Cobordism category of plumbed 3-manifolds and intersection product structures

0.29 MB·English

by  Yoshihiro Fukumoto

Cobordism category of plumbed 3-manifolds and
intersection product structures
9 Yoshihiro FUKUMOTO∗
0
Tottori University of Environmental Studies, Tottori, JAPAN†
0
2
January 26, 2009
n
a
J

6 Abstract
2
Inthispaper,weintroduceacategoryofgradedcommutativeringswithcertainalgebraic

] morphisms,toinvestigatethecobordismcategoryofplumbed3-manifolds. Inparticular,we
T definea non-associative distributivealgebra that gives necessary conditions for an abstract
G morphism between thehomologies of twoplumbed3-manifolds toberealized geometrically
byacobordism. Herewealsoconsiderthehomologycobordismmonoid,andgiveanecessary
.
h condition using w-invariants for the homology 3-spheres to belong to the inertia group
t
a associated tosome homology 3-spheres.
m
[ 1 Introduction

1
v In this paper, we introduce a category of graded commutative rings with certain algebraic mor-
1 phismsinordertoinvestigatethecobordismcategoryofplumbed3-manifolds. Infact,wedefinea
4
non-associativedistributivealgebra,whichweusetogivenecessaryconditionsforalgebraicmor-
9
phismsbetweenthehomologiesoftwoplumbed3-manifoldstoberealizedgeometricallybycobor-
3
. dism. This paper is a generalizationof the paper [2] on homology cobordisms, to generalcobor-
1
dismsbetweenclosed3-manifolds. Moreprecisely,westatetheproblemasfollows. LetC bethe
0 3
9 cobordism category of closed oriented 3-manifolds whose objects M ∈ ob(C3) are 3-manifolds
0 and whose morphisms W ∈ C (M,M′) between two 3-manifolds M and M′ are cobordisms
3
: (W;M,M′). Ontheotherhand,letL beacategorywhoseobjects(H ,•)∈ob(L )aregraded
v 3 ∗ 3
Xi commutative rings of dimension 3 and whose morphisms (L∗;i,i′,•) ∈ L3((H∗,•),(H∗′,•)) be-
tween two objects (H ,•) and (H′,•) are composed of graded modules L of dimension 4 with
∗ ∗ ∗
ar certainproductstructures•andhomomorphismsH∗ →i L∗ ←i′ H∗′ satisfyingcompatibilitycondi-
tionsonproducts. NotethatthereexistsafunctorH :C →L givenbyM 7−→(H (M;Z),•)
∗ 3 3 ∗
and (W;M,M′)7−→(L (W;Z);i ,i′,•), where i and i′ are the induced homomorphisms
∗ ∗ ∗ ∗ ∗
H (M;Z)→i∗ L (W;Z)←i′∗ H (M′;Z),
∗ ∗ ∗
to the Z-module L (W;Z) defined by
∗
L (W;Z)=Im H (M;Z)⊕H (M′;Z)i∗→+i′∗ H (W;Z) .
k k k k
(cid:18) (cid:19)
Let R (W) be a Z-module defined by
∗
R (W;Z)=Ker H (M;Z)⊕H (M′;Z)i∗→+i′∗ H (W;Z)
k k−1 k−1 k−1
(cid:18) (cid:19)
∗ResearchsupportedbyMEXTGrant-in-AidforScientificResearch(18740039)
†E-mail: [email protected]
1
and ∂ ⊕∂′ : R (W;Z) → H (M;Z)⊕H (M′;Z) be the induced homomorphism. Then
∗ ∗ k k−1 k−1
the problem can be stated as follows.
Problem 1 Let M,M′ ∈ ob(C ) be two closed oriented 3-manifolds. Let φ = (L ;i,i′,•) ∈
3 ∗
L ((H (M;Z),•),(H (M′;Z),•))beanalgebraicmorphismofhomologyrings. Thendoesthere
3 ∗ ∗
exist a cobordism (W;M,M′)∈C (M,M′) such that (L (W;Z);i ,i′,•)∼=φ=(L ;i,i′,•).
3 ∗ ∗ ∗ ∗
Note that by the exact sequence of the pair (W,M ⊔M′), the cobordism (W;M,M′) pre-
serves the intersection product structures, that is, the induced morphism (L (W;Z);i ,i′,•)∈
∗ ∗ ∗
L ((H (M;Z),•),(H (M′;Z),•)) must be composed of ring homomorphisms i , i′ with re-
3 ∗ ∗ ∗ ∗
spect to the intersection pairings.
Lemma 2 Let (W;M,M′) be a cobordism between two closed oriented 3-manifolds M and M′.
