www.elsevier.com/locate/top
On string topology of three manifolds
Hossein Abbaspour
∗Centre de Mathématiques, École Polytechnique, 91128 Palaiseau, France Received 8 October 2004; accepted 4 April 2005
Abstract
LetMbe a closed, oriented and smooth manifold of dimensiond. LetLMbe the space of smooth loops inM. In [String topology, preprint math.GT/9911159] Chas and Sullivan introduced the loop product, a product of degree
−don the homology ofLM. We aim at identifying the three manifolds with “nontrivial” loop product. This is an application of some existing powerful tools in three-dimensional topology such as the prime decomposition, torus decomposition, Seifert fiber space theorem, torus theorem.
!2005 Elsevier Ltd. All rights reserved.
Keywords:Free loop space; Prime decomposition; JSJ-decomposition
1. Introduction and statement of main theorem
Throughout this paperMdenotes a connected, oriented smooth manifold unless otherwise stated. We think ofS1the unit circle with amarked point, as the quotientR/Z. Themarked pointofS1is the image of 0 in this quotient. Aloop inMis a continuous map fromS1toM. Thefree loop spaceLM ofMis the space of all loops inM. There is a one-to-one correspondence between the connected components of LM and the conjugacy classes of!1(M). Iff :S1 → Mis a loop inMthen the image of the marked point ofS1is called themarked pointof the loopf. The integral homology ofLMis equipped with the loop product,•, an associative product of degree−d, wheredis the dimension ofM,
• :Hi(LM)⊗Hj(LM)→Hi+j−d(LM).
∗Fax: +33 1 69 33 49 63.
E-mail address:habbaspour@math.polytechnique.fr.
0040-9383/$ - see front matter!2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.top.2005.04.003
We give an informal description of the loop product: given two homology classes inH∗(LM), choose a chain representative for each one of them so that the corresponding sets of marked points intersect transversally. By concatenating the corresponding loops of each chain at the intersection points one obtains a new chain inLM. We refer the reader to[6] for a homotopy theoretic definition of the loop product.
Theorem(Chas–Sullivan[5]). For a connected oriented smooth manifold M,(H∗−d(LM),•)is a graded commutative algebra.
Letp : H∗(LM)→ H∗(M)be the map induced byf %→ f (0)andi :H∗(M) →H∗(LM)be the map induced by the inclusion of the constant loops. Observe that
p◦i=idH∗(M),
hence there is a canonical decompositionH∗(LM)=i(H∗(M))⊕AMwhereAM=Kerp. The projection onAM is denoted bypAM andpAM =idH∗(LM)−p. From now on we identifyH∗(M)with its image underi.
Proposition 1.1 (see Chas and Sullivan [5]). i : H∗(M) → H∗(LM)is a map of graded algebras where the multiplication onH∗(M)is the intersection product denoted by∧.Therefore(H∗(LM),•)is an extension of(H∗(M),∧).
If Mis a closed manifold then the algebra(H∗(LM),•) has aunit, "M, which is the image of the fundamental class ofMunderi.
Definition 1.2. We shall sayMhasnontrivial extended loop productsif the restriction of the loop product toAM is nontrivial. Otherwise we say thatMhastrivial extended loop products.
Definition 1.3. A closed oriented 3-manifoldMis said to bealgebraically hyperbolicif it is aK(!,1) (aspherical) and its fundamental group has no rank 2 abelian subgroup.
Note that every finite cover of an algebraically hyperbolic manifold is also algebraically hyperbolic.
The following is the main theorem of this paper[1]:
Theorem 1.4. Let M be an oriented closed3-manifold. If M is not algebraically hyperbolic then M or a double cover of M has nontrivial extended loop products.
If M is algebraically hyperbolic then M and all its finite covers have trivial extended loop products.
Remark 1.5. Although the main result as stated above concerns the closed 3-manifolds, throughout this paper we identify several classes of 3-manifolds with boundary or which have nontrivial extended loop products (see Sections 3, 5 and 6.1).
Remark 1.6. As we shall see many of our arguments can be generalized to higher-dimensional manifolds.
We have pointed them out with some remarks in Sections 3, 5 and 6. Also we invite the reader to see Example 1.8.
Notation: The based loop space of Mis denoted#M and!ˆ1(M) is the set of conjugacy classes of
!1(M). For$∈!1(M),C$is the centralizer of$in!1(M)and[$]is its conjugacy class. For a conjugacy class[$],(LM)[$]denotes the corresponding connected component ofLM.
