Knotted surfaces in $4$-manifolds by Knot surgery and Stabilization

0.92 MB·

by  Hee Jung Kim

KNOTTED SURFACES IN 4-MANIFOLDS BY KNOT SURGERY AND
STABILIZATION
HEE JUNGKIM
7
Abstract. Given a simply-connected closed 4-manifold X and a smoothly embedded
1
0 orientedsurface Σ,variousconstructionsbased onFintushel-Sternknotsurgery havepro-
2 duced new surfaces in X that are pairwise homeomorphic to Σ, but not diffeomorphic.
We prove that for all known examples of surface knots constructed from knot surgery
n
operations that preservethefundamental group of thecomplement of surface knots, they
Ja become pairwise diffeomorphic after stabilizing by connected summing with one S2ˆrS2.
WhenX is spin,we showin addition thatany surfaces obtained bya knotsurgery whose
9
complementshavecyclicfundamentalgroup becomepairwise diffeomorphicafter onesta-

] bilization by S2ˆrS2.
T
G
1. Introduction
.
h
Let X be a smooth closed 4-manifold and Σ be a smoothly embedded surface. An ‘exotic
t
a embedding’ of a surface Σ in X is a smooth embedding in X that is pairwise homeomor-
m
phic to Σ, but not diffeomorphic. The ‘stabilization’ of given pair pX,Σq is the process
[ of connected summing with a standard manifold pair pS2ˆS2,Hq or pS2ˆrS2,Hq, where

2r 2 2 2
1 S ˆS denotes the non-trivial S bundle over S . The recent work [3] of Auckly, Melvin,
v Ruberman, and the author has constructed the first examples of exotic 2-spheres in closed
7
simply-connected 4-manifolds that become pairwise smoothly isotopic after ‘single’ stabi-
3
2 2
3 lization by pS ˆS ,Hq. In this context, one can ask if this stabilization phenomenon arises
2 to exotic surfaces with higher genus.
0
While a great deal of exotic embeddings in 4-manifolds are known through various con-
.
1 structions [7, 6, 9, 14, 15, 16, 17], interestingly most examples of exotic embeddings for
0
oriented surfaces in simply-connected 4-manifolds derive from the constructions based on
7
3
1 ‘knot surgery’ of Fintushel-Stern [8]. Knot surgery using a knot K in S is the operation
: of removing a neighborhood of a torus T and replacing it by a product of S1 and the ex-
v
i terior of the knot K. Fintushel and Stern provided an effective way to detect the change
X
of diffeomorphism type for knot surgery, showing that the Alexander polynomial of K is
r
a reflected in the Seiberg-Witten invariant for a knot surgered 4-manifold. This allows one to
quickly construct and detect infinite families of exotic smooth structures on a large class of
4-manifolds. Likewise, knot surgery can be used to change a smooth structure of smoothly
embedded surface in a 4-manifold. This approach relies on ‘ambient surgery’ whereby a
given surface Σ is surgered to a new surface Σ pϕq, leaving the ambient manifold X fixed.
K
The rim surgery of Fintushel-Stern [9], author’s twist rim surgery [14], and Finashin’s an-
nulus rim surgery [6] are examples of this technique, underlyingmost examples of smoothly
knotted oriented surfaces in a simply-connected closed 4-manifold.
In the direction of the study of stabilization for exotic smooth structures, Auckly [2] for
2r 2 2 2
S ˆS and Akbulut [1] for S ˆS proved that a simply-connected 4-manifold X and its
2r 2
knot surgered manifold X pϕq become diffeomorphic after single stabilization by S ˆS
K
SupportedbyNRFgrant2015R1D1A1A01059318andBK21PLUSSNUMathematicalSciencesDivision.
Math. Subj. Class. 2010: 57M25 (primary), 57Q60 (secondary) .
1
Hee Jung Kim
2 2
or S ˆS , referred to as 1-stably equivalent with the terminology in [3]; see [4] for the
alternative proof.
Thispaperinvestigatestheanalogousstabilizationquestionforknottedsurfacesproduced
by all of the known constructions based on knot surgery i.e. rim surgery, twist rim surgery,
and annulus rim surgery.
The Wall’s stable h-cobordism theorem [27] states that homotopy equivalent, simply-
connected 4-manifolds become diffeomorphic after stabilization by some finite number of
2 2 2r 2
S ˆS or S ˆS . It also holds for embedded surfaces (up to diffeomorphism of pairs) with
simply connected complements in a 4-manifold that representthe same homology class [21].
And, in fact, all known examples need only one stabilization to be diffeomorphic. So, the
stabilization question for a knot surgered pair pX,Σ pϕqq would be the following. In this
K
paper,wewillusetheterminology ‘surfaceknotgroup’forthefundamentalgroupofsurface
complement in a 4-manifold.