Let(L (W;Z);i ,i′,•)betheinducedhomomorphism H (M;Z)→i∗ L (W;Z)←i′∗ H (M′;Z). If
∗ ∗ ∗ ∗ ∗ ∗
we denote the intersection parings on M and W by •:H (M;Z)⊗H (M;Z)→H (M;Z)
k ℓ k+ℓ−3
and • : R (W;Z)⊗L (W;Z) → L (W;Z), respectively, for any non-negative integers
k+1 ℓ k+ℓ−3
k,ℓ with k+ℓ≥3, then we have i (∂ (η)·θ)=η·i (θ) for any η ∈R (W;Z), θ ∈L (W;Z).
∗ ∗ ∗ k+1 ℓ
In fact, this lemma provides a necessary condition for the existence of cobordisms described
above. However, the following is an example to which Lemma 2 cannot be applied.
Example 3 Let (Γ,ω), (Γ′,ω′) be two Seifert graphs defined by
Γ=(V,E), V ={1,2}, E ={(1,2),(2,1)}
ω ={2;(5,3),(5,3),(5,4)},
1
ω ={2;(9,1),(9,1),(9,4)}.
2
(cid:26)
Γ′ =(V′,E′), V′ ={1,2}, E′ ={(1,2),(2,1)}
ω′ ={2;(5,3),(5,3),(5,4)},
1
ω′ ={1;(9,2),(9,2),(9,2)}.
(cid:26) 2
Note that the associated plumbed 3-manifold M(Γ) can be obtained by plumbing of S1-V-bundles
E → Σ , v ∈ V over the closed oriented V-surfaces Σ with Seifert invariants ω(v) according
v v v
to the graphs Γ, see Appendix 6 or [3]. This plumbed V-manifold can also be described by using
the decorated plumbing graphs of N. Saveliev [12]. Then the homology groups of M(Γ), M(Γ′)
are isomorphic.
H (M(Γ);Z)∼=Z8⊕Z/45⊕Z/675, H (M(Γ);Z)∼=Z8,
1 2
H (M(Γ′);Z)∼=Z6⊕Z/45⊕Z/675, H (M(Γ′);Z)∼=Z6.
1 2
Let {α } ⊂H Σ¯ ;Z beasystemof α,β-cycles ontheunderlying topological
vj v∈{1,2},j∈{1,2,...,2g} 1 v
space Σ¯v of the V-surface Σv of genus g, so that αvj ·αv′j′ = δvv′εjj′ for v,v′ ∈ {1,2} and
(cid:0) (cid:1)
j,j′ ∈{1,2,...,2g}, where
1 (j =2j′′−1, j′ =2j′′, j′′ ∈{1,2,...,g})
εjj′ = −1 (j =2j′′, j′ =2j′′−1, j′′ ∈{1,2,...,g}) .

0 (otherwise)

Wedenotethesamesymbol{α } ⊂H (M(Γ);Z)tobethecorresponding ho-
vj v∈{1,2},j∈{1,2,...,2g} 1
mologyclassesinM(Γ),andfixanisomorphism onthefreepartH (M(Γ);Z)/Tor∼=Z8 byusing
1
this{α }. Foreachα∈H Σ¯ ;Z ,letθ ∈H (M(Γ);Z)bethecorresponding2-cycleobtained
vj 1 v α 2
by using the natural homomorphism defined in Lemma 30.1. Then {θ } ⊂
(cid:0) (cid:1) αvj v∈{1,2},j∈{1,2,...,2g}
2
H2(M(Γ);Z) form a basis, and we can fix an isomorphism H2(M(Γ);Z)∼=Z6 by using {θαvj}.
Let L = 4 L be the graded Z-module defined by
∗ k=0 k
L L =Z, L =Z8⊕Z/45⊕Z/675, L =Z6, L =Z, L =0.
0 1 2 3 4
Let H (M(Γ))→i L ←i′ H (M(Γ′)) be two graded homomorphisms defined by
∗ ∗ ∗
H (M(Γ);Z) →i L ←i′ H (M(Γ′);Z)
1 1 1
↓∼= ↓∼= ↓∼=
Z8 →Φ1 Z8 ←Φ′1 Z6
⊕ ⊕ ⊕ ,
Z/45 →1 Z/45 ←1 Z/45
⊕ ⊕ ⊕
Z/675 →1 Z/675 ←1 Z/675
where
I O O O I O O
2 2
O I O O O O I
Φ1 = O O2 O O , Φ′1 = O I O2 ,
2
 O O O I2   O O O 
   
 1 0 0 0  
I = , O = ,
2 0 1 0 0
(cid:18) (cid:19) (cid:18) (cid:19)
H (M(Γ);Z) →i L ←i′ H (M(Γ′);Z)
2 2 2
↓∼= ↓∼= ↓∼= ,
Z8 →Φ2 Z6 ←Φ′2 Z6
and where
I O O O I O O
2 2
Φ = O I O O , Φ′ = O O I ,
2  2  2  2 
O O I O O O O
2
   
and
H (M(Γ);Z) →i L ←i′ H (M(Γ′);Z)
3 3 3
↓∼= ↓∼= ↓∼= .