Lemma 1.7. For$∈!1(M)andf ∈LMa loop representing$,then there exists a short exact sequence
!2(M)!!1((#M))−→!1((LM)[$], f )−→C$ −→0. (1.1) Example 1.8(see Chas and Sullivan[5]). LetMbe a closed oriented hyperbolic manifold of dimension d!3. SinceMis aspherical, it follows from the long exact sequence associated with the fibration#M !→ LM→p Mthat each connected component ofLMis also aspherical. In order to find the homotopy type of each component(LM)[$], one has to compute the fundamental group of each component. As!2(M)=0, by Lemma 1.7, each component(LM)[$]is aK(C$,1). If$*=1∈!1(M)thenC$ +ZasMis a closed hyperbolic manifold. Hence(LM)[$]is aK(Z,1)and
AM +H∗
#
[$]*=[1]∈ˆ!1(M)
K(Z,1)
.
Therefore the restriction of the loop product toAM is identically zero asAM is concentrated at degree at most 1.
2. 3-Manifolds with finite fundamental group
In this section, we show that closed 3-manifolds with finite fundamental group have nontrivial extended loop products.
The following construction is valid for any oriented manifold M. Let p be the base point of M of dimensiond. Consider the 0-chain inLMthat consists of the constant loop atp. It represents a homology class % ∈ H0(LM). Given a chainc ∈ LM whose set of marked points is transversal top, then the loops incwhose marked points arep, form a chain in the based loop space#M. This induces a map of degree−d
∩ :H∗(LM)→H∗−d(#M),
wheredis the dimension ofM. On the other handH∗(#M)is equipped with the Pontrjagin product.
Lemma 2.1(see Chas and Sullivan[5]). ∩ :(H∗(LM),•)→(H∗−d(#M),×)is an algebra map where
×is the Pontrjagin product. In particular if M is a Lie group then∩is surjective.
Proposition 2.2. S3has nontrivial extended loop products.
Proof. AsS3 is a Lie group, by Lemma 2.1∩is surjective.(H∗(#S3),×)is a polynomial algebra with one generator of degree 2. Letx1 ∈ Hp(#S3) andx2 ∈ Hq(#S3) be two nonzero elements such that p, q >3. Letai ∈ ∩−1(xi)fori=1,2. SinceHi(S3)=0 fori >3 we haveai ∈ AS3. By Lemma 2.1,
∩(a1•a2)= ∩(a1)× ∩(a2)=x1×x2 *=0, thereforea1•a2*=0. "
For an arbitrary homotopy 3-sphere we can obtain some examples of nontrivial loop product by trans- ferring the examples found for 3-sphere in Proposition 2.2, using a suitable homotopy equivalence. The following two lemmata describe and prove the existence of such a homotopy equivalence.
Lemma 2.3. Let M and N be two closed oriented d-manifolds andp ∈ N.Forf : M → N which is transversal to p andf−1(p)consists of only one point,the following diagram commutes:
H∗(LM) −→∩ H∗(#M)
fL
' f#
' H∗(LN ) −→∩ H∗(#N )
(2.1)
f#andfLare the induced maps by f on the homologies of the based loop space and the free loop space.
Proof. We choose p and f−1(p), respectively, as the base point of N and M. We show that (2.1) is commutative at the chain level. Letk:&m→LMbe am-simplex inLM. So for eachx ∈&m
k(x):S1 →M.
Ifk0:&m→M, k0(x)=k(x)(0),
is transversal tof−1(p)then consider the set'= {x ∈&m|k(x)(0)=f−1(p)}.
k|'is a chain in#f−1(p)M and∩(k)=k|'. After composing withf,f ◦k|'is a chain in#pN and (f#◦ ∩)(k)=f ◦k|'.
On the other handfL(k)=f◦kis a chain inLN. Sincek0is transversal tof−1(p)andfis transversal topthusf ◦k0 is transversal top. Sincef−1(p)consists of only one point inMthen
{x ∈&m|((f ◦k)(x))(0)=p} = {x ∈&m|k(x)(0)=f−1(p)} =', therefore,
(∩ ◦fL)(k)=f ◦k|'=(f#◦ ∩)(k). "
Lemma 2.4. Letp1, p2, . . . , pr,be r distinct points in a homotopy3-sphere M. Then there is a homotopy equivalencef :S3 →Mwhich is transversal to allpi’s andf−1(pi)consists of only one point for each i.
Proof. Consider a closed ballD ⊂S3. Letfbe a homeomorphism fromDto a closed ball inD/ ⊂M wherepi ∈D/for alli.
f|!D : !D → !D/ can be extended tof : S3\IntD/ → M\IntD as all the obstructions which are cohomology classes in
Hq+1(S3\IntD/,!D/,!q(M\IntD)) vanish sinceM\I nt D/is contractible.1
Therefore, we have a map f : S3 → M which sends D to D/ homeomorphically and sends the complement ofDto the complement ofD/and in particular it is transversal topi andf−1(pi)consists of only one point inS3. To see thatfis a homotopy equivalence it is enough to observe that it has degree
one. "
Proposition 2.5. A closed simply connected3-manifold M has nontrivial extended loop products.
