New York Journal of Mathematics
New York J. Math. 17 (2011) 383–435.
On small geometric invariants of
3-manifolds
Paul J. Kapitza
Abstract. A small geometric invariant is a nonnegative integer invariant associated with a 3-manifold whose value is bounded above by the
Heegaard genus of the manifold.
Craggs has studied techniques to detect for a given 3-manifold M 3 ,
whether the double 2M = Bd(M? × [−1, 1]) bounds a 4-manifold N that
has the same 3-deformation type as the complement of the interior of a 3ball in M and has a handle presentation with, in some sense, a minimal
number of 1-handles. Here, M? is obtained from M by removing an
open ball. He exhibits a pair of surgery obstructions, whose vanishing
is sufficient for the existence of this type of 4-manifold N and minimal
handle presentation.
We show that for the double of one of the Boileau–Zieschang manifolds, there is a certain handle presentation which, in the absence of the
obstructions studied by Craggs, is reducible to this minimal number of
1-handles and we provide an explicit construction. For this case, the
question of the existence of a minimal handle presentation is reduced to
a study of the obstructions defined by Craggs.
Contents
1.
2.
3.
4.
5.
Introduction
1.1. Notation and conventions
1.2. Presentations and extended Nielsen equivalence
1.3. Historical remarks
1.4. Singular disk systems
1.5. Admissible disk systems
Algebraic co-k-collapsibility
2.1. Algebraic collapsibility
2.2. Algebraic co-k-collapsibility
Computation of the 2-handle presentation
The manifolds of Boileau–Zieschang
A handle presentation for a 4-manifold bounded by 2M1
384
385
385
387
388
388
390
391
395
398
398
400
Received June 15, 2010; revised March 29, 2011.
2000 Mathematics Subject Classification. Primary 57M20; Secondary 57M50, 57R65.
Key words and phrases. Heegaard genus, extended Nielsen genus, Boileau–Zieschang
manifolds.
ISSN 1076-9803/2011
383
384
PAUL J. KAPITZA
6. Derivation of an admissible system for M1
References
409
435
1. Introduction
A geometric invariant of a 3-manifold is a geometrically defined measure
which remains the same across the homeomorphism class of a manifold. The
Heegaard genus of a manifold is one such example.
Craggs [3] studies a geometric invariant for 3-manifolds M defined by
considering certain 4-manifolds N bounded by the double 2M of M . He
looks at handle presentations of N with handles of index at most 2, and
takes the minimum number of 1-handles over all such presentations. This
minimum number is bounded above by the Heegaard genus of M and so is
a small geometric invariant.
It is known that the rank of the fundamental group of an arbitrary 3manifold M 3 and its associated Heegaard genus do not always agree. In
particular, the manifolds of Boileau and Zieschang [2] make up a collection of 3-manifolds for which the Heegaard genus is 3, but the rank of the
fundamental group is 2. See also Schultens and Weideman [11].
There have been efforts to show that for a given 3-manifold M , the 4
manifold N = M? × [−1, 1] has a minimal 2-handle presentation, where the
number of 1-handles is determined by the formal 3-deformation properties
of M? . Here M? is the result of removing the interior of a 3-ball from M .
Craggs [3] uses the extended Nielsen genus en(M ) of the base manifold
M as a measure of the potential minimum number of 1-handles in any
handle presentation associated with an appropriate 4-manifold N bounded
by 2M . The extended Nielsen genus of a 3-manifold is bounded above by the
Heegaard genus of the 3-manifold, and in the case of the Boileau–Zieschang
manifolds it is less than the Heegaard genus. Thus, the extended Nielsen
genus is a small geometric invariant that is sometimes less than Heegaard
genus.
Craggs defines handle presentations for certain 4-manifolds bounded by
the double 2M of M to be minimal if the number of 1-handles is equal
to the extended Nielsen genus en(M ) of M . The geometric realization of
en(M ) as the number of 1-handles in a minimal handle presentation for M
associates in a natural way a pair of framed surgery obstructions {L, T } in
a cube with handles. If these obstructions are always trivial, then minimal
handle presentations always exist, and they provide a new small geometric
invariant that is generally not equal to Heegaard genus.
We examine the thesis that one member of the Boileau–Zieschang family
bounds a 4-manifold with a minimal handle presentation. In this paper,
we construct a 4-manifold of the form M? × [−1, 1] whose associated handle
presentation exhibits an algebraic simplicity which agrees with the extended
Nielsen genus of M .
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
385
Some of the material here is taken from the author’s Ph.D. thesis at
the University of Illinois, Urbana-Champaign. The author wishes to thank
Professor Robert Craggs for his help in directing the thesis and to a referee
for numerous helpful comments on a previous version of this paper.
S
1.1. Notation and conventions. Let K = α eα be a finite connected
CW complex with characteristic maps φα : Dn → K. Here Dn is a topon ) is a homeomorphism onto
logical ball of dimension n such that φα | Int(DS
n
n−1
n
eα , with φα (Bd(D )) ⊂ K
, where K = {eα | dim eα ≤ n} denotes
the n-skeleton of K.
An elementary n-expansion K % L is defined for L = K ∪f Dn , where f
attaches to K all of the boundary of Dn , except one open (n − 1)-cell. An
elementary n-collapse is the inverse of an elementary n-expansion, denoted
as K & L.
We work in the PL category. For a piecewise linear 3-manifold M 3 , a
Heegaard decomposition of M 3 of genus n is a triple, (M ; H, J), where M =
H ∪ J and H and J are handlebodies of genus n with H ∩ J = Bd(H) =
Bd(J). The genus of the decomposition is the genus of the handlebody J.
The Heegaard genus of M 3 , hg(M 3 ), is the minimum value of n obtained
over all Heegaard decompositions of M 3 .
The manifold M 3 exhibits the structure of a CW complex with cells identified with the piecewise linear cells of M in a piecewise linear cell decomposition of M . Every cellular decomposition of M 3 with one 0-cell and one
3-cell defines a Heegaard decomposition of M 3 . Taking M \ N where N is a
regular neighborhood of the 1-skeleton of M results in a handlebody whose
genus is the number of 1-cells in the decomposition.
The cell complex obtained from a Heegaard decomposition of genus n
provides a handle decomposition of the form
"n
# " n
#
[
[
0
1
2
M3 = h ∪
hi ∪
hj ∪ h3
i=1
j=1
where hlm is a three dimensional handle of index l, l = 0, . . . , 3 and n is the
genus of J.
3
If K is a 2-complex, a formal 3-deformation of K, denoted K & L, is a
sequence of polyhedra K = K0 → K1 → · · · → Kn = L, where Ki → Ki+1
results from an expansion (%) or collapse (&) of a piecewise linear cell of
dimension at most three. A complex K is collapsible if K & {?} where {?}
denotes a 0-cell of K.
1.2. Presentations and extended Nielsen equivalence. Let X be a
finite set and R a set of words on X. A group G is defined by the sets X
and R if G ∼
= F/N , where F is the free group on X and N is the normal
subgroup of F normally generated by R. A presentation P = hX | Ri for
G consists of the ordered sets X = {x±1
| i = 1, . . . , m}, the generators
i
386
PAUL J. KAPITZA
of P , and R = {r1 , r2 , . . . , rn }, the defining relators of P . A presentation
P = hX | Ri presents a group G if G is isomorphic to the quotient group,
F/N where F is the free group on the generators x1 , x2 , . . . , xm , and N ⊂ F
is the smallest normal subgroup containing R. The group presented by P
is said to be finitely presented if there exists a presentation in which both
X and R are finite sets. The rank of the group G presented by P , rk(G), is
the minimum number of generators m for X = {x1 , x2 , . . . , xm } required to
present G.
For a 3-manifold M 3 , let K = K 2 in M 3 \ B 3 be defined by
"n # " p
#
[
[
K = e0 ∪
e1i ∪
e2j
i=1
j=1
in which ekα is a k-cell for k = 0, 1, 2 with characteristic maps φkα : Dk → K.
Associated with K is a group presentation of the form
PK = hx1 , x2 , . . . , xn | r1 , r2 , . . . , rp i
where each generator xi is obtained from a 1-cell of K in K 1 \ T (K) for
some chosen maximal tree T (K). For each relator rj there is a 2-cell e2j and
a characteristic map φ2j : D2 → K where φ2j | ∂D2 reads a word rj in the
symbols x1 , x2 , . . . , xn .
A geometric presentation associated with K is a presentation
PK = hx1 , x2 , . . . , xn | r1 , r2 , . . . , rp i
where ri reads the attaching map φi (Bd(e2i )) ⊂ K. The reduced presentation
for P = hx1 , x2 , . . . , xn | r1 , r2 , . . . , rp i is defined to be
|P | = hx1 , x2 , . . . , xn | s1 , s2 , . . . , sp i
where P is a presentation and si ≡ |ri |, where |ri | denotes the freely reduced
form of ri . The reduced presentation associated with K is defined to be
|PK | = hx1 , x2 , . . . , xn | s1 , s2 , . . . , sp i
where PK is the geometric presentation associated with K and si = |ri |,
where |ri | denotes the freely reduced form of ri . In this case, we will also
write |PK | = hx1 , x2 , . . . , xn | |r1 |, |r2 |, · · · , |rp |i to denote the corresponding
abstract presentation with freely reduced relators.
Given a presentation P = hx1 , x2 , . . . , xn | r1 , r2 , . . . , rp i, one may construct a presentation P 0 obtained from P by a finite sequence of elementary
extended Nielsen operations:
(1) For some 1 ≤ j ≤ p, add or delete the trivial relator xx−1 or x−1 x
in rj , leaving rk unchanged for k 6= j.
(2) For some 1 ≤ j ≤ p, replace rj with rj−1 , leaving rk unchanged for
k 6= j.
(3) For some 1 ≤ j ≤ p and some 1 ≤ k ≤ p, replace rj with rj rk , where
k 6= j, leaving ri unchanged for i 6= j.
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
387
(4) For some 1 ≤ j ≤ p, replace rj with w−1 rj w, where w is an element
in F (x1 , x2 , . . . , xn ), leaving rk unchanged for k 6= j.
(5) For an automorphism, α : F (x1 , x2 , . . . , xn ) → F (x1 , x2 , . . . , xn ),
replace rj with α(rj ) for j = 1, . . . , p.
(6) Add xn+1 to the set of generators and rp+1 = xn+1 to the set of
defining relators.
(7) Remove xn from the set of generators and the relator rk = xn from
the set of defining relators, when both occur and xn appears exactly
once among the relators.
Two presentations P and P 0 which are related by a finite sequence of
extended Nielsen operations are said to be extended Nielsen equivalent,
en
en
P ∼ P 0 . It is well known that if P ∼ P 0 then P and P 0 present the same
group. The extended Nielsen operations can be shown to generate a subset
of the Tietze transformations for groups that accounts for all Tietze II operations. The extended Nielsen genus of a presentation P , denoted en(P ), is
defined to be the minimum number of generators in any presentation which
is extended Nielsen equivalent to P .
If K is a 2-complex, the extended Nielsen genus of K, en(K), is defined
to be en(PK ), where PK is the standard reading of a presentation from a
2-complex K. For a 3-manifold M 3 , the extended Nielsen genus of M 3 ,
en(M 3 )is the extended Nielsen genus of any 2-spine of M 3 . See Brown [1],
Kreher and Metzler [8], Young [15] and Wright [14] on the equivalence of extended Nielsen equivalence and formal 3-deformation in both the polyhedral
and the CW categories. These results imply that en(M 3 ) is well-defined.
1.3. Historical remarks. Suppose that M 3 is a 3-manifold with 2-complex spine K, having a geometric presentation PK . It is known that
rk(M ) ≤ en(M ) ≤ hg(M ).
Haken [6] and Waldhausen [12] conjecture that rk(M ) = hg(M ) for all
3-manifolds M . M. Boileau and H. Zieschang [2] exhibit a collection of
Seifert 3-manifolds for which 2 = rk(π1 (M )) < hg(M ) = 3, providing a
counterexample to the conjectures of Waldhausen and Haken.
In an explicit calculation, Montesinos [9] exhibits an extended Nielsen
equivalence between a geometric presentation for π1 (M1 , ?) and a presentation P10 that has 2 generators and 2 relators establishing that en(M1 ) ≤ 2.
Here, M1 is one of the family of manifolds exhibited by Boileau and Zieschang. That en(M1 ) > 1 follows from the the fact that rk(π1 (M1 )) ≤
en(M1 ) . Therefore, when combined with the previous results we have that
2 = en(M1 ) = rk(M1 ) < hg(M1 ) = 3.
The following definitions come from Craggs [4]. Given a sequence of
polyhedra K = K0 → K1 → · · · → Kn = L, where Ki → Ki+1 is an
expansion or collapse of a piecewise linear cell, if there is some polyhedron
X (usually a manifold) such that K(i) ⊂ X for each i, then one says K
388
PAUL J. KAPITZA
deforms to L in X. If M is a manifold and K and L are in the interior
of M , then K deforms to L in M means that K and L have isotopically
embedded regular neighborhoods.
Craggs [4] studies the question, for which 2-complexes K in a 3-manifold
M do the corresponding 2-complexes K × {0} ⊂ M × [0, 1] 3-deform in
M × [0, 1], keeping 1-skeletons fixed, to a 2-complex L ⊂ M × [0, 1] so that
the associated presentation PL is obtained from the presentation PK by
freely reducing relator words? He addresses the following: If M? 3-deforms
in M? × [−1, 1] to a 2-spine complex L such that |PL | has m 1-cells and k
2-cells reading generators, does L 3-deform in M × [−1, 1] to a 2-complex
with m − k 1-cells?
A related question as to whether the 2-complex K × {0} ⊂ M? × [−1, 1]
3-deforms in M? × [−1, 1] to a 2-complex L having at most en(K) 1-cells has
been addressed by Craggs [4] concerning the family of manifolds {Mn }∞
n=1 .
Material necessary for later calculations is contained in the following sections. Section 1.4 describes the basic objects involved, the singular disk systems. Section 1.5 reviews material on singular systems with an admissibility
requirement on the collection of singular disks in the system. Admissible
systems provide a connection between modifications of singular systems and
3-deformations in M? × [−1, 1].
1.4. Singular disk systems. The definitions and results which follow concerning singular and admissible disk systems are due to Craggs.
Definition 1.1. A singular disk system in H is a pair (D, g) where
D=
n
[
Di
i=1
is a finite disjoint union of disks and g : D → H is a proper map such that:
(1) g −1 (Bd((H))) ⊂ Bd(D).
(2) S
The singular set of g is a finite collection of proper disjoint arcs
{Ai1 , Ai2 } such that each pair corresponds to a transverse double
arc intersection.
A singular system is said to be ordinary if g is nonsingular.
Figure 1 illustrates a singular disk system consisting of the 2-cells D =
D1 ∪ D2 . The map g : D → H identifies the arcs A1 ⊂ D1 and A2 ⊂ D2 in
the image.
1.5. Admissible disk systems.
Definition 1.2. A singular system (D, g) in H is said to be an admissible
system if there exists a continuous map : D → {−1, 0, 1} such that:
(1) The map (g, ) : D → H × [−1, 1] defined by (g, )(x) = (g(x), (x))
is an embedding.
(2) If a given disk Di ∈ D contains a singular arc, then (Di ) 6= 0.
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
(D1,A1)
389
g(A1)=g(A2)
(D2 ,A2)
Bd(H)
Figure 1. Singular disk system.
The quantity (Di ) is called the label of the disk Di . We will write
i = (Di ) and place Dii = (g(Di ), (Di )). In particular, if a singular disk
system (D, g) becomes an admissible disk system with the addition of some
map : D → {−1, 0, 1}, then the admissible system will be denoted by the
triple (D, g, ).
In an admissible disk system (D, g, ) in H, there is a natural partition
of the 2-manifold D into three disjoint submanifolds: D+ , D− and D0 ,
corresponding to those disks in D for which = +1, −1, 0 respectively.
