学位の種別: 課程博士審査委員会委員 : （主査）東京大学准教授 逆井 卓也, 東京大学教授 坪井 俊, 東京大学教授 河野 俊丈, 東京大学教授 古田 幹雄, 東京大学准教授 河澄 響矢, 東京大学准教授 北山 貴裕University of Tokyo(東京大学

Nozaki Yuta

野崎 雄太

UTokyo Repository

博士論文（要約）
論文題目 An invariant of 3-manifolds via homology cobordisms
and knots in lens spaces
（ホモロジーコボルディズムを用いた３次元多様体の不変量と
レンズ空間内の結び目）
氏 名 野崎 雄太
My thesis consists of Parts 1 and 2 based on my papers [23] and [21]
respectively. We start giving a background for Part 1. In [1], Alexander
proved that every connected oriented closed 3-manifold X admits an open
book decomposition. In other words, X contains a bered link. Myers [20]
and Gonzalez-Acu~na showed that X admits an open book decomposition
with connected binding (see Rolfsen [24, Chapter 10, Section K]). We dene
the topological invariant op(X) to be the minimal number g such that X
admits an open book decomposition with a page g;1, where g;1 denotes
an oriented compact surface of genus g with one boundary component. The
invariant op(X) is expected to measure how complicated X is. For instance,
op(X) = 0 if and only if X is homeomorphic to the 3-sphere S3. Since the
union of two pages of an open book decomposition is a Heegaard surface,
we have the inequality g(X)  2 op(X), where g(X) denotes the Heegaard
genus of X.
Morimoto [19] started to study genus one bered knots (GOF-knots) in
lens spaces and proved that the lens space L(p; 1) contains a GOF-knot,
but L(19; 2), L(19; 4) and L(19; 7) do not. In [2], Baker completely deter-
mined which lens space L(p; q) contains a GOF-knot, that is, we already
know when op(L(p; q)) = 1 holds. Related researches are also found in
Baldwin [3] and Sekino [27]. However, the computation of op(X) is dicult
in general. Sakasai [25] introduced a homological analogue hc(X) of op(X),
which is dened to be the minimal number g such that X is obtained as
the closure of a homology cobordism over g;1. Roughly speaking, a ho-
mology cobordism over g;1 is a 3-manifold which homologically seems to
be the product manifold g;1  [ 1; 1] (see Goussarov [12], Habiro [13] and
Garoufalidis-Levine [9]). It is easy to see that hc(X) = 0 if and only if X is
an integral homology 3-sphere. Note that hc(X) is related to op(X) through
the inequality op(X)  hc(X).
The purpose of Part 1 is to investigate the topological invariant hc(X).
We rst refer a result due to Sakasai.
Theorem 1 ([25, Remark 6.10]). The invariant hc(X) depends only on the
rst Betti number b1(X) and the isomorphism class of the torsion linking
form X : TH1(X)  TH1(X) ! Q=Z of X, where TH1(X) denotes the
torsion subgroup of H1(X) := H1(X;Z).
Note that hc(X) is not determined only by the isomorphism class of
H1(X). Indeed, for two 3-manifolds Xq = L(5; 1)]L(5; q) (q = 1; 2), we have
hc(X1) = 1 6= hc(X2) = 2 ([25, Remark 6.10]). In fact, X1 and X2 are not
isomorphic since Xq
= (1=5)  (q=5) for q = 1; 2. One of the main results
of the thesis is the following.
1
Theorem 2. The equality hc(X](]2S1  S2)) = hc(X) + 1 holds for every
connected oriented closed 3-manifold X.
Theorems 1 and 2 enable us to reduce the computation of hc(X) to the
computation of hc(Y ) or hc(Y ]S1  S2) for a rational homology 3-sphere
Y with X = Y . Here we focus on lens spaces which are basic examples
of rational homology 3-spheres. The author was informed by Sakasai the
following sucient condition for hc(L(p; q)) = 1 when q is odd: p(p+ 4) or
p(p  4) is a quadratic residue mod q. In fact, we obtain the following.
Theorem 3. The equality hc(L(p; q)) = 1 holds for every lens space L(p; q).
Corollary 4. The following equality holds:
hc(L(p; q)]S1  S2) =
(
1 if q or  q is a quadratic residue mod p,
2 otherwise.
As a consequence of Theorems 1{3 and Corollary 4, if H1(X) = Z=pZr
with torsion linking form (q=p) for some p  2 and q, then
hc(X) =
8><>:
(r + 2)=2 if r is even,
(r + 1)=2 if r is odd, and if q or  q is a quadratic residue mod p,
(r + 3)=2 otherwise.
In order to prove Theorem 3, we nd a surface   L(p; q) of genus one
whose complement L(p; q) n Int( [ 1; 1]) is a homology cobordism. The
following theorem and a well-known fact about binary quadratic forms allow
us to construct a desired surface .
Theorem 5. Let m be an integer and n a positive integer not equal to ve
with m, n coprime. Then there exist innitely many pairs ("; l) 2 f1; 1g
fodd primesg such that the congruence equation nx(x+1)  " mod l has a
solution and l  m mod n holds.
This theorem follows from the Chebotarev density theorem which com-
putes the Dirichlet density of a certain set of primes.
We also discuss the invariant hc(X) from a knot-theoretic point of view.
The main object is a homologically bered knot (see Goda-Sakasai [10]),
in terms of which an alternative denition of hc(X) is given. For example,
Theorem 3 is rephrased as follows: Every lens space contains a homologically
bered knot of genus one. We prove that the homological beredness in a
rational homology 3-sphere Y is characterized by the Alexander polynomial.
2
Theorem 6. Let K  Y be a null-homologous knot. Then, K is homologi-
cally bered if and only if the Alexander polynomial K(t) is monic (up to
sign) and its breadth equals 2g(K).
This theorem allows us to show the following corollary concerning the
connected sum of homologically bered knots, which is an analogue of the
result on bered knots due to Gabai [8].