Z →1 Z ←1 Z
On the other hand, let R = 4 R be the graded Z-module defined by
∗ k=0 k
LR =Ker H ⊕H′ i→−i′ L ,
k k−1 k−1 k−1
(cid:16) (cid:17)
then we have
R =0, R =Z, R =Z6, R =Z8, R =Z.
0 1 2 3 4
Let •:R ⊗L →L =Z and •:R ⊗L →L be bilinear pairings defined by
2 2 0 3 2 1
L L
L 1 2
2 −ε O O O 45ε δ O O
ε O O 2 2
2 , O −ε O O , O 45ε δ O ,
R2  OO εO2 εO  R3  O O2 O −ε2  R3  O O2 O 
2  O O −ε2 O   O O O 
     
0 1    
ε = , δ =1∈Z/675⊂L .
2 −1 0 1
(cid:18) (cid:19)
3
We fix spin structures,
c=((1,1,0),(1,1,0)), c′ =((1,1,0),(0,0,0)),
onM(Γ),M(Γ′),respectively, where0,1aredeterminedbythespinstructuresonpuncturedRie-
mannsurfacesaroundV-singularpoints. Thenthereexistsno(spin)cobordism((W,c˜);(M,c),(M′,c′))
such that (L (W;Z);i ,i′,•)∼=φ=(L ;i,i′,•) and L (W;Z)=H (W;Z).
∗ ∗ ∗ ∗ ∗ ∗
Remark 4 Note that the generators of R =Z6 and R =Z8 correspond to
2 3
R = α α′ ,α α′ ,α α′ ,α α′ ,α 0,α 0 ,
2 11 11 12 12 13 21 14 22 21 22
D E
R = θ θ′ ,θ θ′ ,θ θ′ ,θ θ′ ,θ 0,θ 0,0θ′ ,0θ′ ,
3 α11 α′ α12 α′ α13 α′ α14 α′ α23 α24 α′ α′
11 12 21 22 13 14
D E
where we denote by αα′ (resp. θ θ′ ) the 2(resp. 3)-cycles corresponding to
α α′
(−α)⊕α′ ∈Ker H1⊕H1′ i→+i′ L1 (resp. (−θα)⊕θα′′ ∈Ker H1⊕H1′ i→+i′ L1 )
(cid:16) (cid:17) (cid:16) (cid:17)
with k =2 (resp. k =3). Take a generator δ ,...,δ ,ε¯,δ¯of L ,
1 8 1
hδ ,...,δ i⊕hε¯i⊕ δ¯ ∼=Z8⊕Z/45⊕Z/675∼=L ,
1 8 1
and a generator ρ ,...,ρ of L , (cid:10) (cid:11)
1 6 2
hρ ,...,ρ i∼=Z6 ∼=L .
1 6 2
Then the pairings •:R ⊗L →L =Z and •:R ⊗L →L are described as follows.
2 2 0 3 2 1
• :R ⊗L →L =Z
2 2 0
α α′ ·ρ =1,
11 11 2
α α′ ·ρ =−1,
12 12 1
α α′ ·ρ =1,
13 21 4
α α′ ·ρ =−1,
14 22 3
α 0·ρ =1,
21 6
α 0·ρ =−1
22 5
x·y =0 (otherwise).
• :R ⊗L →L =Z
3 1 0
θα11θα′11 ·δ2 =−1,
θα12θα′12 ·δ1 =1,
θα13θα′21 ·δ4 =−1,
θα14θα′22 ·δ3 =1,
θ 0·δ =−1,
α23 8
θ 0·δ =1,
α24 7
0θα′ ·δ6 =−1,
13
0θα′ ·δ5 =1,
14
x·y =0 (otherwise).
4
• :R ⊗L →L
3 2 1
θα11θα′11 ·ρ2 =−s¯, s¯=45δ¯,
θα12θα′12 ·ρ1 =s¯,
θα13θα′21 ·ρ4 =−s¯,
θα14θα′22 ·ρ3 =s¯,
x·y =0 (otherwise).
As in the paper [2] on the homology cobordisms of plumbed 3-manifolds, we show the above
statements by two approaches. Let M(Γ) (resp. M(Γ′)) be a plumbed 3-manifoldassociatedto
the tree Seifert graph Γ (resp. Γ′) satisfying a certain condition (Ndeg) (Definition 6), and let
φ=(L ;i,i′,•) be an algebraic morphism between H (M(Γ);Z) and H (M(Γ′);Z). Then we
∗ ∗ ∗
construct a distributive algebra R (Γ,Γ′,φ) over Q by using the data (Γ,Γ′,φ) as in Definition
∗
6.