Question 1.1. Suppose that X is simply connected and Σ is an oriented smoothly embed-
ded surface. Let pX,Σ pϕqq be a pair obtained by a knot surgery from pX,Σq. If Σ pϕq
K K
and Σ have the same surface knot group in X then are they 1-stably equivalent?
Thispaperanswersthisquestionforallofthecurrentlyknownconstructions. Theprecise
statements aregiven inSection 2(Theorems A,B,C)afterwediscusstheknowntechniques
for constructing exotic surfaces.
RimsurgeryofFintushelandStern[9]constructedaninfinitefamilyofexoticsmoothem-
beddingfor surfaces with simply-connected complements in a simply-connected 4-manifold.
Finashin used annulus rim surgery [6] for knotting algebraic curves in CP2, and produced
surfaces that are smoothly not isotopic to algebraic curves for degree d ě 5, but the topo-
logical classification of his examples was open. The later work [14] of the author introduced
a method, called twist rim surgery, of knotting surfaces that produced exotic embeddings
for surfaces with cyclic knot groups in a simply-connected 4-manifold. Applied to algebraic
curves in CP2, the twist rim surgery leads to the construction of infinitely many exotic
smooth structures on algebraic curves of degree dě 3. For degrees 1 and 2, the surfaces are
spheres, and it is not easy to distinguish these by Seiberg-Witten invariants. The work of
Rubermanandauthor[15]strengthenedthecriterionfrom[14]fortopological equivalenceof
surfacesbyshowingthatanysurfacesproducedbyaknotsurgerythatpreserveacyclicknot
groupistopologically standard. Asaconsequence, wededucedthatFinashin’sexamplesare
topologically standard. Despite some results abouttheexistence of symplectic, noncomplex
surfaces as well as smooth surfaces without symplectic structures, the main classical source
of examples for smooth embeddings codimension 2 had been complex curves. A subsequent
work [16] extended Gompf’s theorem about the fundamental group of symplectic manifolds
to the relative case, showing that any finitely presented group can be realized as the fun-
damental group of complement of a symplectic surface in a simply-connected symplectic
4-manifold, whereas the fundamental groups of complement of complex curves are quite
restricted. Those examples can be further smoothly knotted by twist rim surgery so that
it has led to a large class of exotic embeddings. Another interesting aspect of twist rim
surgery is that some iteration of the twist rim surgery gives a way of constructing new
smooth surfaces with certain non-abelian finite surface knot group. One consequence is
2 2
that it gave an infinitefamily of exotic surfaces in S ˆS with knotgroup a dihedralgroup
D2p, for any odd p.
2
Knotted surfaces in 4-manifolds by Knot surgery and Stabilization
In this paper, we prove that for all known examples of surface knots constructed from
rim surgery, twisted rim surgery, and annulus rim surgery that preserve their surface knot
2r 2
groups, they become pairwise diffeomorphic after ‘single’ stabilization by pS ˆS ,Hq.
Another result includes an interesting phenomenon in the relative version of stabilization
2 2 2r 2
i.e. connected sum with pS ˆS ,Hq or pS ˆS ,Hq. It is known that for a nonspin simply-
2 2
connected 4-manifold X, essentially due to Wall [26], X #S ˆS is diffeomorphic to X #
2r 2
S ˆS , but surprisingly it is not true for the relative case. Theorem D proves that for
a degree d-curve Σ in CP2, pCP2 # S2ˆS2,Σ q is not even pairwise homeomorphic to
d d
pCP2#S2ˆrS2,Σ q, even when CP2´Σ is not spin i.e. d “ even.
d d
Finally, we show that if a knot surgery pX,Σq Ñ pX pϕq,Σ pϕqq is cyclic, which is
K K
defined to be a surgery preserving a cyclic surface knot group [15], then the pairs are
2r 2
1-stably equivalent by connected summing with pS ˆS ,Hq in the case that X is spin.
Remark 1.2. Note that here we will not impose any extra assumptions on Σ other than
that Σ is an oriented smoothly embedded surface in a simply-connected closed 4-manifold
X. Recall that the constructions of (twisted) rim surgery and annulus rim surgery can
provide exotic embeddings of Σ when Σ is a surface of positive genus and pX,Σq has a non-
trivial relative Seiberg-Witten invariant [8,9,10,25](or arelative Heegaard-Floer invariant
as in the version of Mark [19]).
The main theorems are precisely stated in the next section where it carefully describes
when surface knot groups are preserved for each knotting construction. And it includes the
proof of Theorem D.
2. Main Theorems
Before we state our results, the notions of ‘equivalence’ of embeddings of surfaces in a
4-manifold should be clarified as in [3]:
Definition 2.1. Two smoothly embedded surfaces Σ, Σ1 in a smooth 4-manifold Z are
equivalent if there is an orientation preserving pairwise diffeomorphism of pZ,Σq to pZ,Σ1q.