Corollary 7. If K1  X1 and K2  X2 are homologically bered knots,
then K1]K2 is again a homologically bered knot. Moreover, in the case
where X1 and X2 are rational homology 3-spheres, the converse is true.
We next develop an introduction for Part 2. Note that a homologically
bered knot is, by denition, null-homologous. In contrast, we focus on a
knot K 0 whose homology class [K 0] is a generator of H1(L(p; q)) in Part 2.
Then the preimage  1(K 0) is a knot with free period p, where  : S3 !
L(p; q) is the p-fold cyclic cover.
Let K be a torus knot and p  2 an integer. Hartley [14] created a
decision argument to know whether K can be represented as the preimage
of a knot in the lens space L(p; q) for some q. For example, the trefoil
T3;2 does not appear as the preimage of any knot in L(2; 1) = RP 3. The
Alexander polynomial of torus knots was used in his proof.
Let K be a knot such that the outer automorphism group Out(1(S
3 n
K)) is trivial. For instance, 932, 933 and 24 prime knots with 10 cross-
ings (and their mirror images) satisfy this condition (see Kawauchi [16, Ap-
pendix F.2], Kodama-Sakuma [17, Table 3.1]). Then it follows from Borel's
theorem (Conner-Raymond [7, Theorem 3.2]) that K cannot be represented
as the preimage of any knot in any lens space (see [16, Theorems 10.6.2 and
10.6.6(1)]).
The purpose of Part 2 is to deduce the above facts from a single result.
Let G be a group and p  2 an integer which is not necessarily prime.
We introduce the subgroup Cp(G) generated by the commutators and the
p th power of each element of G. The case p = 2 was studied, for exam-
ple, in Sun [29] and Haugh-MacHale [15], from a purely algebraic point of
view. Moreover, the case where p is prime can be found in Stallings [28] and
Cochran-Harvey [6]. We apply Cp to the fundamental group of the com-
plement of a knot in a 3-manifold. Let  be an integral homology 3-sphere
and let  : ! 0 be a p-fold cyclic cover. The main result of Part 2 is the
following.
3
Theorem 8. Let K 0 be a knot in 0 with connected preimage K =  1(K 0).
Then the image of the induced map  : 1( nK) 1(0 nK 0) coincides
with the subgroup Cp(1(
0 nK 0)).
Considering the case of  = S3 and 0 = L(p; q), we obtain the corollar-
ies below as mentioned before.
Corollary 9. Let m;n; p  2 be integers with gcd(m;n) = 1. Then there
exist an integer q and a knot K 0 in L(p; q) such that  1(K 0) is isotopic to
the torus knot Tm;n or its mirror image if and only if gcd(mn; p) = 1.
Moreover, Chbili [5] determined the above q, and we also observe it via
Corollary 9.
Corollary 10. A knot K in S3 with Out(1(S
3 nK)) = 1 is not represented
as the preimage of any knot in any lens space.
When a knot K is represented as a preimage, there is still the question
of whether K 0 is uniquely determined. Sakuma [26], Boileau and Flapan [4]
independently proved that if the preimages of oriented knots K 01, K 02 are
the same prime knot, then there exists a dieomorphism (L(p; q);K 01) !
(L(p; q);K 02) preserving the orientations both of L(p; q) and K 0i's. Note that
K 01, K 02 are not necessarily ambient isotopic to each other. They also showed
that the same is true for a composite knot under a condition regarding
\slopes". Furthermore, Manfredi [18] recently constructed two knots in a
lens space such that they are not ambient isotopic to each other but their
preimages are the unknot.
Our work is closely related to periodic dieomorphisms of (S3;K) with-
out xed points. Indeed, such a dieomorphism of period p induces a cyclic
cover (S3;K) ! (L(p; q);K 0) for some q, and vice versa. Hartley [14] gave
a list of possible free periods of prime knots up to 10 crossings. Moreover,
Gonzalez-Acu~na and Whitten [11, Chapter 4] proved that any non-torus
two-bridge knot has no free period.
In the thesis, we shall discuss whether the braid group Bn is a C
p-group
for each n, p and study the series of subgroups obtained from a group G by
taking Cp( ) repeatedly. Finally, note that a part of Part 2 is also found in
author's article [22] in the proceedings.
Acknowledgments
First of all, the author wishes to express his thanks to his supervisor Takuya
Sakasai for his various suggestions and warm encouragement. T. Sakasai also
4
played an important role in his undergraduate education and encouraged
him to learn and enjoy topology. He would like to express his gratitude to
Toshitake Kohno, Nariya Kawazumi, Gwenael Massuyeau, Makoto Sakuma,
Hisaaki Endo and Toshiaki Hattori for their helpful suggestions. Also, he
gratefully acknowledges valuable comments of Mutsuro Somekawa and Ippei
Nagamachi from a number-theoretic point of view.
The author would like to thank his friends Motoko Kato, Shunsuke
Tsuji, Adrian Jimenez Pascual, Saiei-Jaeyeong Matsubara-Heo, Yoshinori
Tanimura, Kouki Sato, Kodai Wada, Shun Wakatsuki and Nozomu Sekino
for many discussions. Finally, this work was supported by the Program
for Leading Graduate Schools, MEXT, Japan and JSPS KAKENHI Grant
Number 16J07859.
5
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7
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ホモロジーコボルディズムを用いた３次元多様体の不変量とレンズ空間内の結び目

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ホモロジーコボルディズムを用いた３次元多様体の不変量とレンズ空間内の結び目

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