1. The 10/8-inequality.
WeapplyaV-manifoldversionoftheextendedFuruta-Kametani-10/8-inequalityforclosed
spin4-manifoldswithb >0. The10/8-inequalitycontainstermsdependingonthequadru-
1
ple cup product of the first cohomology on closed 4-V-manifolds. The quadruple products
arecalculatedbyusingthealgebraR (Γ,Γ′,φ)asamapq(Γ,Γ′,φ):R (Γ,Γ′,φ)⊗4 →Q.
∗ ∗
2. The associativity of cup products.
The distributive algebra R (Γ,Γ′,φ) is not necessarily associative by definition. However,
∗
if there exists a cobordismW between M(Γ) andM(Γ′) realizing the algebraicmorphism
φ, then we see that there exists an injective ring homomorphism from R (Γ,Γ′,φ) to the
∗
homologyringH (Z;Q)ofaclosed4-V-manifoldZ obtainedbygluingP (Γ),P (Γ′)andW
∗
along the boundaries M(Γ) and M(Γ′), and hence that R (Γ,Γ′,φ) must be associative.
∗
Therefore,the associativityofthe distributive algebraR (Γ,Γ′,φ) gives anobstructionto
∗
the existence of cobordisms W realizing φ.
Remark 5 As in paper [2], the author does not know examples that can be detected by using the
gauge theory in Approach 1 but cannot be detected by using the associativity of cup products in
Approach 2.
The homology cobordism category is defined to be the category whose objects are closed
3-manifolds and whose morphisms are homology cobordisms. If we take the quotient of the set
ofobjectsbythe homologycobordismrelation,thenweobtainthehomologycobordismmonoid.
In particular, we give a necessary condition, using w-invariants, for the homology 3-spheres to
belong to the inertia group associated to some homology 3-spheres.
The organizationof this paper is as follows. In Section2, weintroducea distributive algebra
R (Γ,Γ′,φ) and state the main theorems and their applications in Approach 1 using the 10/8-
∗
inequality, and in Approach 2 using the associativity of cup products. In Section 3, we recall
the definition of cobordism the category of 3-manifolds and introduce a category of graded
commutative rings with certain algebraic morphisms, which models that of the homology rings.
In Section 5, we recall the definition of the w-invariants, which are integral lifts of the Rochlin
invariants, and give several properties of the invariants under cobordisms and connected sum
operations. Here we also give a necessary condition for the homology 3-spheres to belong to
the inertia group associatedto some 3-manifolds in the homology cobordismmonoid. Finally in
Section6,toproveLemma32,weconsideracobordismoftwoplumbed3-manifoldsandcalculate
the intersection pairings of 3-cycles on closed 4-V-manifolds obtained by gluing 4-V-manifolds
along the boundaries.
5
2 Main theorems and applications
2.1 A distributive algebra constructed from algebraic morphisms
Motivated by the above Lemma 32 below in Section 6, we introduce a distributive algebra
R (Γ,Γ′,φ) using the data (Γ,Γ′,φ).
∗
Definition 6 Let Γ=(V,E,ω), Γ′ =(V′,E′,ω′) be two tree Seifert graphs satisfying the condi-
tion (Ndeg) :
1. theintersection matricesA(Γ), A(Γ′)oftheplumbedV-manifold P (Γ),P (Γ′)withrespect
to the standard basis are non-singular,
2. all the Euler numbers are non-zero e(ω )6=0, e(ω′ )6=0 for all v ∈V and v′ ∈V′.
v v′
Letφ=(L ;i,i′,•)beanalgebraicmorphismbetween(H (M(Γ);Z),•)and(H (M(Γ′);Z),•)
∗ ∗ ∗
in the L category . Then we define a graded commutative distributive algebra R (Γ,Γ′,φ) over
3 ∗
Q to be
R (Γ,Γ′,φ)=Q Z
4
R (Γ,Γ′,φ)=R(Γ,Γ′,φ)⊗Q
3

R (Γ,Γ′,φ)= Q Σ ⊕ Q Σ′ ,
 R21(Γ,Γ′,φ)=L(vΓ∈,VΓ′,φ)v⊗Q v′∈V′ v′
R (Γ,Γ′,φ)=LQ pt L
0
where 
R(Γ,Γ′,φ)=Ker H(Γ)⊕H(Γ′)θ→⊕θ′ H (M(Γ);Z)⊕H (M(Γ′);Z)i→+i′ L ,
2 2 2
L(Γ,Γ′,φ)=Im (cid:16)H(Γ)⊕H(Γ′)λ→⊕λ′ H (M(Γ);Z)⊕H (M(Γ′);Z)i→+i′ L (cid:17),
1 1 1
(cid:16) (cid:17)
as in Lemma30 in Section 6. If we denote theelements corresponding to (−α)⊕α′ ∈R(Γ,Γ′,φ)
formally by θ^θ′ then the product structure on R (Γ,Γ′,φ) is given by
α α′ ∗
Z·x=x (x∈R (Γ,Γ′,φ))
∗
 ^θαθα′′ ·θ]βθβ′′ =− v,v′∈V A(Γ)vv′(αv′ ·βv′)Σv+ v,v′∈V′A(Γ′)vv′(α′v′ ·βv′′)Σ′v,
 ^θiα(αθα′v)′ ··Σθ]βvθ=β′′Σ=v−·foθ^θP]rαβθθaβα′′n′′y·=iα(−,ααiv′()α∈=vH)−,(αθ^Γvα)·θ,α′ββ′v·,,βΣi′′′v(∈=αH′vΣ)(·′vPΓθ]·β′^θ)θα,β′θ′θ^α′=′θ=−′ θi,]′βθ](θαβ′θ′v′′)·,i∈′(Rα′v()Γ=,Γα′,′vφ·)βv′,
α α′ β β′
Note txΣhva·ty·ΣR=v′0(=Γ,(−Γx,′A,yφ(:Γ))iovstvhn′eporttw,niΣseec′v)e·s.sΣa′vr′il=y aAss(oΓc′i)avtvi′vpet,in general.
