Characterization of Y_2-equivalence for homology cylinders

29 Pages·0.49 MB·English

by  Gwenael Massuyeau

CHARACTERIZATION OF Y -EQUIVALENCE
2
FOR HOMOLOGY CYLINDERS
3
0
0 GWE´NAE¨LMASSUYEAUandJEAN-BAPTISTEMEILHAN
2 Laboratoire Jean Leray, UMR 6629 CNRS/Universit´e de Nantes
2 rue de la Houssini`ere, BP 92208, 44322 Nantes Cedex 03, France
n
a [email protected], [email protected]
J

6
ABSTRACT
] For Σ a compact connected oriented surface, we consider homology cylinders over
T
Σ: thesearehomologycobordismswithanextrahomologicaltrivialitycondition. When
G
considereduptoY2-equivalence,whichisasurgeryequivalencerelationarisingfromthe
. Goussarov-Habirotheory,homologycylindersformanAbeliangroup.
h
Inthispaper,whenΣhasoneorzeroboundarycomponent,wedefineasurgerymapfrom
t
a acertainspaceofgraphstothisgroup. Thismapisshowntobeanisomorphism,with
m inversegivenbysomeextensionsofthefirstJohnsonhomomorphismandBirman-Craggs
homomorphisms.
[

Keywords: homologycylinder,finitetypeinvariant,clover,clasper.
2
v Date: March18,2002and,inrevisedform,September 5,2002
9
7 1. Introduction
1
3
0 1.1. Homology cylinders
2
Homologycylindersareimportantobjects inthe theoryoffinite type invariants
0
/ ofGoussarov-Habiro: they havethus appearedinboth [6] and[4]. Letus recallthe
h
definition of these objects.
t
a LetΣbeacompactconnectedorientedsurface. Ahomology cobordism overΣis
m a triple (M,i+,i−) where M is a compact oriented 3-manifold and i± :Σ - M
: are oriented embeddings with images Σ±, such that:
v
i
X (i) i± are homology isomorphisms;
r (ii) ∂M =Σ+∪(−Σ−) and Σ+∩(−Σ−)=±∂Σ±;
a (iii) i+| =i−| .
∂Σ ∂Σ
Homologycobordismsareconsidereduptoorientation-preservingdiffeomorphisms.
When (i−)−1 ◦ (i+) : H (Σ;Z) - H (Σ;Z) is the identity, M is said to be
∗ ∗ 1 1
a homology cylinder. The set of homology cobordisms is denoted here by C(Σ),
and HC(Σ) denotes the subset of homology cylinders. If M = (M,i+,i−) and
2
N =(N,j+,j−) are homology cobordisms, we can define their stacking product by
M ·N :=(M ∪i−◦(j+)−1 N,i+,j−).
This product induces a monoid structure on C(Σ), with HC(Σ) a submonoid. The
unit element is 1 := (Σ×I,Id,Id), where I is the unit interval [0,1] and where
Σ
a collar of Σ± is stretched along ∂Σ×I so that the second defining condition for
homology cobordisms is satisfied.
Habiroin[6, §8.5]outlinedhowhomologycylinderscanserveas apowerfultool
in studying the mapping class groups of surfaces (see [3], [5], [12]). The connection
lies on the homomorphism of monoids
C-
T(Σ) HC(Σ)
sendingeachhintheTorelligroupofΣtothemappingcylinderC =(Σ×I,Id,h)
h
(with, as above, a collar of Σ± stretched along ∂Σ×I).
In the sequel, we restrict ourselves to the following two cases:
(i) Σ = Σ is the standard closed oriented surface of genus g ≥ 0, which here
g
is referred to as the closed case;
(ii) Σ=Σ is the standard compact oriented surface of genus g ≥0 with one
g,1
boundary component, which here is referred to as the boundary case.
The usual notations T = T(Σ ) and T = T(Σ ) for the Torelli groups will be
g,1 g,1 g g
used. Also denote by H the first homology group of Σ with integer coefficients, by
• the intersection form on H and by (x ,y )g a symplectic basis for (H,•).
i i i=1
1.2. Y -equivalence
k
The theory of finite type invariants of Goussarov-Habiro has come equipped
with a topological calculus toolbox: this was called calculus of claspers in [6] or
alternatively clovers in [2]. We will assume a certain familiarity of the reader with
these techniques.