Two smoothly embedded surfaces Σ, Σ1 in a smooth 4-manifold X are n-stably equivalent if
the natural embeddings Σ, Σ1 Ă X#nS2ˆS2 (or nS2ˆrS2) are equivalent in X#nS2ˆS2
2r 2 2 2 2r 2
(or nS ˆS ), but not in X #kS ˆS (or kS ˆS ) for any k ď n´1.
Notethatourconstructedexotic2-spheresin[3]havesimply-connectedcomplements and
they are 1-stably isotopic which is a stronger notion of equivalence of surfaces. It is still
opentoseethedistinctionbetweenequivalenceofsurfacesuptodiffeomorphism andsmooth
isotopy [23, 24], while this issue does not arise in the topological case [20, 22]. Here our
2 2 2r 2
stabilization by S ˆS (or S ˆS ) is taken in the ‘outside’ of embedded surfaces in X, but
there is another notion of stabilization for embedded surfaces, adding an unknotted handle
to the surface. The recent work [5] of Baykur-Sunukjian showed that all constructions of
exotic knotting of surfaces produce surfaces that become smoothly isotopic after adding a
single handle in a standard way.
Let X be a smooth 4-manifold containing a torus T with a trivial normal bundle and let
3
K be a knot in S with its closed complement EpKq. Fintushel-Stern’s knot surgery [8]
1
is the process of removing a neighborhood of T from X and re-gluing S ˆ EpKq via a
1
diffeomorphism ϕ on the boundary to form X pϕq “ X ´ νpTqY S ˆ EpKq. Denote
K ϕ
by µ the boundary of the normal disk of T, and let the meridian/longitude of K be
T
1
µ and λ respectively. Here the gluing map ϕ : BνpTq Ñ S ˆ BEpKq can be chosen
K K
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Hee Jung Kim
by any diffeomorphism such that ϕ µ “ λ . When X is a simply-connected closed 4-
˚ T K
manifold, this operation doesn’t change the homeomorphism type, while it may change its
diffeomorphism type.
Applied to a torus in the exterior of an embedded surface in a closed 4-manifold, the
knot surgery can change embeddings of surfaces in 4-manifolds. We assume that X is a
smooth simply-connected closed 4-manifold, and Σ is an oriented embedded surface in X
throughout the paper. Then the fundamental group π1pX ´Σq is normally generated by a
meridian µΣ of surface. For a surface Σ carrying a non-trivial homology class in X, the first
homology group H1pX ´Σq is always finite cyclic, of order that we will usually write as d.
The process of knotting an embedded surface Σ in X can be obtained by performing knot
surgery on a torus in the exterior X´νpΣq, that usually links with Σ in a simply way, and
then gluing back in the neighborhood of the surface gives a new embedding of Σ in X pϕq
K
with image Σ pϕq. In the case of rim surgery, twist rim surgery, and annulus rim surgery,
K
there is a canonical identification between X and X pϕq so that we can view Σ pϕq as
K K
an embedding in X; see Section 3 for more details of these constructions. In general the
resulting homeomorphism/diffeomorphism type of the new embedding Σ pϕq depends on
K
a choice of torus T, knot K, and gluing map ϕ. Our results show that the surfaces Σ pϕq
K
and Σ are 1-stably equivalent under some circumstances as follows.
Rimsurgerydealswithsurfaceswithsimply-connectedcomplementsinasimply-connected
4-manifold and doesn’t change the fundamental group, so the surfaceΣ pϕq is in fact topo-
K
logicallyisotopictoΣbytheworksin[20,22]. Thefollowingtheoremshowsthestabilization
result for these surfaces.
Theorem A. Suppose that X is a simply-connected closed 4-manifold and Σ is an smoothly
embedded oriented surface with π1pX´Σq“ 1. Let pX,ΣKpϕqq be a pair obtained by a rim
2r 2 2r 2
surgery. Then pX,Σq#pS ˆS ,Hq is pairwise diffeomorphic to pX,Σ pϕqq#pS ˆS ,Hq.
K
1
Finashin’sannulusrimsurgery[6]requiresasuitableannulusM – S ˆI inX toproduce
a new surface via knotting Σ along the annulus. This surgery in his paper is given by an
explicit geometric description of the surgered surface, but in [15] a knot surgery description
for this surgery is provided; see Section 3.3 for this description. It is shown in [6, 15] that
annulus rim surgery preserves the surface knot group when π1pX ´Σq “ Zd, and it turns
out that the surface Σ pϕq is topologically isotopic to Σ by the work in [15, Theorem 1.3].
K
Theorem B. Suppose that X is a simply-connected closed 4-manifold and Σ is an smoothly
embedded oriented surface with π1pX ´Σq “ Zd. Let pX,ΣKpϕqq be a pair obtained by an
2r 2
annulus rim surgery. Then pX,Σq#pS ˆS ,Hq is pairwise diffeomorphic to pX,Σ pϕqq#
K
2r 2
pS ˆS ,Hq.
