Yajima, Takeshi

Osaka City University Repository

Jou rn al of M ath em atics , 
O saka C ity  U n iv e rsity , 
Vol. 13, No. 2.
On the fundamental groups of knotted 2-manifolds  
in the 4-space
By Takeshi YAJIMA 
(Received Dec. 14, 1 9 6 2 )
1. Introduction
Let M be a 2-dimensional manifold imbedded in the 4-dimensional Euclidean 
space R .^ Let &(M) be the fundamental group of R^-M . In the case that M  is 
a spinning sphere S, namely a sphere obtained by rotating an arc about a 2-dimen- 
sional plane, the group S (S ) was investigated by E. Artin [I], E. R. Van Kampen 
[2] and J. J. Andrews and M. L. Curtis [3].
The presentation of S (S ) was discussed by R.H. Fox [4] and S. Kinoshita [5], 
where S is a knotted 2-sphere in general. Their method, the so called moving 
picture method, concerned with the slice knots or the null-equivalent knots, which 
appear as an intersection of S and a 3-dimensional subspace of R'^ .
This paper contains the method of the Wirtinger's presentation of S(M ) by 
the classical projection method as in the knot theory. In this direction the principle 
of the method has been given by S. Kinoshita [6].
As an application of this method, a parallelism between knots in and 
knotted 2-spheres in R  ^ will be discussed.
2. Preliminaries
Let R  ^ be the 4-dimensional Euclidean space with a coordinate system {x,y, 
ZyU). Let R^  be the 3-dimensional subspace of R  ^ defined by u = ^ . With every 
point y,z,u )  of a complex M  in R ,^ we associate the point P*= (x,y,z,0)
and u = u(P). We call P* the trace and u the height of a point P  respectively 
and denote by P= [P*, ^/(P)]. The set of traces of points of M  will be denoted 
by M*. The projection cp \ P P " ^  is defined as usual.
Throughout this paper terminologies are used in the semi-linear point of view. 
Hence complexes are polyhedral and mappings are simplicial.
Let M be a 2-dimensional closed orientable manifold. It is no loss of generality 
to assume the following condition:
(2.1) I f  P i , . . . ,  P  ^ are vertices o f  M, then the system o f points (Pi*,. . . ,  Pm) 
is in general position in R .^
Let P* 6 M*. If there exist at least two points of M  such that their traces
coincide with P*, then we say P* a cutting point of M*. The set of cutting 
points of M* is denoted by and called the cutting of M*.
In virtue of (2.1), 2-dimensional simplexes of Af* have an intersection only 
in the following cases (Fig. I).
64 Takeshi Y a j im a
Fig. I
Hence r(M *) consists of segments, each of whose endpoints belongs to only one 
1-dimensional simplex. Notice that the common vertex of two simplexes in Fig. 
l,(b) is not a point of r(M *). We call such a point a singular cutting point of Af*.
We can also assume the following conditions by a slight modification of 
vertices of M
(2. 2) A segment o f  r(M *) is the intersection o f  just two simplexes.
(2. 3) There exist just three simplexes through a double point o f  r(M *).
Since an endpoint of a segment of r(M *) belongs to only one 1-dimen­
sional simplex as shown in Fig. I, we have:
(2. 4) r (M *) consists o f the following two kinds o f polygons :
(1) closed polygons,
(2) polygonal arcs, whose endpoints are different or coincided singular cutting 
points.
3. The linking
Let iVf be a 2-dimensional closed orientable manifold in R .^ Let / be a conti­
nuous mapping of the unit circle
Ct: X^  -\-y^  = I
into R  ^— M. Put c^ fict). The vertices of c^  may be considered to be in general 
position in R .^
If / can be extended to F  which maps the unit disk
Dt\ x^+y^^l
into R^—M, then we say that c does not link homotopically with M  Conversely, 
if such an extension does not exist, then we say that c links homotopically with M.
max ^^ (P) —minz (^/(0 ))PGM QC-Ci
(3.1) I f  then c does not link homotopically with M.
