Euclidean simplices and invariants of three-manifolds: a modification of the invariant for lens spaces

0.25 MB·English

by  Evgeniy V. Martyushev

EUCLIDEAN SIMPLICES AND INVARIANTS OF
THREE-MANIFOLDS: A MODIFICATION OF THE INVARIANT
7
0 FOR LENS SPACES
0
2
EVGENIYV.MARTYUSHEV
n
a
J Abstract. We propose a three-manifold invariant based on the use of Eu-

5 clidean metric values ascribed to the elements of manifold triangulation. We
1 thus obtain a nontrivial invariant that can, in particular, distinguish non-
homeomorphiclensspaces.

]
T
A
. Introduction
h
t
a It is well knownthat any two triangulationsofa piecewise-linearthree-manifold
m canbe transformedintoeachotherusingthePachnermoves[6]. Ifweconstructan
[ algebraicexpressiondependingonsomevaluesascribedtoamanifoldtriangulation

andinvariantunderthesemoves,wegetavaluethatisindependentofthechoiceof
2
triangulation. ItisnaturalwaytoconstructaPL-manifoldinvariant. Forexample,
v
8 Euler characteristic and Turaev-Virotype invariants can be provedto be invariant
1 using exactly this method.
0 In this paper we present a new invariant of 3-dimensional PL-manifold based
2
on Pachner’s theorem. Its construction is naturally divided into three main parts.
1
First, on a given representation of the fundamental group we build a covering of a
2
3-manifold corresponding to the kernel of this representation. Then, we map the
0
/ covering space into 3-dimensional Euclidean space. In the last, algebraic part, we
h
build an acyclic complex of Euclidean geometric origin. The invariant of all the
t
a Pachner moves is expressed in terms of the torsion of that complex.
m
Such invariant was first constructed by I.G. Korepanov in papers [2, 4] and
: calculated for some 3-manifolds, in particular, for a lens space L(p,q). Recall that
v
i attheendofpaper[4]weproposedageneralformulaforourinvariantforL(p,q)[4,
X
formula (4.1)]. This formula gives, essentially, the squared Reidemeister torsion of
r L(p,q). The main result of this paper is a proof of this formula (theorem 3.2).
a
The paper is organized as follows. In section 1 we prove an auxiliary theorem
thatwill be used to provethe acyclicity ofour complex (theorem1.1). In section2
we define an acyclic complex (theorem 2.4) and prove that its torsion, divided
by some value, is a topological invariant, i.e., it is independent of the Pachner
movesand some details ofits construction(theorem 2.6). In section3 we calculate
our invariant for lens space L(p,q) (theorem 3.2) and show that for this case our
invariantisrelatedtotheReidemeistertorsion(remark3.2). Finally,insection4we
defineandcalculateamodifiedversionofourinvariantforlensspaces(theorem4.1).
The modification uses a nontrivial action of the fundamental group on the acyclic
complex.
ThisworkhasbeenpartiallysupportedbyRussianFoundationforBasicResearch,Grantno.
02-01-06003.
1
2 EVGENIYV.MARTYUSHEV
Acknowledgements. I am grateful to I.G. Korepanov for proposing me this
problem and numerous helpful discussions.
1. Admissible colorings and the map Γ: T˜ R3
→
Let M be a connected closed orientable 3-dimensional manifold. Let T be its
triangulation, i.e. a set ∆ of pairwise disjoint tetrahedra, together with a family
of homeomorphisms, Φ, mapping the set of tetrahedra faces to itself, so that the
identification space ∆/Φ is homeomorphic to M. Note that this definition allows
us to consider non-combinatorialmanifold triangulations, that is, a 3-simplex in it
maynotbedetermineduniquelybythesetofitsvertices. Moreover,simplexofany
dimensionisallowedtoenterseveraltimesintotheboundaryofhigherdimensional
simplex.
As is well known, for each subgroup H of the fundamental group π (M), there
1
existssuchacoveringp: M˜ M thattheinducedhomomorphismp : π (M˜) H
∗ 1
→ →
isisomorphism. Acoveringcorrespondingtothetrivialsubgroupiscalleduniversal.
The universal covering space M˜ is a simply connected orientable 3-manifold. The
fundamental group π (M) acts on M˜ freely and transitively and M˜/π (M)=M.