Suppose (M ; H, J) is a decomposition where (D, g, ) and (D0 , g 0 , 0 ) are
two admissible singular disk systems on H. Then (D0 , g 0 , 0 ) results from
(D, g, ) by an admissible sequence of operations if (D0 , g 0 , 0 ) is obtained
from (D, g, ) by a finite sequence of the following operations and their inverses:
(1) (Bookkeeping): Replace (D, g, ) with the system (D0 , g 0 , 0 ) where
h : M → M is a homeomorphism that takes H onto itself and
g 0 = g ◦ h.
(2) (Level Switch): For Di = (g(Di ), (Di )) where g | Di is nonsingular,
replace (Di ) by 0 (Di ) ∈ {−1, 0, 1}.
(3) (Full Isotopy): Let ht : H × I → H be an isotopy such that h0 = 1H .
Replace (D, g, ) with the system (D0 , g 0 , 0 ) where g 0 = h1 ◦ g.
(4) (Split Isotopy): Replace (D, g, ) with the system (D0 , g 0 , 0 ) where
for some isotopy ht : H × I → H and η ∈ {−1, 1} the following
condition holds:
g 0 | Di = (h1 ◦ g) | Di if (Di ) = η and g 0 | Di = g|Di for (Di ) 6= η.
(5) (Admissible Disk Slide): Replace (D, g, ) with the system (D0 , g 0 , 0 )
by sliding g(Dj ) over g(Dk ) along an arc β where, considered as a
singular system, (D0 , g 0 ) results from (D, g) by a slide of g(Dj ) over
g(Dk ) and either j = k or at least one of these quantities is 0.
(6) (Stabilization): Replace (D, g, ) with the system (D00 , g 00 , 00 ) where
D ⊂ D00 and g = g 00 | D. In this operation, D00 \ D = B 2 is a
nonsingular disk containing a properly embedded arc β. Let N be
a regular neighborhood of β in H ⊂ (M ; H, J) and delete one of
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PAUL J. KAPITZA
the two components of B 2 \ β to produce the system (D00 , g 00 , 00 ) on
(M ; H 0 , J 0 ), where the genus (H 0 ) = genus(J 0 ) = genus(J + 1).
There is a natural association via Craggs [4] between admissible systems
and 2-complexes in M? × [−1, 1], in which each admissible operation induces
an extended Nielsen transformation of the corresponding 2-complex group
presentation.
2. Algebraic co-k-collapsibility
In this section, a property of the words {r1 , r2 , . . . , rp } which are associated with a presentation of a collapsible complex K is examined. A form
for the relators associated with a collapsible complex is presented in terms
of the associated presentation.
The remainder of this section is taken from Whitehead [13]. In what
follows, G = F (x1 , x2 , . . . , xn ) is a free group, W (x1 , x2 , . . . , xn ) is a word
−1
−1
on the symbols X = {x1 , x2 , . . . , xn } ∪ {x−1
1 , x2 , . . . , xn } and x and y are
elements of X.
Definition 2.1. An elementary transformation on a word
W = W (x1 , x2 , . . . , xn )
is either an insertion into W or a deletion from W of a pair of successive
letters of the form xx−1 for x ∈ X.
Definition 2.2. A simple transformation of the first type on a set of words
{W1 , . . . , Wk } in G is a replacement of the form x → xy and x−1 → y −1 x−1
for each occurrence of x or x−1 in {W1 , . . . , Wk }. A simple transformation
of the second type on a set of words {W1 , . . . , Wk } ⊂ G is an elementary
transformation applied to some word in {W1 , . . . , Wk }.
A simple transformation on a set of words {W1 , . . . , Wk } ⊂ G is either a
simple transformation of the first or second type.
Definition 2.3. A simple set of words is a set {W1 , . . . , Wk } of distinct
words derived from an independent set of generators {x1 , x2 , . . . , xn } by a
sequence of simple transformations.
The following results on simple sets and simple transformations will be
used in later sections.
Lemma 2.4. If {W1 , . . . , Wk } is a simple set of words on X, then every
subset is also a simple set. Also, if k < n, any simple set {W1 , . . . , Wk } may
be extended to a simple set {W1 , . . . , Wn }.
Given a simple transformation, say xi → xi xj of the first type, there is
an associated automorphism of G. For fixed i, j, 1 ≤ i, j ≤ n, i 6= j and for
all k 6= i define the map αij : G → G by
αij (xi ) = xi xj
αij (xk ) = xk ,
k 6= i.
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
391
For a word W ∼
= xi11 xi22 . . . xill in G, extend αij to W ∈ G by defining
αij (W ) ∼
= αij (xi1 )1 αij (xi2 )2 . . . αij (xil )l .
Associating the simple transformations with automorphisms of G applied to
the set of words {W1 , . . . , Wk } in this way yields the following result.
Theorem 2.5. The collection {W1 , . . . , Wk } is a simple set of words on
the generating set X if and only if the elements of W correspond to an
independent set of generators in some automorphism of G.
2.1. Algebraic collapsibility. Consider the 2-complex K
"n # " p
#
[
[
K = e0 ∪
e1i ∪
e2j
i=1
j=1
in which ekα is a k-cell for k = 0, 1, 2 together with the characteristic maps
◦
φkα : Dk → K where φkα | Dk is a homeomorphism onto ekα .
For i = 1, . . . , n let xi be the generator associated with e1i . Then for
j = 1, . . . , p, the attaching map associated with the 2-cell e2j yields a word
−1
−1
rj on the symbols X = {x1 , x2 , . . . , xn } ∪ {x−1
1 , x2 , . . . , xn }.
Let PK = hx1 , x2 , . . . , xn | r1 , r2 , . . . , rp i be the geometric presentation
associated with K and suppose that K collapses to a 2-complex K(1) by an
e
elementary collapse. In particular, suppose that K & K(1) by a collapse
across the 1-cell e11 which removes the 2-cell e21 whose associated reading is
given by r1 . Denote the resulting geometric presentation associated with
K(1) in terms of PK by writing
PK(1) = hx̂1 , x2 , . . . , xn | r̂1 , r2 , . . . rp i
where x̂ indicates the removal of quantity x.
Corresponding to the elementary collapse K & K(1) across the 1-cell e11 ,
the set of words {r1 , r2 , . . . , rp } in PK has the following properties:
(1) The symbol x1 or x−1
1 occurs exactly once in the relator r1 .
(2) For 2 ≤ j ≤ p, no rj contains an occurrence of x1 or x−1
1 .
In the case where K & {?} the set of words {r1 , r2 , . . . , rp } in PK will be
called an algebraically collapsible set of words. In Section 2.2, the case where
K & L for a subcomplex L ⊂ K is examined.
To formalize this situation, we introduce the following terminology.
−1
−1
Let W be a collection of words on {x1 , x2 , . . . , xn } ∪ {x−1
1 , x2 , . . . , xn }.
For each 1 ≤ i ≤ n, let νi : W → Z be the function defined by setting νi (r)
equal to the number of occurrences of {x±1
i } in r ∈ W .
We will use the following subscript notation for a nonempty set of words
{r1 , r2 , . . . , rp } on the set X. For ∆ : {1, . . . , p} → {1, . . . , p} an element of
the symmetric group Sp let (j) denote the image under ∆ of the element
j ∈ {1, . . . , p}. That is, define (j) = ∆(j) for ∆ ∈ Sp . For the set of
392
PAUL J. KAPITZA
generators {x1 , x2 , . . . , xn } and for i ∈ {1, . . . , n} let [i] = Γ(i) for Γ ∈ Sn
be the image of i under Γ.
With these conventions, the notation ν[i] (r(j) ) refers to the number of
occurrences of the generator x[i] in the word r(j) under some pair of permutations ∆ and Γ as defined above.
Definition 2.6. An ordered collection of words {r1 , r2 , . . . , rn } on
−1
−1
X = {x1 , x2 , . . . , xn } ∪ {x−1
1 , x2 , . . . , xn }
is called algebraically collapsible if after free reduction, there exist permutations Γ ∈ Sn and ∆ ∈ Sn such that
(
1 for i = j
ν[i] (r(j) ) =
0 for i < j.
A presentation P = hx1 , x2 , . . . , xn | r1 , r2 , . . . , rn i is called algebraically
collapsible if {r1 , r2 , . . . , rn } is an algebraically collapsible collection of words
on X.
In general, we will assume that when given a collection {r1 , r2 , . . . , rn }
of words, any free reduction is performed prior to testing the collection by
Definition 2.6.
Example 2.7. The collection of words {r1 , r2 , r3 } on generators {x1 , x2 , x3 }
−1
3
given by the assignments r1 = x1 , r2 = x−1
1 x3 x2 and r3 = x2 x1 is algebraically collapsible. Let Γ = (1 3) ∈ S3 and ∆ = (1 2 3) ∈ S3 . Then
ν[1] (r(1) ) = 1, ν[1] (r(2) ) = ν[1] (r(3) ) = 0
ν[2] (r(2) ) = 1, ν[2] (r(3) ) = 0
ν[3] (r(3) ) = 1.
Theorem 2.8. Let {r1 , r2 , . . . , rn } be a collection of words on the alphabet
X. Then {r1 , r2 , . . . , rn } is algebraically collapsible if and only if there exist
Γ ∈ Sn and ∆ ∈ Sn where Γ(j) = [j], ∆(i) = (i) such that
r(i) = ui x±1
[i] vi
1≤i≤n
where ui = ui (x[i+1] , . . . , x[n] ) and vi = vi (x[i+1] , . . . , x[n] ).
Proof. Suppose {r1 , r2 , . . . , rn } is algebraically collapsible. Then there exist
Γ ∈ Sn and ∆ ∈ Sn so that
(
1 i=j
ν[i] (r(j) ) =
1 ≤ i, j ≤ n.
0 i<j
For 1 ≤ i ≤ n, ν[i] (r(i) ) = 1 so that r(i) is of the form r(i) = ui x[i] ±1 vi
where ui = ui (x[1] , . . . , x̂[i] , . . . , x[n] ) and vi = vi (x[1] , . . . , . . . , x̂[i] , . . . , x[n] ).
Since ν[m] (r(i) ) = 0 for m = 1, . . . , i − 1 then ui = ui (x[i+1] , . . . , x[n] ) and
vi = vi (x[i+1] , . . . , x[n] ).
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
393
(⇐) Suppose {r1 , r2 , . . . , rn } is given along with ∆ ∈ Sn and Γ ∈ Sn so
that r(i) = ui x±1
[i] vi for ui = ui (x[i+1] , . . . , x[n] ) and vi = vi (x[i+1] , . . . , x[n] ),
1 ≤ i ≤ n. Apply the counting function ν[i] for i = 1, . . . , n to obtain
(
1 i=j
ν[i] (r(j) ) =
1 ≤ i, j ≤ n.
0 i<j
This implies {r1 , . . . , rn } is algebraically collapsible on {x1 , x2 , . . . , xn }.
In general, Γ and ∆ are not unique. For example, the collection of words
{r1 , . . . , rn } on the set of generators {x1 , x2 , . . . , xn } where rj ≡ xj for j =
1, . . . , n is algebraically collapsible for every Γ = ∆ ∈ Sn .
Let F (x1 , x2 , . . . , xn ) be the free group on {x1 , x2 , . . . , xn }. Suppose that
{r1 , r2 , . . . , rn } is a collection of distinct words on F . Recall from Theorem 2.5 that {r1 , r2 , . . . , rn } is a simple set of words if each word rj corresponds to a generator xi under some automorphism σ : F → F . In the notation of Definition 2.6 this is equivalent to the statement that {r1 , r2 , . . . , rn }
is a simple set of words if there exist Γ ∈ Sn and ∆ ∈ Sn so that σ(x[i] ) = r(j)
for i, j = 1, . . . , n.
Lemma 2.9. Suppose that r is a word on X of the form r = uxi v where u
and v are words on the set of generators X \ {xi , x−1
i }and = ±1. Then
there exists an automorphism σ : F → F such that
(
u−1 xi v −1
j=i
σ(xj ) =
j = 1, . . . , n.
xj
j 6= i,
Proof. Let r = uxi v where u and v are words on X \ {xi ∪ x−1
i }. Suppose
that
ls−1
l l
u = xlsls xls−1
. . . xl22 xl11
r
r
r
t−1
v = xr11 xr22 . . . xrt−1
xrrtt .
For k = 1, . . . , s, consider the simple transformation of the first type
−l
defined by xi → xlk k xi . By the remarks following Theorem 2.4 there is an
associated automorphism λlk : F → F where
( −
l
xl k k xi
j=i
λlk (xj ) =
j = 1, . . . , n.
xj
j 6= i
Define λL = λls ◦ λls−1 ◦ · · · ◦ λl2 ◦ λl1 . By construction,
(
u−1 xi
j=i
λL (xj ) =
j = 1, . . . , n.
xj
j 6= i
Similarly, for k = 1, . . . , t, the simple transformation of the first type
−r
defined by xi → xi xrk k may be associated with the automorphism
ρrk : F → F
394
where
PAUL J. KAPITZA
( −
r
xi xrk k
ρrk (xj ) =
xj
j=i
j 6= i
Define ρR = ρr1 ◦ ρr2 ◦ · · · ◦ ρrt . Then
(
xi v −1
j=i
ρR (xj ) =
xj
j=
6 i
Finally, define σ = λL ◦ ρR , so that
(
u−1 xi v −1
σ(xj ) =
xj
j=i
j 6= i
j = 1, . . . , n.
j = 1, . . . , n.
j = 1, . . . , n.
In the case r = ux−1
i v where u and v are words on the set of generators
−1
X \ {xi ∪ xi }, apply the preceding construction to the word v −1 xi u−1 to
obtain an automorphism σ : F → F such that
(
vxi u
j=i
σ(xj ) =
j = 1, . . . , n.
xj
j 6= i,
−1 = u−1 x−1 v −1 .
Then σ(x−1
i ) = (σ(xi ))
i
Theorem 2.10. Suppose that {r1 , r2 , . . . , rn } is an algebraically collapsible
set of words on X. Then there exists an automorphism σ : F → F such that
σ(x[i] ) = r(i) for 1 ≤ i ≤ n.
In particular, if {r1 , r2 , . . . , rn } is an algebraically collapsible set then
{r1 , r2 , . . . , rn } is a simple set of words.
Proof. Let {r1 , r2 , . . . , rn } be an algebraically collapsible set on X. By
Theorem 2.8 there exists Γ ∈ Sn and ∆ ∈ Sn , where (j) = ∆(j) and
[i] = Γ(i), so that r(j) = uj x±1
[j] vj for each j ∈ {1, . . . n} where
uj = uj (x[j+1] , . . . , x[n] )
vj = vj (x[j+1] , . . . , x[n] ).
For each i = 1, . . . n, let σi : F → F be the automorphism of Lemma 2.9
defined by
(
−1
u−1
j=i
i x[i] vi
σi (x[j] ) =
j ∈ {1, . . . , n}.
x[j]
j 6= i
Define σ = σn ◦ · · · ◦ σ1 .
Claim: σ(r(j) ) = x[j] for j = 1, . . . , n.
Let r(j) = uj x[j] vj , where
uj = uj (x[j+1] , . . . , x[n] ),
vj = vj (x[j+1] , . . . , x[n] ).
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
395
By construction, σi (uj ) = uj and also σi (vj ) = vj for i = 1 . . . j. Therefore,
σ(r(j) ) = σn ◦ · · · ◦ σj+1 ◦ σj ◦ · · · ◦ σ1 (uj x[j] vj )
= σn ◦ · · · ◦ σj+1 ◦ σj (uj x[j] vj )
−1
= σn ◦ · · · ◦ σj+1 (uj (u−1
j x[j] vj )vj )
= σn ◦ · · · ◦ σj+1 (x[j] )
= x[j] .
Then the automorphism σ −1 : {x[1] , . . . , x[n] } → {r(1) , . . . , r(n) } exhibits
{r(1) , . . . , r(n) } as images of the generators {x[1] , . . . , x[n] }. By Theorem 2.5
{r(1) , . . . , r(n) } forms a simple set of words.