In order to explore this phenomenon for surface knots with arbitrary knot groups, we
consider twist rim surgery [14, 15, 16], a variation of the Finstushel-Stern’s rim surgery
with additional twists parallel to a meridian and a longitude of a knot K. We write the
meridian twist rim surgery as ‘m-twist rim surgery’ when we wish to indicate the number
of twists applied on the meridian of K, and also denote by Σ pmq the new embedding
K
produced from Σ under the surgery. The way in which m-twist rim surgery affects the
fundamental group of a surface knot depends to some degree on the relation between m
and d, where H1pX ´ Σq – Zd. For example, when m “ ˘1, the twist rim surgery
always preserves the fundamental group of a surface knot; the proof was given for 1-twist
in [16, Proposition 2.3], but it works for ´1-twist in the exactly same way. The 1-twist
rim surgery allows us to construct exotic smooth embeddings for a symplectic surface with
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Knotted surfaces in 4-manifolds by Knot surgery and Stabilization
any finitely presented knot group in a symplectic 4-manifold (see [16, Theorem 3.1, 5.2] for
more details). More generally, Proposition 2.4 in [16] shows when an m-twist rim surgery
preserves the fundamental group of surface knots as shown that for a surface Σ Ă X with
H1pX ´ Σq – Zd, if pm,dq “ 1 and the meridian µΣ has order d in π1pX ´ Σq, then
π1pX ´ Σq – π1pX ´ ΣKpmqq. This criterion is used to produce infinitely many exotic
2 2
embeddings in S ˆS with knot group a dihedral group D2p for any odd p [16, Theorem
5.1]. Note that when π1pX ´Σq “ Zd, the surface ΣKpmq with pm,dq “ 1 is topologically
isotopic to Σ; see [15, Theorem 1.3]. But for arbitrary surface knot groups, when the knot
K is chosen carefully, Σ pmq is equivalent to Σ up to smooth s-cobordism; see [16] for
K
more details. In all cases that surface knot groups under twist rim surgery are preserved,
we show that Σ pmq and Σ are 1-stably equivalent:
K
Theorem C. Suppose that the surface ΣĂ X has H1pX ´Σq– Zd, and let π1pX ´Σq be
any group G. Then the following is true.
2r 2 2r 2
(1) pX #S ˆS ,Σq is pairwise diffeomorphic to pX #S ˆS ,Σ p˘1qq.
K
(2) If pm,dq “ 1 and µΣ has order d in π1pX ´Σq then pX #S2ˆrS2,Σq is pairwise
2r 2
diffeomorphic to pX #S ˆS ,Σ pmqq.
K
Now, we give a simple proof to show an interesting phenomenon in this relative stabi-
lization. Wall’s stabilization result [26] for a nonspin simply-connected 4-manifold X shows
2 2 2r 2
that X #S ˆS is diffeomorphic to X #S ˆS , but interestingly it fails as follows:
Theorem D. Let Σ be a degree d-curve in CP2. Then the pair pCP2 # S2ˆS2,Σ q is
d d
‘not’ pairwise homeomorphic to pCP2#S2ˆrS2,Σ q.
d
Proof. If d is odd then it is obvious since CP2´Σ is spin. But, we will show that the pairs
d
are still not homeomorphic in the case that dis even so that CP2´Σ is not spin. We claim
d
that there is no immersed sphere S í pCP2 ´Σ q#S2ˆS2 such that its self intersection
d
2
number rSs is odd. The Hopf sequence shows
π2ppCP2´Σdq#S2ˆS2q ÝhÑH2ppCP2´Σdq#S2ˆS2q Ñ H2pπ1ppCP2´Σdq#S2ˆS2qÑ 0.
Since π1ppCP2´Σdq#S2ˆS2q“ Zd and H2pZdq“ 0, the Hurewicz map h is onto, so every
element of H2ppCP2 ´Σdq#S2ˆS2q is represented by an immersed sphere. Let’s assume
thatthereexistsanimmersedsphereS inpCP2´Σ q#S2ˆS2withoddintersectionnumber.
d
Then the homology class rSs can be written by krCP1s`naSa`nbSb in H2pCP2#S2ˆS2q,
2 2
where Sa and Sb denote the first and second generators of H2pS ˆS q respectively. It gives
rSs ¨ rΣ s “ kd that must be zero, so k “ 0. This implies that there is no odd class in
d
H2ppCP2´Σdq#S2ˆS2q, but there is in H2ppCP2´Σdq#S2ˆrS2q. (cid:3)
Remark 2.2. It is worth pointing out that there is no odd class in CP2 ´Σ even when
d
CP2´Σd is not spin. To understandthis, firstnote that thenonzero element α P H2pCP2´
2
Σdqhasα “ 0asshownintheaboveproof,soitgivesQCP2´Σd “ 0. Infactthehandlebody
picture of CP2 ´Σ shows that there are 2g 0-framed 2-handles, and a 1-framed 2-handle
d
which is d-times linked with a 1-handle. For d “ even, there is a Z2-homology class β of the
1-framed 2-handle and over Z2, the intersection form is given by r1s. By the Wu formula,
w2pCP2´Σdq vanishes on α, but has value 1 on the Z2-homology class β.