∗
By Lemma 29, 30 in Section 6 below, we have the following
Theorem 7 Let Γ = (V,E,ω), Γ′ = (V′,E′,ω′) be two tree Seifert graphs satisfying the condi-
tion (Ndeg), and let φ = (L ;i,i′,•) be a morphism between homology rings (H (M(Γ);Z),•),
∗ ∗
(H (M(Γ′);Z),•) in the L category. If there exists a cobordism (W;M(Γ),M(Γ′)) realizing
∗ 3
φ = (L ;i,i′,•), then there exists an injective ring homomorphism R (Γ,Γ′,φ) → H (Z;Q),
∗ ∗ ∗
and hence R (Γ,Γ′,φ) must be an associative ring.
∗
Remark 8 The V-manifold Z can be regarded as a rational homology manifold, and hence the
homology ring H (Z;Q) can be defined over the rationals Q.
∗
6
2.2 The 10/8-inequality and the quadruple cup products
The11/8-conjecture,duetoY.Matsumoto,statesthatforanyclosedspin4-manifoldthesecond
Bettinumberisgreaterthanorequalto11/8 timestheabsolutevalueofthesignature. Aweaker
inequality, called the 10/8-inequality, was first proved by M. Furuta using a technique based on
thefinite-dimensionalapproximationoftheSeiberg-Wittenequation. Thisinequalitywasproved
undertheassumptionthatthefirstBettinumberiszero,butthisconditioncanalwaysberealized
by surgeries. However, by dealing with the first cohomology of closed 4-manifolds, M. Furuta
and Y. Kametani improved the 10/8-inequality for closed spin 4-manifolds with positive first
Betti numbers by considering Pin(2)-equivariant maps between sphere bundles over the Jacobi
toriof 4-manifoldsconstructedfromthe finite-dimensionalapproximationof the Seiberg-Witten
equation [9]. Their result is based on the joint work of by M. Furuta, Y. Kametani, H. Matsue,
and N. Minami [8] on the stable homotopy version [7] of the Seiberg-Witten invariants. The
improved inequality contains terms that come from the quadruple cup product structures on
closed spin 4-manifolds.
Inthepaper[2],weextendedtheFuruta-Kametani-10/8-inequalitytothecaseofV-manifolds.
For various definitions concerning V-manifolds, see [11].
Let ((M,c),(X,cˆ)) be a pair consisting of a closed 3-manifold M with spin structure c and
a closed spin 4-V-manifold X with V-spin structure cˆsatisfying ∂(X,cˆ)=(M,c). Then we can
define an integral lift of the Rochlin invariant, which we call the w-invariant (Definition 19),
w((M,c),(X,cˆ))≡−µ(M,c) mod16.
The w-invariantwas defined injoint workwith M.Furuta onapplicationsof the 10/8-inequality
[4]. In fact, we proved the vanishing of the V-indices of the Dirac operators on closed 4-V-
manifolds X with b±(X) ≤ 2, which implies the homology cobordism invariance of w in a
2
certain class of homology 3-spheres. In joint work with M. Furuta and M. Ue [5], and in the
extensive work of N. Saveliev [12], it is shown that the Neumann-Siebenmann invariant for
plumbed homology3-spheresis equalto the w-invariantfor some auxiliaryplumbed V-manifold
and its V-spin structure. Recently, M. Ue proved that the Ozsv´ath-Szab´o correction term is
equal to the Neumann-Siebenmann invariant (and hence the w-invariant) for a large class of
plumbed rational homology 3-spheres [15].
We introduce the following quadruple product structure.
