrics have nonpositive curvature, in the
sense of polyhedral differential geometry. It is easy to show that only the three ge
ometries indicated admit such metrics. However this is sufficient to describe generic
3manifolds. We quickly review the basic strategy for classifying 3manifolds.
By Kneser [25] and Milnor [28], there is essentially a canonical way of decomposing
allY closed 31nanifold into a finite connected sum, i.e. M = M 1 #M2 # ... #Mk.
P˜a.ch Mi is prime, i.e. if M is expressed as a connected sum, M = P#Q, then
c'ither P or Q is a 3sphere. It is elementary to show that if N is a prime 3manifold,
t.hen either N is a 2sphere bundle over 51 or N is irreducible, meaning that every
('lllbedded 2sphere in N bounds a 3cell. From now on we will always assume
I,hat 3manifolds under consideration are irreducible and orientable, to simplify the
(Iiscussion.
Aitchison & Rubinstein: An introduction to polyhedral metrics
128
Suppose V is a closed, embedded surface in M and V is not a 2sphere or projective
plane. Then V is called incompressible if the map '7rl(V) ˜ 1r}(M) of fundamental
groups induced by the embedding is onetoone. By Dehn's lemma and the loop
theorem [31], an orientable V is incompressible if and only if whenever D is a 2disk
embedded in M with D nV = aD (the boundary of D), then aD is contractible in
V. We will call a 3manifold Haken if it contains such an imcompressible surface.
The characteristic variety of a Haken 3:manifold M comes from the work of W. Jaco
and P. Shalen [23] and also K. Johannson [24]. A Seifert fibered 3manifold N has
a foliation by circles. Each leaf has a foliated neighborhood which is a solid torus
D X SI. The foliation can be described by gluing the ends together of D x [0,1],
where (z,O) is identified with (exp(21riq/p)z, 1). Note that here p, q are relatively
prime positive integers and D is viewed as the unit disk in the complex plane. The
leaves are the unions of finitely many arcs of the form {z} X [0, 1]. Suppose that N ,.:
is a compact orientable Seifert fibered 3manifold and aN is a nonempty collection ˜'
of tori. If N is embedded in M, we say that N is incompressible if every torus of
aN is incompressible in M. ,
)
The characteristic variety Theorem ([23], [24]) can now be sununarized. If M is i
Haken, then either 1˜1 is a Seifert fiber space or M has a maximal incompressible
Seifert fibered submanifold N. N is unique up to isotopy and every map f: N' + M 1
where N' is Seifert fibe˜"ed and I.: 1r1 (N') + 1rl (M) is onetoone, can be homotoped
to have image in N. N is calle<.l the characteristic variety of M.
If M is a Seifert fiber space, then we can call M itself the characteristic variety. ,
Thurston's uniformization theorem [33], [34], [35] then shows that if M is a Haken 3 :
manifold with empty characteristic variety then M has a metric of constant negative
curvature, i.e. is hyperbolic. Also if M is Haken, not Seifert fibered and has a
nonempty characteristic variety N, then M  N has aÂ· complete metric which is
hyperbolic. Also N admits a metric of type H2 X R, or R3 , i.e. of nonpositive
curvature. R. Schoen andÂ·P. Shalen (unpublished) have shown that if N is of type
H2 x R and M  N has finite volume then M admits a Riemannian metric of
nonpOSItIve curvature.
Finally the Geometrization Programme conjectures that if M is irreducible, has
infinite fundamental group and is neither Raken nor Seifert fibered, then M always
should admit a hyperbolic metric. So metrics of nonpositive curvature should occur
on most 3manifolds which are irreducible and have infinite fundamental group.
Our approach is to describe polyhedral metrics with nonpositive curvatures on
Aitchison & Rubinstein: An introduction to polyhedral metrics 129
several large classes of 3manifolds. Our constructions include various branched
coverings over knots and links, Heegaard splittings, groups generated by reflections
(Coxeter groups), surgery on knots and links and singular incolnpressible surfaces.
The last method seems to have special significance. We are able to derive a strong
topological rigidity result. Assume a 3manifold M admits a polyhedral metric of
nonpositive curvature coming from decomposing M into regular Euclidean cubes.
If M' is irreducible and there is a homotopy equivalence between M and M' then
M is homeomorphic to M'.