Inparticular,letusrecallthat,fork ≥1aninteger,theY -equivalence1istheequiv-
k
alence relation generated by surgery on connected clovers of degree k. Following
Habiro in [6], we can then define a descending filtration of monoids
C(Σ)⊃C (Σ)⊃C (Σ)⊃···⊃C (Σ)⊃···
1 2 k
where C (Σ) is the submonoid consisting of the homology cobordisms which are
k
Y -equivalentto the trivialcobordism1 . Note the following fact, a proofofwhich
k Σ
has been inserted in §4.
Proposition 1.1. If Σ=Σ or Σ , then HC(Σ)=C (Σ).
g g,1 1
As mentioned by Habiro, we can show from the calculus of clovers that for every
k ≥1, the quotient monoid
C (Σ):=C (Σ)/Y
k k k+1
isanAbeliangroup. Inparticular,C (Σ)istheAbeliangroupofhomologycylinders
1
over Σ up to Y -equivalence. This group is the subject of the present paper.
2
1Thisequivalencerelationiscalled(k−1)-equivalence in[4],andAk-equivalencein[6].
3
For k ≥ 2, Habiro gives a combinatorial upper bound for the Abelian group
C (Σ). Precisely, he defines A (H) to be the Abelian group (finitely) generated
k k
by unitrivalent graphs of internaldegreek, with cyclic orientationat eachtrivalent
vertex and whose univalent vertices are labelled by elements of H and are totally
ordered. These graphs are considered modulo the well-known AS, IHX, multilin-
earity relations, and up to some “STU-like relations” dealing with the order of the
univalent vertices. In the closed case, some relations of a symplectic type can be
added. Then, there is a surjective surgery map
A (H) ψ--k C (Σ)
k k
sending each graph G to (1 ) , where G˜ is a clover in the manifold 1 with G as
Σ G˜ Σ
associated abstract graph, whose leaves are stacked from the upper surface Σ×1
according to the total order, framed along this surface and embedded according to
the labels of the corresponding univalent vertices. The fact that ψ is well-defined
k
also follows from the calculus of clovers.
Asforthecasek =1,Habirodoesnotdefineanyspaceofgraphsbutannounces
the following isomorphisms
C (Σ )≃Λ3H ⊕Λ2H ⊕H ⊕Z
(1.1) 1 g,1 (2) (2) 2
C (Σ )≃Λ3H/(ω∧H)⊕Λ2H /ω ⊕H ⊕Z
(cid:26) 1 g (2) (2) (2) 2
where H =H ⊗Z and where
(2) 2
g
ω = x ∧y ∈Λ2H
i i
i=1
X
is the symplectic element. This fact has been used afterwards in [12].
The goal of this paper is to prove these isomorphisms, in a diagrammatic way, by
again defining a surgery map
A (P) ψ-1 C (Σ).
1 1
The space of graphs A (P) and the map ψ appear to be meaningfully different
1 1
from A (H) and ψ for k > 1, making thus the case k = 1 exceptional. Indeed,
k k
their definition will involve both the homology group H and Spin(Σ), the set of
spin structures on Σ.
1.3. The Abelianized Torelli group
-
We denote by Ω the set of quadratic forms with • : H ×H Z as
g (2) (2) 2
associated bilinear form, namely
q-
Ω = H Z : ∀x,y ∈H , q(x+y)−q(x)−q(y)=x•y .
g (2) 2 (2)
n o
Note that Ω is an affine space over H , with action given by
g (2)
∀q ∈Ω ,∀x∈H , x·q :=q+x•(−).
g (2)
-
Thus, among the maps Ω Z , there are the affine functions, and more gen-
g 2
erally there are the Boolean polynomials which are defined to be sums of products
4
of affine ones (see [8, §4]). These polynomials form a Z -algebra denoted by B ,
2 g
which is filtered by the degree (defined in the obvious way):
B(0) ⊂B(1) ⊂···⊂B .
g g g
For instance, B(1) is the space of affine functions on Ω ; the constant function
g g
-
1 : Ω Z sending each q to 1 and, for h ∈ H, the function h sending each q
g 2
to q(h) are affine functions. Note the following identity:
(1.2) ∀h ,h ∈H, h +h =h +h +(h •h )·1∈B(1).