Finally, we focus on the case that π1pX ´Σq is a cyclic group Zd, and investigate the
stabilization problemofknotsurgery. Astheterminology in[15], ifaknotsurgerypX,Σq Ñ
pXKpϕq,ΣKpϕqq satisfies that π1pX ´ Σq – π1pXKpϕq ´ ΣKpϕqq is cyclic then the knot
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Hee Jung Kim
surgery is called a cyclic surgery. In [15, Theorem 1.2], Ruberman and the author showed
that for any pair pX,Σq that X is simply-connected and Σ is an embedded surface with
π1pX ´ Σq – Zd, if a knot surgery pX,Σq Ñ pXKpϕq,ΣKpϕqq is cyclic then there is a
pairwise homeomorphism pX,Σq Ñ pX pϕq,Σ pϕqq. Thus, it is natural to ask the 1-stable
K K
equivalence for the cyclic knot surgery. We answer for this question in the case that X is
spin:
Theorem E. Let X be a simply-connected, closed, spin 4-manifold and Σ be an embed-
ded oriented surface with π1pX ´ Σq – Zd. Suppose that the knot surgery pX,Σq Ñ
pX pϕq,Σ pϕqq is cyclic. Then pX,Σq is pairwise diffeomorphic to pX pϕq,Σ pϕqq af-
K K K K
2r 2
ter one stabilization with pS ˆS ,Hq.
Remark 2.3. Inthecontrasttothewell-knownstabilizationtheoremsforsimply-connected
4-manifolds,therelativestabilizationforcyclicknotsurgerydoesn’tseemtogiveanygeneral
2 2 2r 2
statement for a choice of pS ˆS ,Hq or pS ˆS ,Hq in the case of a nonspin4-manifold X.
Our main argument for stabilization results will rely on proving the 1-stable equivalence
of surface knots ΣKipϕq and ΣKi`1pϕq for two knots Ki and Ki`1 related by one crossing
change. At each stage of crossing change that will make any knot to an unknot, one
cannot assert that the pairs pX pϕq,Σ pϕqq and pX pϕq,Σ pϕqq become pairwise
Ki Ki Ki`1 Ki`1
2 2 2r 2
diffeomorphic after one stabilization with only pS ˆS ,Hq or with only pS ˆS ,Hq when
2 2
X ´Σ is nonspin. This issue arises because pX pϕq,Σ pϕqq#pS ˆS ,Hq may not be
2Kri 2 Ki
pairwisediffeomorphictopX pϕq,Σ pϕqq#pS ˆS ,HqasseenintheproofofTheoremD.
Ki Ki
3. Knot surgery constructions to change embeddings in 4-manifolds
Let X be a simply-connected closed 4-manifold and Σ be an embedded oriented surface.
3.1. Twist rim surgery. Let R be a torus with R ¨R “ 0 (called a rim torus) that
α α α
is the preimage in BνpΣq of a closed curve α Ă Σ. Identify the neighborhood νpαq of the
1 2 1 3 1
curve α in X with S ˆI ˆD “ S ˆB where νpαq in Σ is S ˆI. In this trivialization,
2 2
let β be a pushed-in copy of the meridian circle t0uˆBD Ă I ˆD , so it is isotopic to a
1 3
meridian of Σ. Then the rim torus R can be written as αˆβ Ă S ˆpB ,Iq and we will
α
2 1 3
identify a neighborhood νpR q of R with αˆpβ ˆD q Ă S ˆpB ,Iq. Let K be a knot
α α
3
in S with its exterior EpKq, and µ ,λ denotes a pair of meridian-longitude of K. The
K K
m-twists and n-rolls of rim surgery on pX,Σq is defined by
1
pX,Σ pϕqq “ pX,Σq´νpR qY S ˆEpKq.
K α ϕ
1
Here the gluing map ϕ :BνpR q Ñ S ˆBEpKq is the diffeomorphism determined by
α
(1) ϕ pα1q “ mµ `nλ `rS1s, ϕ pβ1q“ µ , and ϕ pµ q“ λ
˚ K K ˚ K ˚ R K
with respect to a basis tα1,β1,µRu for H1pBνpRαqq and trS1s,µK,λKu for H1pS1ˆBEpKqq,
where α1,β1 are the pushoffs of α, β into BνpR q and µ denotes a meridian of the rim
α R
torus.
Such a gluing corresponds to the spinning construction of the rim surgery of Fintushel-
Stern i.e. m “ n “ 0, adding a combination of m-fold twist spinning [28] and n-fold roll
spinning [11, 18]. It is useful to specify these twists by classical diffeomorphisms that give
equivalent descriptions for the twisted rim surgery.