Proof. Letp∈Mandf :S3 →Mbe the homotopy equivalence provided by Lemma 2.4 forr =1.
finduces the homotopy equivalencesf# : #S3 → #M andfL : LS3 → LM. By Lemma 2.3 the following diagram is commutative:
H∗(LS3) −→∩S3 H∗(#f−1(p)S3)
fL
' f#
' H∗(LM) −→∩M H∗(#pM)
(2.2)
Sincef#andfLare isomorphisms and∩S3 is surjective, therefore∩M is also surjective. Letx1 *=0 ∈ Hm(#M)2 andx2 *= 0 ∈ Hn(#M)wheren, m >3 andai ∈ ∩−M1(xi),i=1,2. Then,ai ∈ AM since Hk(M)=0 for allk >3. By Lemma 2.1,
∩(a1•a2)= ∩(a1)× ∩(a2)=x1×x2 *=0, hencea1•a2 *=0. "
Now we construct some examples of nontrivial loop product for a 3-manifold with finite fundamental group. The homology classes are obtained by pushing forward the classes introduced in Proposition 2.5 for the universal cover.
Proposition 2.6. If M is a closed oriented3-manifold with finite fundamental group then M has nontrivial extended loop products.
Proof. LetM˜ be the universal cover ofMandqbe the covering map andr=degree(q)= |!1(M)|<∞. Choosep ∈M and let{p1, p2, . . . , pr} ∈q−1(p)wherepi’s are distinct.
SinceM˜ is homotopy equivalent toS3, by Lemma 2.4 there is a homotopy equivalencef :S3 → ˜M which is transversal at eachpi andf−1(pi)consists of only one point inS3 inM. Letmi=f−1(pi), i=1,2, . . . , r. Hence(q◦f )−1(p)= {m1, m2, . . . , mr}andq◦f is transversal top. We choosem1as the base point ofS3.
1This can be proved using Mayer–Vietrois exact sequence.
2H∗(#M)!H∗(#S3)!Z[x]and degreex=2.
The covering mapq : ˜M →Mcomposed byfinduces the maps (q◦f )L:H∗(LS3)→H∗(LM)
and
(q◦f )#:H∗(#m1S3)→H∗(#pM), where the latter is an isomorphism.
We claim that
H∗(LS3) r.−→∩S3 H∗(#m1S3)
(f◦q)L
' (f◦q)#
' H∗(LM) −→∩M H∗(#pM)
(2.3)
is commutative. To prove this, note thatS3 has a group structure hence we have the homeomorphism LS3 +S3×#S3,and
H∗(LS3)+H∗(S3)⊗H∗(#S3). (2.4)
There are two types of homology classes inH∗(LS3)and we verify the commutativity for the both cases:
(1) The classes that correspond to homology classes inH0(S3)⊗H∗(#S3)under isomorphism (2.4).
These classes can be represented by cycles inLS3whose sets of marked points consist of a single point inS3. Therefore they are mapped to 0 by∩S3, as we can choose a chain representative whose set of marked point, consisting of only one points, is different from allmi’s. Therefore they are mapped to 0 by (f ◦q)#◦(r.∩S3).
On the other hand their images under(f ◦q)L have the set of marked point consisting of only one point which is different fromp. Thus, they are mapped to zero under∩M ◦(f ◦q)Land the diagram commutes.
(2) The classes that by isomorphism (2.4) correspond to the homology classes inH3(S3)⊗H∗(#S3).
Such an element(∈H∗(LS3)corresponds to"⊗b∈H3(S3)⊗H∗(#S3)for someb∈H∗(#S3)and"
is the fundamental class ofS3. Then
∩S3(()=b so
(f ◦q)#◦(r∩S3()=r(f ◦q)#(b)
and this is exactly∩M◦(f ◦q)#(b). Note that underf◦q,rcopies ofbwhich are based at each one of mi’s, come together. Therefore∩M ◦(f ◦q)L(()=r(f ◦q)#(b).
To finish the proof, considera1, a2 ∈ H∗(LS3) as constructed in the proof of Proposition 2.2. We have(q ◦f )L(ai)∈ AM,i=1,2 and from the commutativity of (2.3) and the fact that(q◦f )#is an isomorphism it follows that
(q◦f )L(a1)•(q◦f )L(a2)*=0. "
(1,b)
(1,p) )
*(S2) +
Fig. 1.S1×S2.
3. 3-Manifolds with non-separating 2-sphere or 2-torus
In this section, we prove the 3-manifolds with a non-separating sphere or 2-torus have nontrivial extended loop product. We recall that the action of the unit circle on LMinduces a map of degree 1,
&:H∗(LM)→H∗+1(LM).
Proposition 3.1. S1×S2has nontrivial extended loop products.