Note that Mostow rigidity shows that if M and M' are both hyperbolic and ho˜oÂ
topy equivalent then M and M' are isometric. Here we only need to assume one
of the manifolds has a special polyhedral metric of nonpositive curvature. Notice
also that from a purely topological point of view, F. Waldhausen [39] proved that
if M is Haken, M' is irreducible and M is homotopy equivalent to M' then M and
M' are homeomorphic. In our case we do not suppose M or M' is Haken. Since
A. Hatcher [191' has shown most surgeries on links yield nonHaken 3manifolds,
it is most useful to drop the assumption of existence of embedded incompressible
surfaces.
To deal with surgery on knots and links, we discuss also polyhedral metrics of
nonpositive curvature on knot and link complements. One example is given in
detail a two component link in Rp3 (real projective 3space) which is obtained by
identifying the faces of a single regular ideal cube in H3. We show how this link lifts
to a simple four component link in S3. Also by branched coverings we get an infinite
collection of examples formed from gluing cubes with some ideal vertices. We sketch
the interesting result that nearly all surgeries on all components of such links give
amanifolds which satisfy the conclusions of the topological rigidity theorem.
fn this paper, our aim is to give an overview of this subject. Arguments are only
summarized. For. more details, the reader is referred to [1], [2], [3]. In the final
section, we give some extremely optimistic conjectures. It would certainly be pos
sible to study polyhedral metrics adapted to the other five geometries. Especially
ill view of the work already done by R. Hamilton [15], [16] and P. Scott [30], this
(loes not appear to be quite as significant as the case of nonpositive curvature.
We would like to thank J. Hass, C. Hodgson, P. Scott and G. Swarup for many
Il(˜lpful conversations.
!il POLYHEDRAL METRICS OF NONPOSITIVE CURVATURE
We begin by discussing examples of such metrics in,. dimension two, Le. on surfaces.
Aitchison & Rubinstein: An introduction to polyhedral metrics
130
The boundary of the regular Euclidean cube defines a polyhedral metric on the
2sphere. At each vertex, the three squares give total dihedral angle of 311"/2. If we
attribute a positive curvature of 1r/2 at each vertex, then the total curvature is 41r,
as given by GaussBonnet. Similarly if a flat torus is formed as usual by identifying
opposite edges of a regular square, then the dihedral angle at the vertex is 21r and
so the vertex has zero curvature. Finally if an octagon has edges identified by the
word aba1b1cdc1d 1, in cyclic order, the result is a closed orientable surface of
genus two. We find it very useful to ascribe dihedral angles of 11"/2 at each vertex
of the octagon. There is a natural way to do this, so that the metric is Euclidean
except at the center of the octagon. Join the midpoint of each edge to the center
of the octagon. This divides the polygon into eight quadrilaterals. We view each
of the latter as a regular Euclidean square. The result is the dihedral angle at the
center is 47r, so there is curvature of  21r there. Similarly after identification, the
single vertex has dihedral angle 47r and curvature 21r as well. Again the total
curvature agrees with GaussBonnet.
Note that a key feature of such metrics is that geodesics diverge at least linearly.
If a line segment meets a point p where the dihedral angle is 211" + k, for k > 0,
then it is easy to see that all possible geodesic extensions form a cone subtending
an angle k at p. So if two points move out along geodesic rays emanating from a
single point, then the points travel away from each other at least at linear speed. In
fact, working in the universal cover of the octagon surface described above, we see
that such geodesic rays actually diverge exponentially. (See also M. Gromov [13].)
Definition. A polyhedral metric of nonpositive curvature on a closed orientable
surface is a metric which is locally Euclidean except at a finite number of points,
where the dihedral angle is greater than 21r.
Suppose a closed orientable ndimensional manifold M is formed by gluing together
the codimension one faces of a finite collection of compact Euclidean polyhedra
˜l, ˜2,." ,!J t . Assume that the face identifications are achieved by Euclidean
isometries. If Q is an rdimensional face of ˜i, let Bnr be a small ball of dimension
n  r which is orthogonal to Q and is centered at a point x in int Q. We call oBnr
the link of Q and denote it by lk(Q). Clearly the metric on the (n  r  I)sphere
lk(Q) changes only by a homothety if we vary x and the size of Bnr. At any
point of lk(Q) n int ˜i or lk(Q) n int F, where F is a codimension one face of Ei,
it is easy to see that lk(Q) is locally spherical. So lk(Q) is locally spherical away
from a codimension two cOlnplex. We again require geodesic rays which meet Q
Aitchison & Rubinstein: An introduction to polyhedral metrics 131
orthogonally to diverge as in the surface case.
Definition. M has a polyhedral metric with non.. positive curvature if every closed
embedded geodesic loop in lk(Q) has length at least 211", for every face Q, assuming
that the metric on lk(Q) is scaled to have curvature one at the locally spherical
points.