3
Consider self-deffeomoprhisms denoted by τ and ρ of pS ,Kq that correspond to twists
2
parallel to a meridian and a longitude of K respectively. Let BEpKqˆI “ KˆBD ˆI be
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Knotted surfaces in 4-manifolds by Knot surgery and Stabilization
a collar of BEpKq in EpKq under a suitable trivialization with 0-framing. Identify K with
S1 – R{Z and then the twist map τ is given by
(2) τpθ,eiψ,tq “ pθ,eipψ`2πtq,tq for pθ,eiψ,tq P K ˆBD2ˆI
and otherwise, τpyq “ y.
Similarly,aroll,ρ,isobtainedfromρpθ,eiψ,tq “ pθ`t,eiψ,tqbyextendingastheidentity
3
on the rest of pS ,Kq. Although a roll can also produce exotic embeddings, we will only
deal with an m-twist rim surgery in this paper since ameridian twist is sufficiently usefulto
construct all desired smoothly knotted surfaces. Most of thearguments for the stabilization
result of the m-twist rim surgery be easily modified to address the rolling as well.
3 3 3 3
Writing pS ,Kq “ pB ,K qYpB ,K q where pB ,K q is an unknotted ball pair, we
` ´ ´
3
regard τ as an automorphism of pB ,K q. Since the rim torus R lies in a neighborhood
` α
1 3
of the curve α, the twisted rim surgery performed in νpαq – S ˆpB ,Iq gives rise to the
3
mapping torus of pB ,K q with monodromy given by the twist map τ. So the m-twisted
`
rim surgery on pX,Σq can be written as follows;
1 3 1 3
(3) pX,ΣKpmqq “ pX,Σq´S ˆpB ,IqYB S ˆτm pB ,K`q.
In doing any rim surgery (twisted or otherwise) we assume that α Ă Σ is a curve for
which there is a framing of νpΣq along α such that the pushoff of α into BνpΣq is null-
homotopic in X ´Σ. But we don’t assume that α is a non-separating curve on Σ, which
is necessary to distinguish the diffeomorphism type of Σ pmq from that of Σ with Seiberg-
K
Witten invariant. Note from [14, Lemma 2.2] that if α bounds a disk in Σ, the surface
Σ pmq is the connected sum of Σ with the m-twist spun knot Kpmq of Zeeman [28]. Our
K
stabilization results include this example as well.
3.2. Twisted rim surgery and the surface knot group. As mentioned in Section 2,
˘1-twist rim surgery always preserves surface knot groups [16, Proposition 2.3], and also
Proposition2.4in[16]showswhenanm-twistrimsurgerypreservesthefundamentalgroup.
Here we will revisit Proposition 2.4 with more elementary argument (compare the proof
in [16]) since it explicitly provides the presentation of the fundamental group for later use.
Proposition 3.1 (Proposition 2.4 in [16]). Let π1pX ´Σq be any group G. Suppose that
the surface Σ Ă X has H1pX ´Σq “ Z{d and the meridian µΣ has order d in π1pX ´Σq.
If pm,dq “ 1 then π1pX ´ΣKpmqq is isomorphic to G.
Proof. In order to investigate a presentation of π1pX ´ΣKpmqq, we first consider the de-
composition of X ´Σ pmq induced from (3):
K
1 3 1 3
(4) X ´ΣKpmq “ X ´Σ´S ˆpB ,IqYB S ˆτm pB ´K`q.
Choosing a base point ˚ at the intersection of two components in this decomposition (4),
we get the following diagram from the van Kampen theorem;
1 3
(5) π1pX ´Σ´S ˆpB ,Iqq
❦❦❦❦❦❦i❦1❦❦❦❦❦❦❦❦55 ❖❖❖❖❖❖j❖1❖❖❖❖''
1 3
π1pS ˆpBB ´ttwo❙❙p❙o❙❙in❙❙t❙si❙u2❙q❙q❙❙❙❙)) ♣♣♣♣j♣2♣π♣♣1♣p♣X♣77 ´ΣKpmqq
1 3
π1pS ˆτm pB ´K`qq
7
Hee Jung Kim
1 3
In the diagram, each map is obviously induced by an inclusion and π1pS ˆ pBB ´
1
ttwo pointsuqq is generated by two elements rS s and µ. So, the relations in a presentation
of π1pX ´ ΣKpmqq are given by i1rS1s “ α1 which is trivial by the assumption that the
pushoff α1 of α is null-homotopic in X ´Σ, and i1pµq is a meridian µΣ of Σ in π1pX ´Σ´
1 3
S ˆpB ,Iqq – π1pX ´Σq. So, it leads the presentation for π1pX ´ΣKpmqq as follows:
1 3
(6) xπ1pX ´Σq˚π1pS ˆτm pB ´K`qq | 1 “δ,µΣ “ µKy,
1 1 3
where δ “ rS s, µK “ rµKs denote generators of π1pS ˆτm pB ´ K`q,˚q in Figure 1.
