Variations on Fintushel-Stern Knot Surgery on 4-manifolds

12 Pages·0.17 MB·

by  Selman Akbulut

Proceedings of 8th G¨okova
Geometry-Topology Conference,
pp,1 – 12
Variations on Fintushel-Stern Knot Surgery on
2
0 4-manifolds
0
2
n Selman Akbulut 1
a
J
Abstract. We discuss some consequences Fintushel-Stern ‘knot surgery’ operation
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on4-manifoldscomingfromitshandlebodydescription. Wegivesomegeneralizations
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ofthisoperationandgiveacounterexample totheirconjecture.

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T
G
. 1. Introduction
h
at Let X be a smooth 4-manifold and K ⊂ S3 be a knot, In [FS] among other things
m Fintushel and Stern had shown that the operation K → X of replacing a tubular
K
neighborhood of imbedded torus in X by (S3−K)×S1 could results change of smooth
[
structure of X. In [A] an algorithm of describing handlebody of XK in terms of the
1
handlebody of X was described. In this article we will give some corollaries of this
v
construction, and present a counterexample to conjecture of Fintushel and Stern which
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5 was overlookedin [A]. First we need to recall the precise description of XK: Recall that
1 the first picture of Figure 1 is T2×D2, and the second one is the cusp C (i.e. B4 with
1 a 2-handle attached along the trefoil knot with the zero framing). Clearly the cusp C
0 contains a copy of T2×D2.
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0
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h 0 0
t
a
m
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i
X
r
a
Figure 1
1991 Mathematics Subject Classification. 57R65,58A05,58D27.
1SupportedinpartbyNSFfundDMS9971440.
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In[FS]animbeddedtorusT2 ⊂X iscalledac-imbeddedtorusifithasacuspneighbor-
hood in X, i.e. T2 ֒→C ֒→X as in Figure 1. Now let N ≈K×D2 be the trivialization
of the open tubular neighborhood of the knot K in S3 given by the 0-framing. Let
ϕ : ∂(T2×D2) → ∂(K ×D2)×S1 be any diffeomorphism with ϕ(p×∂D2) = K ×p,
where p∈T2 is a point, then define:
X =(X −T2×D2)⌣ (S3−N)×S1
K ϕ
LetSpin (X)bethesetofSpin structuresonX,e.g. ifH (X)hasno2-torsionthen.
c c 1
Spin (X)={a∈H2(X;Z)|a=w (TX)mod2}
c 2
Recall that Seiberg-Witten invariant SW of X is a symmetric function
X
SW :Spin (X)→Z
X c
It is known that the function SW is nonzero on the complement of a finite set B =
X
{±α ,±α ,..,±a } which is called the set of basic homology classes classes. By setting
1 2 n
α =0 and t =exp(α ), the function SW is usually written as a single polynomial
0 j j X
n
SW = SW (α )t
X X j j
Xj=0
Now if T is a c-imbedded torus in X , and [T] be the homology class in H (X ;Z)
2 K
induced from T2 ⊂ X, and t = exp(2[T]), and ∆ (t) the Alexander polynomial of the
K
knot K (as a symmetric Laurent polynomial), then Fintushel and Stern [FS] theorem
says:
Theorem 1.1. SW =SW .∆ (t)
XK X K
Recallthatin[A]the algorithmofdrawingthehandlebodyofX fromX isdescribed
K
asfollows: Firstweidentify the core circles ofthe 1-handlesofthe handlebody ofS3−K
core circles K
Figure 2
Then when we see an imbedded cusp C in the handlebody of X as in the first picture
ofFigure3,wechangeittothe secondpictureC˜ ofFigure3. This meansthatwechange
oneofthe1-handlesofT2×D2insideofC tothe“slice1-handle”obtainedfromK#(−K)
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(i.e. remove the obvious slice disk which K#(−K) bounds from B4), and connect the
core circles of the knots K and −K by 2-handles as shown in Figure 3. More precisely,
there is a diffeomorphismbetween the boundaries of manifolds C and C˜ of Figure 3, and
theoperationX →X correspondstocuttingoutT2×D2 fromX andgluingthesecond
K
manifold of Figure 3 by this diffeomorphism(in the figureK is drawnas the trefoil not).
-1
-1 0
-K
0
-1
0
-1
K
Figure 3
Since the attaching circles of the other 2-handles of X could tangle to the boundary
of T2×D2, it is important to indicate where the various linking circles of the boundary
are thrown to by the diffeomorphism of Figure 3. This is indicated in Figure 6.
Recall that since 3- and 4- handles of four manifolds are attached in the canonical
way, to describe a 4-manifold it suffices to describe its 1- and 2- handle structure. So,
in order to visualize (S3−K)×S1, which is obtained by by identifying the two ends of
(S3−K)×I,itsufficestovisualize(B3−K )×I withitsendsidentified,whereK ⊂B3
0 0
isaproperlyimbeddedarcwiththeknotK tiedonit(therestisa3-handle). Thesecond
picture of Figure 4 gives the handlebody picture of (B3−K )×I.