Corollary 2.11. If P = hx1 , x2 , . . . , xn | r1 , r2 , . . . , rn i is an algebraically
collapsible presentation then P is extended Nielsen equivalent to the empty
presentation.
Proof. Let P = hx1 , x2 , . . . , xn | r1 , r2 , . . . , rn i be an algebraically collapsible presentation. Then there exists Γ ∈ Sn and ∆ ∈ Sn , with ν[i] (r(j) ) = 1
if i = j and ν[i] (r(j) ) = 0 where i < j for 1 ≤ i, j ≤ n.
Since P is algebraically collapsible, Lemma 2.10 implies there exists an
automorphism σ : F (x1 , x2 , . . . , xn ) → F (x1 , x2 , . . . , xn ) where σ(x[i] ) = r(i)
for 1 ≤ i ≤ n.
Then
P = hx1 , x2 , . . . , xn | r1 , r2 , . . . , rn i
en
∼ hx[1] , . . . , x[n] | r(1) , . . . , r(n) i
en
∼ hx[1] , . . . , x[n] | x[1] , . . . , x[n] i
en
∼ h− | −i.
2.2. Algebraic co-k-collapsibility. As in the previous section, let
"n # " p
#
[
[
e1i ∪
e2j
K = e0 ∪
i=1
j=1
and let PK = hx1 , x2 , . . . , xn | r1 , r2 , . . . , rp i be the geometric presentation associated with K. Suppose that L ⊂ K is a subcomplex of K for
which K & L. In this case, if the corresponding elementary collapses are
across the 1-cells {e11 , e12 , . . . , e1p−k } for some k ≥ 0 which remove the 2-cells
{e21 , e22 , . . . , e2p−k }, then the corresponding presentation associated with L is
PL = hx̂1 , . . . , x̂p−k , . . . , xn | r̂1 , . . . , r̂p−k , rp−k+1 . . . , rp i
where k ≤ n.
Definition 2.12. A collection of words {r1 , r2 , . . . rq } for 0 ≤ k ≤ q on
−1
−1
the letters {x1 , x2 , . . . , xn } ∪ {x−1
1 , x2 , . . . , xn } for 0 < q ≤ n is called
396
PAUL J. KAPITZA
algebraically co-k-collapsible if after free reduction, there exists Γ ∈ Sn and
∆ ∈ Sq such that
1
i=j
ν[i] (r(j) ) =
1 ≤ i, j ≤ q − k.
0
i<j
A presentation P = hx1 , x2 , . . . , xn | r1 , r2 , . . . , rp i for p ≤ n is said
to be algebraically co-k-collapsible if {r1 , . . . , rp } is an algebraically co-kcollapsible collection of words on X.
Lemma 2.13. If {r1 , r2 , . . . , rp } is algebraically co-k-collapsible, 0 ≤ k < p,
then there exists a subset of cardinality p − k which forms an algebraically
collapsible set.
Proof. Let {r1 , r2 , . . . , rp } be algebraically co-k-collapsible with 0 ≤ k < p.
Then there exists Γ ∈ Sn and ∆ ∈ Sp such that
1
i=j
ν[i] (r(j) ) =
1 ≤ i, j ≤ p − k.
0
i<j
From Definition 2.6 it follows directly that {r(1) , . . . , r(p−k) } is algebraically
collapsible on X.
Lemma 2.14. Let {r1 , r2 , . . . , rp } for p ≤ n be a collection of words on
the generating set X. Then {r1 , r2 , . . . , rp } is algebraically co-k-collapsible if
and only if there exist Γ ∈ Sn and ∆ ∈ Sp where Γ(j) = [j], ∆(i) = (i) such
that
r(i) = ui x±1
1≤i≤p−k
[i] vi
where ui = ui (x[i+1] , . . . , x[n] ) and vi = vi (x[i+1] , . . . , x[n] ).
Proof. (⇒) Let {r1 , r2 , . . . , rp } be algebraically co-k-collapsible. Then Lemma 2.13 implies there exists Γ ∈ Sn and ∆ ∈ Sp and an algebraically collapsible subset {r(1) , . . . , r(p−k) }. Theorem 2.8 applied to this subset implies
the result.
(⇐) Suppose given {r1 , r2 , . . . , rp }, ∆ ∈ Sp and Γ ∈ Sn so that r(i) =
ui x±1
[i] vi for ui = ui (x[i+1] , . . . , x[n] ) and vi = vi (x[i+1] , . . . , x[n] ), 1 ≤ i ≤ p−k.
Apply the counting function ν to obtain
1
i=j
ν[i] (r(j) ) =
1 ≤ i, j ≤ p − k.
0
i<j
Then {r(1) , . . . , r(p−k) } ⊂ {r1 , . . . , rp } is algebraically collapsible so that
{r1 , . . . , rp } is an algebraically co-k-collapsible set on X.
Lemma 2.15. Suppose that P = hx1 , x2 , . . . , xn | r1 , r2 , . . . , rp i for p ≤ n is
algebraically-co-k collapsible. Then P is extended Nielsen equivalent to
0
0
P 0 = hx[p−k+1] , . . . , x[n] | r(p−k+1)
, . . . , r(p)
i
0
0 } on {x
for Γ ∈ Sn , ∆ ∈ Sp , and words {r(p−k+1)
, . . . , r(p)
[p−k+1] , . . . , x[n] }.
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
397
Proof. Let P = hx1 , x2 , . . . , xn | r1 , r2 , . . . , rp i for p ≤ n be algebraically
co-k-collapsible. Then there exists Γ ∈ Sn and ∆ ∈ Sp , with ν[i] (r(j) ) = 1 if
i = j and ν[i] (r(j) ) = 0 where i < j for 1 ≤ i, j ≤ p − k.
By Lemma 2.10 there exists an automorphism σ : F → F where σ(r(i) ) =
x[i] for 1 ≤ i ≤ p − k. For Γ ∈ Sn as above we have
(1)
P = hx1 , x2 , . . . , xn | r1 , r2 , . . . , rp i
en
∼ hx[1] , . . . , x[p−k] , x[p−k+1] , . . . , x[n] | r(1) , . . . , r(p−k) , . . . r(p) i
en
∼ hx[1] , . . . , x[p−k] , x[p−k+1] , . . . , x[n]
| x[1] , . . . , x[p−k] , σ(r(p−k+1) ), . . . , σ(r(p) )i.
Claim:
(x[1] , . . . , x[p−k] , σ(r(p−k+1) ), . . . , σ(r(p) ))
en
0
0
∼ (x[1] , . . . , x[p−k] , r(p−k+1)
, . . . , r(p)
)
0 = r 0 (x
where r(i)
(i) [p−k+1] , . . . , x[n] ) for p − k + 1 ≤ i ≤ p.
Proof. We argue by induction on the number of words k in the set
{σ(r(p−k+1) ), . . . , σ(r(p) )}.
If k = 0, then {r1 , r2 , . . . , rp } is algebraically collapsible on X. Corollary 2.11 implies that Equation (1) is extended Nielsen equivalent to the
presentation
hx[p+1] , . . . , x[n] | −i.
For k > 0, choose σ(r(j) ) ∈ {σ(r(p−k+1) ), . . . , σ(r(p) )} for some j, where
p − k + 1 ≤ j ≤ p. For some α ∈ {1, . . . , p − k}, suppose that x[α] is the first
occurrence in σ(r(j) ) of a member of {x[1] , . . . , x[p−k] }. Then σ(r(j) ) = ux[α] v
where u = u(x[p−k+1] , . . . , x[n] ). From Equation (1) we obtain
(x[1] , . . . , x[α] , . . . , x[p−k] , σ(r(p−k+1) ), . . . , σ(r(j) ), . . . , σ(r(p) ))
en
∼ (x[1] , . . . x[α] , . . . , x[p−k] , σ(r(p−k+1) ), . . . , ux[α] v, . . . , σ(r(p) ))
en
∼ (x[1] , . . . x[α] , . . . , x[p−k] , σ(r(p−k+1) ), . . . , x[α] vu, . . . , σ(r(p) ))
en
∼ (x[1] , . . . x[α] , . . . , x[p−k] , σ(r(p−k+1) ), . . . , uv, . . . , σ(r(p) )).
Then the number of occurrences of elements of {x[1] , . . . , x[p−k] } has been
reduced by one in the word σ(r(j) ). Continuing for a finite number of such
occurrences results in a word rj0 = rj0 (x[p−k+1] , . . . , x[n] ), which reduces k
by 1. The induction hypothesis then implies the existence of the extended
Nielsen equivalent presentation,
0
0
hx[1] , . . . x[p−k] , x[p−k+1] , . . . , x[n] | x[1] , . . . , x[p−k] , r(p−k+1)
, . . . , r(p)
i
which in turn is equivalent to
en
0
0
∼ hx[p−k+1] , . . . , x[n] | r(p−k+1)
, . . . , r(p)
i
398
PAUL J. KAPITZA
0
0 } are words on {x
such that {r(p−k+1)
, . . . , r(p)
[p−k+1] , . . . , x[n] }.
3. Computation of the 2-handle presentation
We adopt the following terminology concerning minimal handle presentations [3].
Definition 3.1. Let M 3 be a 3-manifold and let N be a 4-manifold with
Bd(N ) = 2M . A minimal handle structure for N (relative to the boundary
2M ) is a handle presentation for N of the form
0
H=h ∪
en(M
[ )
i=1
h1i
∪
q
[
h2j ,
j=1
where:
(1) H has one 0-handle and en(M ) 1-handles.
(2) If KH is a 2-complex associated with H, then KH formally 3-deforms
to M? .
We establish a partial result in support of the following conjecture:
Conjecture 3.2 ([3]). Let M be a 3-manifold. Then there exists a 4-manifold N with boundary 2M , and there is a minimal handle presentation for
N.
Details concerning the manifold M1 are discussed in the following section.
Recall that en(K) is the minimum number of generators on the presentation PK which is achievable by formal three deformations on K, whereas
en(M 3 ) is the minimum number of generators in any 2-complex L which 3deforms to a 2-complex spine K of M 3 . Here, en(PK ) = en(K) = en(M 3 ).
We consider the problem of reducing the number of 1-handles in M? ×[−1, 1],
to obtain a handle presentation of M? × [−1, 1] for which the number of
1-handles is strictly less than hg(M 3 )for one of a family of manifolds introduced by Boileau–Zieschang.
4. The manifolds of Boileau–Zieschang
Recall that a presentation P for a three manifold group π1 (M ) is said to
be geometric if there is a 2-spine K of M? , so that P is the presentation
given by
P = PK = hx1 , x2 , . . . , xn | r1 , r2 , . . . , rn i
where ri reads the attaching map φi (Bd(e2i )) ⊂ K. The Heegaard genus of
a 3-manifold M 3 is defined as the minimum number of 1-handles geometrically realizable in any Heegaard decomposition of M 3 . For each 1-handle in
a Heegaard decomposition, there is a free generator in a geometric presentation for π1 (M ). This implies that a lower bound for the number of 1-handles
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
399
in M 3 in any Heegaard decomposition is given by rk(π1 (M )). That is, if
M 3 is a 3-manifold, then
rk(π1 (M 3 )) ≤ hg((M )).
(2)
Boileau and Zieschang [2] exhibit a family of manifolds {Mi }∞
i=1 for which
the inequality (2) is strict, that is
Theorem 4.1 ([2]). There exists a family of 3-manifolds {Mi }∞
i=1 such that
for all i ≥ 1,
2 = rk(π1 (Mi )) < hg(Mi ) = 3.
The proof of the theorem proceeds by exhibiting particular Heegaard
decompositions of genus 3 and reducing the number of generators to 2 by
algebraic techniques. A discussion of this occurs in Montesinos [10].
One member of this family will be denoted throughout the rest of this
paper as M1 . A geometric presentation for M1 is given by Montesinos as
(3)
−1 2
−1 3
−3
−1
PM1 = hx1 , x2 , x3 | x3 x1 x3 x−1
1 , x2 x1 x2 x1 , (x3 x2 x1 ) (x2 x1 ) i.
Using the given presentation for π1 (M1 ), the following theorem [10] verifies
that the extended Nielsen genus of M13 is 2, so
(4)
2 = rk(π1 (M1 )) = en(M1 ) < hg(M1 ) = 3.
The derivation following the statement of the next theorem is included for
reference. It is referenced in Section 5 to calculate a handle presentation for
M1? × [−1, 1] whose associated presentation is algebraically co-2-collapsible.
Theorem 4.2 ([10]). Let M13 be the manifold of Boileau–Zieschang with
presentation PM1 as given above. Then the extended Nielsen genus of PM1
is 2.
−3
−1
Proof. Let r1 = x3 x1 x3 x−1
1 and r2 = x2 x1 x2 x1 . Then,
−1 2
3
(r1 , r2 , (x3 x2 x−1
1 ) (x2 x1 ) )
en
−1 2
−1 −2 2
3
∼ (r1 , r2 , (x3 x2 x−1
1 ) (x2 x1 ) (x2 x1 ) x1 )
en
−1
−1 −1
−1
∼ (r1 , r2 , x3 x2 x−1
1 x3 (x3 x1 x3 x1 )x2 x1 x3 x2 x1 )
en
−1
−1
−1 3 −1
∼ (r1 , r2 , x3 x2 x−1
3 x1 x2 (x2 x1 x2 x1 )x1 x3 x2 x1 )
en
−1 2
−1 −1 −1
∼ (r1 , r2 , x3 x2 x−1
3 x2 x1 x3 (x3 x1 x3 x1 )x2 x1 )
en
−1
−1
−1
∼ (r1 , r2 , x3 x2 x−1
3 x2 x1 x3 (x3 x1 x3 x1 )x1 x2 x1 )
en
−1
−1 3 −1
∼ (r1 , r2 , x3 x2 x−1
3 x2 x3 x1 x1 (x1 x2 x1 x2 )x2 x1 )
en
−1
−3
−1 4
3 −1
∼ (r1 , r2 , x3 x2 x−1
3 x2 x3 x1 x2 (x2 x1 x2 x1 )x1 )
en
−1
−1
−1
2 −2
∼ (r1 , r2 , x3 x2 x−1
3 x2 x3 x2 x1 (x1 x2 x1 x2 x1 )x1 )
en
−1 −1
2
∼ (r1 , r2 , x−1
1 (x2 x3 x2 x3 x2 x3 x2 )).
400
PAUL J. KAPITZA
Substituting into PM1 , we obtain
−1
−3
−1 3
−1 2
PM1 = hx1 , x2 , x3 | x3 x1 x3 x−1
1 , x2 x1 x2 x1 , (x3 x2 x1 ) (x2 x1 ) i
en
−1
−3 −1
−1 −1
2 −1
∼ hx1 , x2 , x3 | x3 x1 x3 x−1
1 , x2 x1 x2 x1 , x1 (x2 x3 x2 x3 x2 x3 x2 ) i
en
−1
−1 −1
2 −1
2
∼ hx2 , x3 | x3 (x2 x3 x2 x−1
3 x2 x3 x2 ) x3 (x2 x3 x2 x3 x2 x3 x2 ),
−1
−1 −1
2
2 3
x2 (x2 x3 x2 x−1
3 x2 x3 x2 )x2 (x2 x3 x2 x3 x2 x3 x2 ) i.
So, the extended Nielsen genus of PM1 is at most 2. Since the genus must
be at least the rank of the group, it must be equal to 2.
When inequality (2) is strict for a manifold M 3 , it follows that no cell
decomposition of M 3 can result in a 2-spine K having exactly one 0-cell,
rk(π1 (M 3 )) 1-cells and rk(π1 (M 3 )) 2-cells. From such a spine, a Heegaard
decomposition could be constructed with genus rk(π1 (M 3 )).
Thus, for the Boileau–Zieschang manifold M13 , Theorem 4.2 yields that
en(M13 ) = 2 < hg(M13 )
so that no handle decomposition consisting of exactly one 0-handle, two
1-handles, two 2-handles and one 3-handle exists.
