The topological interpretation of the core group of a surface in S^4

a rX iv:mat h /43475v1[mat h.GT]26M ar24The topological interpretation of the core group of a surface in S 4by J´o zef H.Przytycki 1and Witold Rosicki 2Abstract.We give a topological interpretation of the core group invariant of a surface embedded in S 4[F-R,Ro].We show that the group is isomorphic to the free product of the fundamental group of the double branch cover of S 4with the surface as a branched set,and the infinite cyclic group.We present a generalization for unoriented surfaces,for other cyclic branched covers,and other codimension two embeddings of manifolds in spheres.It is shown in [Ya-1](compare [Ka-1,C-S-2,Ya-2])that the fundamental group of the complement of an oriented surface,M ,in S 4allows a Wirtinger type presentation.In [Ro]the core group invariant of M was introduced (following [F-R,Joy])and the topological interpretation of the group was promised.In the first section we give this interpretation following the result of Wada’s [Wa]for classical knots and the proof of Wada result presented in [Pr].In the second section we generalize our results to unoriented surfaces,and codimension two embeddings of closed n -manifolds in S n +2.1Surfaces in S 4Let M be an oriented surface embedded in S 4=R 4∪{∞}.Let p :R 4→R 3be a projection such that the restriction to M ,p |M is a general position map.Define the lower decker set (“invisible”set)[C-S-1,C-S-2]as A ={x ∈M |∃y ∈M,p (x )=p (y )and x is the lower point,that is x is further than y in the direction of the projection }.The set A separates M into regions (“visibility”regions),that is arc connected components of M −A .Theorem 1.1([Ya-1])The fundamental group of S 4−M (ΠM =π1(S 4−M ))has the following Wirtinger type presentation.Generators of thegroup,x1,x2,...,x n,correspond to regions of M.Relations of the group correspond to double point arcs and are of the form x i x p x−1i x−1q or x i x−1p x−1i x q,depending on the orientations of p(M)at double point arcs.Here x i corresponds to the higher region of the projection and x p,x q to lower regions.Similarly to the groupΠM,we define the core group,G M of a surface M in S4.Definition1.2([Ro])The core group,G M of a surface M in S4,has the following,quandle type,presentation.Generators of the group,y1,y2,...,y n, correspond to regions of M.Relations of the group correspond to double point arcs and are of the form y i y−1p y i y−1q.Here y i corresponds to the higher region of the projection and y p,y q to lower regions.The operation y p∗y i=y i y−1p y i defines a quandle.We will interpret topologically the core group as follows.It is the analogue of the Wada result for classical links[Wa].Theorem1.3The core group,G M of a surface M in S4,is isomorphic to the free product of the fundamental group of the branched double cover of S4with branching set M,and the infinite cyclic group.That is G M=π1(M(2))∗Z where M(2)is the considered double branched cover.Below we will prove a generalization of the theorem for any cyclic branched cover,following the exposition of the classical case in[Pr].Let f:M(k)→S4be a cyclic k fold branched covering with a branch set M.More precisely M(k)is defined as follows:Let V M be a tubular neighborhood of M in S4.It has a structure of a2-disc bundle(over M). For a point x∈M the boundary of the disk D x of the bundle is called a meridian of M(at x),denoted byµx.It has the natural orientation for oriented M.Thefirst homology group of S4−M is freely generated by meridians of M(one for each connected component of M).Thus,we have