Proof. Letbandpbe two distinct points inS2. We choose(1, p)as the base point ofS1×S2. The map x %→(x, p),x ∈S1, gives rise to an element,of!1(S1×S2).
Consider the map+:S1 →S1×S2defined by+(x)=(x, b). Note that+as a loop with the marked point(1, b), represents a homology class-∈H0((L(S1×S2))[,]).
Let*:S2→S1×S2be the map defined by*(y)=(1, y). The images of+and*intersect exactly at (1, b). We write*(S2)as a union of circles, any two of them having only the point(1, p)in common. This gives rise to a one dimensional family of loops inS1×S2(seeFig. 1). Note that the free homotopy type of the loops of this one-dimensional family is the one of the trivial loop. One can compose the loops of this family with a fixed loop whose marked point is(1, p)and modify their free homotopy type. Suppose that we have done this modification with a fixed loop which does not meet+and represents a nontrivial element" ∈!1(S1×S2)where"*=,. This new one-dimensional family of loops represents a homology class.∈H1((L(S1×S2))["]).
We prove that pAS1×S2(!-)• pAS1×S2(!.) *= 0 which implies that S1 ×S2 has nontrivial loop products. SincepAS1×S2(!-)•pAS1×S2(!.)belongs toH0(L(S1×S2)), it can be expressed as a sum of conjugacy classes with+1 or−1 as the coefficients. Indeed it equals±[,"] ± [,] ± ["] ± [1]. Since 1,,and"are distinct therefore three terms out of four are distinct and hence there cannot be a complete cancellation. "
Corollary 3.2. An oriented 3-manifold with a non-separating 2-sphere has nontrivial extended loop products.
Proof. The proof is similar to the one of Proposition 3.1. The only property we used in the proof of Proposition 3.1 was that there was a non-separating two sided 2-sphere.3 "
3Two sided means that the normal bundle is trivial (see[9]).
M
T +
Fig. 2. NonseparatingT.
Remark 3.3. Corollary 3.2 can be generalized to higher-dimensional manifolds with a nonseparating codimension one sphere. The proof is a direct generalization of the one of Corollary 3.2.
Proposition 3.4. An oriented3-manifold M with a non-separating two sided incompressible4 torus has nontrivial extended loop products.
Proof. Let * : S1 ×S1 → T ⊂ M be a homeomorphism. We set *(1,1) as the base point of M.
Consider the one-dimensional family of loops*t defined by*t(s)=*(t, s)(longitudes ofTinFig. 2).
This 1-family of loops represents a homology class.inH1((LM)[h]), wherehis the element of!1(M) represented by*1. Now consider a closed simple curve+ : S1 → M which meetsTtransversally at exactly one point*(1,1). Note that+represents an elementg∈!1(M)and also gives rise to a homology class-∈H0((LM)[g]).
We show thatpAM(!-)•pAM(!.)*=0∈H0(LM).Similar toS1×S2we havepAM(!-)•pAM(!.)=
±[gh] ± [h] ± [g] ± [1].
To prove the claim, it is sufficient to show that[1],[h]and[g]are distinct. SinceTis!1-injective then [h] *= [1]. Note that the loop+intersectsTexactly at one point hence the intersection product of the two homology classes (inM) represented by+ andTare nontrivial and in particular the homology classes are nontrivial, therefore[g] *= [1]. Similarly[g] *= [h]because the intersection of the homology classes represented byTand*1 is zero asThas a trivial normal bundle. "
Remark 3.5. A similar argument shows that an-dimensional manifold with a non-separating two sided
!1-injective(n−1)-dimensional torus has nontrivial extended loop product.
4. Closed Seifert manifolds
In this section we consider the closed Seifert manifolds. We refer the reader to[8,9]for an introduction to Seifert manifolds.
Remark 4.1. An orientable Seifert manifoldMmay not be oriented as a fibration but it always has a double cover which is Seifert and is orientable as a fibration or in other words its base surface is orientable.
This double coverM˜ can be described as following:
M˜ = {(m, o)|m∈M &oan orientation for the fiber passing throughm}.
4!1-injective (see[9]).
The covering mapM˜ →q Misq(m, o)=m.The Seifert fibration ofM˜ is obtained by pulling back the one ofM. IfMhas a boundary then each boundary component, a torus, gives rise to 2 boundary components, both a torus. IfMhasgcross caps thenM˜ has genusg−1.
Proposition 4.2. If M is a closed oriented Seifert3-manifold then M or a double cover of M has nontrivial extended loop products.
Proof. By Remark 4.1 we may assume that the base surface of M is orientable. Also we assume
!1(M)!Z2, since Proposition 2.6 deals with this case.
Lethbe the generator of!1(M)corresponding to the normal fiber according (5.1). There is a natural 3-homology class/M inH3((LM)[h])that has a representative whose set of marked points is exactlyM.