Associated with the relations, this presentation becomes the following:
(7) xπ1pX ´Σq˚π1pB3´K`q| µΣ “ µK,µ´KmgµmK “ g ,@g P π1pB3´K`qy.
If pm,dq “ 1 and µdΣ “ 1, it obviously gives π1pX ´Σq.
µK “rµK,˚s
˚
δ“rS1,˚s
Figure 1. Generators of π1pS1ˆτm pB3´K`q,˚q
(cid:3)
1 3
Remark 3.2. Note that in the diagram, if the image of π1pS ˆτm pB ´K`qq in π1pX ´
Σ pmqqis acyclic subgroupgenerated bythemeridianof Σ pmq, thenthepresentation (6)
K K
for π1pX ´ΣKpmqq readily leads to the group π1pX ´Σq. The choice of the parameter m
in Proposition 3.1 makes this case. This property enables one to see that the stabilization
result of Baykur-Sunukjian in [5] can be extended to knotted surfaces produced by the m-
twist rim surgery with the choice of m in Proposition 3.1. They showed that for m “ 1 or
any m with pm,dq “ 1 in the case of π1pX´Σq“ Zd, the m-twist surgered surfaces become
smoothly isotopic by adding one trivial handle. The same argument in [5, Section 3.2] can
work for the case of Proposition 3.1 by adding a handle at a crossing of the knotted arc
1 3
K` in S ˆτm pB ,K`q to unknot it crossing by crossing, where it must be checked that
1 3
the attached handle is trivial at each stage. It follows from that π1pS ˆτm pB ´K`qq is
a cyclic subgroup generated by the meridian of Σ pmq; see [5, Lemma 3] for more details.
K
3.3. Annulus rim surgery. Suppose that there is a smoothly embedded annulus Mp–
1
S ˆI, where I denotes an interval r´1,1s) in X such that M meets Σ normally along BM
so that M X Σ “ BM are two curves α´1 and α1 on Σ. We assume that Σ´ tα´1,α1u
1 2 1 3
is connected. Choose a trivialization νpMq Ñ pS ˆIqˆD – S ˆB such that M –
1 1
pS ˆIqˆt0u and νpMq|Σ – S ˆf, where f denotes a disjoint union of two unknotted
2 3
segments BI ˆ I Ă I ˆ D “ B , a part of the boundary of a trivially embedded band
3 1 1 1 3
b “ I ˆI in B (See Figure 2). So, M is identified with S ˆI ˆt0u in S ˆb Ă S ˆB .
3
Denote by m a meridian of b in B and let T be a torus in νpMq corresponding to
b
1 1 3
S ˆm Ă S ˆpB ,fq. Knot surgery along this torus T produces a new surface Σ pϕq.
b K
8
Knotted surfaces in 4-manifolds by Knot surgery and Stabilization
1 1 1
The simplest gluing ϕ : BνpTq Ñ S ˆ BEpKq, given by rS s ÞÑ rS s, m ÞÑ µ , and
b K
µ ÞÑ λ , provides the Finashin’s annulus rim surgery. This operation obviously yields a
T K
3
band b Ă B by knotting the band b along K and let f be the pair of arcs bounding b .
K K K
Here the framing of b is chosen the same as the framing of b. So the resulting manifold of
K
1 3 1 3
the annulus rim surgery performed on νpMq– S ˆpB ,fq becomes S ˆpB ,f q and we
K
write a new pair as follows:
1 3 1 3
(8) pX,Σ pϕqq “ pX,Σq´S ˆpB ,fqYS ˆpB ,f q
K K
This construction can be further modified by twists along a meridian and a longitude of K,
but we will stick to Finashin’s construction; see [15] for other modifications.
M
m
b
b
I M
α1 α´1 I
Σ
pB3,fq
Figure 2. T “ S1ˆm Ă S1ˆpB3,fq– νpMq
b
Notethatwhenπ1pX´Σq “ Zd,any(twistedorotherwise)annulusrimsurgerypreserves
surface knot groups [6], [15, Proposition 3.3].
4. Basic Construction
In order to get our main theorems, for two knots K and K1 related by a single crossing
change we will show the 1-stable equivalence on surface knots ΣKpϕq and ΣK1pϕq produced
by knot surgery. The complete proof for each knotting construction will be given in Sec-
tion 5.1, 5.2, 5.3 but in this section we will first discuss some constructions and properties
that will be used repeatedly in the proofs of our stabilization results.
K K1
c ˘1
Figure 3. ˘1-Dehn surgery along c at a crossing
SupposethattwoknotsK,K1 inS3 differbyasinglecrossingchange, sothattheknotK1
is obtained by performinga ˘1-Dehn surgery along a curve c around an oppositely oriented
crossing of K as in Figure 3. Let pXKpϕq,ΣKpϕqq, pXK1pϕq,ΣK1pϕqq be two pairs obtained
by a knot surgery along a torus T Ă X ´Σ and gluing map ϕ along the knots K and K1
respectively. Then we begin by showing that these pairs are related by a torus surgery:
9
Hee Jung Kim
Lemma 4.1. A log transform of multiplicity ˘1 on pX pϕq,Σ pϕqq produces the pair
K K
pXK1pϕq,ΣK1pϕqq.