0
Identifyingtheendsof(B3−K )×I (upto3-handles)correspondstoattachinganew
0
1-handle,and2-handles,wherethe new2-handlesareattachedalongthe1-handlesofthe
twoboundarycomponentsof(B3−K )×I asindicatedinFigure5(morespecificallythe2-
0
handlesareattachedalongtheloopsconnectingthecorecirclesoftheknotcomplements).
ToseethediffeomorphismofFigure3(i.e. toseethattheboundaryofthefirstpicture
inFigure 5is standard),wesimply removethe dotonthe “slice”1-handle(i.e. turn itto
a 2-handle) and slide it over the two 2-handles (as indicated by the arrows) in the first
picture of Figure 5. This gives the second picture of Figure 5. After sliding 2-handles
over each other of second picture of Figure 5, and cancelling the resulting S2×D2 with
the 3-handle we obtain T2×D2. Also, to see the inverse boundary diffeomorphism from
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K
0
Figure 4
0 0
0
0
Figure 5
T2 ×D2 to the first picture in Figure 5, we remove the dot from the 1-handle of the
second picture of Figure 5 and perform the handle slides indicated by the arrows.
Now putting these together in Figure 6 we see where the boundary diffeomorphism
takes various linking circles of ∂(T2 ×D2). In particular the linking circle c of the 2-
handle is thrown to the loop which corresponds the zero push-off of K in K#(−K).
Figure 7 is the same as the second picture of Figure 6 except that the slice disk
complement, which K#(−K) bounds, is drawn more concretely. Also note that, though
our discussion is for general K, for the sake of concreteness, we have drawn our figures
by taking K to be the trefoil knot.
2. Applications
In[FS]FintushelandSternconjecturedthatifX isthe Kummer surfaceK3,thenthe
associationK →X gives an injective map from the isotopy classes of knots K in S3 to
K
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b
b
a
c
c
0
0
0
a
Figure 6
0
0
b
0
a
Figure 7
the set of diffeomorphism classes of smooth structures on X. The following provides a
counterexample to this conjecture:
Theorem 2.1. XK =X−K
Proof. There is an obvious self-diffeomorphism of the second picture in Figure 3 (i.e.
(S3−K)×S1) exchangingroles of K and −K ; i.e. the diffeomorphisminduced by 180o
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rotation of R3 around the y-axis. It is easily check that this diffeomorphism extends to
the interior of (S3−K)×S1, implying the desired result.
The following says that all smooth manifolds X obtained from X by using from dif-
K
ferent knots K become standardafter single stabilization. This resultwas independently
observed by Auckly [Au].
Theorem 2.2. X #(S2×S2)=X#(S2×S2)
K
Proof. X #(S2×S2) isobtainedby surgeringanyhomotopicallytrivialloop(with the
K
correct framing). We choose to surger X along the trivially linking circle of its slice
K
1-handle (the knot K#(−K) with a dot). This corresponds to turning the slice 1-handle
to a 0-framed 2-handle (i.e. replace the dot with 0 framing), hence we are free to isotop
the attaching circle of this 2-handle to the standard position as indicated in Figure 5.
In particular, this makes the boundary diffeomorphism between the two handlebodies of
Figure 5 extend to a 4-manifold diffeomorphism. So, Surgered X is diffeomorphic to
K
the surgered X which is X#(S2×S2).
Notethatthoughweindicatedthe argumentforthe trefoilknotK inourpictures,the
sameapplies for ageneralK (i.e. in Figure5 the knotK#(−K)unknots in the presence
of the 2-handles)
NoticethatX canbeviewedasX =X =(X−T2×D2)⌣ (S3×S1−U),where
K K f ϕ
U is an open tubular neighborhood of an imbedded torus f : S1×S1 → S3×S1, with
Image(f)=K×S1. The map f is induced fromthe obviousimbedding K×I →S3×I
by identifying the ends. More generally to any concordance s from K to itself, we can
associate an imbedding of a torus f :S1×S1 ֒→S3×S1, hence getting map
s
Diffeomorphism classes of
C(K)−→
(cid:26) smooth structures onX (cid:27)
definedbys→Xs,whereC(K)={s:S1×I ֒→B3×I |s|S1×0 =s|S1×1 =K }. Itisan
interestingquestionthathowthediffeomorphismclassofX dependsontheconcordance
s
class s of K? The following says that the above map is not injective.
Theorem 2.3. If K is the trefoil knot, there is s∈C(K#(−K)) such that X =X
s
Proof. Let s be the concordance of K#(−K) to itself, given by connected summing the
twoobviousslicediscswhichtwocopiesofK#(−K)boundinB4 asinthesecondpicture
of Figure 8. Now if we use the product concordance s from K#(−K) to itself, i.e. the
0
first picture of Figure 8, our algorithm says that changing the cusp neighborhood by
(S3−K#(−K))×S1 isgivenbythehandlebodyofFigure9,whichisthesameasFigure
10 (where the slice 1-handle is drawn as a usual handlebody). Whereas if we use the
concordances, describedabove,we getFigure11. Byanisotopywe see that Figure11is
diffeomorphic to Figure 12 which is diffeomorphic to Figure 13, and Figure 13 is isotopic
toFigure14. ByhandleslidesindicatedinFigures14and15weobtainthesecondpicture
of Figure 15. By cancelling a 1-and 2- handle pairs we get the first picture of Figure 16.