Tlteorem (CartanHadamard). If M has a polyhedral metric with nonpositive
curvature, then the universal cover of M is diffeomorphic to Euclidean space.
In [7], manifolds with such metrics are called CartanHadamard spaces. Note that
if a 3manifold has a nonpositively curved polyhedral metric then it is irreducible,
as can be seen by lifting an embedded 2sphere to the universal cover.
To finish this section we give two important methods for presenting 3manifolds
with these metrics. This viewpoint works well in all dimensions but is particularly
useful in dimension three.
I. Cubings witlt nOltpositive curvature. Suppose M is obtained by gluing
together faces of a finite collection of regular Euclidean cubes, all with the same
edge length. There are two conditions to achieve a polyhedral metric of nonpositive
eurvature on M from such a cubing:
(a) Each edge must belong to at least four cubes. (This is equivalent to the link
of the edge has length at least 271", as each cube contributes 71"/2.)
(b) Let F, F',F" be three faces of cubes at a vertex v. Assume the faces have
edges ei, ei, ei' respectively at v, for i = 1,2. Finally suppose all edges
are oriented with v as initial point and for (i)(iii) below, identifications are
orientationpreserving. Then we exclude the following three types of gluings
of edges.
(i) el is identified to e2.
e˜ e˜).
(ii) el (resp. e2) is identified to (resp.
(iii) el (resp. e˜, e˜) is identified to e˜ (resp. e˜, e2)'
Itcmarks. 1) Condition (b)(iii) above does not apply if F, F', F" are all faces of
a single cube and el = e˜, e˜ = e˜, e˜' = e2.
132 Aitchison & Rubinstein: An introduction to polyhedral metries
2) It turns out that an embedded closed geodesic of length less than 21r can
only arise on Ik(v) if one of the gluings (i), (ii), (iii) occurs.
II. Generalized cubings with nOllpositive curvature. There is an obvious
rotation of order three about the diagonal of a cube. The quotient space (orbifold)
is homeomoprhic to a ball and is locally Euclidean, except along the diagonal,
where the dihedral angle is clearly 21r/3. Suppose we now take a cyclic branched
cover of this orbifold, with branch set the diagonal and having degree d ˜ 3. The
resulting space is called a flying saucer. It can be viewed as having top and bottom
hemispheres, dividing up the 2sphere boundary into two disks. Each disk consists
of d Euclidean squares with a common vertex (the two ends of the diagonal). For
example, if d = 4 then each hemisphere has the same induced metric as for the
octagon at the beginning of this section. If d = 3 we get a cube back. Note that
all the dihedral angles at the edges of a flying saucer are still 1t" /2 and all faces are
squares.
We can now define a generalized cubing with nonpositive curvature exactly as for :'1
cubings. The conditions which are needed f9r the gluing of faces are identical. Here ˜
M is constructed by attaching together faces of a finite number of flying saucers, .:;
where all edges have the same length. Note again that some of these flying saucers
could be cubes.
Â§2 CONSTRUCTING 3MANIFOLDS WITH NICE METRICS
Our first result is a general criterion for constructing polyhedral metrics of non .'
positive curvature by gluing together a finite set of compact 3dimensional Euclidean :˜
polyhedra ˜1, ˜2, â€¢â€¢â€¢ ,E t â€¢ Assume that for each Ej, no loop C on a˜i crosses at ':'
most three edges at one point each, unless C bounds a disk on oE i containing a
vertex of degree three and C meets the three edges ending at v.
identifying˜
TheorelD 1. Suppose a closed orientable 3manifold M is obtained by
faces ofE 1 ,E 2 , â€¢â€¢. ,E t . Assume:
(i) Each face of Ei has at least four edges.
(ii) Each edge in M belongs to at least four of the Ei.
(iii) Let v be a vertex of M and F, F', F" any three faces at v. Suppose th˜˜
same gluings of edges of these faces, as in I(b) of the previous section, are:
excluded. Note that of course F, F', F" could be all the faces of some E i at
v. Then M has a polyhedral metric of nanpositive curvature.
Aitchison & Rubinstein: An introduction to polyhedral metrics 133
Remarks. This result is proved by dividing each Ei into cubes (respectively flying
saucers) if every vertex of Ei has degree three (resp. degree three or greater). We
illustrate this with the example of the WeberSeifert hyperbolic dodecahedral space
[40].
Example. Construct a regular hyperbolic dodecahedron with all dihedral angles
21r/5. Identify opposite pairs of faces by rotation of 31r/5. Then all edges have
degree five in the resulting manifold M which thus has a hyperbolic metric.