Proof. We first recall that the knot surgered pair is defined as follows:
1
pX pϕq,Σ pϕqq “ pX,Σq´νpTqY S ˆEpKq.
K K ϕ
1 1
Here, we denote T a torus S ˆc in S ˆEpKq of this decomposition, where c is a curve
c
1 2
at a crossing of K as in Figure 3. We identify a neighborhood νpT q with S ˆpcˆD q,
c
2
where cˆD is a neighborhood of c in EpKq. Then the ˘1-log transform parallel to the
1
curve c on T in pX pϕq,Σ pϕqq can be described by the identity in the S direction times
c K K
the ˘1-Dehn surgery along c Ă EpKq. Note that this process hasn’t change the boundary
1
S ˆBEpKq, so the resulting manifold is obtained by performing a torus surgery along T
c
1 1
in S ˆ EpKq, denoted by pS ˆ EpKqq , and gluing back in pX,Σq ´ νpTq along their
Tc
boundaries via ϕ. This manifold is easily identified with pXK1pϕq,ΣK1pϕqq. It follows from
thatthereisanobviousdiffeomorphismfrompS1ˆEpKqq toS1ˆEpK1qwhichcarrieseach
Tc
1 1 1
element in a basis trS s,µK,λKu of H1pBpS ˆEpKqqTcq to each element in trS s,µK1,λK1u
of H1pS1ˆBEpK1qq respectively.
(cid:3)
Remark 4.2. When the knot surgery pX,Σq Ñ pX,Σ pϕqq is an ambient surgery such
K
as (twisted) rim surgery and annulus rim surgery, the above torus surgery on pX,Σ pϕqq
K
provides a new embedding ΣK1pϕq in X, which is given by the knot surgery performed on
pX,Σq along the same torus and gluing to produce Σ pϕq. And, it can be observed that
K
π1pXK1pϕq ´ ΣK1pϕqq – π1pXKpϕq ´ ΣKpϕqq from the proof in Lemma 4.1, so if a knot
surgery pX,Σq Ñ pX,ΣKpϕqq does not change the surface knot group π1pX ´ Σq under
some suitable circumstances then the new embedding ΣK1pϕq also preserves its knot group.
4.1. Fiber sum and gluing map. A torus surgery of pX pϕq,Σ pϕqq along T can be
K K c
1 3
described as a fiber sum of pX pϕq,Σ pϕqq and S ˆS : Let T be a standardly embedded
K K u
1 1 3 3
torus S ˆu in S ˆS , where u is an unknot in S . Then we write the torus surgered
manifold in Lemma 4.1 as a fiber sum along tori T and T :
c u
1 3
(9) pXK1pϕq,ΣK1pϕqq “ pXKpϕq,ΣKpϕqqTc#Tu S ˆS .
In order to describe the gluing map f : BνpT q Ñ BνpT q carefully, identify a tubular
u c
1 3 1 2 2
neighborhood νpT q in S ˆ S with S ˆ pu ˆ D q, where u ˆ D is a neighborhood
u
νpuq of u in S3. Let a “ S1 ˆ tptu Ă S1 ˆ u “ T and a1, u1 be the pushoffs of a,
u
u into BνpTuq. Then ta1,u1,rBD2su forms a basis for H1pBνpTuqq. Similarly, under an
1 2 1 1 1
identification νpT q – S ˆpcˆD q in S ˆEpKq, let γ “ S ˆtptu Ă S ˆc “ T and
c c
m “ t˚uˆpt˚uˆBD2q, l “ t˚uˆpcˆt˚uq bea meridian-longitude pair of c. So tγ1,m ,l u
c c c c
gives a basis for H1pBνpTcqq, where γ1 is a pushoff of γ into BνpTcq. As described in the
1 2 1 2
proofofLemma4.1, thegluingmapf : S ˆpuˆBD qÑ S ˆpcˆBD qis adiffeomorphism
determined by
(10) f pa1q“ γ1, f pu1q“ ˘m `l , and f prBD2sq “ ˘l .
˚ ˚ c c ˚ c
In our present purpose, it is important to keep track of the gluing map in this fiber sum,
from which we can determine the framing arising in our proof of the stabilization result.
4.2. Cobordism. In this section, we will construct a cobordism W whose upper boundary
1 3 1 3
ispX pϕq,Σ pϕqq # S ˆS frompX pϕq,Σ pϕqq\S ˆS . Theproofof stableequiv-
K K Tc Tu K K
alence for surfaces ΣKpϕq and ΣK1pϕq will come from the middle level of the constructed
cobordism W.
10