Proof. Let (0, r) be the polar coordinate of Di, where and
Let Cl/2 be a circle of r=l/ 2 . Take a positive number h such that
h>
Put
F(Q ,r)==lf(Q r, u(f(Q))-^2a - r ) h l  l / 2^ r^ l.
Since F{Q,rT=f<iQy, F  is a continuous mapping of D t-D t /2 into R^-M. It is
obvious that F(ci/2 ) is null-homotopic in the half-space defined by u^h^-minQect
^(/(Q))- Hence c is null-homotopic in R^—M.
Consequently if c links with M  homotopically, then we have 
Suppose that consists of two points and Let A t^c and
be the points such that We define sgn^l* as follows:
-I if u(iAt)>u(iA2),
On the fundam ental groups o f  knotted 2 -manifolds in the 4 ‘Space 65
 ^ '.—I if u(iAt)<u{A2').
Sgn is defined in the same way.
(3.2) I f  Sgn A^ • Sgn then c does not link with M homotopically.
Proof. Suppose that sgn A*==sgn 5 * = + 1 , The same proof as (3.1) assures 
the statement.
In the case that sgn^* = sgn5 * = - I ,  take a negaive number h' such that
¥ < - min u^P) —max u(f(Q ))P^ M Q6Ci
instead of h in the proof of (3.1).
(3. 3) SgnA^-Sgn 5 * = —I, then c links with M homotopically.
Proof. Suppose that sg n ^ *= + l and s g n ^ *= -L  Assume that c does not 
link homotopically with M. Then there exists an extension F  of / over Dt such 
that F{Dt)C.R^ — M. Since consists of two points A  ^ and 5* , F(Z>i)*oM*
contains a cutting of polygonal arc, whose endpoints are A  ^ and Therefore 
there exist an arc a^  connecting At and Bt on F(D t), and an arc ^2 connecting 
Ao and B 2 on M  such that a f = a$.
Let P t^ at and ^2 6 ^ 2  be two variable points such that P^ = Pf. If Pt = At 
and P2 = A 2, then we have u(Pt)>u(P2). If Pt = Bt and P2 = ^ 2, then we have 
uCPt')<u(P2). Therefore there exist points P^ on a^  = a$ and P o i Po2 € « 2  
such that Poi=PoI=^o* and ^^ (Poi) = ^ /(Po2). Hence 
Poi = Po2- This contradicts the assumption.
We have the following corollary from (3. 2 ).
(3. 4) Let C be a continuous image o f an arc 
Ct in R^—M. Let A  ^ and P* be successive points 
o f  c*r~\M* on C*, where A* and B* can be connected
by an arc on M *—r{M*'). I f  sgn A* = sgn B *, Fig. 2
66 Takeshi Y a j im a
then and can be cancelled, and vice versa. 
4: The fundamental groups
In virtue of the conditions in § 2, r(M *) separates M* into several domains 
, 2 k, each of which has an orientation induced by the orientation of M  
We represent the orientation of 2^ * by a small vector v,- such that the direction 
of Yi coincides with the direction of a right-handed screw twisting along the 
orientation of I f .
Let T* be a simple arc of r(M *). From (2.2) there exist domains 
2 % 2’ +^! such that 7* is a common boundary of these domains. Suppose that 
I / O Vi =T.-, are arcs in i?4 such that rf=r^j = T^ \ If
then we call 2’/U2’/+i the over surface, and 2'yU2’y+i the under surface. To 
represent the relation of these surface, we use the following notations, cancelling 
the vector of the under surface (Fig. 3).
u (y d > u (jj) U(TiXUCTj)
Fig. 3
The direction of the vector corresponds to the orientation of the over surface.
For each Hf, we take a small circle cf such that I fr\ c f  consists of points 
AJ, B f  where sgn A f= ^ l, S g n B f= -I ,  and 2 * * = O for We define the
orientation of cf such that it coincides with the direction of v/ at the point Af. 