Now the dodecahedron can be decomposed into twenty cubes as follows. Join the
center of each face to the midpoints of all edges of the face. This divides each
pentagonal face into five quadrilaterals (squares  c.f. the octagon example in
Â§1). Now join the center of each face to the center of the dodecahedron. The
neighborhood of every original vertex on the boundary of the dodecahedron is three
squares. when these three squares are "coned" to the center of the dodecahedron,
the result is a cube. It is easy to check that every edge of these cubes has degree
four or five in M, checking lea) in Â§1. On the other hand, there is a unique vertex
in M coming from the vertices of the dodecahedron. This vertex has a link in M
which has the structure of an icosahedron (the dual tesselation to the dodecahedral
tesselation of H3 ). It is easy to check that the other vertices have links which
are either octahedral or are formed from ten triangles arranged like two pentagons
joined along their boundaries. Therefore I(b) follows immediately and M has a
polyhedral metric of nonpositive curvature.
'fhe next construction of nice metrics can be viewed as dual to Theorem 1.
As is wellknown, every closed orientable 3manifold M can be constructed by identi
f'ying the boundaries of two 3dimensional handlebodies, by an orientationreversing
homeomorphism. The genu3 is the number of handles and such a decomposition
is called a Heegaard splitting of M. There is no procedure known for constructing
hyperbolic metrics from such splittings.
Il(˜fore stating the result, we need to define Heegaard diagrams. Assume M = YUY',
\vhere Y, Y' are handlebodies of genus 9 with BY = BY' = L, a closed orientable
Hurface. let D 1 , D 2 , â€¢ â€¢â€¢ , D m (resp. Di, ... ,D˜) be a collection of disjoint meridian
di./{k3 for Y (resp. Y ' ). This means that Di (resp. Dj) is properly embedded in Y
(n'Hp. V') with aDi (resp. aDj) a noncontractible loop in BY (resp. aY' = BY =
/1). We say that the collection D t , D 2 , â€¢â€¢â€¢ ,D m is full if int Y  D 1 ..â€¢ D m is a set of
4 .l)(˜n 3cells. The meridia.n disks will never be chosen so that Di and Dk are parallel
Aitchison & Rubinstein: An introduction to polyhedral metrics
134
in Y, for i =1= k. Finally denote aDi by Ci, aDj by Cj and let C = {el , C2 , â€¢â€¢â€¢ ,em},
c' = {C˜,C˜, ... ,C˜}, v= {D 1 ,D2 , â€¢â€¢â€¢ ,Dm }, 1)' = {D˜,D˜, ... ,D˜}.
Definition. The triple (L, C, C') is called a Heegaard diagram for M if both collec
tions of meridian disks V and V' are full.
Obviously M can be assembled from the Heegaard diagram by attaching 2handles
to L along C and C', then filling in by 3handles. Without loss of generality, assume
C and C' are isotoped to be transverse and to have minimal intersection, Le. no pair
of arcs in C and C' have common endpoints and bound a 2gon in L.
Suppose R is the closure of a component of L  C  C'. We call R a region of the
Heegaard diagram. A boundary component of R can be viewed as a polygon, with '
an equal number of arcs coming from C and C'.
Theorem 2. Suppose a Heegaard diagram (L,C,C') for M satisnes the following
conditions:
(a) Every curve Ci (resp. Cj) meets C' (resp. C) at least four times. Moreover,
every noncontractible loop on L must intersect CUC' in at least four points.
(b) Every region of the Heegaard diagram is a disk and has at least six boundary ;1",'
edges.
,1
1
f
Then M has a polyhedral metric of nonpositive curvature. Moreover M has a
.˜
cubing (resp. generalized cubing) if all regions are hexagons and every curve Cj
meets C exactly in four points (resp. regions have six or more edges and OJ nc has
at least four points).
Remarks. Note that we are not assuming that any region R is an embedded disk. :˜
So an edge may occur twice in the boundary of R. Also splittings as in the theorem
are often reducible, i.e. have trivial handles. So there can be curves Ci and OJ in'
the diagram which cross exactly once.
Examples. 1) Heegaard diagra.ms corresponding to cubings come from torsion
free subgroups of finite index in the Coxeter group constructed from the
tesselation of the hyperbolic plane by the regular rightangled hyperbolic
hexagon. Note that such subgroups yield similar tHings of Riemann surfaces
by rightangled hexagons, which is condition (b) in the theorem, in the case
of a cubing. For more information, see also [5].