a k-fold cyclic covering of S4−M given by an epimorphism H1(S4−M)→Z k which sendsµx to1.This covering can be uniquely extended to the cyclic k fold branched covering with a branch set ly,we define the cover on each(meridian)disk D x as given by the function p:D′x→D x,p(z)=z k where z is a complex coordinate of the disks.LetΠk M=π1(M(k)).2Proposition1.4Consider the epimorphismˆg k:π1(S4−M)→Z k given by ˆg k(x i)=1for any i.ThenΠk M=ker(ˆg k)/(x i)k.Proof:Consider a regular neighborhood of M in S4and its normal disk bundle.Each disk in the bundle is covered by a disk(the center being a branch point).As noted before the covering can be described by the map z→z k where z is a complex coordinate of the disk.In the Wirtinger presentation of the fundamental group,generators correspond to some meridians(that is, boundaries of disks of the normal bundle).Consider the unbranched k-covering of S4−M obtained by removing the branch set.The epimorphismˆg:π1(S4−M)→Z k corresponding to the covering sends each meridian to1∈Z k,(ˆg(µx)=1).The fundamental group of the unbranched cover is equal to kerˆg.To get a branched cover from the unbranched one we have tofill the cover of every meridian.On algebraic level we add relationsµk x=0.Because every meridian is conjugated to a generator x i,the proposition is proved.Theorem1.5Π(k)M ∗Z∗...∗Zk−1=Π(k)M⊔S2has the following quandle type de-scription(in the disjoint sum M⊔S2,S2is the unknotted component that can be separated by a3-sphere from M).To every(visible)region there correspond k−1generatorsτj(y i)(0≤j<k−1)and to each double point arc correspond k−1relationsτj(y pτ(y i)τ(y−1q)y−1i) orτj(y iτ(y p)τ(y−1i)y−1q)depending on a local orientation.Lemma1.6Let F n+1={x1,x2,...,x n+1|}be a free group of n+1gener-ators,x1,x2,...,x n+1.Consider the epimorphism g k:F n+1→Z k such that g k(x i)=1for any i.Let F(k)=ker(g k),and let¯F(k)=F(k)/(x k i).Let τ:F n+1→F n+1be an automorphism,defined byτ(x i)=x n+1x i x−1n+1and y i=x i x−1n+1.Then(i)F(k)is a free group of nk+1generators:x k n+1andτj(y i)for0≤j≤k−1,1≤i≤n.(ii)¯F(k)is a free group on n(k−1)generatorsτj(y i)for0≤j<k−1, 1≤i≤n.Furthermore y iτ(y i)...τk−1(y i)=1for any i.3(a)T T 44~(8)(b)Fig.1(The covering ˜T (8)4→T 4)Proof:(i)F n +1=π1(T n +1)where T n +1is a graph of one vertex and n +1loops (Fig.1(a)).g k :F n +1→Z k determines a k-fold covering space ˜T (k )n +1of the graph T n +1as show in Fig.1(b).Of course,the Euler charac-teristic χ(T n +1)=−n and χ(˜T (k )n +1)=−kn .Thus,π1(˜T (k )n +1)is freelygenerated by nk +1generators.We can easily identify free genera-tors of π1(˜T (k )n +1)=kerg k ;in particular,g k (x k n +1)=g k (τj (y i ))=0for 1≤i ≤n and 0≤j ≤k −1.(ii)y i τ(y i )τ2(y i )...τk −1(y i )=(x i x −1n +1)(x n +1x i x −2n +1)...(x k −1n +1x i x −k n +1)=x k i x −k n +1.Thus,in F (k )n +1/(x k n +1)we have y 1...τk −1(y i )=x k i and furthermore¯F (k )n +1={F (k )n +1|x k n +1,x k i ;1≤i ≤n }={x k n +1,τj (y i )|x k n +1,y i τ(y i )...τk −1(y i ),1≤i ≤n,0≤j ≤k −1}=={τj (y i )|y i τ(y i )...τk −1(y i ),1≤i ≤n,0≤j ≤k −1}={τj (y i )|1≤i ≤n,0≤j <k −1}as required.We will use an extension of Lemma 1.6for a group with Wirtinger type relations.Let G ={F n |r 1,...,r m }={x 1,...,x n |r 1,...,r m }where any relation is of the4form r=x i xǫp x−1i x−ǫq forǫ=±1,and1≤i,p,q≤nWe have G∗Z={x1,...,x n+1|r1,...,r m}.The epimorphism g k:F n+1→Z k yields the epimorphismˆg k:G∗Z→Z k.The epimorphismˆg k is well defined because g k(r i)=0.We use Lemma1.6tofind a presentation of G(k)=kerˆg k and¯G(k)=G(k)/(x k i).Lemma1.7(i)G(k)={F(k)|τj(r s),1≤s≤m,0≤j≤k−1}.