1 1 ∼
Simplicial structure T of manifold M naturally induces a simplicial structure T˜
of M˜: every simplex from T can be lifted to T˜ (in general, in many different ways
according to the action of π (M)).
1
Let us ascribe a real number λ > 0 to every edge e T, connecting vertices
ij ij
∈
i and j, so that the following inequality holds for every tetrahedron of complex T
with edges (e ,e ,e ,e ,e ,e ):
ij ik il jk jl kl
0 1 1 1 1
1 0 λ2 λ2 λ2
(cid:12) ij ik il(cid:12)
(cid:12)1 λ2 0 λ2 λ2 (cid:12)>0. (1)
(cid:12) ij jk jl(cid:12)
(cid:12)1 λ2 λ2 0 λ2 (cid:12)
(cid:12) ik jk kl(cid:12)
(cid:12)1 λ2 λ2 λ2 0 (cid:12)
(cid:12) il jl kl (cid:12)
(cid:12) (cid:12)
Recall that this condition(cid:12)guarantees an existenc(cid:12)e of non-degenerate Euclidean
(cid:12) (cid:12)
tetrahedron with edge lengths (λ ,λ ,λ ,λ ,λ ,λ ) and this tetrahedron is
ij ik il jk jl kl
unique up to isometries, cf. [1].
Further, let us ascribe a sign + or to every tetrahedron τ T. For each edge
− ∈
a, belonging to ith tetrahedron of complex T, we denote by ϕ(i) its inner dihedral
a
angle at this edge, taken with the sign ascribed to ith tetrahedron. This angle is a
function of λ .
...
Definition 1.1. Defect angle ω at an edge a is a quantity
a
ω = ϕ(i) mod 2π,
a − a
i
X
where the sum is taken over all tetrahedra sharing the edge a.
Letussaythatwearegivenanadmissible coloring onT,iftoeveryedgee T
ij
∈
a number λ is ascribed so that the condition (1) holds, and to every tetrahedron
ij
from T a sign is ascribed so that the defect angle at every edge vanishes.
Note that an admissible coloring on T induces an admissible coloring on T˜ in a
naturalway: to everyedge from the preimage of some e T we ascribe the same
ij
∈
number λ , and to every tetrahedron from the preimage of some τ T we ascribe
ij
∈
the same sign as to τ.
EUCLIDEAN SIMPLICES AND INVARIANTS OF THREE-MANIFOLDS 3
Theorem 1.1. Let T˜ be a simplicial complex for the universal covering space M˜.
Let an admissible coloring be given on T˜. Then, there exists a continuous map
Γ: T˜ R3 such that l = λ , i,j, where l is the length of Γ(e ). Any other
ij ij ij ij
→ ∀
map Γ′: T˜ R3 for the same admissible coloring can be obtained from Γ by an
orientation→preserving isometry of R3.
Proof. Let τ be a certain tetrahedron of T˜ and let (λ ,λ ,λ ,λ ,λ ,λ ) be
1 12 13 14 23 24 34
anadmissiblecoloringofitsedges. DenotebyΓ : τ R3 anarbitraryembedding
1 1
of this tetrahedron in R3 such that l = λ . Let τ→be an adjacent tetrahedron,
ij ij 2
i.e. τ τ is 2-dimensionalsimplex. Then, there existsanembedding Γ : τ R3
1 2 2 2
∩ →
such that Γ = Γ on τ τ . This embedding is unique: the tetrahedra Γ (τ )
2 1 2 1 1 1
∩
andΓ (τ )areonthe samesideoronthe differentsidesfromtheircommon2-face,
2 2
if the signs of τ and τ are the same or different respectively. We say that Γ
1 2 2
continue Γ by means of τ .
1 2
Let us take an arbitrarytetrahedron τ T˜ and consider a sequence of pairwise
i
∈
adjacent tetrahedra starting from τ , that is, a sequence s = (τ ,...,τ ), where
1 i 1 i
τ τ isa2-simplexforeachconsecutivepair(τ ,τ )andforallj 1,...,i .
j j+1 j j+1
Fin∩ally, define Γ : τ R3 as a continuation of Γ by means of the se∈qu{ence s .}
si i → 1 i
Lemma 1.2. The map Γ is independent of a sequence s , i.e., if s′ is any other
si i i
sequence of adjacent tetrahedra joining τ1 and τi, then Γsi =Γs′i =Γi.