5. A handle presentation for a 4-manifold bounded by 2M1
To obtain information about minimal handle structures for 4-manifolds N
bounded by 2M , we examine handle decompositions of M1? × [−1, 1] using
a handle calculus for handle presentations with no handles of index greater
than 2.
A handle presentation H is normal if the attaching spheres for the 2handles are contained in (∂J) × {−0.75, 0.75}.
See Craggs [3] for a treatment of normal handle presentations, and Craggs
[4, 5] for material concerning algebraic cancellation, linking obstructions and
the free reduction problems.
Definition 5.1. A normal handle presentation H for a 4-manifold N with
boundary 2M∗ is algebraically minimal (relative to the boundary 2M∗ ) provided:
(1) The handle presentation H has no handles of index greater than 2.
(2) All but en(M? ) of the 1-handles can be canceled algebraically.
3
(3) If KH is a 2-complex naturally associated with H, then M∗ & KH .
Note that if H is a handle presentation for M1? × [−1, 1] which is algebraically minimal, then in the absence of any linking obstructions, Theorem
A, Craggs [5] implies that H is reducible to a minimal handle structure for
M1 .
The remainder of this section is devoted to establishing an explicit description of an algebraically minimal handle presentation.
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
401
Theorem 5.2. There exists an algebraically minimal normal handle presentation H for a 4-manifold N with boundary 2M1 .
We calculate a handle presentation H whose associated presentation is
algebraically co-2-collapsible. This will imply that H is an algebraically
minimal handle presentation.
Unless stated otherwise, all handle presentations for M1? × [−1, 1] are
assumed to have handles of index at most two.
We introduce a sequence of admissible operations that will be used extensively in what follows. Suppose that (D, g) is a singular system with
members including Di and Dj . If a push is performed on Di along an arc
β which encounters Dj the resulting system may be modeled by an appropriately chosen admissible system. Figure 2 illustrates one such possibility.
Here, the arc β, and the relator paths rk = Bd(Dk ) ∩ Bd(J) for k = i, j are
illustrated.
To describe the corresponding operations as an admissible sequence of operations, we introduce the following notation: Let (ri , i ) denote the relator
curve within an admissible system (D, g, ), that is, let
(ri , i ) = Bd(g(Di , i )) ∩ Bd(J)
where i ∈ {−1, 0, 1} is the label associated with g(Di ). Let (ri , i ) → (ri , 0i )
denote a change of label corresponding to a level change, and denote an
admissible slide of g(Di , i ) over g(Dj , j ) by the notation (ri , i ) y (rj , j ).
The next lemma states that the configuration of Figure 2 may be obtained
entirely within an admissible context.
Lemma 5.3. Suppose that A is an admissible system having (ri , i ) and
(rj , j ) as relators. Let β be an arc joining a point of (ri , i ) with a point
of (rj , j ) and let N be a regular neighborhood of β in Bd(J) which fails
to intersect the other arcs of the system. Then there exists an admissible
system A0 and a sequence of admissible operations taking A to A0 which
result in the configuration given by Figure 2.
Proof. The result consists of calculating a suitable sequence of admissible
operations. Let (ri , i ), (rj , j ) and β be given for some admissible system
A.
Begin by stabilizing as indicated in Figure 3 to obtain relators (rk , 0) and
(rl , 0) on the generators y1 and y2 respectively. Then the label changes
(rl , 0) → (rl , j ),
(rk , 0) → (rk , i )
allow the nonsingular slides,
(rj , j ) y (rl , j ),
(ri , i ) y (rk , i ).
Since the sliding operations leave their respective targets nonsingular, we
may adjust labels again resulting in the pair
(rl , j ) → (rl , 0), (rk , i ) → (rk , 0).
402
PAUL J. KAPITZA
β
rj
rj
β
ri
ri
rn
Y3
Y2
rj
rl
rl
Y4
Y3
Y1
( rn, 0 )
( r l , −εk)
Y2
( r i , εi )
( r l , −εk)
Y4
( r k, εk)
Y1
ri
Y3
Y2
Y4
( r m, 0 )
rj
r k,
Y1
ri
( r j , εj )
Y2
Y4
rm
rk
Y3
( r k, εk)
Y1
( r j , εj )
( r i , εi )
Figure 2. A singular push as an admissible system.
This situation forms the basis for Step 2, indicated in the upper right hand
corner of Figure 3.
Stabilizing again, we obtain relators (rn , 0) and (rm , 0) on the generating
symbols y3 and y4 . Set (rk , 0) → (rk , k ), where k 6= 0 is some choice of
label, and set (rl , 0) → (rl , −k ), performing a feeler push along the arc β.
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
( rl , 0 )
( rn, 0 )
Y2
Y2
β
Y1
Y3
Y2
( r j , εj )
( r j , εj )
( r i , εi )
( r i , εi )
Y3
( r l , −εk)
Y2
β
Y1
Y1
( r k, εk)
Y4
( r k, 0 )
Y1
( r j , εj )
( r i , εi )
Y4
( r m, 0 )
Y3
( r j , εj )
403
( rn, 0 )
( r l , −εk)
Y2
Y4
( r k, εk)
( r i , εi )
( r l , −εk)
Y1
Y3
Y2
Y4
( r m, 0 )
( r k, εk)
Y1
( r j , εj )
( r i , εi )
Figure 3. Admissible operations to obtain a singular feeler push.
This allows the following sequence of admissible moves:
(rn , 0) → (rn , k ), (rm , 0) → (rm , −k )
(rk , k ) y (rn , k ), (rl , −k ) y (rm , −k )
(rn , k ) → (rn , 0), (rm , −k ) → (rm , 0).
The resulting configuration is illustrated in the lower section of Figure 3
which is the desired result.
Lemma 5.4. There exists an admissible system A1 , having the following
properties:
(1) A1 is admissibly equivalent to the geometric presentation PM1 .
(2) PA1 = hx1 , x2 , x3 , y1 , . . . , y92 | r1 , r2 , . . . , r95 i, where ri is given in
Table 1.
(3) PA1 is extended Nielsen equivalent to P (M1 ).
Proof. Section 6 contains a derivation of the admissible system A1 . Geometric readings are presented at intermediate stages ending in an explicit list
of the relators of the corresponding presentation PA1 . Table 1 represents the
404
PAUL J. KAPITZA
reduced form of this final entry. Each admissible operation induces an extended Nielsen transformation of the original presentation PM1 . Therefore,
the third part of the lemma follows immediately.
Table 1: PA1 with relators {r1 , r2 , . . . , r95 }.
r10
r20
r30
r4+
r5−
r60
r70
r8+
r9−
0
r10
r9−
0
r10
0
r11
+
r12
−
r13
0
r14
0
r15
+
r16
−
r17
0
r18
0
r19
+
r20
−
r21
0
r22
0
r23
+
r24
−
r25
0
r26
0
r27
+
r28
−
r29
0
r30
0
r31
+
r32
−
r33
0
r34
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
x3 x1 x3 x49 x−1
1
−3
x2 x−1
1 x2 x1
y10 y8 y1 y5 y4 y17
−1
−1 −1
y29 y37
y3 y57 y73
y1
−1
y4 y2
y9−1 y2
−1 −1
y16
y3
−1 −1
y6 y18 y8
−1
−1 −1
y7 y5 y33
y22
y6
−1
−1 −1
y7 y5 y33
y22
y6
y7−1
−1
y9
y11
−1
−1 −1 −1
y10
y12 x3 y41 y25
y77 y53
y12
−1 −1
y15
y11
−1
y15
y13
−1 −1 −1
y45 x3 y14 y16
y14
x3 y13
−1
y17
y26 y19
−1
−1
y
x−1
20 y61 y69 y18
2
y20
−1
−1
y65
y85 y21 y19
−1
y23 y21
−1
−1
y24
x2 y89 y81
y22
y24
−1
x2 y23
−1
y28 y26
−1
y27
y25
y27
−1
y30
y28
−1
y32
y30
−1
y31
y29
y31
Continued on next page
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
Table 1 (continued )
0
r35
+
r36
−
r37
0
r38
0
r39
+
r40
−
r41
0
r42
0
r43
+
r44
−
r45
0
r46
0
r47
+
r48
−
r49
0
r50
0
r51
+
r52
−
r53
0
r54
0
r55
+
r56
−
r57
0
r58
0
r59
+
r60
−
r61
0
r62
0
r63
+
r64
−
r65
0
r66
0
r67
+
r68
−
r69
0
r70
0
r71
+
r72
−
r73
0
r74
0
r75
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
−1
y34
y32
−1
y36 y34
−1
y35
y33
y35
−1
y38
y36
−1
y40
y38
−1
y39
y37
y39
−1
y42
y40
−1
y44
y42
−1
y43 y41
y43
−1
y44
y46
−1
y46
y48
−1
y45
y47
y47
−1
y50
y48
−1
y50
y52
−1
y51 y49
y51
−1
y52
y54
−1
y56 y54
−1
y53
y55
y55
−1
y56
y58
−1
y58
y60
−1
y59 y57
y59
−1
y60
y62
−1
y64 y62
−1
y63
y61
y63
−1
y66
y64
−1
y68
y66
−1
y67 y65
y67
−1
y70
y68
−1
y72 y70
−1
y71
y69
y71
−1
y74
y72
Continued on next page
405
406
PAUL J. KAPITZA
Table 1 (continued )
+
r76
−
r77
0
r78
0
r79
+
r80
−
r81
0
r82
0
r83
+
r84
−
r85
0
r86
0
r87
+
r88
−
r89
0
r90
0
r91
+
r92
−
r93
0
r94
0
r95
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
−1
y76
y74
−1
y75 y73
y75
−1
y78
y76
−1
y80 y78
−1
y79
y77
y79
−1
y82
y80
−1
y84
y82
−1
y83
y81
y83
−1
y86
y84
−1
y88 y86
−1
y85
y87
y87
−1
y88
y90
−1
y92
y90
−1
y89
y91
y91
x2 x−1
1 x2 y92
Lemma 5.5. The presentation PA1 (M1 ) presents an algebraically co-2-collapsible complex.
Proof. Given PA1 (M1 ) as presented in Lemma 5.4, we claim that the subset
{r3 , . . . , r95 } is algebraically collapsible on {x1 , x2 , x3 , y1 , . . . , y92 }.
To see this, examine Table 2 which presents the relators from Table 1
according to the following convention: The general entry,
(i)
(i) r(i)
= u(i) x[i] v(i)
[x[i] ]
corresponds to the permutations Γ ∈ S95 and ∆ ∈ S93 so ∆(i) = (i),
Γ(i) = [i]. In addition, inspection of Table 2 demonstrates that ui =
ui (x[i+1] , . . . , x[95] ) and vi = vi (x[i+1] , . . . , x[95] ) for all i = 1, . . . , 93. Lemma 2.14 then directly implies that {r1 , . . . , r95 } forms an algebraically co-2collapsible set on {x1 , x2 , x3 , y1 , . . . , y95 }.
Theorem 5.2. Given M1? × [−1, 1], Lemma 5.4 implies that there exists
an admissible system A1 representing M1? × [−1, 1] whose presentation is
given by PA1 = hx1 , x2 , x3 , y1 , . . . , y92 | r1 , r2 , . . . , r95 i. By Theorem 4.2, the
extended Nielsen genus of PM1 is 2. Therefore, Lemma 5.5 implies that PA1
presents an algebraically co-en(M1 )-collapsible presentation.
From Lemma 2.15 there is an automorphism σ : F → F where σ(r(i) ) =
x[i] , where x[i] is an element of the ordered collection (x1 , x2 , x3 , y1 , . . . , y92 )
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
407
for 1 ≤ i ≤ 93, so that
PA1 = hx1 , x2 , x3 , y1 , . . . , y92 | r1 , r2 , . . . r95 i
en
∼ hx[1] , . . . x[95] | σ(r1 ), σ(r2 ), σ(r(1) ), . . . , σ(r(93) )i
en
∼ hx[1] , . . . x[95] | r10 , r20 , x[1] , . . . , x[93] i.
There exists an admissible system A2 for M1? × [−1, 1] and a sequence of
admissible systems which take A1 to A2 having a presentation
PA2 = hx[1] , . . . x[95] | r10 , r20 , x[1] , . . . , x[93] i.
Table 2: Sequence of collapses in order by relator number
and generator.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
0
r95
+
r92
0
r91
+
r88
0
r87
+
r84
0
r83
+
r80
0
r79
+
r76
0
r75
+
r72
0
r71
+
r68
0
r67
+
r64
0
r63
+
r60
0
r59
+
r56
0
r55
+
r52
0
r51
+
r48
0
r47
+
r44
0
r43
+
r40
0
r39
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
x2 x−1
1 x2 y92
−1
y92 y90
−1
y88
y90
−1
y88 y86
−1
y84
y86
−1
y82
y84
−1
y82 y80
−1
y78
y80
−1
y78 y76
−1
y74
y76
−1
y72
y74
−1
y72
y70
−1
y70 y68
−1
y66
y68
−1
y64
y66
−1
y62
y64
−1
y62 y60
−1
y60
y58
−1
y58 y56
−1
y56
y54
−1
y54 y52
−1
y52
y50
−1
y50 y48
−1
y48
y46
−1
y46
y44
−1
y44
y42
−1
y42 y40
−1
y40
y38
−1
y38 y36
Continued on next
[x1 ]
[y92 ]
[y90 ]
[y88 ]
[y86 ]
[y84 ]
[y82 ]
[y80 ]
[y78 ]
[y76 ]
[y74 ]
[y72 ]
[y70 ]
[y68 ]
[y66 ]
[y64 ]
[y62 ]
[y60 ]
[y58 ]
[y56 ]
[y54 ]
[y52 ]
[y50 ]
[y48 ]
[y46 ]
[y44 ]
[y42 ]
[y40 ]
[y38 ]
page
408
PAUL J. KAPITZA
Table 2 (continued )
+
(30) r36
0
(31) r35
+
(32) r32
0
(33) r31
+
(34) r28
+
(35) r20
(36) r30
−
(37) r13
−
(38) r29
0
(39) r30
−
(40) r45
0
(41) r46
0
(42) r14
−
(43) r57
0
(44) r58
−
(45) r81
0
(46) r82
(47) r8+
−
(48) r21
−
(49) r73
0
(50) r74
−
(51) r65
0
(52) r66
0
(53) r22
0
(54) r10
(55) r4+
−
(56) r33
0
(57) r34
−
(58) r41
0
(59) r42
(60) r70
−
(61) r17
0
(62) r18
−
(63) r49
0
(64) r50
−
(65) r61
0
(66) r62
−
(67) r77
0
(68) r78
(69) r9−
−
(70) r25
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
−1
y36
y34
−1
y34 y32
−1
y32
y30
−1
y30 y28
−1
y28
y26
−1
y26 y19
y17
y10 y8 y1 y5 y4 y17
−1 −1 −1
−1
y77 y53
y12 x3 y41 y25
y10
−1
y27 y25
y27
−1
y43
y41
y43
y12
−1
y53
y55
y55
−1
y77
y79
y79
−1
y8
y6−1 y18
−1
−1 −1
y18
x2 y20 y61 y69
−1
y71 y69
y71
−1
y61
y63
y63
y20
y6
−1 −1
−1
y1
y3 y57 y73
y29 y37
−1
y29
y31
y31
−1
y37
y39
y39
−1 −1
y16
y3
−1 −1 −1
y45 x3 y14 y16
y14
−1
y47
y45
y47
−1
y59
y57
y59
−1
y75
y73
y75
−1 −1
−1
y22
y7 y5 y33
−1
−1
y24
x2 y89 y81
y22
Continued on next
[y36 ]
[y34 ]
[y32 ]
[y30 ]
[y28 ]
[y26 ]
[y17 ]
[y10 ]
[y25 ]
[y27 ]
[y41 ]
[y43 ]
[y12 ]
[y53 ]
[y55 ]
[y77 ]
[y79 ]
[y8 ]
[y18 ]
[y69 ]
[y71 ]
[y61 ]
[y63 ]
[y20 ]
[y6 ]
[y1 ]
[y29 ]
[y31 ]
[y37 ]
[y39 ]
[y3 ]
[y16 ]
[y14 ]
[y45 ]
[y47 ]
[y57 ]
[y59 ]
[y73 ]
[y75 ]
[y5 ]
[y22 ]
page
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
Table 2 (continued )
0
(71) r26
−
(72) r93
0
(73) r94
−
(74) r85
0
(75) r86
0
(76) r11
−
(77) r37
0
(78) r38
(79) r5−
(80) r60
+
(81) r12
0
(82) r15
+
(83) r16
0
(84) r19
0
(85) r23
−
(86) r69
0
(87) r70
−
(88) r89
0
(89) r90
+
(90) r24
0
(91) r27
−
(92) r53
0
(93) r54
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
y24
−1
y91
y89
y91
−1
y83
y81
y83
y7−1
−1
y35
y33
y35
y4 y2−1
y9−1 y2
−1
y11
y9
−1 −1
y15
y11
−1
y15
y13
x3 y13
−1
−1
y85 y21 y19
y65
−1
y67 y65
y67
−1
y85
y87
y87
−1
y21
y23
−1
x2 y23
−1
y51
y49
y51
409
[y24 ]
[y89 ]
[y91 ]
[y81 ]
[y83 ]
[y7 ]
[y33 ]
[y35 ]
[y4 ]
[y2 ]
[y9 ]
[y11 ]
[y15 ]
[y13 ]
[y19 ]
[y65 ]
[y67 ]
[y85 ]
[y87 ]
[y21 ]
[y23 ]
[y49 ]
[y51 ]
Let H be the handle presentation for M1? × [−1, 1] whose associated presentation is given by PA2 . Then with 2 = en(PA1 ) = en(PA2 ), and for
3 ≤ j ≤ 95 where rj = r(i−2) , rj freely reduces to x[i] after the change of
basis, so that H is algebraically minimal.