1 2 1 2 1 2 1 2 g
Another example of Boolean polynomial is the quadratic Boolean function
g
α= x ·y ,
i i
i=1
X
which is known as the Arf invariant. For any basis (e )2g for H, there is an
i i=1
isomorphism of algebras:
Z [t ,...,t ]
2 1 2g
(1.3) B ≃
g t2 =t
i i
sending 1 to 1 and e to t .
i i
Recall now from [8], that the many Birman-Craggs homomorphisms can be
summed up into a single homomorphism
β β B(3)
T - B(3) or T - g ,
g,1 g g α·B(1)
g
accordingtowhetheroneisconsideringtheboundarycaseortheclosedcase. Recall
also from [9] that the first Johnson homomorphism is a homomorphism
η η Λ3H
T 1- Λ3H or T -1 .
g,1 g
ω∧H
Form the following pull-back:
Λ3H × B(3) - B(3)
Λ3H(2) g g
q
? ?
Λ3H - Λ3H ,
(2)
−⊗Z
2
where the map q is the canonical projection B(3) - B(3)/B(2) followed by the
g g g
isomorphismB(3)/B(2) ≃Λ3H whichidentifiesthecubicpolynomialh h h with
g g (2) 1 2 3
h ∧h ∧h (this is well-defined because of (1.2) and (1.3)).
1 2 3
We denote by S the subgroupof this pull-back correspondingto ω∧H ⊂Λ3H and
α·B(1) ⊂ B(3). Johnson has shown in [10] that, under the assumption g ≥ 3, the
g g
homomorphisms η and β induce isomorphisms
1
Tg,1 (η1,β-) Λ3H × B(3) and Tg (η1,β-) Λ3H ×Λ3H(2) Bg(3).
T′ ≃ Λ3H(2) g T′ ≃ S
g,1 g
5
Remark 1.2. Notethat,becauseof(1.3),thecodomainsofthesemapsarerespec-
tivelynon-canonicallyisomorphictoΛ3H⊕Λ2H ⊕H ⊕Z andΛ3H/(ω∧H)⊕
(2) (2) 2
Λ2H /ω ⊕H ⊕Z .
(2) (2) (2) 2
1.4. Statement of the results
In §2, we will construct the space of graphs A (P) and the surgery map ψ :
1 1
-
A (P) C (Σ). Spin structures play a prominent role in their definitions.
1 1
Observe that, C (Σ) being an Abelian group, the mapping cylinder construction
1
induces a group homomorphism
T(Σ) C
-
C (Σ).
T(Σ)′ 1
AspointedoutbyGaroufalidisandLevinein[3]and[12],Johnsonhomomorphisms
-
and Birman-Craggs homomorphisms factor through C :T(Σ) HC(Σ). These
extensions will be detailed in §3.
Next, we will specify in §4 an isomorphism ρ : A (P) - Λ3H × B(3) and
1 Λ3H(2) g
the following two theorems will be proved from the previous material.
Theorem 1.3. In the boundary case, the diagram
ψ C T
1 - (cid:27) g,1
A (P) C (Σ )
1 1 g,1 T′
g,1
@ (cid:0)
@ (cid:0)
@ (η1,β) (cid:0)
ρ @ (cid:0) (η1,β)
@R ? (cid:9)(cid:0)
Λ3H × B(3)
Λ3H(2) g
commutes and all of its arrows are isomorphisms, except for the two maps starting
from T /T′ when g <3.
g,1 g,1
Theorem 1.4. In the closed case, the diagram
A (P) ψ C T
1 1 - (cid:27) g
C (Σ )
ρ−1(S) 1 g T′
g
@ (cid:0)
@ (cid:0)
@ (η1,β) (cid:0)
ρ @ ? (cid:0) (η1,β)
@RΛ3H × B(3)(cid:9)(cid:0)
Λ3H(2) g
S
commutes and all of its arrows are isomorphisms, except for the two maps starting
from T /T′ when g <3.
g g
Note that Theorem 1.3 and Theorem 1.4 together with Remark 1.2, give Habiro’s
isomorphisms (1.1), which are non-canonical. Also, we will easily deduce the fol-
lowing.
Corollary 1.5. For Σ=Σ or Σ , let M and M′ be two homology cylinders over
g,1 g
Σ. Then, the following assertions are equivalent:
(a) M and M′ are Y -equivalent;
2
6
(b) M and M′ are not distinguished by degree 1 Goussarov-Habiro finite type
invariants;
(c) M and M′ are not distinguished by the first Johnson homomorphism nor
Birman-Craggs homomorphisms.