Proof. Wehaveasequences connectingτ andτ . Itcanbegeometricallyimagined
i 1 i
as a broken line whose every straight segment connects the barycenters of two
adjacent tetrahedra. We must show that if s′ is any other sequence connecting τ
i 1
and τ , then Γ = Γ′, where Γ′ : τ R is a continuation of Γ by means of a
i 1 1 1 1 → 3 1
closed path s (s′)−1.
i◦ i
RecallthatwearegivenanadmissiblecoloringonT˜. Itfollowsthatallthedefect
angles are zeros. Let n tetrahedra τ ,...,τ share an only edge e, where τ and
k1 kn ki
τ share a face for all i. Denote by α a closed path (τ ,...,τ ,τ ). Since the
ki+1 k1 kn k1
defect angle at e is zero, then Γ =Γ′ , where Γ′ : τ R is a continuation of
k1 k1 k1 k1 → 3
Γ by means of α.
k1
Using Seifert-Van Kampen theorem, one canshow that a closed path s (s′)−1
mayberepresentedasafiniteproductofpathslikeα. Hencelemma1.2follio◦ws.i (cid:3)
Thus, for each tetrahedron τ T˜ we have an embedding Γ : τ R3 so that
i i i
two embeddings Γ and Γ agree o∈n τ τ , i,j. Now we define Γ: T˜→ R3 by
i j i j
∩ ∀ →
Γ(x)=Γ (x) x τ , i.
i i
⇐⇒ ∈ ∀
In order to complete the proof of theorem 1.1, we need the following well-known
result from general topology.
Lemma 1.3. Let X = A B, where A and B are the closed subspaces of X. Let
∪
f: A Y and g: B Y be continuous. If f = g on A B, then a function h
→ → ∩
defined by
f(x), x A,
h(x)= ∈
(g(x), x B,
∈
is continuous.
Lemma 1.3 and definition of Γ imply that it is continuous.
4 EVGENIYV.MARTYUSHEV
The last statement of the theorem concerns the arbitrariness in the choice of
embedding Γ . Let Γ′ be another embedding of τ in R3. Denote by E(3) the
1 1 1
groupoforientationpreservingmotionsof3-dimensionalEuclideanspaceR3. Then,
there is an element h E(3) such that Γ′(τ ) = hΓ (τ ). It follows from the
∈ 1 1 1 1
construction of Γ that Γ′(T˜) = hΓ(T˜). Hence, Γ is independent of a choice of the
first tetrahedron τ up to multiplying by an element of E(3). This completes the
1
proof of theorem 1.1. (cid:3)
Let τ˜ and τ˜′ be preimages of an only tetrahedron τ T with respect to the
∈
coveringprojection. Then, there is anelement g π (M) such that τ˜′ =gτ˜. Since
1
∈
the edge lengths of τ˜ and τ˜′ are the same, there exists an element h E(3) such
∈
that Γ(τ˜′) = hΓ(τ˜). Thus we have a mapping ρ: g h. Obviously, ρ(g g ) =
1 2
7→
ρ(g )ρ(g ), hence ρ: π (M) E(3) is a homomorphism.
1 2 1
→
Denote Γ′ = h Γ, where h E(3). Then, ρ′ = h−1 ρ h, where ρ′ is a
◦ ∈ ◦ ◦
representationcorrespondingtoΓ′. Recallthatthe representationsρ,ρ′: π (M)
1
→
E(3) are called equivalent in this case. So, the following proposition holds.
Proposition 1.4. For any admissible coloring of T˜, there is a unique class [ρ] of
equivalent representations π (M) in E(3).
1
Conversely, given a representation
ρ: π (M) E(3), (2)
1
→
one can construct a continuous map Γ : T˜ R3 as follows.
ρ
→
First of all, note that all the tetrahedra in the triangulation of orientable space
M˜ canbe orientedconsistently,i.e.,foreverytetrahedron,wecanorderits vertices
up to even permutations. For example, we suppose the vertex orderings ABCD
and EABC of two adjacent tetrahedra to be consistent.