The loop associated with a point inMwhich is on a normal fiber, is the fiber passing through the point.
The loop passing through a point on a singular fiber as a map is a multiple of the singular fiber.
We havepAM(/M)=/M −pM(/M)=/M −"M where"M is the unit ofH∗(LM).
Consider a simple curvelrepresenting an element$∈!1(M),$*=h,1. This is possible since we are assuming!1(M)*=Z2. Note that[h] *= [$]ashis a central element. Consider the 0-homology class$ˆ in H0((LM)[$])represented byl. We claim thatpAM(/M)•pAM($ˆ)*=0.
In expandingpAM(/M)•pAM($ˆ)∈H0(LM)we obtain four terms,±[$.h],±[h],±[$]and±[1]. Note that[h] *= [$]sinceh*= $andhis in the center. As there are at least three different conjugacy classes, any linear combination of them with coefficient±1 is nonzero. ThereforeMhas nontrivial extended loop products. "
5. Seifert manifolds with boundary
By torus decomposition[10,11], Seifert manifolds with incompressible boundary are among the build- ing blocks of 3-manifolds. In this section, we provide various examples of Seifert manifold with boundary which have nontrivial extended loop product. We introduce the following notation as they will crucial from now on.
Notation: LetMbe an oriented Seifert manifold withpexceptional fibers andbboundary components.
If S, the base surface of M, is orientable, then it has genus g!0 otherwise S is nonorientable and g is the number of cross caps in S. The fundamental group of M has the following presentation (see [8,9]):
(1) IfSis orientable and of genusg,
!1(M)= (
a1, b1, . . . , ag, bg, c1, . . . , cp, d1, d2. . . db, h|aiha−i 1=h, bihb−i 1=h, cihc−i 1=h, dihd−i 1=h,
c$iih0i =1, )g i=1
[ai, bi] )p i=1
ci
)b i=1
di=1
*
, (5.1)
c1 )2
)1 c2 c3
Fig. 3.)1,)2and a trivialized fibration over)1.
(2) IfSis nonorientable and hasg >0 cross caps,
!1(M)= (
a1, . . . , ag, c1, . . . , cp, d1, d2. . . db, h|aiha−i 1=h−1, cihc−i 1=h1i, dihd−i 1=h,
c$iih0i =1 )g i=1
ai2 )p i=1
ci
)b i=1
di=1
*
, (5.2)
where 0<0i<$i are integers and1i= ±1 andhcorresponds to a normal fiber. Thedi’s correspond to the boundary components. Eachci can be represented by a loop on the base surface going around the singular fiber once,$i is the multiplicity of the singular fiber corresponding tociand we say that it has type($i,0i). Note that1h2is a normal subgroup of!1(M)and it is central ifSis orientable.
Throughout this sectionM is a compact oriented Seifert manifold with orientable base surface unless otherwise stated (see Corollary 5.6),b!1 andp/denotes the number of the singular fibers of multiplicity greater than 2. We assume5 thatb+p!3.
We present two main ideas for proving that a Seifert manifold has nontrivial extended loop product, Two curves argumentandChas’figure eight argument.
The following table indicates the cases where we can apply these arguments.
Argument p+b!4 Two curves argument p/+b!3 Chas’ figure eight argument 5.1. p+b!3(Two curves argument)
Proposition 5.1. Let M be a compact oriented Seifert manifold withp >2andb!1.Then M has nontrivial extended loop products.
Proof. Letc1,c2andc3be the generators of!1(M)corresponding to three singular fibers. Consider two simple curves)1and)2, on the base surface, away from singular fibers and representing respectively the free homotopy class[c1c2]and[c2c3]. Moreover,)2can be chosen such that it has exactly 2 intersection points with)1.
Since)1is away from the singular point, the fibration can be trivialized over)1(seeFig. 3). Therefore, we obtain a mapf :S1×S1 →M, wheref (0,·)=)1.
5This assumption will be justified in Section 10.
c1
c2 c3
c1
c2 [c1c2c3c2]
c3
c1 c2
[c1c22c3]
c3
Fig. 4. Two curves argument.
Note thatfgives rise to a homology class21 ∈ H1((LM)[c1c2]), by declaringf (t,0),t ∈ S1, as the set of marked points andf (t, s),s ∈S1, as the loop passing throughf (t,0). On the other hand the loop )2gives rise to a homology class22 ∈H0((LM)[c2c3]).
We show thatpAM(!21)•pAM(!22)*=0∈H0(LM)which proves the proposition.
The sets of marked points of!21 ∈H2((LM)[c1c2])and!22∈H1((LM)[c2c3])intersect at two points.
Each point contributes four terms in the expansion ofpAM(!21)•pAM(!22). All in all eight terms emerge while calculatingpAM(!21)•pAM(!22). Each is represented by a conjugacy class in!ˆ1(M) and a± sign. The conjugacy classes are
[c1c2c3c2],[c1c2],[c2c3],[1] and [c1c22c3],[c1c2],[c2c3],[1].