Lemma 9 Let Γ=(V,E,ω), Γ′ =(V′,E′,ω) be two tree Seifert graphs and let φ=(L ;i,i′,•)
∗
be a morphism between the homology rings (H (M(Γ)),•), (H (M(Γ′)),•) in the L category.
∗ ∗ 3
Then a quadruple product q(Γ,Γ′,φ):R (Γ,Γ′,φ)⊗4 →Q is calculated to be
3
θ^θ′ ·θ]θ′ ·θ]θ′ ·θ]θ′
α α′ β β′ γ γ′ δ δ′
=− A(Γ)vv′(αv·βv)(γv′ ·δv′)+ A(Γ′)vv′(α′v·βv′)(γ′v′ ·δv′′)
v,v′∈V v,v′∈V′
X X
By Lemma 32, we generalize the Theorem in [2] to obtain
Theorem 10 Let Γ=(V,E,ω), Γ′ =(V′,E′,ω′) be two tree Seifert graphs such that
1. b±(Γ)+b∓(Γ′)≤2m+2,
2. theintersection matricesA(Γ), A(Γ′)oftheplumbedV-manifold P (Γ),P (Γ′)withrespect
to the standard basis are non-singular, and
3. the Euler numbers are non-zero e(ω )6=0, e(ω′ )6=0 for all v ∈V and v′ ∈V′.
v v′
7
Letφ=(L ;i,i′,•)beamorphismbetweenthehomologyrings(H (M(Γ);Z),•),(H (M(Γ′);Z),•)
∗ ∗ ∗
intheL category. Supposethattheassociatedplumbed3-manifoldswithspinstructures(M(Γ),c),
3
(M(Γ′),c′) are spin cobordant (M(Γ),c) ≃φ (M(Γ′),c′) for some compact spin 4-manifold
(W,c˜)
(W,c˜)inducinganalgebraicmorphismφ=(L ;i,i′,•)∼=(L (W;Z);i ,i′,•)suchthatL (W;Z)=
∗ ∗ ∗ ∗ ∗
H (W;Z). If there exists an injective homomorphism
∗
h:H1 ♯m T4;Z →R(Γ,Γ′,φ)⊂R (Γ,Γ′,φ)
i=1 i 3
such that (cid:0) (cid:1)
h(x)·h(y)·h(z)·h(w)≡ x∪y∪z∪w, ♯m T4 mod 2,
i=1 i
for any x,y,z,w∈H1 ♯mi=1Ti4;Z , where the(cid:10)Ti4’s are m-cop(cid:2)ies of th(cid:3)e(cid:11)4-torusT4. Then we have
w(cid:0)((M(Γ),c(cid:1)),(P(Γ),cˆ))=w((M(Γ′),c′),(P(Γ),cˆ′)).
Example 11 Let Γ and Γ′ be two Seifert graphs in the above Example 3. Let Γ˜ =Γ ♯ s·({0},∅)
be the Seifert graph consisting of the disjoint union of Γ and s-vertices for s ≤2, with no edges
and with Seifert invariants
ω˜(v)=ω(v), v ∈V,
ω˜ (0)=ω ={0;(a ,b ),...,(a ,b )}, 1≤r ≤s.
ℓ 1 1 n n
We also define Γ˜′ = Γ′♯s·({0},∅) and the Seifert invariants ω˜′ by ω˜′(v) = ω′(v) for v ∈ V′
and ω˜′(0) = ω. Suppose that one or more of the a ’s are even for ω˜(0), so that the asso-
i
ciated disk V-bundle admits a V-spin structure. Then the plumbed 3-manifold M(Γ˜) is the
connected sum (M(Γ),c) ♯ s · (Σ,c ) of the plumbed 3-manifold (M(Γ),c) and s-copies of
Σ
the Seifert rational homology 3-sphere (Σ,c ) of Seifert invariant ω˜(0). The plumbed 4-V-
Σ
manifold P(Γ˜) is the boundary connected sum (P (Γ),cˆ) ♮ s·(E,c ), where E is the associated
E
disk V-bundle of the S1-fibration Σ. Suppose the Euler number e(ω) is positive then we have
b±(P (Γ)♮ s·E)+b∓(P (Γ′)♮ s·E) ≤ 2+s. Now the quadruple product associated to the 3-
2 2
cocycles θ^θ′ ,θ^θ′ ,θ^θ′ ,θ^θ′ ∈R(Γ,Γ′,φ) satisfies
α11 α′ α12 α′ α13 α′ α14 α′
11 12 21 22
θ^θ′ ·θ^θ′ ·θ^θ′ ·θ^θ′ =−5≡1 mod 2
α11 α′ α12 α′ α13 α′ α14 α′
11 12 21 22
In fact, by noting that
i(θα11)=i′(θα′′11), i(θα12)=i′(θα′′12), i(θα13)=i′(θα′′21), i(θα14)=i′(θα′′22),
and
A(Γ)−1 =A(Γ′)−1 = 2 −3 ,
−3 6
(cid:18) (cid:19)
and applying Lemma 32, we obtain
q(Γ,Γ′,φ) θ^θ′ ⊗θ^θ′ ⊗θ^θ′ ⊗θ^θ′
α11 α′ α12 α′ α13 α′ α14 α′
11 12 21 22
=−A(Γ)1(cid:16)1(α ·α )(α ·α )+A(Γ′)12(α′ ·α′(cid:17))(α′ ·α′ )
11 12 13 14 11 12 21 22
=−2·1·1+(−3)·1·1
=−5≡1 mod 2.