It is obvious that each c f  defines a equivalent class of ci in R^-M .
(4.1) Ci and Cj are homotopic in R^  — M.
Proof. If i= j, then the statement is obvious. Let us prove that c- and c-+± 
in Fig. 3 are homotopic in R^  — M.
Suppose that is the under surface. Let T be a tube such that T* =
consists of two longitudes /3^= B fB,
Deform T  such that u {ai)> u{a{)  and /^(/?^ X^z(Ag). Then we have
On the fundamental groups o f knotted 2 -manifolds in the 4-space 67
r\
SjXjI^y+i- Deform T  so far as ^/( i^)<^ (^ 2^) but ^^(ai)>2/(^2). Then we have 
T cR ^ -M . Hence and c:,+i are homotopic in R^-M . Other cases are proved 
successibly.
Take a base point O in R^—M"^ . Let wf be an arbitrary path connecting O 
and an arbitrary point P f of cf. We define the signs of points wf.r^M^ be all +1.
Denote the closed path
O — >  PT - ^ P f  — >  O
w f  C f  w f
by . It is obvious that the equivalent class of the closed path a- in R ^-M  
corresponding to of does not depend on the choice of wf and c,*.
(4.2) Theorem. (Ti, ...,<7  ^ form a generator system o f  S (M ) with the base 
point 0 .
Proof. Suppose that w is an arbitrary oriented closed path in - M  with 
the base point 0. Let P* be a point of We make a  ^ correspond to P*
in the following manner:
1) If Sgn P * = + l, then P* — > I,
2) If S g n  P* = — I and the direction of v • coincides with the direction of 
at the point P*, then P* — >
3) If Sgn P * = —I and v^. and w"^  have the opposite directions at the point 
P*, then P* — > (7-.
Thus a word w{a) corresponds to w. It is obvious from (3. 4) that a representa­
tive of W {a) is equivalent to w.
(4.3) I f  is the over surface, then we have the following relations:
(1) =
(2) a-aj^ = l,
where e = + l  or —I according as the direction o f  the vector of the over surface 
coincides or not tmth the direction S/*—
Proof. (I) is obvious from (4.1). Let us prove (2) in the case of  ^= + 1. 
Let T be the tube in the proof of (4.1). Take a curve /+1 connecting P- 
and P- I i on T. The closed path
^  "',-+I ,-+I ^  <=i ^  «’,•„■+1 „  “',•+I ^
O — > p.+i — > p. — > Pi — > P,+1 — > O
is represented by ajOiaj .^ It is obvious that this closed path is homotopic to 
(Fig. 5). Hence
6 8  Takeshi Y a jim a
Fig. 5
(4. 4) Theorem. The relations (4. 3) corresponding to all arcs o f  V{M^) form  
a system o f  defining relations o f F{M).
Proof. If there exist no singular cutting points, then the statement is obvious. 
Suppose that there exist some singular cutting points. Let be a closed path 
which is null-homotopic in R^-M. There exist continuous mappingsV, F  such 
that f{c t)= d , F ( D i ) - M  and F\c±=f, where Ct and Di are as in §3. By
a slight modification of F  we have F (D i)*C i? ^ -(r (M *)-r (M *)) . Hence a can 
be represented as a consequence of relations (4. 3).
5, Spheres in
Let  ^ be a knot in A construction of a 2-sphere S in whose funda­
mental group S(S) is isomorpic to f?(^), was given in [3] by rotating an arc 
along a plane in Let us discuss the same problem by the projection method.
(5.1) Let k he a knot in R^ , There exists a torus T^  in R  ^ such that 
is isomorphic to S(^).
T ,)
On the fundamental groups o f knotted 2 -manifolds in the 4-space 69
Proof. In the Wirtinger's presentation, the defining relations of S(M ) are 
given in the same form as the defining relations of g(^). So we construct Tk in 
the following correspondence (Fig. 6), where tubes represented by dotted lines, 
which show that they go through the other tubes, correspond to the cross points 
of the under-going arcs of k. The inessential generators are omitted.