6. Derivation of an admissible system for M1
This section details a calculation of a 2-complex spine and corresponding
2-handle presentation for M1? ×[−1, 1] following the derivation of Montesinos
presented in Theorem 4.2. The calculation consists of generating a series of
admissible systems to produce a 2-complex whose associated presentation
is algebraically co-2-collapsible. Plate B1 shows the 2-spine presentation
given by the first equation of Theorem 4.2, and commences by performing
a nonsingular slide. The corresponding reading is recorded below it.
The diagrams which follow Plate B1 represent the effect of the projection
maps
p+ : J × {+1} → J × {0}
p− : J × {−1} → J × {0}
and
410
PAUL J. KAPITZA
and are presented using an admissible representation. A high resolution
collection of plates is available at [7] in addition to those presented here.
Each diagram is accompanied by a table at each stage of the calculation
which corresponds to the relators of the complex whose presentation is given
by
Pn = hx1 , x2 , x3 , y1 , y2 , . . . , yn | r1 , r2 , . . . rn+3 i.
−3
where r1 = x3 x1 x3 x−1
1 and r2 = x2 x1 x2 x1 are the relators of PM1 as given
in Equation (3). The generating symbols corresponding to the 1-handles are
taken from the set {x1 , x2 , x3 , y1 , y2 , . . . }, where the generators {x1 , x2 , x3 }
correspond to the generators of π1 (M1 ) and {y1 , y2 , . . . } are introduced by
repeated stabilizations as in Lemma 5.3.
To convey the information associated with the admissible system at each
stage, we adopt the following notational conventions:
(1) If ri corresponds to a 2-handle attachment in J˙ × [ 21 , 1], it will be
recorded as r+ . Similarly, a 2-handle attachment in J˙ × [−1, − 1 ]
i
2
will be recorded as ri− and those nonsingular members of the disk
system will be denoted as ri0 . In terms of the admissible disk system
structure this implies that ri+ = (ri , +1), ri− = (ri , −1), and ri0 =
(ri , 0).
(2) The basepoint of each relator curve ri is denoted as ?i . This symbol
is located near the line segment denoting the starting position of the
associated reading (the initial segment of ri ).
(3) For noninitial segments, the mth line segment of curve ri is labeled
i.m. If i.k denotes the terminal segment of ri , additionally this
segment will contain the basepoint. When space is available, the
terminal segment may contain the symbols i.k and i.1 in addition
to the basepoint marker ?n . However, the terminal segment and
the segment containing the basepoint are always assumed to be the
same.
(4) The admissible slide construction of Lemma 5.3 is used to realize
2-handle slides geometrically in M1? × [−1, 1]. Segments which correspond to the demonstration of Theorem 4.2 are underlined as they
are first encountered in the derivation.
The plates which follow have been carefully checked for accuracy. However, it remains possible that a mislabeled segment or an out of sequence
segment numbering has been overlooked. In this case, the reader should
proceed with the logical indexing that the particular situation calls for.
Reading for Plate B1.
r10 = x3 x1 x3 x−1
1
−3
r20 = x2 x−1
1 x2 x1
−1 2
3
r30 = (x3 x2 x−1
1 ) (x2 x1 )
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
411
1.4
3.6
1.2
3.3
3.13
2.5
2.4
+
3.11
X1
2.2
+
+
X2
1.4
X3
3.9
1.3
1.2
3.8
3.5
3.2
2.5
2.7
2.6
*3
2.1
*2
X1
3.10
2.3
X2
3.8
X3
3.5
3.2
3.12
2.5
2.5
3.14
1.1
3.14
3.4
3.4
3.7
3.7
1.5
1.5
Figure 4. Plate B1.
Reading for Plate B2.
r10 = x3 x1 x3 x−1
1
−3
r20 = x2 x−1
1 x2 x1
−1 2
−1 −2 2
3
r30 = (x3 x2 x−1
1 ) (x2 x1 ) (x2 x1 ) x1
Reading for Plate B3.
r10 = x3 x1 x3 x−1
1
−3
r20 = x2 x−1
1 x2 x1
2
r30 = (x3 x2 x−1
1 ) x3 x2 x1
*1
1.3
412
PAUL J. KAPITZA
1.4
1.2
3.6
3.3
3.13
3.15
2.5
2.4
+
3.11
X1
3.20
3.17
2.2
+
+
X2
1.4
X3
3.9
1.3
3.19
1.2
3.8
3.5
3.2
2.5
*3
2.6
2.7
3.18
-
*2
2.1
3.10
X1
3.16
2.3
-
X2
3.8
-
3.5
X3
3.2
3.12
2.5
3.14
2.5
3.14
3.14
3.14
3.4
1.1
3.4
3.7
3.7
1.5
1.5
Figure 5. Plate B2.
Reading for Plate B4.
r10 = x3 x1 x3 x−1
1
−3
r20 = x2 x−1
1 x2 x1
−1
r30 = x3 x2 x−1
1 x3 y1 x2 x1 x3 y4 x2 x1
r4+ = y3 y1−1
r5− = y4 y2−1
r60 = y2
−1
−1 −1
r70 = (x−1
3 x1 x3 x1 )y3
*1
1.3
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
413
1.4
3.6
1.2
3.3
2.4
2.5
+
+
+
X1
X2
2.2
1.4
X3
3.10
1.3
3.9
1.2
3.8
3.5
3.2
*3
2.7
2.6
*2
2.1
-
-
-
X1
X3
X2
2.3
1.1
2.5
*1
3.4
3.4
3.7
3.7
1.5
Figure 6. Plate B3.
Reading for Plate B5.
r10 = x3 x1 x3 x−1
1
−3
r20 = x2 x−1
1 x2 x1
r30 = x3 x2 y1 x2 x−1
1 x3 y4 x2 x1
−1
−1
r4+ = x−1
1 x3 x3 x1 y3 y1
r5− = y4 y2−1
r60 = y2
−1 −1
r70 = x−1
3 x 1 y3
1.3
3.10
414
PAUL J. KAPITZA
7.4
7.4
1.4
7.3
7.3
3.7
1.2
3.3
7.1
2.4
2.5
+
+
+
X1
X2
2.2
1.4
X3
3.12
1.3
7.4
3.11
1.2
*7
3-
4-
7.5
*5
3+
4.2
4+
2+
*6
3.9
*4
27.5
3.5
1-
3.2
5.2
1+
3.10
*3
3.12
3.6
2.7
2.6
*2
2.1
-
-
-
X1
X3
X2
2.3
1.1
2.5
*1
3.4
3.4
3.8
3.8
7.2
7.2
Figure 7. Plate B4.
Reading for Plate B6.
r10 = x3 x1 x3 x−1
1
−3
r20 = x2 x−1
1 x2 x1
r30 = x3 x2 y8 y1 x2 y5 x−1
1 x 3 y4 x 2 x 1
−1
−1
r4+ = x−1
1 x 3 x 3 x 1 y3 y1
r5− = y4 y2−1
r60 = y2
−1 −1
r70 = x−1
3 x1 y3
r8+ = y6−1 y8
r9− = y7−1 y5
0
r10
= y6
−1
−1 3
0
r11
= (x−1
2 x1 x2 x1 )y7
1.5
1.3
7.4
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
415
7.2
7.2
4.5
1.4
*7
3-
7.1
1.2
*4
1-
3.3
3.5
4.3
2.5
2.4
+
+
+
X1
X2
2.2
X3
1.4
3.10
1.3
7.2
3.9
1.2
42-
7.3
7.3
3.7
*5
4.3
3+
4.6
2+ *6
3.2
2
4+ 5.
1+
3.8
*3
3.4
*2
2.7
2.6
X3
-
-
X1
X2
2.3
1.3
7.2
1.1
2.5
3.6
3.10
2.1
4.2
4.2
3.6
4.4
*1
4.4
1.5
Figure 8. Plate B5.
Reading for Plate B7.
r10 = x3 x1 x3 x−1
1
−3
r20 = x2 x−1
1 x2 x1
r30 = x3 x2 y8 y1 y5 x3 y4 x2 x1
−1
−1
r4+ = x−1
1 x3 x3 x1 y3 y1
r5− = y4 y2−1
r60 = y2
−1 −1
r70 = x−1
3 x1 y3
r8+ = y6−1 y8
−1
−1
r9− = x−1
1 y7 x2 x2 y5 x1
0
r10
= y6
0
2 −1
r11
= x1 x−1
2 x 1 y7
416
PAUL J. KAPITZA
7.2
7.2
1.4
4.5
*4 1-
3.4
2.5
5+
*9
6-
7+
+
3.3
8.2 89.2
*11
7-
4.3
+
+
2.4
X1
7.1
1.2
3.6
56+ *10
8+
*8
11.7
11.6
*7
33.7
1.4
X3
X2
11.3
2.2
3.12
1.3
7.2
3.11
1.2
427.3
7.3
3.9
*5
4.3
3+
4.6
4+
6
2+ *
3.2
5.2
1+
11.6
3.10
2.7
2.6
*2
-
2.1
X3
-
11.4
X1
X2
2.3
11.2
1.1
2.5
11.6
11.6
4.2
4.2
*1
3.8
3.8
4.4
4.4
Figure 9. Plate B6.
Reading for Plate B8.
r10 = x3 x1 x3 x−1
1
−3
r20 = x2 x−1
1 x2 x1
r30 = x3 y10 x2 y8 y1 y5 x3 y4 x2 x1
−1
−1
r4+ = x−1
1 x 3 x 3 x 1 y3 y1
r5− = y4 y2−1
r60 = y9−1 y2
−1 −1
r70 = x−1
3 x1 y3
r8+ = y6−1 y8
−1
−1
r9− = x−1
1 y7 x2 x2 y5 x1
0
r10
= y6
0
2 −1
r11
= x1 x−1
2 x1 y7
3.12
*3
3.5
11.5
1.5
1.3
7.2
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
417
7.2
7.2
1.4
4.5
*7
39.7
*4 1-
2.5
11.5
7.1
1.2
3.4
*9
8+
*8
6+ *10
3.3
8.2 8-
6-
9.1
9.4
+
X3
X2
11.2
1.4
+
+
2.4
X1
4.3
2.2
9.4
3.10
1.3
7.2
3.9
1.2
3.7
427.3
4.3
4.6
6
2+ *
3.2
5.2
4+
7.3
*5
3+
1+
3.8
11.5
3.5
11.4
2.7
7-
X3
X2
2.3
*11
-
-
11.3
X1
9.3
2.5
2.5
9.2
11.5
7+
4.2
1.1
4.2
9.6
3.10
9.5
2.1
*2
-
2.6
*3
5-
3.6
5+
*1
4.4
4.4
1.5
Figure 10. Plate B7.
−1
+
y9
= y11
r12
−
−1
r13
= y12
y10
0
r14
= y12
−1 −1
−1
0
r15
= (x−1
3 x1 x3 x1 )y11
Reading for Plate B9.
r10 = x3 x1 x3 x−1
1
−3
r20 = x2 x−1
1 x2 x1
r30 = y10 x2 y8 y1 y5 y4 x2 x1
−1
2
r4+ = x−2
1 x 1 y3 y1
−1
r5− = y4 x1 x−1
1 y2
r60 = y9−1 y2
−1
r70 = x−1
3 y3
r8+ = y6−1 y8
1.3
7.2
418
PAUL J. KAPITZA
1.4
4.5
15.5
*7
3-
7.1
15.3
9.7
1-
*4
*9
3.5
8+
*10
6+
8.2
6-
1.2
8-
3.4
*8
4.3
2.5
9.4
9.1
+
2.4
X1
11.5
11.2
2.2
+
+
X2
X3
1.4
9.4
15.1
3.11
15.2
1.3
7.2
3.10
15.3
1.2
3.8
4*5
27.3
7.3
4.3
3+
4.6
2+
*6
5.2
4+
6.2
911.5
3.6
12-
5*2
2.7
11.4
2.6
X1
2.3
*11
*12
12+
X2
3.3
12.2
11-
2.1
11.3
13.2
9+
9.5
3.2
10-
1+
3.9
*15
*14
-
11+
*3
X3
10+ *13
15.5
9.3
7-
2.5
9.2
11.5
7+
*1
4.2
4.2
9.6
4.4
1.5
15.4
7.2
1.3
15.2
3.11
1.5
15.4
7.2
1.3
15.2
3.11
Figure 11. Plate B8.
−1
−1 −1
2
r9− = x−2
1 y7 x 1 x 2 x 2 x 1 y5 x 1
0
r10
= y6
−1
0
r11
= x−1
2 x 1 y7
−1
+
y9
= y11
r12
−1 −1 −1
−
x3 x1 x1 y10
r13
= y12
0
r14
= y12
−1
0
r15
= x−1
3 x1 y11
Reading for Plate B10.
r10 = x3 x1 x3 x−1
1
−3
r20 = x2 x−1
1 x2 x1
r30 = y10 x2 y8 y1 y5 y4 x2 x1
−1
2
r4+ = x−2
1 x 1 y3 y1
−1
r5− = y4 x1 x−1
1 y2
r60 = y9−1 y2
4.4
3.7
5+
1.5
15.4
7.2
1.3
15.2
3.11
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
419
9.2
1.4
15.3
4.5
15.3
*7
3-
7.1
9.11
3.4
*4
1-
1.4
2.5
15.1
*9
8+
*10
6+
8.2
6-
1.2
8-
3.3
*8
9.1
2.5
+
9.4
X2
*11
7-
+
+
2.4
X1
9.2
*3
9.6
2.2
X3
10-
3.9
13.5
9.6
4.2
13.2
9.10
4.4
13.3
1.3
5.3
9.8
5-
5.3
15.3
2.6
11.3
3.8
3.8
3.5
13.4
13.4
4.3
1+
4.3
9.9
13.6
5+
*5
4-
6.2
2-
*15
12.2
9-
11-
4.3
4.6
4.3
13.4
13.4
3+
7+
1.2
5.4
5.4
7.2
9.3
9.7
9.7
5.2
5.2
2+
*6
4+
2.5
3.7
9.2
13.2
4.2
129.7
4.4
*14
*12
2.7
2.6
-
-
11.2
X1
12+
2.1
*2
X2
2.3
9.10
9+
3.2
-
10+ *13
11+
13.3
X3
15.3
9.5
2.5
2.5
9.2
9.2
1.5
4.4
*1
4.2
4.2
9.10
9.10
4.4
1.5
15.2
15.2
7.2
7.2
1.3
1.3
13.3
13.3
4.4
9.10
4.2
4.4
9.10
4.2
Figure 12. Plate B9.