Definition 1.2. A fundamental family of simplices in T˜ is such a family of
simplices of T˜ that over each simplex of T lies exactly one simplex of this famFily.
Let us fix a consistent orientation on the set of all tetrahedra from T˜ and also
the fundamental family for T˜.
F
LetvbeavertexbelongingtoT andletv˜ beitspreimageinthefundamental
family. Then we place the orbit of v˜, i.e. the∈sFet gv˜ g π (M) , in R3 so that if
1
{ | ∈ }
v˜ = gv˜ , where g π (M), then Γ (v˜ ) = ρ(g)Γ (v˜ ). Thus, to each vertex of T
2 1 1 ρ 2 ρ 1
we assigna set ofp∈oints in R3 whose coordinatesarerelatedby the motions ofR3.
Remark 1.1. The verticesfrom aremappedinR3 inarbitraryway. However,we
requirethe configurationofthesFe vertices in R3 to obey the following conditions of
general position:
the volumes of all tetrahedra from Γ( ) are nonzero;
• F
the rank of ∂ωi possesses the maximal value at the point ω = 0, i;
• ∂lj i ∀
here, ωi is th(cid:16)e def(cid:17)ect angle at ith edge, lj is the length of jth edge, i and j
run over all the edges from .
F
Under Γ , each simplex σk T˜ of nonzero dimension k is mapped into the
ρ
convexlinear shellof its vertex i∈magesin R3. Thatis, if v , ..., v are the vertices
0 k
k
of σk, then Γ (σk)= m Γ (v ), where m +...+m =1 and m 0, i.
ρ i ρ i 0 k i
≥ ∀
i=0
P
EUCLIDEAN SIMPLICES AND INVARIANTS OF THREE-MANIFOLDS 5
Hence,everyedgee T˜isassignedthelengthofΓ (e) R3,andevery3-simplex
ρ
∈ ∈
τ T˜ is assignedthe orientedvolume of Γ (τ). Similarly, we can also speak about
ρ
∈
dihedral angles between 2-simplices in the triangulation.
Under Γ , the orientation of a tetrahedron either coincides with the orientation
ρ
of R3 or does not. In the first case we take the volume of the tetrahedron and all
its dihedral angles with the sign +, in the second case with the sign .
Thus, given representation ρ, we have found an admissible colori−ng on T˜: for
everyedgeweascribethelengthofitsimageinR3,foreverytetrahedronweascribe
the sign of its volume in R3.
Before constructing the invariant, we need to recall some formulas of Euclidean
geometry.
1.1. FormulasofEuclideangeometry. Wearegoingtowriteoutsomeformulas
forderivatives ∂ωa,takenunderconditionthatthelengthsofalledges,exceptb,are
∂lb
fixed. These formulas can be easily proved by direct calculation using definitions
and elementary geometric relations like cosine rule, see [4] for details.
Clearly,we obtain nonzeroderivativesonly if a andb lie ina single tetrahedron.
Consider several cases.
1st case: Edges a=AB and b=CD are skew in a tetrahedron ABCD and
there is no other tetrahedron in the triangulation containing both edges a
and b:
∂ω l l
AB AB CD
= , (3)
∂l −V
CD ABCD
where V is the volume of ABCD multiplied by 6.
ABCD
2nd case: Edgesa=AB b=BC belongtothecommonfaceoftwoadjacent
tetrahedra and, again, there is no other tetrahedron in the triangulation
containing both edges a and b:
∂ω V
AB CAED
=l l . (4)
AB BC
∂l V V
BC ABCD EABC
3rd case: Edgea=b=DE is commonforexactly threetetrahedraABED,
BCED andCAED, andnooneofthesetetrahedracontainthis edgemore
than one time:
∂ω V V
DE = l2 ABCD EABC . (5)
∂l − DE V V V
DE ABED BCED CAED
4th case: Edgea=b=DE iscommonfor>3tetrahedra,andagainnoone
of these tetrahedra contain this edge more than one time.