Thus in calculatingpAM(!21)•pAM(!22)mod 2, only two terms remain, namely[c1c22c3]and[c1c2c3c2] (seeFig. 4). To prove the claim it is sufficient to show that these two conjugacy classes are different. For this, consider the group
H= 1c1, c2, c3, d|c1c2c3d1=1, c$ii =1,1#i#32,
where d1 is a generator corresponding to a boundary component. IndeedH is the free product of the groupsZ$1∗Z$2∗Z$3.
Consider the homomorphism * : !1(M) → H which is the identity onc1, c2, c3, d1 and sends the other generators to the trivial element ofH. By (5.1)*is well defined.
Note that the images ofc1c22c3andc1c2c3c2under the*are not conjugate inHasc1c22c3andc1c2c3c2 are cyclically different reduced words. Therefore [c1c22c3] and[c1c2c3c2] are two different conjugacy classes of!1(M). "
Remark 5.2. In the proof of the Proposition 5.1 one can replace a singular fiber by a boundary component, in other words a boundary component works as well as a singular fiber of multiplicity zero. To be more explicit, the curves)1 and)2 can go around a boundary component of the base surface instead of the singular fiber. In the free productH, one of the finite cyclic groups is replaced by a infinite cyclic group.
The rest of the proof remains the same.
Corollary 5.3. Let M be a compact oriented Seifert manifold with p singular fibers and b boundary components. Ifp+b >3then M has nontrivial extended loop products.
Proof. Ifp!3 then this is just Proposition 5.1. Ifp <3, then by Remark 5.2, in the proof of Proposition 5.1 one replaces the generator of!1(M) coming from a singular fiber with the one corresponding to a boundary component. "
5.2. p/+b!3(Chas’ figure eight argument)
The main idea of the proof of this case is due to Moira Chas. She found the idea while reformulating a conjecture of Turaev (see[4]) on Lie bi-algebras of surfaces, characterizing non self-intersecting closed curves.
Proposition 5.4. Let M be a compact oriented Seifert manifold withp=2singular fibers and b=1 boundary component. Suppose that none of its singular fibers has multiplicity2.Then M has nontrivial extended loop products.
Proof. Letc1 andc2 be generators of!1(M) corresponding to the singular fibers. Let)1 be a smooth curve on the base surface, with one self-intersection point, away from singular fibers and representing the free homotopy class[c1c−21]. Similarly, consider a curve)2on the base surface, away from the singular fibers, with one intersection point and representing the free homotopy class[c−11c2]. Moreover,)2can be chosen such that it has exactly 2 intersection points with)1.
The fibration can be trivialized over the)1since it is away from the singular point. Therefore, we obtain a mapf :S1×S1→M, wheref (0,·)=)1. This map gives rise to a homology class21 ∈H1(LM), by declaringf (t,0),t ∈S1, as the set of marked points. The loop passing throughf (t,0)isf (t, s),s ∈S1. All the loops of this family have the free homotopy type[c1c−21]. The loop)2gives rise to a homology class22 ∈H0(LM). We claim that
pAM(!21)•pAM(!22)*=0∈H0(LM) which proves the proposition.
The sets of marked points of!21∈H2(LM)and!22∈H1(LM)intersect exactly at two points. Each point contributes four terms in the expansion ofpAM(!21)•pAM(!22). All in all eight terms emerge in the expansion ofpAM(!21)•pAM(!22). Each one of them is represented by a conjugacy class in!ˆ1(M) and a±sign. These conjugacy classes are
[c1c2c1−1c−21],[c−11c2],[c1c2−1],[1] and [c1c2−1c−11c2],[c1−1c2],[c1c−21],[1].
Calculating pAM(!21) • pAM(!22)mod 2, only two terms remain, namely [c1c2c−11c2−1] and [c1c−21c1−1c2](seeFig. 5).
To prove the claim it is sufficient to show that these two conjugacy classes are different. For this, consider the group
H= 1c1, c2, d1|c1c2d1=1, c$ii =1, i=1,22 +Z$1∗Z$2, whered1 corresponds to one of the boundary components.
c1
c1
c1 c2
c2 c2
[c1c−1 2 c1−1c2] [c1c2 c1−1 c2 −1 ]
Fig. 5. Figure eight argument.
Consider the homomorphism*:!1(M)→H which is the identity onc1, c2, d1and sends the other generators of!1(M)to the trivial element ofH. It follows from (5.1) that*is well defined.
Since$i *= 2, i=1,2, thenci *= ci−1 inH. The images ofc1c2c1−1c−21andc1c−21c−11c2 under*are not conjugate as they are cyclically different. Thereforec1c2c−11c−21andc1c−21c−11c2are not conjugate in
!1(M)either. "
Corollary 5.5. Let M be a compact oriented Seifert manifold withp/ +b!3.Then M has nontrivial extended loop products.