'Si+i
Si+l=SjSiSj^
L+I
Sj^SiSj
Fig. 6
It is obvious that " {^Tk) is isomorphic to S(^).
Now let us construct a knotted 2-sphere Sk in from Tk as follows. Let 
P  be an arbitrary point of k. Take a meridian circle c on Tk corresponding to 
the point P. Cut the torus Tk into a tube by a plane through c, and add 
two disks to the terminals of T l Then we get a knotted sphere in R .^ We 
say that Tk and are similar to k.
(5. 2) Theorem. I f  Sk is similar to k, then ^(Sk) is isomorphic to S(/0.
Proof. Suppose that the presentation of g(^) is given as follows:
Generators : (5i, . . . ,  Sn)
r.s'i =  .s!;s2sr,"'‘
Relations: (Mk) .^........ (.^,=-4:1)
Let P be a point of a segment of the projection of k, and Q, R be the 
endpoints of Let Sm and Sm be the subsegment of such that Sm = QP and
70 Takeshi Y a j im a
Sm=PR. If we take a system of generators (si , . . .  ,Sm, Sm, .^. ,Sn) instead of (s i , . . . ,  
Sm, ...,Sn), then we have relations (R i) replacing 5^  in (Rk) by s'm or s / and a 
new relation Sm = sJ! as a system of defining relations of S(^). By a geometrical 
consideration, we can prove that the relation s'm^sJ! is an induced relation of the 
relations of (R'k).
On the other hand the presentation of '^(Sk) is given by the generators (at, 
. . . ,  (^ m Om, .. • ,(^ n) and relations corresponding to (Ri)- Hence is isomorphic
to ^(k).
Example I.
Generators: Si, S2, S3.
Relations:
S±=S3S2Ss^
S2=SiS3St^
S3=S2SlS2^
Relations: I (72 =(y 1(^ 301^
Fig. 7
The first relation of ^(S) in Fig. 7 is cancelled. We can prove that the projec­
tion S* in Fig. 7 is deformed into the projection S'* in Fig. 8 by a deformation 
of S into S' in R .^
It is worthy of notice that if T is not a similar torus of knots, then S(S)  is 
not always isomorphic to S(T)  as shown in Example 2.
A
Example 2. We get the torus T in Fig. 9 by changing the relation of heights 
of the torus in Fig. 7. If we cut the torus T  by the plane A, then we get the 
same sphere as in Fig. 8. But if we cut T by the plane B, then we get a 
sphere which is the same as Example 10, p. 135 in [4]. Obviously the funda­
mental groups of these spheres are not coincide.
References
[1 ]  E . Artin, Zur Isotopie zweidimensionaler Flachen in i?4 , Abh. Math. Sem. Univ. Hamburg 
4 (1925), 174-177.
[2 ]  E.R . Van Kampen, Zur Isotopie zweidimensionaler Flachen in i?4 , Abh. Math. Sem. Univ. 
Hambrug 6 (1927), 216.
[3 ]  J . J . Andrews and M. L . Curtis, Knotted 2-spheres in the 4-sphere, Ann. of Math., v. 70, No
3, (1956), 565-571.
[ 4 ]  R.H. Fox, Topology of 3-manifolds, edited by M.K. Fort Jr ., Prentiee Hall (1962), 133.
[5 ]  S. Kinoshita, On the Alexander polynomials of 2-spheres in a 4-sphere, Ann. of Math., v.74, 
No. 3 (1961), 518-531.
[ 6 ]  S. Kinoshita, Alexander polynomials as isotopy invariants, I, Osaka Math. Jour., v. 10, No. 
2 (1958), 263-271.
On the fundamental groups o f knotted 2 -manifolds in the 4 -space 71


On the Fundamental groups of knotted 2-manifold in the 4-space

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On the Fundamental groups of knotted 2-manifold in the 4-space

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