We write out this formula only for the case of four tetrahedra (the gen-
eralization for greater number of tetrahedra is obvious):
∂ω l2 V V V V
DE = DE ABCD EABC + AFBD EAFB . (6)
∂l − 6 V V V V V V
DE (cid:18) ABED BCED CAED AFED FBED BAED(cid:19)
Here, the right-handside consistsoftwo terms: the firstone is the sameas
in (5), and the second one is obtained by replacing B F and C B.
→ →
5th case: Actually, our definition of triangulation admits such gluings be-
tween faces of tetrahedrawhere different combinationsof abovementioned
cases appear. Namely, the edge b may
lie opposite to edge a (1st case) in more than one tetrahedron;
•
6 EVGENIYV.MARTYUSHEV
belongtothesame2-dimensionalfaceasa(2ndcase),andagainthere
•
may be more than one such faces;
coincide with a (as in the 3rd and 4th case),
•
and these possibilities (as examples show) do not exclude one another.
Thus, there may be several “ways of influence” of dl on dω . Clearly, the
b a
terms corresponding to these ways sum together.
2. Acyclic complex and invariant of 3-manifold
2.1. Generalities on acyclic complexes and their torsions. We shortly re-
mind basic definitions from the theory of algebraic complexes, see [8] for details.
LetC ,C ,...,C befinite-dimensionalR-vectorspaces. Wesupposethateach
0 1 n
C is based, that is has distinguished basis. Then, linear mapping f : C C
i i i+1 i
→
can be identified with matrix.
Definition 2.1. The sequence of vector spaces and linear mappings
C =(0 C fn−1 C ... C f0 C 0) (7)
n n−1 1 0
−→ −−−→ −→ −→ −→ −→
is called a complex if Imf Kerf for all i = 1,...,n 1. This condition is
i i−1
⊂ −
equivalent to f f =0 for all i.
i−1 i
Definition 2.2. The space H (C) = Kerf /Imf is called the ith homology of
i i−1 i
the complex C.
Definition 2.3. The complex C is said to be acyclic if H (C) = 0 for all i. This
i
condition is equivalent to rankf =dimC rankf for all i.
i−1 i i
−
Suppose that the sequence (7) is an acyclic complex. Let be an ordered set
i
C
of basis vectors in C and let be a subset of basis vectors belonging to the
i i i
B ⊂C
space Imf .
i
Denoteby f anondegeneratetransitionmatrixfromthebasisinspaceCokerf =
Bi i i+1
C /Imf to the basisinspaceImf . Byacyclicity,suchamatrix reallyexists.
i+1 i+1 i
Hence, f is a principal minor of the matrix f obtained by striking out the rows
Bi i i
correspondingtovectorsof andthecolumnscorrespondingtovectorsof .
i+1 i i
B C \B
Definition 2.4. A quantity
n−1
τ(C)= (det f )(−1)i+1 (8)
Bi i
i=0
Y
is called the torsion of acyclic complex C.
Theorem 2.1 ([8]). Up to a sign, τ(C) does not depend on the choice of subsets
.
i
B
Remark 2.1. The torsion τ(C) does depend on the distinguished basis of C . If
i
one performs change-of-basistransformationin every space C with nondegenerate
i
matrix A , then the torsion τ(C) is multiplied by
i
n
(detA )(−1)i+1.
i
i=0
Y
From now on we suppose the fundamental group π (M) to be finite. Since any
1
real representation of a finite group is equivalent to the orthogonal one, then Imρ
is actually a subgroup in SO(3).
EUCLIDEAN SIMPLICES AND INVARIANTS OF THREE-MANIFOLDS 7
Definition 2.5. For a given representation ρ: π (M) E(3), the centralizer of
1
→
the subgroup Imρ is a set of elements g E(3) so that gρ(h)g−1 = ρ(h) for all
∈
h π (M). The centralizer is a subgroup in E(3), and we will denote it by E(3) .
1 ρ
W∈e denote by e(3) the Lie algebra of E(3) . So,
ρ ρ
e(3) = u e(3) Ad u=u, h π (M) ,
ρ ρ(h) 1
{ ∈ | ∀ ∈ }
where Ad = Ad ρ: π (M) Aut(e(3)) and Ad: g Ad is the adjoint repre-
ρ 1 g
◦ → 7→
sentation.
Let us take a geometrically natural basis of the whole Lie algebra e(3) of E(3).