Proof. If p/!2 then this is just Proposition 5.4. Otherwise, by Remark 5.2, a boundary component works like a singular fiber with multiplicity zero. So in the proof of Proposition 5.4 we should replace the generators of!1(M) due to the singular fibers with the generators corresponding to two boundary components. "
Corollary 5.6. Let M be a compact oriented Seifert manifold with a non-orientable base surface and b!2boundary components,then M has nontrivial extended loop products.
Proof. The proof is similar to the one for Proposition 5.4 but with a slight modification. Note that a boundary component serves our purposes as well as a singular fiber. Moreover the restriction of the fibration to the boundary components is oriented. Thus it can be trivialized over a simple curve representing the free homotopy type [d1d2−1] where d1 and d2 are generators of !1(M) contributed by two boundary components. The rest of the proof is similar to the one of Proposition 5.4. Cal- culating mod 2 we get two terms [d1d2d1−1d2−1] and [d1d2−1d1−1d2]. We must prove that these two conjugacy classes are different. If a1 is a generator of !1(M) due to a cross cap then consider the group
H= 1d1, d2, a1|d1d2a12=12 + 1d12 ∗ 1a12 +Z∗Z
and the homomorphism3:!1(M)→Hwhich is the identity ond1, d2, a1and sends the other generators of!1(M)to the trivial element. It follows from (5.2) that3is well defined.
The images of d1a21d1−1a1−2 and d1a1−2d1−1a12 in the free product H are not conjugate. Therefore d1d2d1−1d2−1andd1d2−1d1−1d2are not conjugate in!1(M). "
) Σ
p
M10 + M20
Fig. 6.M=M1#M2.
6. Connected sum of 3-manifolds
In this section we consider the connected sum of 3-manifolds. We shall show how one can obtain some examples of nontrivial loop products in this kind of manifold. For this part, the author has benefited from conversations with numerous colleagues[2].
Proposition 6.1. Suppose thatM=M1#M2whereMi’s are3-manifolds of nontrivial fundamental group.
Then M has nontrivial extended loop products.
Proof. Let 4 ⊂ M be the 2-sphere that separates the two components M10 and M20, where Mk0, for k∈ {1,2}, isMkwith a ball removed. Just like Section 3.2, the 2-sphere4gives rise to a one-dimensional family of loops which have the same marked point p ∈ M. We setp to be the base point of M(Fig.
6). The loops in this one-dimensional family have the free homotopy type of the trivial loop. In order to modify their free homotopy type, one can compose the loops of this one-dimensional family with a fixed loop whose marked point isp. Suppose that we have done this modification using a fixed loop)(Fig. 6) which represents a nontrivial elementh∈!1(M). The new one-dimensional family of loops represents a homology class.∈H1((LM)[h]). Now consider a simple smooth curve+:S1 →M which intersects 4exactly at 2 points and has the free homotopy type[x1x2] wherexi *= 1∈ !1(Mi),i=1,2 (Fig. 6).
Note that!1(M)=!1(M1)∗!1(M2)andx1x2is regarded as an element of this free product. We choose +such that it does not intersect). As a loop,+represents a homology class -∈ H0((LM)[x1x2]). We show that there exist some choices ofx1, x2andhsuch thatpAM(!-)•pAM(!.)*=0∈H0(LM).
In expanding pAS1×S2(!-)•pAS1×S2(!.) we get eight terms. By passing to mod 2 only two terms remain, namely[x1x2h]and[x2x1h]. Now we must show that there exist some choices ofhsuch that these two conjugacy classes are different. Indeedh=x1x2is a convenient choice since
[x1x2h] = [x1x2x1x2] and [x2x1h] = [x2x1x1x2] = [x12x22]
and the reduced wordsx1x2x1x2andx12x22are cyclically different. "
Remark 6.2. We shall point out that in Proposition 6.1 M is not required to be closed. Moreover, Proposition 6.1 and its proof can be generalized to manifolds of any dimensiond!3. In the proof,4 would be replaced with a(n−1)-sphere and the homology classes-and., would be of degree 0 and n−2.
7. An injectivity lemma
Suppose thatMis a closed oriented 3-manifold andTis a collection of incompressible separating tori inM. LetM\T=M1∪M2∪ · · · ∪Mn. We associate with(M,T)a tree of groups(G, T )(see[17]and Appendix A.2 for the definitions).