−1 −1
r70 = x−1
3 y16 y3
r8+ = y6−1 y8
−1
−1 −1
2
r9− = x−2
1 y7 x 1 x 2 x 2 x 1 y5 x 1
0
r10
= y6
−1
0
r11
= x−1
2 x 1 y7
+
−1
r12
= y11
y9
−
−1 −1 −1
r13
= y12
x3 x1 x1 y10
0
r14
= y12
−1
−1
0
r15
= x−1
3 y15 x1 y11
+
−1
r16
= y13
y15
−
−1
r17
= y14
y16
0
r18
= y14
0
r19
= (x3 x−1
1 x3 x1 )y13
Reading for Plate B11.
r10 = x3 x1 x3 x−1
1
420
PAUL J. KAPITZA
15.4
1.4
2.5
15.3
19.2
3-
4.5
*7
9.2
1.4
15.1
9.11
1.2
*4
3.4
1-
*9
8+
*10
6+
8.2
6-
19.4
3.3
8-
7.1
*8
2.5
9.1
9.4
X2
*11
7-
19.2
+
+
2.4
X1
X3
2.2
*3 / 3.9
10-
9.6
+
9.2
13.5
9.6
4.2
13.2
9.10
4.4
13.3
1.3
19.5
5.3
9.8
5-
15.3
2.6
5.3
11.3
3.8
3.5
3.8
13.4
13.4
4.3
1+
4.3
9.9
5+
4.3
13.4
1.2
19.4
4.6
3+
7+
3.6
4-
*15
*5
6.2
2-
4.3
9-
12.2
11-
13.4
5.4
5.4
7.3
9.7
9.7
5.2
9.3
5.2
2+
*6
4+
2.5
3.7
9.2
13.2
4.2
129.7
2.6
*2
-
12+
2.1
-
11.2
X1
X2
2.3
4.4
*14
*12
2.7
9.10
9+
3.2
13.3
-
10+ *13
11+
X3
15.3
1.3
7.2
9.5
16+
*17
2.5
15.2
2.5
13+
15+
*16
9.2
9.2
*19
14+
*1
4.2
4.2
19.3
9.10
*18
9.10
4.4
4.4
1419.3
1.5
1.5
15.3
15.3
15-
16-
7.3
7.3
19.5
19.5
1.3
1.3
13.3
13.3
4.4
4.4
9.10
4.2
9.10
4.2
Figure 13. Plate B10.
−3
r20 = x2 x−1
1 x2 x1
r30 = y10 y8 y1 y5 y4 x2 x1
−1
2
r4+ = x−2
1 x1 y3 y1
−1
r5− = y4 x1 x−1
1 y2
r60 = y9−1 y2
−1 −1
r70 = x−1
3 y16 y3
r8+ = y6−1 x−1
2 y8
−1 −1
2
r9− = x−2
1 y7 x1 x1 y5 x1
0
r10
= y6
−1
0
r11
= x−1
2 x 1 y7
+
−1
r12
= y11
y9
−
−1 −1 −1
r13
= y12
x3 x1 x1 y10
0
r14
= y12
−1
−1
0
r15
= y15
x1 y11
17.2
16.2
13-
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
421
9.2
*19
19.1
15.3
3-
4.5
1.4
*7
16.1
9.9
*3
19.5
616.3
+
+
2.4
X1
3.8
2.5
13+
19.3
8.2
*10
6+
*8
9.1
2.5
9.2
10-
8+
7.1
1.2
*9
3.3
1-
*4
15.3
1.4
9.4
7-
+
8.2
X3
X2
*11
2.2
13.5
4.2
13.2
9.8
4.4
13.3
1.3
19.4
5.3
9.6
5-
5.3
15.3
2.6
11.3
3.7
3.4
3.7
13.4
13.4
4.3
1+
4.3
9.7
5+
4.3
4.6
3.5
4-
*16
*5
6.2
2-
4.3
13.4
9-
12.2
*15
11-
15+
13.4
1.2
3+
19.3
7+
5.4
5.4
9.5
7.3
9.5
5.2
5.2
9.3
2+
*6
4+
2.5
3.6
9.2
13.2
4.2
12-
9.8
9+
4.4
*14
*12
2.7
2.6
*2
X1
X2
11.2
2.3
12+
9.5
2.1
8.3
8-
X3
3.2 10+ *13
11+
13.3
15.3
1.3
16.2
7.2
9.5
16+
*17
16.2
2.5
2.5
9.2
9.2
*1
14+
4.2
4.2
*18
9.8
19.2
9.8
4.4
4.4
1.5
1419.2
1.5
17.2
15.2
15-
15.2
1316.4
7.3
7.3
19.4
1.3
16-
19.4
1.3
13.3
13.3
4.4
4.4
9.8
4.2
9.8
4.2
Figure 14. Plate B11.
−1
+
r16
= x−1
3 x3 y13 y15
−1
−
y16
= y14
r17
0
r18
= y14
0
r19
= x−1
1 x3 x1 y13
Reading for Plate B12.
r10 = x3 x1 x3 x−1
1
−3
r20 = x2 x−1
1 x2 x1
r30 = y10 y8 y1 y5 y4 y17 x2 x1
−1
2
r4+ = x−2
1 x1 y3 y1
−1
r5− = y4 x1 x−1
1 y2
r60 = y9−1 y2
−1 −1
r70 = x−1
3 y16 y3
−1
r8+ = y6−1 x−1
2 y18 y8
−1 −1
2
r9− = x−2
1 y7 x1 x1 y5 x1
422
PAUL J. KAPITZA
9.2
2.5
15.3
19.1
15.3
1.4
3-
4.5
*7
1.4
7.1
9.9
*19
16.1
*9
3.3
*4
1-
8+
*8
6-
19.3
16.3
2.4
X1
9.4
7-
+
+
23.6
+
X2
*11
23.2
3.9
*3
8.2
9.1
2.5
23.5
9.2
10-
13+
1.2
*10
6+
X3
2.2
13.5
4.2
13.2
9.8
4.4
13.3
1.3
19.4
5.3
9.6
5-
5.3
15.3
2.6
23.4
11.3
3.8
3.4
3.8
13.4
13.4
4.3
1+
4.3
9.7
5+
3.5
4.3
13.4
1.2
4.6
3+
4-
*16
*5
6.2
2-
4.3
9-
12.2
*15
11-
15+
13.4
13.2
15.3
19.3
7+
5.4
5.4
7.3
9.5
9.5
5.2
5.2
9.3
2+
12-
*6
9+
4+
2.5
23.5
*14
*12
12+
3.6
9.2
9.2
3.7
4.2
3.2 10+ *13
8-
9.8
11+
2.7
23.4
2.6
*2
2.1
23.3
X1
11.2
2.3
X2
4.4
15.3
9.5
17+
*20
8.3
18+ *21
20+ *22
20- 21.2
13.3
X3
8.4
17-
18-
1.3
16.2
7.2
23.1
20.2
23.7
19+
19-
16+
*17
*23
16.2
9.5
2.5
2.5
14+
*1
19.2
4.2
9.8
23.5
23.5
9.2
4.2
*18
9.2
9.8
4.4
4.4
1419.2
17.2
1.5
1.5
15.2
15.2
15-
7.3
7.3
19.4
1.3
16.4
13-
1619.4
1.3
13.3
13.3
4.4
9.8
4.2
4.4
9.8
4.2
Figure 15. Plate B12.
0
r10
= y6
−1
0
r11
= x−1
2 x1 y7
−1
+
y9
= y11
r12
−1 −1 −1
−
x3 x1 x1 y10
= y12
r13
0
r14
= y12
−1
−1
0
x1 y11
r15
= y15
+
−1
r16
= x−1
3 x3 y13 y15
−
−1
r17
= y14
y16
0
r18
= y14
0
r19
= x−1
1 x3 x1 y13
+
−1
r20
= y19
y17
−
−1
r21
= y20
y18
0
r22
= y20
−1
0
3 −1
r23
= (x1 x−1
2 x1 x2 )y19
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
423
9.2
2.5
15.3
1.4
15.3
1.4
17.3
17.3
13.4
13.4
4.5
4.5
9.9
9.9
4.3
4.3
13.6
13.6
17.1
3-
4.7
17.3
16+
*7
*17
9.11
*4
3.3
1-
8+
23.6
+
X1
9.2
6-
18+
9.5
X2
*11
7-
+
+
20.1
2.4
19+
23.2
3.8
*3
8.2
*10
6+
*8
19.2
9.1
2.5
23.5
10-
1.2
*21
*9
X3
2.2
13.7
4.2
13.2
9.10
4.6
13.3
1.3
19.3
5.3
9.7
*20
5.3
5-
2.6
15.3
11.3
9.3
17+
3.7
3.7
3.4
13.6
4.3
13.6
4.3
1+
9.9
9.9
4.5
13.4
1.2
19.2
4.5
13.4
1.2
19.2
4.8
3+
13.2
9.6
5.2
7.2
15.3
5.4
5.4
9.6
5.2
2+
12-
*6
9+
4+
2.5
23.5
*14
*12
3.6
12+
9.2
9.2
9.10
11+
9.3
*2
2.7
2.6
2.1
23.3
X1
23.4
7+
9.4
11.2
15.3
4.6
9.6
X2
2.3
4.2
3.2 10+ *13
8-
3.6
20+ *22
21.2
20- 21.3
13.3
X3
8.3
17-
18-
1.3
20.2
23.1
19-
17.4
*23
9.6
2.5
2.5
*1
23.5
4.2
14+
23.5
9.2
9.2
9.10
*18
4.2
9.10
14-
4.6
4.6
17.5
17.2
15-
13.5
4.4
9.8
5+
3.5
4-
*5
2-
6.2
9-
12.2
*15
11-
15+
*16
13+
*19
19.1
16.2
13-
16-
4.4
13.5
17.2
1.5
1.5
15.2
15.2
7.2
7.2
19.3
19.3
1.3
1.3
13.3
13.3
4.6
4.6
9.10
9.10
4.2
4.2
Figure 16. Plate B13.
Reading for Plate B13.
r10 = x3 x1 x3 x−1
1
−3
r20 = x2 x−1
1 x2 x1
r30 = y10 y8 y1 y5 y4 y17 x1
−1
3
r4+ = x−3
1 x1 y3 y1
−1
r5− = y4 x1 x−1
1 y2
r60 = y9−1 y2
−1 −1
r70 = y16
y3
−1
r8+ = y6−1 y18
y8
−1 −1
3
r9− = x−3
1 y7 x1 x1 y5 x1
0
r10
= y6
424
PAUL J. KAPITZA
15.3
1.4
15.3
9.2
2.5
1.4
17.3
17.3
13.4
4.5
9.10
4.3
13.6
13.4
4.5
9.10
4.3
13.6
17.1
4.7
3-
17.3
16+
*7
*17
9.12
1-
*4
27.3
2.5
8+
6+
*8
23.7
X1
9.2
18+
6-
27.2
9.6
X2
27.2
7-
*11
23.2
+
+
20.1
2.4
19+
3.8
13.7
10-
19.2
8.2
*10
9.1
+
23.6
*3
1.2
*21
*9
3.3
X3
2.2
27.6
4.2
13.2
9.11
4.6
13.3
1.3
19.3
5.3
9.8
5-
*20
5.3
15.3
2.6
23.5
27.4
9.3
17+
3.7
3.7
3.4
13.6
4.3
13.6
4.3
1+
9.10
9.10
4.5
13.4
1.2
19.2
4.5
13.4
1.2
19.2
4.8
3+
13.2
9.7
5.2
7.2
15.3
5.4
5.4
9.7
5.2
2+
12-
*6
27.3
9+
4+
2.5
23.6
*14
*12
3.6
12+
9.2
9.2
4.2
3.2 10+ *13
8-
9.11
11+
4.6
15.3
27.5
9.3
X1
2.6
27.4
*2
2.7
-
23.5
23.4
2.1
21+
9.4
7+
22+
24+
*26
*25
22-
24- 25.2
9.7
-
3.6
X2
23.3
21-
*24
11.2
20+ *22
21.2
9.5
23+
23-
27.3
20.2
2.3
17.4
*23
27.3
2.5
23.6
4.2
1.3
*27
2.3
23.1
9.7
2.5
X3
18-
19-
24.2
27.7
13.3
-
8.3
17-
20- 21.3
9.2
9.11
4.6
*1
14+
23.6
9.2
4.2
9.11
4.6
*18
1417.5
17.2
15-
13.5
16.2
4.4
9.9
5+
3.5
4-
*5
2-
6.2
9-
12.2
*15
11-
15+
*16
13+
*19
19.1
13-
16-
4.4
13.5
17.2
1.5
1.5
15.2
15.2
7.2
7.2
19.3
1.3
19.3
1.3
13.3
13.3
4.6
4.6
9.11
9.11
4.2
4.2
Figure 17. Plate B14.
−1
0
r11
= x−1
2 y7
−1
+
y9
= y11
r12
−
−1 −1 −2 2
r13
= y12
x3 x1 x1 y10
0
r14
= y12
−1
−1
0
r15
= y15
x1 y11
+
−1
r16
= y13
y15
−
−1 −1
r17
= x−1
1 x1 x3 y14 y16
0
r18
= y14
0
r19
= x3 x1 y13
+
−1
r20
= y19
y17
−
−1
r21
= x−1
2 y20 y18
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
425
9.2
9.2
23.13
23.13
2.5
25.4
2.5
25.4
23.9
25.8
23.9
25.8
13.1
13.1
4.13
4.13
21.9
21.9
23.3
23.3
21.5
4.9
21.5
4.9
13.5
13.5
1.4
1.4
1.4
17.3
17.3
13.8
13.8
4.5
4.5
9.10
9.10
4.3
4.3
13.10
13.10
17.1
17.1
4.7
17.3
3-
16+
*7
*17
1.4
9.12
1.2
*9
21.12
*21
19.2
3.3
8+
1-
*4
8.2
*10
6+
*8
9.1
23.14
20.1
19+
25.8
23.9
2.4
+
25.4
2.5
25.3
3.8
13.11
4.2
X2
23.8
25.9
+
+
25.3
23.8
X1
23.13
9.2
*3
10-
18+
6-
25.9
X3
21.1
9.11
4.6
13.7
1.3
17.5
23.6
13.6
13.4
9.6
4.10
*11
21.6
7-
23.2
27.2
23.6
21.8
4.12
9.3
2.2
24.3
*24
13.2
27.2
21+
23+
13.6
9.8
*20
12-
5-
25.7
23.10
25.7
23.10
25.5
25.5
2.6
2.6
23.12
*14
17+
23.12
12+
9.3
3.7
3.7
3.4
13.10
13.10
4.3
4.3
9.10
9.10
4.5
13.8
1+
4.5
13.8
1.2
1.2
17.6
4.14
13.3
4.11
21.7
21.10
23.4
21.4
23.1
4.8
17.7
*23
23.15
20.2
19-
3.6
17-
*6
5.2
4+
15.2
*12
9+
2+
16.2
15-
11+
13-
21.7
3+
4.11
13.3
17.6
7.2
1625.8
25.8
23.9
23.9
25.4
*13
25.4
2.5
2.5
23.13
23.13
17.7
9.7
9.7
3.2
4.2
10+
4.2
8-
25.6
4.12
-
4.12
21.8
21.8
23.2
21.6
4.10
21-
23-
*2
23.7
11.2
7+
*22
20+
20-
8.3
18-
13.7
X3
21.3
21.2
13.7
-
8.3
X2
2.1
2.7
24.2
9.4
9.11
4.6
9.7
-
27.1
25.6
X1
23.2
21.6
4.10
13.4
*27
24.4
4.6
4.6
9.7
13.2
23.11
9.11
9.11
21.3
13.2
9.2
9.2
13.12
21.11
23.5
1.3
13.4
*26
*25
22+
9.5
24+
22-
24-
25.10
25.2
17.5
23.7
24.2
25.2
17.4
2.3
2.3
23.5
23.5
9.7
9.7
17.5
25.8
25.8
23.9
23.9
25.4
25.4
*1
14+
2.5
2.5
23.13
23.13
*18
9.2
9.2
4.2
4.2
9.11
9.11
4.6
4.6
17.2
13.9
4.4
9.9
6.2
*5
3.5
5+
4-
9-
2-
*15
12.2
*16
13+
15+
11-
19.1
*19
4.4
13.9
17.2
17.5
1.5
1.5
14-
18.1
18.1
1.3
1.3
13.7
13.7
4.6
4.6
9.11
9.11
4.2
4.2
Figure 18. Plate B15.