Thisbasisconsistsofthreetranslationsdx,dy anddz alongtheCartesianaxesand
three rotationsdϕ , dϕ and dϕ aroundthese axes. Then, we alsohave a basis in
x y z
e(3) induced by restriction.
ρ
Definition2.6. Arepresentationρ: π (M) E(3)iscalledabelian (resp. trivial)
1
→
if its image is abelian (resp. trivial) subgroup of E(3).
Proposition 2.2. Let e(3) be a 2-dimensional subalgebra of e(3) generated by the
2
differential of screw motion along a fixed axis in R3 and let o be the trivial algebra
Lie. Then,
e(3), if ρ is trivial;
e(3) = e(3) , if ρ is abelian; (9)
ρ  2
o, if ρ is non abelian.
Proof. If ρ is trivial, then the centralizer of Imρ is E(3), and hence, e(3)ρ =e(3).
Let ρ be abelian. Then, Imρ consists of rotations around the only axis z in R3.
There exist two types of motions in E(3) which commute with every element of
Imρ: arotationaroundz andtranslationalongthisaxis. Hence,e(3) isgenerated
ρ
by the differential of a screw motion along z.
Let ρ be non abelian. In this case, Imρ consists of rotations around at least
twononparallelaxes. The existenceofsuchaxesfixes the coordinatesysteminR3.
Therefore, E(3) =1 and e(3) =o. (cid:3)
ρ ρ
Denote by N the full amount of i-dimensional simplices in .
i
We define a 3N0-dimensional vector space R3xN0 as a spaceFof all mappings of
tahteΓv.erTthiceesbafrsoisminF(dtox)Ri3s.geDneenroatteedbyby(dax)ch=oicTeΓRof3xNco0otrhdeintaatnegseynsttesmpacine Rto3.R3xFNo0r
convenience, we choose the Cartesian coordinates.
A linear mapping f : e(3) (dx) is defined as follows. Under f , the six
1 ρ 1
generators dx, dy, dz, dϕ , dϕ →and dϕ of e(3) are mapped into the differentials
x y z
of a vertex A according to the evident formula:
dx 0 dϕ dϕ x dx
A z y A
−
dy = dϕ 0 dϕ y + dy , (10)
A z x A
  −    
dz dϕ dϕ 0 z dz
A y x A
−
where x , y and z arethe coordinates of A.    
A A A
Let e be any edge of the fundamental family. Let L = 1l2, where l is the
∈F e 2 e e
Euclideanlength of Γ(e) R3. Define N -dimensionalvector space RN1 as a space
of all mappings of the ed∈ges from to R1. The subscript L stresses tLhat the basis
of this space consists of the vectorFs (L ,...,L ). Denote by (dL) = T RN1 the
1 N1 Γ L
tangent space to RN1 at Γ.
L
8 EVGENIYV.MARTYUSHEV
Consider a linear mapping f : (dx) (dL). By definition, f sends the given
2 2
→
differentials of two vertices A and B to
dL =(x x )(dx dx )+(y y )(dy dy )+(z z )(dz dz ). (11)
AB B A B A B A B A B A B A
− − − − − −
Since the bases of e(3) , (dx) and (dL) are distinguished, then the mappings f
ρ 1
and f are uniquely determined by their matrices which we denote by the same
2
letters.
Put Ω = ωe, where ω is the defect angle at Γ(e) (definition 1.1). Similarly as
e le e
we have defined the space (dL), let us define the vector space (dΩ) = T RN1: its
Γ Ω
basis consists of the vectors (dΩ ,...,dΩ ).
1 N1
Let us number the edges of in a certain order and consider the matrix f =
3
F
∂Ωa , whereaandbrunoverthe edgesof andthe partialderivativesaretaken
∂Lb F
a(cid:16)t ω1(cid:17)= ... = ωN1 = 0. The matrix elements of f3 are defined by formulas (3),
(4), (5) and (6). In particular, these formulas imply that f is symmetric. We can
3
obtain the same result directly.
Lemma 2.3. The matrix f is symmetric, i.e. fT =f .