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K # -K K # -K
Figure 8
Then by a 2-handle slide, and cancelling an unknotted 0-framed 2-handle by a 3-handle,
we obtain the last picture of 16 which is the cusp C. So we proved C = C, but since
s
C ⊂X and every self diffeomorphism of ∂C extends to C we conclude X =X
s
Remark 2.1. This theorem says that taking different elements s ∈ C(K) can result
changing the smooth structure of X . For example, if take any c-imbedded torus in a
s
smooth manifold X with SW 6= 0, and K is the trefoil knot, and if s ∈ C(K#(−K))
X 0
is the product concordance,then by Theorem 1.1
SWXs0 =SWXK#(−K) =SWX.∆K#(−K) 6=SWX
hence X 6=X. But on the other hand Theorem 2.3 says that there is s∈C(K#(−K))
s0
with X = X, so X 6= X . In particular, this shows that the concordances s and s
s s0 s 0
are different. This gives a hope the that hard to distinguish knot concordances might be
distinguished by the Seiberg-Witten invariants of the associated manifolds X .
s
Remark 2.2. Lets∈C(K),andf :S1×S1 ֒→S3×S1bethecorrespondingimbedding
s
One can ask whether Theorem 1.1 generalizes to SW = SW .∆ (t)?, where ∆ (t) is
Xs X s s
the Alexander polynomial associated to this imbedding.
2.1. A twisted version of XK
Another version of the operation X → X that was previously introduced in [CS],
K
which, in a sense, is the square root of this operation: Let K is an invertible knot, i.e.
an orientation preserving involution τ : R3 → R3 (e.g. 1800 rotation) restricts to K as
an involution with two fixed points, and let N be the open tubular neighborhood of K.
Then we can form the following S3−N bundle over S1:
(S3−N)טS1 =(S3−N)×[0,1]/(x,0)∼(τ(x),1)
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Define a D2-bundle over the Klein bottle K2 by C∗ =S1×D2×[0,1]/(x,0)∼(τ(x),1).
Then(S3−N)טS1 andC∗ havethesameboundaries,andsoifX isasmooth4-manifold
with C∗ ⊂X, we can construct
X∗ =(X −C∗)⌣ (S3−N)טS1
K ϕ
where ϕ : ∂C∗ → ∂(K ×S1)טS1 is a diffeomorphism with ϕ(p×∂D2) = K ×p. The
operation X → X∗, is a certain generalization of the Fintushel and Stern operation
K
X → X done using a ‘Klein bottle’ instead of a torus. This operation was previously
K
discussed in [CS]. By using the previous arguments one can see see that the handlebody
picture of the operation X →X∗ is given by Figure 17. The first picture of Figure 17 is
K
C∗ andthe secondis(S3−N)טS1. The restofX∗ isobtainedbysimplybydrawingthe
K
images of the additional handles under the diffeomorphism ϕ : ∂C∗ → ∂(S3−N)טS1.
Forconvenience,inFigure17theimagesofthelinkingcirclesa,b,cunderϕareindicated.
Now, call an imbedded Klein bottle K2 ⊂X c-imbedded Klein bottle, if
K2 ⊂C∗ ⊂U ⊂X
where U is either one of the manifolds of Figure 18, and π (U) → π (X) injects (notice
1 1
π (U) = Z ). Then it is easy to see that the obvious 2-fold cover X˜ → X contains
1 2
a cusp C (so it contains a c-imbedded T2), and the operation X → X∗ lifts to the
K
usual Fintushel-Stern knot surgery operation X˜ → X˜ (done using this T2). Hence if
K
SW 6=0and∆ (t)6=0,theoperationX →X∗ changesthesmoothstructureofX,i.e.
X˜ K K
X 6= X∗. For example, X can be a manifold with boundary, which is 2-fold covered by
K
a Stein manifold X˜ (so X˜ compactifies into a closed symplectic manifold which Theorem
1.1 applies). It is easy to check that the first manifold of Figure 18 is such an example.
-1
0
0
0
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Figure 9
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0 -1
0
0
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0
-1
Figure 10
-1
0 0
0 0 0
--11
Figure 11
References
[A] S.Akbulut,A Fake Cusp and a Fishtail.TJM,vol23,no.1(1999), 19-31
[Au] D. Auckly, Families of four dimensional manifolds that become mutually diffeomorphic after one
stabilization.GT/0009248
[CS] S.CappellandJ.Shaneson, On4-dimensional s-cobordisms.JDG,vol22(1985), 97-115
[FS] R.FintushelandR.Stern,Knots, links, and 4-manifolds.Invent.Math.134,no.2(1998), 363–400.
MSU,Deptof Mathemtics,E.Lansing, MI,48824,USA
E-mail address: [email protected]
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Figure 12
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Figure 13
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