The vertices ofTare in a one-to-one correspondence withMi’s. Two verticesviandvj are connected by an edge if the correspondingMiandMj are glued along a torus. With each vertexviwe associate the fundamental group of!1(Mi)and with each edge we associate the fundamental group of the corresponding torus. As the tori are incompressible there are two monomorphisms from each edge group to the vertex groups of the corresponding vertices, namely ifeis an edge (a torus inM) andva vertex ofe(someMi
s.t.T ⊂!Mi) then there is an homomorphism fev :Ge→Gv
which is indeed the map induced by inclusion i:!1(T )→!1(Mi).
By Van-Kampen theorem,!1(M)is the amalgamation ofGv’s alongGe’s (see Appendix A.2).
Combining all these with Lemma A.5 in Appendix A, we have,
Lemma 7.1(Injectivity lemma). Let M,TandMi’s be as above. Suppose that[a]and[b]are two distinct conjugacy classes in!1(Mi)such that a is not conjugate to the image of any element of the edge groups in!1(Mi).Then a and b represent distinct conjugacy classes of!1(M).
The injectivity lemma will be very useful to show that certain examples of nontrivial loop products in a submanifold ofMgive rise to nontrivial loop products inM.
8. Gluing Seifert manifolds along tori
In this section we consider the gluing of the Seifert manifolds along tori. We prove that the result of the gluing Seifert manifolds along incompressibleseparatingtori has nontrivial extended loop products except in certain cases for which there are double covers that have nontrivial extended loop products.
The idea of the proof is to apply either thetwo curves argumentor thefigure eight argumentto a Seifert piece that fulfills the conditions as indicated by the table in Section 5. Then we show that the examples of nontrivial loop product in the piece give rise to examples of nontrivial loop product in the manifold, and for that we benefit from the injectivity lemma, Lemma 7.1.
When we cannot apply either the two curves argument or the figure eight argument, we consider an appropriate double cover of the manifold which either has a nonseparating torus or one of the two arguments can be applied to a Seifert piece of its torus decomposition[10,11].
Notation: LetMbe a closed 3-manifold andT*= ∅be a collection of tori inMand M\T=M1∪M2∪ · · · ∪Mn,
where all theMi’s are oriented Seifert manifolds with incompressible boundary.
Letbi!1 andpidenote, respectively, the number of boundary components and singular fibers ofMi
andAi ⊂ Z+ is the set of all the multiplicities of the singular fibers ofMi. The base surface ofMi is denotedSiand the genus ofSi isgiifSiis oriented otherwisegi is the number of cross caps inSi.
We assume that all tori in T are separating since in Section 3 we proved that 3-manifolds with nonseparating torus have nontrivial extended loop products. We assumebi+pi!3 ifSiis orientable.
Proposition 8.1. Let M be a closed3-manifold described as above. Then M or a double cover of M has nontrivial extended loop products.
Proof. We assume thatgi=0 whenSiis orientable, as ifgi>0 then there is a non-separating torus inM and this case has already been studied in Section 3. ThenMbelongs to one of the following categories:
(1) bi!3 andSiis orientable for somei. In this case we use the figure eight argument. Note that ifbi>3 we can also use the two curves argument.
(2) bi!2 andSiis nonorientable for somei, for this case we also use the figure eight argument.
(3) pi+bi>3 andSi orientable for somei, we can apply the two curves argument.
(4) bi +pi =3, 1#bi#2, Si orientable and 2∈/Ai for some i. In this case we use the figure eight argument.
(5) For alli,bi+pi=3, 1#bi#2,Si orientable and 2∈Ai for alli. For this case we consider a double cover ofM.
(6) n= |T|, n!2:
• bi=2,pi=1,Ai= {2}andSi is orientable fori *=1, n;
• b1=bn=1;
• S1is nonorientable;
• IfSnis orientable thenpn=1 and 2∈An.
In this case also we will find a double cover with nontrivial extended loop products.
We verify the statement for each case.
(1) bi!3 and Si orientable for some i: Suppose that b1!3 and S1 orientable. By Corollary 5.5 M1
has nontrivial extended loop products. Letd1,d2 andd3 be the generators of!1(M1)contributed by 3 boundary components. Consider the homology classes21 and22 ∈ H1(LM1)constructed in the proof of Corollary 5.5, they can be regarded as the elements ofH∗(LM). We claim that
pAM(!21)•pAM(!22)*=0∈H0(LM).
Mod 2 only two terms survive,[d1d2d1−1d2−1]and[d1d2−1d1−1d2],which by the proof of Corollary 5.5 are two different conjugacy classes of!1(M1). So if we show thatd1d2d1−1d2−1is not conjugate to any element of the fundamental group of one of the components of!M1, then by the injectivity lemma[d1d2d1−1d2−1] and[d1d2−1d1−1d2]are different as the conjugacy classes of!1(M).
Suppose that
[d1d2d1−1d2−1] = [hrds], (8.1) wherehis the generator of!1(M1)that corresponds to a normal fiber ofM1anddis a generator of!1(M1) due to a boundary component.
