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(1)l. '. Singularities of n-manifold in (n+2)-space '. '. '. Hirotaro TOKUDA. DePartment of Mathematics . (Received Aug. 31, 1966). -tt ;tIntroduction t. '. By the concept of V.KA.M. GuGENHEiM [2, 3]*, R.H. Fox and J.W･ MiLNoR gave the definition of the non-singular of the singularities of a 2bMya"tikfeOmidseM/v2esin[14r:pan. ilf-03i]d. M` and that of the siice type of the singularities. Belowaregiventhedetails:- ' .. ' Let M2 be an oriented 2-manifold imbedded as a subcomplex in an oriented. combinatorial 4-manifold M` such that OM2cOM4 and intM2cintM`. Let N denote the star neighborhood in M` of interior point x of M2, and set. S==ON and K=M2A6N, and the.oriented knot type rc of the imbedding of K in S will be called the singularity of the imbedding at x. When rc is of trivia! type, we may define that x is･a non-singular Point, that the singularity is O or that M is loc,ally flat at x.. rc is called a slice hnot if and only if it spans a non-singular 2-disk which. lies completely within one of the two half-spaces bounded by a hyperplane f. ofWethe euclidean 4-space R4. ･ ,: .･ . have generaliezed these defiriitions into the ciosedi combinatorial. n-. manifold in euclidean (n+2)-space Rn"2 and proved the following three theo-. rems :- . t"-. t /ttl.tt.. THEoREM 1. Let M be an oriented closed n-manifold (n l2) in an oriented ezaclidean (n+2)-sPace R. And let L, K be the simPlicial Partitions of R, M, resPectively, such that K is a subconzPlex of L. ･Set an r-simPlex of Kby AT=v,v, ･･･ v.+, (r=1, 2, ;･･,n-2) and 'the (r-1)-simPlex viv2･･･vK-ivK+i･･･ vr+i by A?K-)'. ij the singularity of M imbedded' at A?Kn)i is O, then the same can b2 Said Tof HEAorriEM 2. If, untier the' firgt halyr oj) the'hypothesis of theor2m i, the. si"gzalarity of Min R imbedded at Ar'i is slice tNPe, then the same can b2. said of Ar. . , THEoREM 3. The converses of theorem 1 and 2 are not true.. pape*r.The. tt t tttt. bracketed number in.dicates the num6er of refernce given at the end of this. '.

(2) H. ToKuDA. 36. 1. Singularities of r-simplex of Min R Let R be an oriented euclidean (n+2)-space with L as its simplicial parti-. tion a.nd let M be an oriented closed n-manifold in R (nl2) with K as its simplicial partition such that K is a subcomplex of L. Let A?k;=viv2･･･DK･･･v. (r =1, 2, ･･･,n-2) be (r-1)-simplex of K, where v'. K indicates that vK was taken off from the set {vi, v,, ･･･,v,+,}. Then Ar=vKA?K-)'. isar-simplex of K. i. Polyhedra hav:/ng isomorphic partitions will be said to be eqztivalent [2: p. 30]. An n-element will be defined as a polyhedron which is equivalent to an n-simplex [2: p. 30]. Then the polyhedron 11le(Ar, L)l of the link of A" in L is (n-r+1)-sphere, and 11le(Ar, K)l of the link of Ar in K is '(n-r-1)-sphere. The polyhedron. ". -. lst(vK, 1le(vK+i, L)l of the star of vK in the link of vK-, in L is (n+1)-element.. Since R is oriented, [st(vKHi, L)l has the orientation of R. Since the orientation of 11k(vK-i, L)1 is induced by that of lst(vK--,, L)l and the orientation of ]1le(vK-ivK, L)1=l1le(vK, lk(vKmi, L))1 