(ii)¯G(k)={¯F(k)|τj(r s),1≤s≤m,0≤j<k−1}={τj(y i)|τj(r s),1≤s≤m,0≤j<k−1,1≤i≤n}whereτj(x i x p x−1i x−1q)=τj(y iτ(y p)τ(y−1i)y−1q)andτj(x i x−1p x−1i x q)=τj(y i y−1pτ−1(y−1i)τ−1(y q)). Proof:(i)Because r s∈kerg k then the relations of kerˆg k are of the form wr i w−1for w∈F n+1.We observe that if g k(w)=u(0≤u≤k−1)then w=x u n+1w′where w′∈kerg k.Therefore relations of G(k)=kerˆg k are of the formτu(r s)for0≤u≤k−1,1≤s≤m.These yield the presentation of G(k).We have:x i x p x−1i x−1q=(x i x−1n+1)(x n+1(x p x−1n+1)x−1n+1)(x n+1(x n+1x−1i)x−1n+1)(x n+1x−1q)=y iτ(y p)τ(y−1i)y−1qx i x−1p x−1i x q=(x i x−1n+1)(x n+1x−1p)(x−1i x n+1)(x−1n+1x q)=y i y−1p(τ−1(y i))−1τ−1(y q)(ii)Adding relations(x i)k reduces F(k)to¯F(k)and G(k)to¯G(k)so¯G(k)= {¯F(k)|τj(r s),1≤s≤m,0≤j≤k−1}We can eliminate the relation τk−1(r s)(expressing it using other relations).We use the identityy iτ(y i)...τk−1(y i)=1in¯G(k)(in particularτk(y i)=y i).We can write our relationsτu(y iτ(y p)τ(y−1i)y−1q)asτu+1(y p y−1i)=τu(y−1i y q).We assume it holds for0≤u<k−1and we will see thatτk−1(r s)=1as well:τk−1(y iτ(y p)τ(y−1i)y−1q)=τk−1(y i)τk(y p)τk(y−1i)τk−1(y−1q)==τk−1(y i)y p y−1iτk−1(y−1q)=τk−1(y i)y p(y−1i y q)τ(y q)...τk−2(y q)==τk−1(y i)y pτ(y p)(τ(y−1i)τ(y q))τ2(y q)...τk−2(y q)==τk−1(y i)y pτ(y p)τ2(y p)(τ2(y−1i)τ2(y q))τ3(y q)...τk−2(y q)==τk−1(y i)y pτ(y p)τ2(y p)τ3(y p)...τk−1(y p)τk−1(y−1i)=1 as required.5Theorem1.5follows from Proposition1.4and Lemma1.7.In particular, Theorem1.3follows because for k=2we have y iτ(y i)=1soτ(y i)=y−1i.In the next section,we show that the recent result of Kamada[Ka-2]allows us to extend Theorem1.5to any closed oriented n-manifold in S n+2.Similarly Theorem1.3can be generalized to any closed unoriented n-manifold in S n+2. 2Higher dimensional case:M n⊂S n+2.We can extend our results to the case of a closed n-manifold M in S n+2(not necessary oriented or connected).First we need an existence of the Wirtinger type presentation of the fundamental group of S n+2−M for oriented M, described in[Ka-2]and its variant for unoriented M.Theorem2.1([Kamada])Let p:R n+2→R n+1be a projection and M be in general position with respect to the projection.We can think that the base point is very high above R n+1 (say at∞∈S n+2).We have n−1dimensional strata being the closure of the(invisible)set{x∈M|∃y∈M,p(x)=p(y)and x is the lower point}. These strata cut M into n-dimensional regions(“visibility regions”).LetΠM denote the fundamental group of S n+2−M(ΠM=π1(S n+2−M)).(1)Assume that M is oriented.ThenΠM has the following Wirtinger type presentation.Generators of the group,x1,x2,...,x n,correspond to regions of M(and can be visual-ize by joining the base point b to any meridian of the region;meridians are oriented so elements ofΠM are well defined).Relations of the group correspond to double point strata and are of the form x i x p x−1i x−1q or x i x−1p x−1i x q,depending on orientations of p(M)at double point strata.Here x i corresponds to the higher region of the projection and x p,x q to lower regions.(2)Consider now M not necessary oriented.Let generators,x1,x2,...,x nbe chosen as before(one for each region).Because a region may be unorientable we have to make choices here.However,modulo relations x2i=1(for1≤i≤n),the group has Wirtinger type presentation.That is:ΠM/(x2i)has a presentation as in(1)with additional relations x2i=1.6In the unoriented case meridians have no preferred orientation so only the branched double cover is uniquely defined,so we can interpret the core group of the embedded unoriented codimension two submanifold of S n+2.Kamada’s theorem allows us to generalize Theorems1.3and1.5.The proof,as before,bases on Lemma1.7(k=2yields an unoriented case),so we omit it.Let p:R n+2→R n+1be a projection and M be an n-dimensional closed submanifold of R n+2(and S n+2=R n+2∪∞)being in general position with respect to the ing notation of Theorem2.1we have. Theorem2.2Let M(k)denote the cyclic branched regular k-covering of S n+2with an oriented manifold M as a branched set,andΠ(k)Mits funda-mental group.ThenΠ(k)M ∗Z∗...