9.2
9.2
23.13
23.13
95.1
2.5
25.4
25.4
23.9
25.8
95.1
23.9
25.8
13.1
13.1
4.13
4.13
21.9
21.9
23.3
23.3
21.5
21.5
4.9
4.9
13.5
13.5
1.4
1.4
1.4
17.3
17.3
13.8
13.8
4.5
4.5
9.10
9.10
4.3
4.3
13.10
13.10
17.1
17.1
4.7
3-
16+
R7
25+
R17
29+
R29
9.12
26+
R9
21.12
3.3
8+
1R4
25.3
3.8
X2
23.8
25.9
25.9
9.6
7-
R11
23.6
26-
+
25.3
23.4
X1
23.13
R37
R35
32+
34+
31+
R36
28.2
38+
R40
32.2
34-
29-
25-
36.2
R44
38-
40.2
33-
46+
44.2
50+
R52
R55
54+
45-
2.2
1.3
17.5
13.4
95.4
4.10
1.2
21.6
19.2
23.2
+
27.2
21.8
9.3
24.3
R24
13.2
27.2
21+
17.3
X3
4.12
1.4
23+
13.6
9.8
R20
12-
5-
25.7
23.10
25.7
23.10
25.5
25.5
95.2
2.6
2.6
23.12
R14
17+
23.12
12+
9.3
3.7
3.7
3.4
13.10
13.10
4.3
9.10
4.3
9.10
4.5
13.8
1+
4.5
13.8
1.2
1.2
17.6
4.14
13.34.11
21.7
21.10
23.4
21.4
23.1
4.8
17.7
R23
23.15
20.2
19-
3.6
17-
R6
5.2
4+
15.2
R12
9+
2+
15-
11+
16.2
13-
21.7
3+
4.11
13.3
17.6
7.2
1625.8
25.8
23.9
25.4
23.9
25.4
95.7
R13
95.7
2.5
2.5
23.13
23.13
17.7
9.7
9.7
3.2
4.2
10+
8-
9.7
4.12
-
4.12
21.8
21.8
23.2
23.11
21.6
4.10
-
R27
24.4
21-
23-
27.1
25.6
X1
23.2
21.6
4.10
13.4
R2
2.1
23.7
11.2
7+
9.7
20+
R22
20-
18-
13.7
X3
21.3
21.2
13.7
-
8.3
X2
95.3
2.7
24.2
9.4
9.11
4.6
4.6
4.6
13.2
25.6
4.2
9.11
9.11
21.3
13.2
9.2
9.2
13.12
21.11
23.5
8.3
1.3
13.4
R26
R25
22+
24+
9.5
22-
24-
25.10
25.2
17.5
23.7
24.2
25.2
17.4
95.5
2.3
2.3
23.5
23.5
9.7
9.7
17.5
25.8
25.8
23.9
23.9
25.4
25.4
R1
95.7
14+
95.7
2.5
2.5
23.13
23.13
R18
9.2
9.2
4.2
4.2
9.11
9.11
4.6
4.6
17.2
13.9
4.4
9.9
6.2
R5
3.5
5+
4-
2-
9-
R15
12.2
11-
R16
15+
13+
R19
19.1
4.4
13.9
17.2
17.5
1.5
1.5
14-
18.1
18.1
1.3
1.3
13.7
13.7
4.6
4.6
9.11
9.11
4.2
54-
4.2
Figure 19. Plate B16.
R60
60+
62+
58-
64+
60.2
66+
57-
62-
64.2
R71
70+
61-
72+
68.2
65-
R76
70-
76+
72.2
78+
69-
74-
76.2
R83
82+
73-
84+
78-
86+
77-
82-
R93
88+
R91
90+
87+
84.2
81-
87.2
86-
92+
95.1
91+
91-
89.2
84-
R92
R94
87-
85.2
83.2
R88
R90
83-
8080.2
89+
R89
R87
83+
81.2
79.2
R84
R86
79-
76-
85+
R85
80+
79+
77.2
75.2
R80
R82
75-
72-
81+
R81
R79
75+
R78
73.2
68-
77+
R77
74+
71-
71.2
66-
R75
71+
69.2
67.2
R72
R74
67-
64-
73+
R73
68+
67+
65.2
63.2
R68
R70
63-
60-
69+
R69
R67
63+
61.2
59.2
R64
R66
59-
56-
65+
R65
R63
59+
R62
56.2
53-
49-
21.1
4.6
13.7
58+
57.2
55.2
5252.2
61+
R61
R59
55-
50-
23.6
9.11
56+
55+
53.2
51.2
48-
R56
R58
51-
48.2
57+
R57
52+
51+
R54
49.2
46-
41-
37-
R51
47-
47.2
44-
53+
R53
48+
47+
45.2
42-
R48
R50
43-
43.2
40-
49+
R49
R47
44+
43+
R46
41.2
39.2
36-
45+
R45
42+
39-
37.2
35.2
32-
R43
R42
35-
33.2
30-
40+
39+
R38
31-
31.2
28-
41+
R41
R39
36+
35+
R34
29.2
95.6
+
95.1
2.5
13.11
R32
30+
27-
37+
33+
R33
R31
28+
R30
2.4
4.2
R28
27+
18+
6-
20.1
19+
25.8
23.9
9.2
R3
10-
8.2
R10
6+
R8
9.1
23.10
25.4
R21
88.2
85-
93.2
88-
91.2
90-
9292.2
89-
95.7
426
PAUL J. KAPITZA
33.2
37.5
41.8
44.4
41.14
37.11
33.8
32.4
32.4
33.8
37.11
41.14
44.4
41.8
37.5
33.2
32.4
33.8
37.11
41.14
44.4
41.8
37.5
33.2
4.3
37.3
41.6
44.6
41.16
37.13
32.2
9.3
40.4
41.20
44.10
41.2
4.7
40.2 44.12
13.9
4.3
37.3
41.6
44.6
41.16
37.13
32.2
9.3
40.4
41.20
44.10
41.2
4.7
40.2 44.12
13.9
13.9
40.2
44.12
4.7
41.2
44.10
41.20
40.4
9.3
32.2
37.13
41.16
44.6
41.6
37.3
4.3
*20
20.4
28.2
26-
17+
10-
3.7
*3
9-
12.2
11-
14-
57+
61.9
*61
59+
59-
57-
3+
16-
*62
15+
*18 14+
15.3/*15
*1
67.2
66-
64.2
64-
60-
63.2
62-
*16 13+ 19.3 /*19
17.5
16.3
72.2
70-
75-
*78
75+
71.2
68-
72-
74-
75.2
73+
73-
*24
2.2
95.2
27.2
76-
79.2
78-
80-
80.2
7-
*11
7+
9.8
22-
83.2
82-
84.2
X2
85.9
91.2
92-
89+
*93
91+
*94
88-
90-
91-
89-
93.2
87.2
84-
25.3
21.5
95.4
2.4
92.8
86-
*86
84.2
83+
83-
81-
85.8
23.7
81+
+
25.11
23.7
29.6
33.7
37.10
41.13
44.3
41.9
37.6
33.3
29.2
9.6
24.3
95.2
2.2
23.7
9.9
81.11
76.10
81.11
81.8
76.6
81.2
25.10
*85/85.1
81.2
76.6
81.8
85.7
88.8
23.7
88.2
92.4
25.4
92.4
2.4
21.5
23+
9.6
27.2
95.2
2.2
23.7
25.11
9.6
29.6
33.7
37.10
41.13
44.3
41.9
37.6
33.3
29.2
13.13
4.4
37.2
41.5
44.7
41.17
37.14
36.5
9.2
36.3
41.19
44.9
41.3
4.6
13.11
8.2
18+
6-
6+
*10
8.4
8+
13.11
4.6
4.12
13.6
1.5
17.3
76.3
73.5
68.3
23.4
68.7
65.5
61.6
57.7
53.7
49.7
44.15
49.3
53.3
57.3
61.2
9.5
21.2
29.3
33.4
37.7
41.10
44.2
41.12
37.9
33.6
29.5
21.4
2.5
92.5
25.5
92.3
88.3
23.8
88.7
85.6
81.7
76.7
81.3
85.2
25.9
13.2
4.16
4.16
17.4
1.2
1.4
13.2
25.9
85.2
81.3
76.7
81.7
85.6
88.7
23.8
88.3
92.3
25.5
92.5
2.5
21.4
29.5
33.6
37.9
41.12
44.2
41.10
37.7
33.4
29.3
21.2
9.5
24.2
25.7
11+ 15.2
12- 13.8
13.4
13.4
12.3/*12
17.7
15- 16.2
X3
+
13- 19.2
44.13
49.10
53.10
57.10
60.3
55-
55+
4.16
76.3
73.5
68.3
23.4
68.7
65.5
61.6
57.7
53.7
49.7
44.15
49.3
53.3
57.3
61.2
4.12
13.6
1.5
17.3
47-
51-
47+
51+
49.1
53.1
57.1
4.12
53+
49+
45+
*66
63+
4.12
13.6
1.5
17.3
61.6
57.7
53.7
49.7
44.15
49.3
53.3
57.3
61.2
63-
9.5
21.2
29.3
33.4
37.7
41.10
44.2
41.12
37.9
33.6
29.5
21.4
2.5
92.5
25.5
92.3
88.3
23.8
88.7
85.6
81.7
76.7
81.3
85.2
25.9
13.2
4.16
13.2
25.9
85.2
81.3
76.7
81.7
85.6
88.7
23.8
88.3
92.3
25.5
92.5
2.5
21.4
29.5
33.6
37.9
41.12
44.2
41.10
37.7
33.4
29.3
21.2
9.5
9.5
21.2
29.3
33.4
37.7
41.10
44.2
41.12
37.9
33.6
29.5
21.4
2.5
92.5
25.5
92.3
88.3
23.8
88.7
85.6
81.7
76.7
81.3
85.2
25.9
13.2
4.16
*74
71+
71-
73.6
40.3
41.21
44.11
*41
4.8
44.11
40.3
*17 / 17.1
13.13
4.4
37.2
41.5
44.7
41.17
37.14
36.5
9.2
36.3
41.19
44.9
41.3
Figure 20. Plate B17.
69+
9+
44.13
*57
*7
9.5
21.2
29.3
33.4
37.7
41.10
44.2
41.12
37.9
33.6
29.5
21.4
2.5
92.5
25.5
92.3
88.3
23.8
88.7
85.6
81.7
76.7
81.3
85.2
25.9
13.2
4.16
21.14
*73
76.3
73.5
68.3
23.4
68.7
65.5
61.6
57.7
53.7
49.7
44.15
49.3
53.3
57.3
61.2
*65
21.10
4.12
13.6
1.5
17.3
13.11
4.6
61+
53- 13.7 12+ *14
2+ 6.3/*6
60.3
57.9
53.9
49.9
*58
*49
*53
3-
49-
4.14
4.14
4+ 5.2
17.7
61.2
57.3
53.3
49.3
44.15
49.7
53.7
57.7
61.6
23.4
4.16
73.6
76.3
68.7
68.3
*54
7.3
16+
37-
17- 3.6
57.12
53.12
49.12 45-
17.7
13.4
4.14
21.12
19- 20.3
49.1
53.1
57.1
4.12
61.2
57.3
53.3
49.3
44.15
49.7
53.7
57.7
61.6
65.6
68.7
*50
4.9
39+
39-
65.3
21.12
44.13
21.9
*42
37+
4.6
13.11
61.4
*23/23.13
49.9
65.8
29.6
33.7
37.10
41.13
44.3
41.9
37.6
33.3
29.2
21.18/ *21
9.4
32.3
33.9
37.12
41.15
44.5
41.7
37.4
*33/33.1
4.2
37.4
41.7
44.5
41.15
37.12
32.3
9.4
29+
31+
31-
*34c
37.12
29-
1-
*4
33.10
33.9
3.3
32.3
4.19
13.13
4.4
37.2
41.5
44.7
41.17
37.14
36.5
9.2
36.3
41.19
44.9
41.3
68.1
53.9
61-
23.4
73.6
76.3
69-
68.3
4.16
76.5
76.11
9.10
73.8
21.15
+
24.5
X1
9.11
21.16
23.5
21.8
25.8
85.3
81.4
76.8
81.6
85.5
88.6
23.9
88.4
92.2
25.6
92.6
2.6
5-
3.4
1+
23.12
21+
20.2
19+
28.3
57.5
67+
65+
*69/69.3
57.9
76.9
81.5
85.4
21.17
8-
18-
9.10
77-
10+
81.12
8.3
23.10
60.3
76+
9.10
3.2
77.3
*77
*8/8.1
53.5
*70
92.7
88.5
85-
78+
80+
82+
84+
86+
79.3/*79
25.7
13.16/*13
*28
25.3
95.4
2.4
21.5
44.13
49.10
53.10
57.10
60.3
57.12
53.12
49.12
1.2
65.3
61.4
57.5
53.5
73.3
69.2 67-
90+
87-
87+
85+
21-
88.9/*88
*83
89.2
88.5
85.4
81.5
80.3/*80
31.3
17.7
13.4
4.14
21.12
23.2 65-
92+
88+
25.7
20+ 21.7
20-
87.3/*87
*90
92.9/*92
44.17 44+
46+ 48.3*48 48+
60+
62+ 64.3
64+
66+
50+ 52.3
52+
54+ 56.3
56+
58+ 60.5
47.3/*47
51.3/*51
*60
63.3/*63
*64
67.3/*67
*52
55.3/*55
*56 59.3/*59
53.5
53.5
57.5
57.5
61.4
61.4
65.3
65.3
68.1
*68
68.5
9.7
24.2
2.7
89.3/*89
77.2
49.5
-
21.6
23.6
23-
23.11
*84/84.3
*31
13.10
44.12
49.11
53.11
57.11
60.4
57.11
53.11
49.11
1.3
17.6
13.5
4.13
57.8
53.8
49.8
44.14
49.2
53.2
57.2
4.11
61.1
57.2
53.2
49.2
44.14
49.8
53.8
57.8
61.7
65.7
68.8
65.7
21.8
23.5
21.16
73.7
68.2
73.7
76.4
4.17
76.4
76.12
49.5
22+
24+
24-
X2
24.4
32.5
29.6
33.7
37.10
41.13
44.3
41.9
37.6
33.3
29.2
13.13
4.4
37.2
41.5
44.7
41.17
37.14
36.5
9.2
36.3
41.19
44.9
41.3
-
*72/72.3
*76/76.13 74+
72+
70+
68+
*75/75.3
*71/71.3
79-
79+
23.6
9.10
81.12
76.11
2.3
88.1
*32
26+
35.3
28+
*35
30+
36.7
32+
*36
34+
39.3
36+
38+
40+
42+
X1
77+
76.5
13.2
25.9
85.2
81.3
76.7
81.7
85.6
88.7
23.8
88.3
92.3
25.5
92.5
2.5
21.4
29.5
33.6
37.9
41.12
44.2
41.10
37.7
33.4
29.3
21.2
9.5
81.9
81.10
95.3
91.3/*91
41+
6.2
81.13 / *81
*25/25.12
36-
2-
*39
40.5
37.8
33.5
29.4
21.3
9.6
4.6
13.11
95.5/*95
34-
*5/5.3
*40
25+
4-
X3
21.3
29.4
33.5
37.8
*43
43.3
41.11
27+
3.5
61.8
4.18
*44 44.19
33.8
37.11
41.14
44.4
41.8
37.5
33.2
29.1
13.14
4.3
37.3
41.6
44.6
41.16
37.13
36.6
9.3
36.2
41.20
44.10
41.2
23.3
68.4
73.4
76.2
73.2
21.13
4.15
13.3
17.8
60.2
23.5
21.8
25.8
85.3
81.4
76.8
81.6
85.5
88.6
23.9
88.4
92.2
25.6
92.6
2.6
88.6
23.9
88.4
92.2
25.6
92.6
2.6
41.11
88.1
58-
1.6
9.12 5+
4.10
85.5
33.2
37.5
41.8
44.4
41.14
37.11
33.8
32.4
*26
45.3
33+
13.9
40.2
44.12
4.7
41.2
44.10
41.20
40.4
9.3
32.2
37.13
41.16
44.6
41.6
37.3
4.3
33.2
37.5
41.8
44.4
41.14
37.11
33.8
32.4
44.12
4.7
41.2
44.10
41.20
40.4
9.3
32.2
37.13
41.16
44.6
41.6
37.3
4.3
*22
56-
17.2
37.17
17.6
1.3
7.2
25.8
85.3
81.4
76.8
81.6
40.2
29.7
21.11
65.2
61.3
57.4
53.4
49.4
44.16
49.6
53.6
57.6
61.5
65.4
68.6
21.8
23.5
21.16
*2
39.2
1.3
17.6
13.12
17.2
13.12
4.5
4.5
1.3
17.6
13.5
4.13
21.11
65.2
61.3
57.4
53.4
49.4
44.16
49.6
53.6
57.6
61.5
65.4
68.6
23.3
68.4
73.2
76.2
73.2
21.13
4.15
13.3
17.8
60.2
25.2
*30
4.7
59.2
60.2
2.1
35.2
*45
56.2
*46
43+
55.2
54-
52.2
52-
51.2
50-
48.2
48-
46-
44-
47.2
45.2
43-
13.9
41-
43.2
44.12
44.18
42-
40-
40.2
44.18
13.5
4.13
21.11
65.2
61.3
57.4
53.4
49.4
44.16
49.6
53.6
57.6
61.5
65.4
68.6
23.3
68.4
73.4
76.2
73.2
21.13
4.15
13.3
*82
27-
4.7
44.12
13.9
38-
33-
35-
35+
4.3
37.3
41.6
44.6
41.16
37.13
32.2
9.3
40.4
41.20
44.10
41.2
32-
4.3
37.3
41.6
44.6
41.16
37.13
32.2
9.3
40.4
41.20
44.10
41.2
4.7
40.2 44.12
76.12
*27
29.8
28.2
*29
36.4
37.15
41.18
44.8
41.4
*37
4.5
13.12
17.2
31.2
29.9
*9
37.15
28-
4.3
37.3
41.6
44.6
41.16
37.13
32.2
9.3
40.4
41.20
44.10
41.2
4.7
40.2 44.12
13.9
4.5
13.12
17.2
1.6
37.16
30-
32.4
33.8
37.11
41.14
44.4
41.8
37.5
33.2
1.3
17.6
1.3
17.6
32.4
33.8
37.11
41.14
44.4
41.8
37.5
33.2
17.2
13.12
4.5
41.4
44.8
41.18
36.4
*38
25-
13.15
13.15
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
0
r22
= y20
0
3 −1
r23
= x1 x−1
2 x1 y19
Reading for Plate B14.