Aitchison & Rubinstein: An introduction to polyhedral metrics 135
2) Generally, suppose as in Theorem 1 that M is formed by identifying faces
of E 1 , E 2 , â€¢ .â€¢ ,Et â€¢ Let us truncate each Ei by chopping off a piece at each
vertex v by a plane chosen to pass through each edge e at v at 3/4 of the
distance along e away from v. Then we obtain a truncated polyhedron E as i
illustrated in Figure 1 for the cube.
\.
L
\.
\.
\.
\.
\.
\.
˜
Figure 1
'I'Itis shows how to obtain the surface L (the hexagonal faces in Figure 1). y' is
the union of the truncated polyhedra E and Y is the closure of M  Y ' . The disks
i
r)1 are the quadrilateral faces in Figure 1 and the disks V are dual to the edges
uf the polyhedra Ei. Moreover, to go from the Heegaard diagram to the cubing
Ei
f H' generalized cubing, we merely reverse the process of truncation, i.e. expand
IJnek to Ei.
t\ (lother standard method of constructing 3manifolds is via branched coverings. W.
'l'hllrston [37] has introduced the concept of a universal knot or link Â£, in 8 3 , which
I˜l disjoint union of embedded circles so that every closed orientable 3manifold is
iiolllC branched cover of S3 over Â£. H. Hilden, M. Lozano and J. Montesinos [20],
I:˜ II have shown that various knots and links are universal. Among these are the
\/Vllit.(˜head link, Borromean rings, Figure 8 knot and the 52 knot.
Aitchison & Rubinstein: An introduction to polyhedral metrics
136
I':,"
3
Theorem 3. Suppose [, is a link in 8 and a 3manifold M is constructed which
is a branched coverings of S3 over Â£', with all components of the branch set having
degree at least d. Then M has a poly˜edra1 metric of nonpositive curvature if [,
is the Whitehead link or 52 knot and d = 4, Â£, is the Figure 8 knot and d = 3
or Â£, is the Borromean rings and d = 2. Moreover if Â£, is the Whitehead link, 52
knot or Borromean rings (resp. Figure 8 knot) then all such M have cubings (resp. }.
generalized cubings) as in Â§1.
Relnarks. We sketch the argument for the Whitehead link t. The hyperbolic
dodecahedral space M has a cyclic group action of order five given by rotation of '
the dodecahedron about an axis through the centers of opposite faces. This is shown
in [40] to exhibit M as a 5fold cyclic branched cover of 53 over Â£. Clearly this t˜!. ,:;
action permutes the twenty cubes of M described previously. So we see that 53
1
can be built from four cubes. Also.c is contained in the edges of these cubes and ˜J
each edge in Â£, has dihedral angle 1r /2. All other edges in 53 still have degree four t
or five. We conclude that any branched cover of 8 3 over Â£, where all components .˜
have degree at least four, yields a cubing with nonpositive curvature.
It is also sometimes useful to construct metrics by gluing together hyperbolic poly
hedra and also to look at branched coverings for polyhedral hyperbolic metrics.
Suppose [, is a simple link which is not a 2bridge link, torus link or Montesinos
link. (Here we include knots as examples of links.) Also assume that if an embedded
2sphere S meets [, in four points, then S bounds a 3cell B with Â£, n B consisting
of two unknotted arcs. Then by the orbifold theorem of W. Thurston [36] (see also
[22]), the 2fold branched cover M of S3 over [, has a hyperbolic metric and the
covering transfonnation is an isometry. Projecting to 53, we conclude that 8 3 has
a polyhedral hyperbolic metric with singular set {, and dihedral angle 7r along Â£.
Using the smoothing technique of [14], we easily deduce the following result.
Theorem 4. Any branched cover of S3 over Â£, where all compo17ents of the branch
set have degree at least two, has a Riemannian metric with strictly negative curva
ture.
Corollary. Any such a branched cover has universal cover R3 , by the Cartan
Hadamard theorem.
An important class of hyperbolic 3manifolds are surface bundles over the circle with
pseudoAnosov monodromy (see [38]). We describe two closely related constructions
of such bundles with polyhedral metrics of nonpositive curvature.
Aitchison & Rubinstein: An introduction to polyhedral metrics 137
The first is given in the article of D. Sullivan [32] on Thurston's work. The 2fold
cyclic branched cover of 8 3 over the Borromean rings Â£ is a flat 3manifold M. In
Figure 2, 8 3 is formed by folding a cube along six edges bisecting the faces. This
gives a polyhedral metric which is flat except along Â£, where the dihedral angle is
'Jr. The fiat 3manifold M has a fibering as a 2torus bundle over a circle. The fibers
are easily seen to be transverse to the preimage Â£ of Â£.
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