3 3 3
Proof. Since f = 1 ∂ωa , it suffices to prove the symmetric form of matrix
3 lalb ∂lb
∂ωa . Let us find(cid:16)the Hes(cid:17)sian of the function H(l ,...,l ) = l ω at the
∂lb 1 N1 k k
k
(cid:16)point(cid:17)ω1 =...=ωN1 =0: P
∂ ∂H ∂ N1 ∂ω ∂ω
k a
= ω + l = .
a k
∂l ∂l ∂l ∂l ∂l
b (cid:18) a(cid:19) b k=1 a ! b
X
N1
The second equality holds since l dω = 0 (it is a consequence of well-known
k k
k=1
Schl¨afli’s differential identity [5, pP. 281–295]). Hence, the claim follows. (cid:3)
Denote by e(3)∗ and (dx)∗ vector spaces whose distinguished bases are dual to
ρ
the ones of e(3) and (dx) respectively. This means that if, say, d=(d ,...,d ) is
ρ 1 k
a basis of e(3) , then a basis d∗ =(d∗,...,d∗) of e(3)∗ is defined by
ρ 1 k ρ
1, i=j,
d∗(d )=
i j (0, i=j.
6
Theorem 2.4. The sequence of vector spaces and linear mappings
0 e(3) f1 (dx) f2 (dL) f3=f3T (dΩ) −f2T (dx)∗ f1T e(3)∗ 0 (12)
−→ ρ −→ −→ −−−−→ −−−→ −−→ ρ −→
is an acyclic complex (definition 2.3).
Remark 2.2. Recall that we restrict ourselves to the case of finite fundamental
group. An example of space S2 S1 shows that for manifolds with an infinite
×
fundamental groupthere can exist a representationρ such that (12) is not acyclic.
However,it turns out that for this case one can define an acyclic complex like (12)
as well (see [3]).
Proof of theorem 2.4.
Acyclicity at e(3) (= injectivity of f ) holds because the vertices of the funda-
ρ 1
mental family are in general position (remark 1.1).
EUCLIDEAN SIMPLICES AND INVARIANTS OF THREE-MANIFOLDS 9
Acyclicityat(dx). Theelementsofe(3) generateglobalinfinitesimaltranslations
ρ
of the vertices from which do not change the lengths, according to (10). It
F
follows that Imf Kerf . Conversely, if the vertex coordinates take arbitrary
1 2
⊂
infinitesimalincrementssothatthedifferentialsofallthelengthsremainzero,then
this corresponds to an element of the algebra e(3). It can be shown by direct
calculation that if h is such an element of π (M) that u = Ad u, then any
1 ρ(h)
nonzero element u e(3)/e(3) generates infinitesimal incre6 ments of the vertices
ρ
∈
for which dL = 0, where A is a vertex from . Clearly, it is impossible if
A,hA
6 F
dL =0 for every e in . Hence, Kerf Imf and the acyclicity at (dx) follows.
e 2 1
F ⊂
Acyclicity at (dL). The elements of factor-space (dx)/e(3) generate infinites-
ρ
imal increments of the edge lengths, but all the defect angles remain zero, i.e.
Imf Kerf . To prove the converse, we must show that if all L take infinitesi-
2 3 j
⊂
malincrementsdL sothatalldΩ =0,thendL ’saregeneratedbysomeelements
j i j
of (dx). Let us first prove the “finite” version of this statement.
Lemma 2.5. Let the lengths of the edges from take small but finite increments
∆l so that all ∆ω = 0. Then, there are suFch points in R3 with coordinates
j i
(x′,y′,z′) that
k k k
∆L =l ∆l =(x x )(∆x ∆x )+(y y )(∆y ∆y )+(z z )(∆z ∆z ),
j j j k s k s k s k s k s k s
− − − − − −
(13)
where ∆x =x′ x .
k k− k
Proof. If∆l aresmallenough,then allthe determinantsfrom(1)remainpositive.
j
Withacondition∆ω =0, ithismeansthatl +∆l giveanadmissiblecoloringon
i j j
∀
T andinducedadmissible coloringonT˜. By theorem1.1, there exists acontinuous
mapΓ′: T˜ R3 suchthatl′ =l +∆l , j,wherel′ isthelengthofΓ′(e )ofsome
→ j j j ∀ j j
edge e . Then, Γ′ maps all the vertices from the fundamental family to points in
j
R3 which are the ones we looked for. (cid:3)
This lemma implies that for the sequence
R3N0 F2 RN1 F3 RN1, (14)
x −→ L −→ Ω
the inclusion KerF ImF holds. Note that f =dF and f = dF . Therefore,
3 2 2 2 3 3
⊂
taking the differential of this sequence and taking into account the conditions of
general position from remark 1.1, we get Kerf Imf , hence the acyclicity at
3 2
⊂
(dL) follows.