r10 = x3 x1 x3 x−1
1
−3
r20 = x2 x−1
1 x2 x1
r30 = y10 y8 y1 y5 y4 y17 x1
−1
3
r4+ = x−3
1 x1 y3 y1
−1
r5− = y4 x1 x−1
1 y2
r60 = y9−1 y2
−1 −1
y3
r70 = y16
−1
y8
r8+ = y6−1 y18
−1 −1 −1
3
r9− = x−3
1 y22 y7 x1 x1 y5 x1
0
r10
= y6
−1
0
r11
= x−1
2 y7
−1
+
y9
= y11
r12
−1 −1 −2 2
−
x3 x1 x1 y10
= y12
r13
0
r14
= y12
−1
−1
0
x1 y11
r15
= y15
−1
+
y15
= y13
r16
−1 −1
−
= x−1
r17
1 x1 x3 y14 y16
0
r18
= y14
0
r19
= x3 x1 y13
+
−1
r20
= y19
y17
−
−1
r21
= x−1
2 y20 y18
0
r22
= y20
0
3 −1
r23
= x1 x−1
2 y21 x1 y19
+
−1
r24
= y23
y21
−
−1
r25
= y24
y22
0
r26
= y24
−1
−1
0
= (x2 x−3
r27
1 x2 x1 )y23
427
428
PAUL J. KAPITZA
Reading for Plate B15.
r10 = x3 x1 x3 x−1
1
−3
r20 = x2 x−1
1 x2 x1
r30 = y10 y8 y1 y5 y4 y17 x1
−3 3 −1
3
r4+ = x−3
1 x1 y3 x1 x1 y1
r5− = y4 y2−1
r60 = y9−1 y2
−1 −1
r70 = y16
y3
−1
r8+ = y6−1 y18
y8
−1 −1 −1
3
r9− = x−3
1 y22 y7 x1 x1 y5 x1
0
r10
= y6
−1
0
r11
= x−1
2 y7
−1
+
y9
= y11
r12
−
2 −1 −1 −2 2
= x−2
r13
1 x1 y12 x3 x1 x1 y10
0
r14
= y12
−1 −1
0
y11
r15
= y15
−1
+
y15
= y13
r16
−1 −1 −1
−
= x−1
r17
1 x1 x3 y14 x1 x1 y16
0
r18
= y14
0
r19
= x3 y13
−1
+
y17
= y19
r20
−3 3 −1
−1
−
r21
= x−1
2 y20 x1 x1 x1 x1 y18
0
r22
= y20
−1
−3
0
2 −1
r23
= x31 x−1
1 x1 x2 x2 x1 y21 x1 y19
+
−1
r24
= x−1
1 x1 y23 y21
−
−1
3 −1
r25
= y24
x2 x−3
1 x1 x2 y22
0
r26
= y24
−1
0
r27
= x2 y23
Reading for Plate B16.
r10 = x3 x1 x3 y49 x−1
1
−3
r20 = x2 x−1
1 x2 x1
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
r30 = y10 y8 y1 y5 y4 y17 x1
−1
−1 2
−1
−2 2 −1
−1
r4+ = x−2
1 y29 x1 x1 y37 x1 y3 x1 y57 x1 x1 y73 x1 y1
r5− = y4 y2−1
r60 = y9−1 y2
−1 −1
r70 = y16
y3
−1
r8+ = y6−1 y18
y8
−1 −1 −1
−1 2
r9− = x−3
1 y22 y7 x1 x1 y5 x1 y33 x1
0
r10
= y6
−1
0
r11
= x−1
2 y7
−1
+
y9
= y11
r12
−1
−1
−
2 −1 −1 −1 −1
= y77 x−2
r13
1 x1 y53 y12 x3 x1 y41 x1 x1 y25 x1 y10
0
r14
= y12
−1 −1
0
y11
r15
= y15
−1
+
y15
= y13
r16
−1 −1 −1 −1
−
= x−1
r17
1 x1 y45 x3 y14 x1 x1 y16
0
r18
= y14
0
r19
= x3 y13
−1
+
y17
= y26 y19
r20
−1
−2 2 −1
−1
−1
−
= x−1
r21
2 y20 x1 x1 y61 x1 x1 y69 x1 x1 y18
0
r22
= y20
−2
−1
−1
−1
2 −1
0
x1 x−1
r23
= x21 y65
1 x1 x2 x2 x1 y85 x1 y21 x1 y19
−1
+
r24
= x−1
1 x1 y23 y21
−1
−1
−2 2 −1
−
x2 x−1
= y24
r25
1 y89 x1 x1 y81 x1 x2 y22
0
r26
= y24
−1
0
r27
= x2 y23
+
−1
r28
= y28
y26
−
−1
r29
= y27
y25
0
r30
= y27
−1
0
r31
= y30
y28
+
−1
r32
= y32
y30
−
−1
r33
= y31
y29
429
430
PAUL J. KAPITZA
0
r34
= y31
−1
0
r35
= y34
y32
+
−1
r36
= y36
y34
−
−1
r37
= y35
y33
0
r38
= y35
−1
0
r39
= y38
y36
+
−1
r40
= y40
y38
−
−1
r41
= y39
y37
0
r42
= y39
−1
0
y40
r43
= y42
−1
+
y42
= y44
r44
−1
−
y41
= y43
r45
0
r46
= y43
−1
0
y44
r47
= y46
−1
+
y46
= y48
r48
−1
−
y45
= y47
r49
0
r50
= y47
−1
0
y48
r51
= y50
−1
+
y50
= y52
r52
−1
−
y49
= y51
r53
0
r54
= y51
−1
0
y52
r55
= y54
−1
+
y54
= y56
r56
−
−1
r57
= y55
y53
0
r58
= y55
−1
0
r59
= y58
y56
+
−1
r60
= y60
y58
−
−1
r61
= y59
y57
0
r62
= y59
−1
0
r63
= y62
y60
+
−1
r64
= y64
y62
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
−
−1
r65
= y63
y61
0
= y63
r66
−1
0
r67
= y66
y64
+
−1
r68
= y68
y66
−
−1
r69
= y67
y65
0
r70
= y67
−1
0
r71
= y70
y68
+
−1
r72
= y72
y70
−
−1
r73
= y71
y69
0
r74
= y71
−1
0
y72
r75
= y74
−1
+
y74
= y76
r76
−1
−
y73
= y75
r77
0
r78
= y75
−1
0
y76
r79
= y78
−1
+
y78
= y80
r80
−1
−
y77
= y79
r81
0
r82
= y79
−1
0
y80
r83
= y82
−1
+
y82
= y84
r84
−1
−
y81
= y83
r85
0
r86
= y83
−1
0
r87
= y86
y84
+
−1
r88
= y88
y86
−
−1
r89
= y87
y85
0
r90
= y87
−1
0
r91
= y90
y88
+
−1
r92
= y92
y90
−
−1
r93
= y91
y89
0
r94
= y91
−1
−1
0
r95
= (x−2
1 x2 x1 x2 x1 )y92
431
432
PAUL J. KAPITZA
Reading for Plate B17.
r10 = x3 x1 x3 y49 x−1
1
−3
r20 = x2 x−1
1 x2 x1
r30 = y10 y8 y1 y5 y4 y17
−3 3 −1 −1
3 −1
r4+ = y29 x−3
1 x1 y37 y3 y57 x1 x1 y73 y1
r5− = y4 y2−1
r60 = y9−1 y2
−1 −1
r70 = y16
y3
−1
y8
r8+ = y6−1 y18
−1
−1 −1 −1
r9− = x31 x−3
1 y22 y7 x1 x1 y5 y33
0
r10
= y6
0
r11
= y7−1
−1
+
y9
= y11
r12
−2 2 −1
−
2 −1 −1 −1
= y77 x−2
r13
1 x1 y53 y12 x3 y41 x1 x1 y25 y10
0
r14
= y12
−1 −1
0
y11
r15
= y15
−1
+
y15
= y13
r16
−1 −1 −1 −1
−
= x−1
r17
1 x1 y45 x3 y14 x1 x1 y16
0
r18
= y14
0
r19
= x3 y13
−1
+
y17
= y26 y19
r20
−1
−1
−2 2 −1
−
2 −1 −1
r21
= x−2
1 x1 x2 y20 x1 x1 y61 x1 x1 y69 x1 x1 y18
0
r22
= y20
−1 3 −1
−1
0
r23
= y65
x1 x1 x1 x−3
1 y85 y21 y19
+
−1
r24
= x−1
1 x1 y23 y21
−
−1
3 −1
r25
= y24
x2 y89 x−3
1 x1 y81 y22
0
r26
= y24
−1
0
r27
= x2 y23
+
−1
r28
= y28
y26
−
3 −1
r29
= x−3
1 x1 y27 y25
0
r30
= y27
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
−1
0
r31
= y30
y28
+
−1
r32
= y32
x1 x−1
1 y30
−
4 −1
r33
= x−4
1 x1 y31 y29
0
r34
= y31
−1
0
r35
= y34
y32
+
−1 −2 2
r36
= y36
x1 x1 y34
−
−1
3 −3 −1
r37
= x31 x−3
1 x1 x1 x1 x1 y35 y33
0
r38
= y35
−1
0
r39
= y38
y36
−1
+
x1 x−1
= y40
r40
1 y38
−
3 −4 4 −3 3 −1
= x−3
r41
1 x1 x1 x1 x1 x1 y39 y37
0
r42
= y39
−1
0
y40
r43
= y42
−3 −1
+
2 −2
= x1 x31 x−3
r44
1 x1 x1 x1 x1 x1 y44 y42
−1
−
y41
= y43
r45
0
r46
= y43
−1
0
y44
r47
= y46
−1
+
y46
= y48
r48
−1
−1 −1
−2 2
−
= x1 x−1
r49
1 x1 x1 x1 x1 y47 x1 x1 y45
0
r50
= y47
−1
0
y48
r51
= y50
−1
+
y50
= y52
r52
−1
−2 2
−1 −1
−
= x1 x−1
r53
1 x1 x1 x1 x1 y51 x1 x1 y49
0
r54
= y51
−1
0
r55
= y54
y52
+
−1
r56
= y56
y54
−
−2 2
−1 −1
−1
r57
= x1 x−1
1 x1 x1 x1 x1 y55 x1 x1 y53
0
r58
= y55
−1
0
r59
= y58
y56
+
−1 −1
r60
= y60
x1 x1 y58
−
3 −1
r61
= x−3
1 x1 y59 y57
433
434
PAUL J. KAPITZA
0
r62
= y59
−1
0
r63
= y62
y60
+
−1
r64
= y64
y62
−
−1
2 −1
r65
= x−2
1 x1 y63 x1 x1 y61
0
r66
= y63
−1
0
r67
= y66
y64
+
−1 −3 3
r68
= y68
x1 x1 y66
−
−1
r69
= y67
y65
0
r70
= y67
−1
0
y68
r71
= y70
−1
+
y70
= y72
r72
−1
−
2 −1
= x−2
r73
1 x1 y71 x1 x1 y69
0
r74
= y71
−1
0
y72
r75
= y74
−3 −1 −1
+
= x31 x−1
r76
1 x1 x1 y76 x1 x1 y74
−1
−
y73
= y75
r77
0
r78
= y75
−1
0
y76
r79
= y78
−1
+
y78
= y80
r80
−1
−1 −1
−2 2
−
= x1 x−1
r81
1 x1 x1 x1 x1 y79 x1 x1 y77
0
r82
= y79
−1
0
r83
= y82
y80
+
−1
= y84
y82
r84
−
3 −1
r85
= x−3
1 x1 y83 y81
0
r86
= y83
−1
0
r87
= y86
y84
+
−1 −3 3
r88
= y88
x1 x1 y86
−
−1
r89
= y87
y85
0
r90
= y87
−1
0
r91
= y90
y88
+
−1
r92
= x31 x−3
1 y92 y90
ON SMALL GEOMETRIC INVARIANTS OF 3-MANIFOLDS
435
−
−1
r93
= y91
y89
0
= y91
r94
0
r95
= x2 x−1
1 x2 y92
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Berry College
pkapitza@berry.edu
This paper is available via http://nyjm.albany.edu/j/2011/17-18.html.