∗Zk−1=Π(k)M⊔S nhas the following quandletype description.To every region corresponds k−1generatorsτj(y i)(0≤j<k−1)and to each double point strata correspond k−1relationsτj(y pτ(y i)τ(y−1q)y−1i)or τj(y iτ(y p)τ(y−1i)y−1q)depending on a local orientation.Theorem2.3Let M(2)denote the double branched regular2-covering of S n+2with an unoriented(possibly not orientable)manifold M as a branchedset.Thenπ1(M(2))∗Z=Π(2)M⊔S n has the following core(quandle type)de-scription.Generators of the group,y1,y2,...,y n,correspond to regions of M.Relations of the group correspond to double point strata and are of the form y i y−1p y i y−1q. In the case of a3-manifold in S5,one can check invariantness of our groups, defined combinatorically,using the Roseman moves[Ros-1].3Acknowledgment.We would like to thank Prof.L.Kauffman for his encouragement to work on the above problems after the talk of the second author at the Knots in Hellas98conference.7References[CJKLS]J.S.Carter, D.Jelsovsky,S.Kamada,ngford,M.Saito, Quandle cohomology and state-sum invariants of knottedcurves and surfaces,Preprint,April1999.[C-S-1]J.S.Carter,M.Saito,Knot diagrams and braid theories in di-mension4,Real and complex singularities(S˜a o Carlos,1994),112–147,Pitman Res.Notes Math.Ser.,333,Longman,Har-low,1995.[C-S-2]J.S.Carter,M.Saito,Knotted surfaces and their diagrams, Mathematical Surveys and Monographs,55,American Math-ematical Society,Providence,RI,1998.xii+258pp.[F-R]R.Fenn,C.Rourke,Racks and links in codimension two,J.Knot Theory Ramifications,1(4),1992,343–406.[Joy] D.Joyce,A classifying invariant of knots:the knot quandle, Jour.Pure Appl.Alg.,23(1982),37-65.[Ka-1]S.Kamada,A characterization of groups of closed orientable surfaces in4-space,Topology33(1),1994,113–122.[Ka-2]S.Kamada,Wirtinger presentations for higher dimensional manifold knots obtained from diagrams,preprint,1999. [Pr]J.H.Przytycki,3-coloring and other elementary invariants of knots,Banach Center Publications,Vol.42,Knot Theory,1998,275-295.[Ros-1] D.Roseman,Reidemeister-type moves for surfaces in four-dimensional space,Banach Center Publications,Vol.42,KnotTheory,1998,347-380.[Ros-2] D.Roseman,Projections of codimension two embeddings, Knots in Hellas’98,Series on Knots and Everything,Vol.24,Proceedings of the International Conference on Knot Theoryand its Ramifications,Delphi1998,World Scientific,Septem-ber2000,380-410.8[Ro]W.Rosicki,Some simple invariants of the position of a surface in R4,Bull.Polish Acad.Sci.Math.46(4),1998,335–344.[Wa]M.Wada,Group invariants of links,Topology31(2),1992, 399–406.[Ya-1]T.Yajima,On the fundamental groups of knotted2-manifolds in the4-space.J.Math.Osaka City Univ.13,1962,63–71. [Ya-2]T.Yajima,Wirtinger presentations of knot groups,Proc.Japan Acad.46(1970),no.10,suppl.to461970no.9,997–1000.[Yo]K.Yoshikawa,The order of a meridian of unknotted Klein bottle,Proc.Amer.Math.Soc.126,1998,3727-3731.J´o zef H.PrzytyckiDepartment of Mathematics,George Washington Universityand University of Maryland,College Park.e-mail:przytyck@Witold RosickiInstitute of Mathematics,Gda´n sk Universitye-mail:wrosicki@math.univ.gda.pl9。