Acyclicity at other spaces. Recall that Kerf =Imf is equivalent to
3 2
rankf =dim(dL) rankf .
3 2
−
Taking into account that dim(dL)=dim(dΩ) and rank( fT)=rankf , we get
− 2 2
rank( fT)=dim(dΩ) rankf ,
− 2 − 3
thatmeanstheacyclicityat(dΩ). Likewise,theacyclicityat(dx)∗ ande(3)∗ follows
ρ
from the acyclicity at (dx) and e(3) respectively. Theorem 2.4 is proven. (cid:3)
ρ
Denoteby , and arbitraryorderedsetsofbasisvectorsinthespacese(3) ,
0 1 2 ρ
C C C
(dx) and (dL) respectively. Let be a subset of basis vectors belonging
i i
B ⊂ C
to Imf . Denote by f such a principal minor of the matrix f that its rows
i Bi i i
correspondtothevectorsfrom anditscolumnscorrespondtothevectors
i−1 i−1
C \B
10 EVGENIYV.MARTYUSHEV
from . Let = . Then, according to definition 2.4, the torsion of the
i 3 2 2
B B C \B
complex (12) looks like
τ = (detB2f2)2(−1)rankf2.
det f (det f )2
B3 3 B1 1
By the acyclicity,
( 1)rankf2 =( 1)dim(dx)−dime(3)ρ =( 1)3N0 =( 1)N0,
− − − −
where the second equality is due to proposition 2.2. It follows that
τ = (detB2f2)2(−1)N0 . (15)
det f (det f )2
B3 3 B1 1
Besides, the form of (12) implies that the torsion τ is independent of the ordering
on , i.
i
C ∀
Put
τ
I (M)= , (16)
ρ
( V)
−
where V is the volume of tetrahedron multiplied by 6, and the product is taken
Q
over all tetrahedra of .
F
Theorem 2.6. For a given representation ρ, the quantity I (M) is an invariant
ρ
of the manifold M.
Proof is by a series of lemmas. We have to show that I (M) is independent of
ρ
various details of its construction. First, it does not depend on the choice of the
fundamental family . Second, using formula (5) and Pachner’s theorem, one can
F
show that I (M) does not depend on the choice of the triangulation of M (see [2]
ρ
for details of proof). Finally, I (M) is independent of Γ, i.e. it does not matter in
ρ
whichpoints of Euclideanspace we place the vertices fromthe fundamental family
(see remark 1.1).
Lemma 2.7. I (M) does not depend on the choice of the fundamental family .
ρ
F
Proof. First, note that the corresponding simplices from the different fundamental
familiescanbeobtainedfromeachotherbyanactionofacertainelementofπ (M).
1
It follows that the lengths, dihedralangles and tetrahedronvolumes do not change
under replacing the fundamental family with another one. Therefore, the bases in
(dx) and (dx)∗ can only change and they must stay dual to each other.
Denote by A,B SO(3N ) the change-of-basis matrices in the spaces (dx) and
0
∈ ] T
(dx)∗ respectively. Then, f = f A and fT = B−1fT = f = ATfT.
2 2 − 2 − 2 − 2 − 2
Hence, B = (AT)−1. That is, the torsion of (12) is divided by (detA)2 = 1
(see remark 2.1) and the claeim follows. e (cid:3)
Lemma 2.8 ([2]). I (M) does not depend on the triangulation of M.
ρ
Lemma 2.9. I (M) is independent of Γ, i.e., it does not matter in which points
ρ
of Euclidean space we place the vertices from .
F
Proof. Let v be an arbitrary vertex of T and v˜ . We are going to show that
usingthePachnermovesonecanmoveO =Γ(v˜)∈inFanyotherpointofR3 insucha
waythatalltheremainingverticesfromΓ( )donotchangetheirpositions. Then,
F
the claim will follow from lemma 2.8.