On the other hand, the w-invariants are calculated to be
w((M(Γ),c) ♯ s·(Σ,c ),(P (Γ),cˆ) ♮ s·(E,c ))
Σ E
−w((M(Γ′),c′) ♯ s·(Σ,c ),(P (Γ′),cˆ′) ♮ s·(E,c ))
Σ E
=w((M(Γ),c),(P (Γ),cˆ))−w((M(Γ′),c′),(P (Γ′),cˆ′))
=12−(−4)=166=0
8
and 2+s ≤ 2+2. Hence by Theorem 10, we see that there exists no cobordism (W,c˜) between
(M(Γ),c) ♯ s·(Σ,c ) and (M(Γ′),c′) ♯ s·(Σ,c ) inducing φ = (L ;i,i′,•) for s ≤ 2 such
Σ Σ ∗
that L (W;Z)=H (W;Z). Note that the difference is divisible by 16 and hence this cannot be
∗ ∗
detected by using the Rohlin invariant.
2.3 Associativity of intersection products on homology
On the other hand, we can prove the above statement in Approach 2. Motivated by Lemma 32,
we introduce the following triple product.
Definition 12 Let Γ = (V,E), Γ′ = (V′,E′) be two Seifert graphs and let φ = (K ;i,i′,•) be
∗
a morphism between homology rings (H (M(Γ);Z),•), (H (M(Γ′);Z),•) in the category L .
∗ ∗ 3
Then a triple product t(Γ,Γ′,φ):R (Γ,Γ′,φ)⊗3 →R (Γ,Γ′,φ) is defined and calculated to be
3 1
t(Γ,Γ′,φ) θ^θ′ ⊗θ]θ′ ⊗θ]θ′
α α′ β β′ γ γ′
≡ θ^θ′ (cid:16)·θ]θ′ ·θ]θ′ −θ^θ′ (cid:17)· θ]θ′ ·θ]θ′
α α′ β β′ γ γ′ α α′ β β′ γ γ′
=(cid:16) A(Γ)vv′(cid:17)(αv·βv)i(γv′)−(cid:16) A(Γ′)vv(cid:17)′(α′v·βv′)i′(γv′′)
v,v′∈V v,v′∈V′
X X
− A(Γ)vv′(βv·γv)i(αv′)+ A(Γ′)vv′(βv′ ·γv′)i′(α′v′)
v,v′∈V v,v′∈V′
X X
∈R (Γ,Γ′,φ).
1
In fact, we obtain the following criterion by Lemma 32.
Theorem 13 Let Γ = (V,E,ω), Γ′ = (V′,E′,ω′) be two tree Seifert graphs satisfying the con-
dition (Ndeg).
Let M(Γ) and M(Γ′) be the associated plumbed 3-manifolds, and φ = (L ;i,i′,•) be a mor-
∗
phism between homology rings (H (M(Γ)),•), (H (M(Γ′)),•) in the L category. If the triple
∗ ∗ 3
product is not zero,
t(Γ,Γ′,φ) θ^θ′ ⊗θ]θ′ ⊗θ]θ′ 6≡0∈R (Γ,Γ′,φ)
α α′ β β′ γ γ′ 1
(cid:16) (cid:17)
forsometripleθ^θ′ ,θ]θ′ ,θ]θ′ ∈R(Γ,Γ′,φ),thenthereexistsnocobordism W betweenM(Γ)
α α′ β β′ γ γ′
and M(Γ′) inducing φ such that L (W;Z)=H (W;Z).
∗ ∗
Example 14 In the above Example 3, we obtain
t(Γ,Γ′,φ) θ^θ′ ⊗θ^θ′ ⊗θ^θ′ =5δ 6=0.