-
Finally, if an embedding Σ ⊂ Σ is fixed, there is an obvious “filling-up”
g,1 g
-
map C (Σ ) C (Σ ), through which the commutative diagrams of Theorem
1 g,1 1 g
1.3 and Theorem 1.4 are compatible. The reader is referred to §4 for a precise
statement.
2. Definition of the Surgery Map ψ
1
In this section, we define the space of graphs A (P) and the surgery map ψ
1 1
announced in the introduction.
2.1. Special Abelian groups and the A functor
1
Let us denote by Ab the category of Abelian groups. An Abelian group with
special element isapair(G,s)whereGisanAbeliangroupands∈Gisoforderat
most2. WedenotebyAb thecategoryofspecialAbeliangroupswhosemorphisms
s
aregrouphomomorphismspreservingthespecialelements. Wenowdefineafunctor
Ab A-1 Ab
s
inthefollowingway. For(G,s)anobjectinAb ,A (G,s)isthefreeAbeliangroup
s 1
generatedbyY-shapedunitrivalentgraphs,whosetrivalentvertexisequippedwith
a cyclic orderonthe incident edgesand whose univalent vertices arelabelled by G,
subject to some relations. The notation
Y[z ,z ,z ]
1 2 3
will stand for the Y-shaped graph whose univalent vertices are colored by z , z
1 2
and z ∈ G in accordance with the cyclic order, so that our notation is invariant
3
under cyclic permutation of the z ’s. The relations are the following ones:
i
Antisymetry (AS) : Y[z ,z ,z ]=−Y[z ,z ,z ],
1 2 3 2 1 3
Multilinearity of colors : Y[z +z ,z ,z ]=Y[z ,z ,z ]+Y[z ,z ,z ],
0 1 2 3 0 2 3 1 2 3
Slide : Y[z ,z ,z ]=Y[s,z ,z ],
1 1 2 1 2
where z ,z ,z ,z ∈ G. For (G,s) f- (G′,s′) a morphism in Ab , A (f) maps
0 1 2 3 s 1
each generator Y[z ,z ,z ] of A (G,s) to Y[f(z ),f(z ),f(z )]∈A (G′,s′).
1 2 3 1 1 2 3 1
7
-
Example 2.6. The map [G (G,0)] makes Ab a (full) subcategory of Ab . It
s
follows from the definitions that the following diagram commutes:
- -
Ab Ab
s
@
@
@ A1
Λ3(−) @
R@ ?
Ab.
Non-trivial examples will be given in the next paragraph. For future use, note
that this category has an obvious pull-back construction extending that of Ab:
-
(G ,s )× (G ,s ) (G ,s )
1 1 (G,s) 2 2 2 2
f
2
? ?
-
(G ,s ) (G,s)
1 1
f
1
where(G ,s )× (G ,s )isthe subgroupofG ×G consistingofthose (z ,z )
1 1 (G,s) 2 2 1 2 1 2
such that f (z )=f (z ), and with special element (s ,s ).
1 1 2 2 1 2
2.2. Spin structures and the special Abelian group P
In this paragraph, let M be a compact oriented 3-manifold endowed with a
Riemannian metric, and let FM be its bundle of oriented orthonormalframes:
i p
- - --
SO(3) E(FM) M.
Lets∈H (E(FM);Z) be the image by i ofthe generatorofH (SO(3);Z)≃Z .
1 ∗ 1 2
Recall that M is spinnable and that Spin(M) can be defined as
Spin(M)= y ∈H1(E(FM);Z ), <y,s>=6 0 ,
2
which is essentially independ(cid:8)ent of the metric. The manifold M(cid:9)being spinnable, s
is not 0 (and so is of order 2).
Now, Spin(M) being an affine space over H1(M;Z ) with action given by
2
∀x∈H1(M;Z ),∀σ ∈Spin(M), x·σ :=σ+p∗(x),
2
we can consider the space
A(Spin(M),Z )
2
of Z -valued affine functions on Spin(M). For instance, 1 ∈ A(Spin(M),Z ) will
2 2
-
denote the constant map defined by σ 1.
There is a canonical map
κ-
A(Spin(M),Z ) H (M;Z ).
2 1 2
For f ∈A(Spin(M),Z ), the homology class κ(f) is defined unambiguously by
2
∀σ,σ′ ∈Spin(M), f(σ′)−f(σ)=<σ′/σ,κ(f)>∈Z ,
2
8
where σ′/σ ∈H1(M;Z ) is definedby the affine actionofH1(M;Z ) onSpin(M).
2 2
Another canonical map is
e-
H (E(FM);Z) A(Spin(M),Z )
1 2
-
sending a x to the map defined by σ < σ,x >. Next lemma gives us a nice
understanding of the special Abelian group (H (E(FM);Z),s).
1
Lemma 2.7. a) The following diagram of special groups is a pull-back diagram:
e
-
(H (E(FM);Z),s) A(Spin(M),Z ),1
1 2
(cid:0) (cid:1)
p κ
∗
? ?
-
(H (M;Z),0) (H (M;Z ),0).
1 1 2
−⊗Z
2
b) Let t be the map
t-
Oriented framed knots in M H (E(FM);Z)
1
which adds to a(cid:8)ny oriented framed knot K an(cid:9)extra (+1)-twist, and next sends it to
the homology class of its lift in FM. Then,
(i) t is surjective;
(ii) t = t if and only if K and K are cobordant as oriented knots in M
K1 K2 1 2
and if their framings with respect to a surface with boundary (K )∪(−K )
1 2
then differ from each other by an even integer;
(iii) if K ♯K denotes the band connected sum of K and K , then t =
1 2 1 2 K1♯K2
t +t ;
K1 K2
(iv) the k-framed trivial oriented knot (k ∈Z) is sent by t to k·s.
Proof. We begin by proving a). The commutativity of the diagram of special
groups is easy to verify. By functoriality, we get a map
(H1(E(FM);Z),s) (p∗,-e) (H1(M;Z),0)×(H1(M;Z2),0) A(Spin(M),Z2),1 .
The Serresequence associatedto the fibrationFM givesfo(cid:0)r homologywith int(cid:1)eger
coefficients:
i p
- -∗ -∗ -
0 H (SO(3);Z) H (E(FM);Z) H (M;Z) 0.
1 1 1
The bijectivity of (p ,e) follows from the exactness of this sequence.
∗
We now prove b) and we begin with assertion(iv). Let K be a trivial k-framed
oriented knot, let ∗∈K and let e=(e ,e ,e )∈p−1(∗) be the framing of K at ∗.
1 2 3
We denote by K˜ the lift of K to FM. Then, as a loop in E(FM), K˜ is homotopic
to the loop in the fiber p−1(∗) defined by
-
[0,1]∋t R (e),
2π(k+1)t
where R (with θ ∈ R) denotes the rotation of oriented axis directed by e and
θ 3
angle θ. From an appropriate description of the generator of π (SO(3)) ≃ Z , it
1 2
follows that K˜ =(k+1)·s∈H (E(FM);Z), and assertion (iv) then follows.
1
h i
9
Letusmakeanobservation. LetK beanyorientedframedknotinM;sincethe
framing of K determines a trivialization of its normal bundle in M, it allows us to
restrictanyspin structureonM to K. Recallnowthat the cobordismgroupΩSpin
1
is isomorphic to Z (with generator given by S1 endowed with the spin structure
2
induced by its Lie group structure: see [11, p. 35, 36]). The following observation
then makes sense:
(2.4) ∀σ ∈Spin(M), e(t )(σ)=(K,σ| ) ∈ΩSpin ≃Z ,
K K 1 2
and can be derived from an appropriate characterization of the spin structures on
the circle (see [11, p. 35, 36]).
Let now K and K be some disjoint oriented framed knots in M. There is an
1 2
obviousgenus0surface withboundaryK ♯K ∪˙(−K )∪˙(−K ). Then, accordingto
1 2 1 2
(2.4), we have e(t ) = e(t )+e(t ). Also, p (t )= [K ♯K ]= [K ]+
K1♯K2 K1 K2 ∗ K1♯K2 1 2 1
[K ]= p (t )+p (t ), and so by a), we obtain that assertion (iii) holds for K
2 ∗ K1 ∗ K2 1
and K .
2
We now justify assertion (ii). According to a), t = t if and only if p (t ) =
K1 K2 ∗ K1
p (t ) and e(t ) = e(t ). Also, the condition p (t ) = p (t ) holds if and
∗ K2 K1 K2 ∗ K1 ∗ K2
onlyifK andK arehomologousinM. Inthiscase,letS beanembeddedoriented
1 2
surface in M such that ∂S =K ∪˙(−K ). Let k be the framing of K with respect
1 2 i i
to S and let K′ be the oriented framed knot obtained from K by adding an extra
i i
(−k )-twist, so that the framing of K′ is given by S. Then, according to (2.4),
i i
we have e tK′ = e tK′ . Moreover, applying assertions (iii) and (iv), we obtain:
1 2
e tKi′ = e(cid:0)(tK(cid:1)i)+k(cid:0)i·s.(cid:1)We conclude that e(tK1) = e(tK2) if and only if k1 and
k are equal modulo 2, proving thus assertion (ii).