α11 α′ α12 α′ α13 α′ 3
11 12 21
(cid:16) (cid:17)
In fact, by Definition 12, we have
t(Γ,Γ′,φ) θ^θ′ ⊗θ^θ′ ⊗θ^θ′
α11 α′ α12 α′ α13 α′
11 12 21
=A(Γ)11(cid:16)(α ·α )i(α )−A(Γ′)12(α′ (cid:17)·α′ )i′(α′ )
11 12 13 11 12 21
−A(Γ)11(α ·α )i(α )+A(Γ′)11(α′ ·0)i′(α′ )+A(Γ′)21(0·α′ )i′(α′ )
12 13 11 12 11 21 11
=2·1·i(α )−(−3)·1·i′(α′ )=5i(α )
13 21 13
=5δ 6=0.
3
Hence by Theorem 13 we see that there exists no cobordism W between M(Γ) ♯ s · Σ and
M(Γ′) ♯ s·Σ inducing φ=(L ;i,i′) such that L(W;Z)=H (W;Z) for any rational homology
∗ ∗
3-sphere Σ and non-negative integer s.
9
3 Cobordism category of 3-manifolds and an abstract cat-
egory of graded commutative rings
Inthis section,wegiveaprecisedefinitionoftheL category,acategoryofgradedcommutative
3
ringwithcertainalgebraicmorphismswhichwasintroducedinordertoinvestigatethecobordism
category C of 3-manifolds.
3
3.1 Cobordism category of 3-manifolds
Let (C ,⊔,∅) be the category whose set of objects ob(C ) is the set of all disjoint unions M
3 3
of closed oriented 3-manifolds and whose set of morphisms C (M,M′) between M ∈ ob(C )
3 3
and M′ ∈ ob(C ) is the set of all cobordisms (W;M,M′) between M and M′, that is, W is a
3
compact oriented smooth 4-manifold with boundary ∂W ∼=M ⊔(−M′). If M, M′, and M′′ are
three objects in C , then the composite operation on morphisms C (M,M′)×C (M′,M′′) →
3 3 3
C3(M,M′′), (W,W′) 7−→ W ∪M′ W′ is defined by gluing the 4-manifolds W and W′ along the
boundarycomponentM′. Thereexistsabifunctor⊔:C ×C →C definedbythedisjointunion
3 3 3
⊔ : ob(C )×ob(C ) → ob(C ), (M ,M ) 7−→ M ⊔M and ⊔ : C (M ,M′)×C (M ,M′) →
3 3 3 1 2 1 2 3 1 1 3 2 2
C (M ⊔M ,M′ ⊔M′), (W ,W )7−→W ⊔W , and the empty set ∅∈ ob(C ) defines the unit
3 1 2 1 2 1 2 1 2 3
element. Hence (C ,⊔,∅) defines a monoidal category.
3
LetCspinbethecategorywhosesetofobjectsob(Cspin)isthesetofalldisjointunions(M,c)of
3 3
closedoriented3-manifoldswithspinstructurescandwhosesetofmorphismsCspin((M,c),(M′,c′))
3
between(M,c)∈ob(Cspin)and(M′,c′)∈ob(Cspin)isthesetofallspincobordisms((W,c˜);(M,c),(M′,c′))
3 3
between (M,c) and (M′,c′), that is, (W,c˜) is a compact spin smooth 4-manifold with boundary
∂(W,c˜)∼=(M,c)⊔(−(M′,c′)). Then the monoidalcategoryof 3-manifoldswith spin structures
(Cspin,⊔,∅) is defined similarly.
3
3.2 A category of graded commutative rings
Let (L ,⊕,0) be a category whose set of objects ob(L ) is the set of all pairs (H ,•) composed
3 3 ∗
of
1. a graded Z-module H∗ = 3k=0Hk of dimension 3 such that H0 ∼=Zc for some c∈Z≥0,
2. a graded commutative proLduct •:H ⊗H →H , i.e. α ·β =(−1)(3−k)(3−ℓ)β ·α
k ℓ k+ℓ−3 k ℓ ℓ k
for α ∈H , β ∈H , satisfying the following conditions:
k k ℓ ℓ
(a) • : Hk ⊗H3−k → H0 ∼= Zc →ε Z induces an isomorphism H3−k ⊗Q ∼= (Hk ⊗Q)∗,
where ε:Zc →Z is given by ε(⊕c m )= c m ,
i=1 i i=1 i
(b) H ∼=Zc andthe elementµ∈H ∼=Zc correspondingto ⊕c 1∈Zc is a unit element
3 3 P i=1
with respect to the product •:H ⊗H →H ,
k 3 k
and whose set of morphisms L ((H ,•),(H′,•)) between two objects (H ,•) and (H′,•) is
3 ∗ ∗ ∗ ∗
the set of all quadruples (L ;i,i′,•) composed of
∗
1. a graded Z-module L = 4 L of dimension 4 such that L ∼=Zd for some d∈Z ,
∗ k=0 ∗ 0 ≥0
2. two homomorphisms H →Li L ←i′ H′ such that
∗ ∗ ∗
(a) L =Im H ⊕H′ i→+i′ L ,
k k k k
(cid:16) (cid:17)
10