2(cid:0) (cid:1)
Let x∈ H (E(FM);Z), then p (x) ∈H (M;Z) can be realized by an oriented
1 ∗ 1
knot K in M: we give it an arbitrary framing. By construction, p (t −x) = 0 ∈
∗ K
H (M;Z), andso by exactnessofthe Serresequence,t −x=ε·swith ε∈{0,1}.
1 K
By possibly band-summing K with a trivial (+1)-framed knot when ε = 1, and
accordingtoassertion(iii)and(iv),the framedknotK canbe supposedtobe such
that t =x; this proves assertion (i). (cid:3)
K
We now restrict ourselves to the 3-manifold M = 1 = Σ×I where Σ can be
Σ
Σ or Σ . The inclusion i+ :Σ ⊂ - 1 , with image Σ+, induces an isomorphism
g g,1 Σ
betweenH andH (M;Z)andabijectionbetweenSpin(Σ)andSpin(M). Asshown
1
byJohnsonin[7],thereisanalgebraicwaytothinkofSpin(Σ). Indeed,thereexists
a canonical affine isomorphism
≃-
Spin(Σ) Ω ,
g
sendinganyspinstructureσ toaquadraticformq whichcanbedefinedasfollows.
σ
Letx∈H =H (Σ;Z )berepresentedbyanorientedsimpleclosedcurveonΣ+;
(2) 1 2
by framing it along Σ+ and pushing it into the interior of 1 , we get a framed
Σ
oriented knot K in 1 . Then,
Σ
(2.5) q (x)=e(t )(σ×I) ∈Z .
σ K 2
10
Therefore, according to Lemma 2.7 a), (H (E(F1 );Z),s) is canonically isomor-
1 Σ
phic to the special Abelian group defined by the pull-back construction
e
(H,0)× B(1),1 - B(1),1
(H(2),0) g g
(cid:16) (cid:17) (cid:16) (cid:17)
p κ
? ?
-
(H,0) H ,0
(2)
−⊗Z
2
(cid:0) (cid:1)
whose projections are denoted by p and e, and where κ is the composite
≃
B(1) -- B(1)/B(0) - H .
g g g (2)
The last isomorphism here identifies h with h for all h ∈ H (this is well-defined
(2)
by (1.2) and (1.3)). We define the special Abelian group P to be
P =(H,0)× B(1),1 ,
(H(2),0) g
(cid:16) (cid:17)
and A (P) is the space of graphs announced in the introduction.
1
Remark 2.8. Thus, any element z of P can be written as
z = h,h+ε·1 ∈P,
with h∈H and ε∈{0,1}. Observe(cid:0)also the fo(cid:1)llowing. Suppose that there exists a
simple oriented closed curve in Σ+ with homology class h. Let K be the push-off
of this curve, framed along Σ+, with an extra ε-twist. Then, it follows from (2.5)
that t =z ∈P ≃H (E(F1 );Z).
K 1 Σ
Remark 2.9. Accordingto the proofofLemma2.7, the Serresequencefor homol-
ogy associated to the bundle F1 gives the following short exact sequence:
Σ
p
- - - -
0 Z P H 0,
2
-
whereZ isinjectedintoP bysending1to(0,1). Themaps:H P definedby
2
s(h)= h,h is a set-theoretic section. Accordingto (1.2), the associated2-cocycle
-
H×H Z isthe mod2reducedintersectionformofΣ. Thus,P isisomorphic
2
to H ⋊(cid:0)Z w(cid:1)ith crossed product defined by
2
(h ,ε )·(h ,ε )=(h +h ,ε +ε +h •h ).
1 1 2 2 1 2 1 2 1 2
The element h,h+ε·1 ∈P corresponds to (h,ǫ)∈H ⋊Z .
2
(cid:0) (cid:1)
2.3. The surgery map ψ
1
In this paragraph, Σ is allowed to be Σ or Σ and the surgery map ψ :
g g,1 1
-
A (P) C (Σ) is constructed by means of calculi of clovers.
1 1
Convention2.10. Here,weadopt Goussarov’sconventionforthesurgerymeaning
of Y-graphs and clovers [4], [2].