is induced by that of l1le(vK-,, L)l. There-. fore, llle(vK-ivK, L)l has the orientation inherited from that of R. And the same can be said of the orientation of 11le(vK-ivK,K)1. That is the reason why' the orientations of 11le(AT, L)land1lk(Ar, K)l are induced from those of R and M, respectively (refer to p. 38: Fig. 1.).. Let S"K"i==11k(vK, L)1, and it will be the oriented (n+1)-sphere,. SeK-,r'2==llk(A?k)`,L)1, ,, theoriented(n-r+2)-sphere,. sn-r+i=11k(Ar,L)[, ,, theoriented(n-r+1-)sphere, and let. Tk-i==Ilk(vK, K)I, and it will be the oriented (n-1)-sphere,. T7K-)r=llle(A?kS,K)I, ･,, theoriented(n-r)-sphere,. '. Tn-r-i==llk(Ar,K)1, ,, theoriented(n-r-1)-sphere. Then the sphere pair (Tn-"',S"-r"i) is an (n-r-1,n-r+1)-knot. The singularity of M in R imbedded at Ar is given by the knot type (Tn-T-i, Sn-r+i)* of (Tn-r-i, sn-rbi) [4: p. 26]. The knottype is clearly a combinatOrial invari-. ". ant of R, M, A. An r-sphere Sr is called flat in Sr+2 if Sr is congruent [3: p. 129] in Sr+2 to a boundary of (r+1)-simplex [3: pp. 129-130]. DEFINITIoN 1. So long as Sr is flat in S'"2, we can de.fine that the singularity of M in R imbedded at Ar is O, or that Ar is a non-singular simPlex. DEFiNITIoN 2. I7Vhen M is closed, Mis called locally .flat if the singularitN. of Min Rimbedded at eaeh r-simPlex of K (r=O,1,･･･,n-2) is O. On the other hand, when M has a boundary, M is called locally fla4 so long as every. r-simPlex of intK is non-singular in the above sense and that of OK nonsingular in the sense of Pzanctured knot [4: p. 26].. j.

(3) l. Singularities of n-manifoJd in (n+2>-space. 37. DEFINITIoN 3. .l[f there is a locallyflat (lz-r)-element B"-r lying comPletely. on the outside or inside of S"'r+i in Sznvr+2 with Tn-'-i as the boundary, then the singularity of Ar is called a silice tyPe. .. 2. Lemmas and their proof In order to prove the first two' of these theorems, we shall prove the. fo!lowing lemmas :- ' LEMMA 1. Sn-r+i==Sz+iAS?Ke)r+2==S?+iASs+iA･t･ASge++'l. LEMMA 2.. (i) Let E?-r+2=lst(vK,S?K-)r+2)l, Es'r+2 =cl(S?K-)r+2-E?-r+2), then E,n.r,+2 (iim--. 1, 2) are each (n-r+2)-element. (ii) S?Kh)r'2 is divided into two (n-r+2)-elements-E?-r'2 and Esm'+2 by S".+i. LEMMA 3. T?K-)r is divided into two (n-r)-elements-Fi"-r=st(vK, Tei)')I and Fs-e =cl(TeKrv)7-Fin-r) by Sk+i, so that intF,n･-rAintE:er+2 and 6F,n-r==OFs-r c OE?'r+2 == 6Es-r+2.. REMARK: Let Ar (r=:1, 2, ･･･,n) be the r-simPlex of an n-manijCold M in euclidean (n+2)-sPace and let P,, P, be two Points of int Ar. Th2n their singularities will be one and the same [4: p. 27. proposition 3].. The following definition of the link must be kept in mind: The linle lk(A, L) of any simplex A in L is a set of simplices of L, which are faces opposite to A in some simplex of st(A, L) [2: p. 30].. PRooF of Lemma 1. We will prove the following to solve lemma1:,. (a) lk(A', L)clk(vK, L)Alle(AfKfi)',L) ,' (b) 1le(vK, L)Alk(A?k',i, L)clle(/lr, L). PRooF of (a). If B is a P-simplex of lk(A", L), B,vK and A?Km)i are each in the general position. Then BArK')i is a (P+r)-simplex of st(Alk-)i,L) and BvK is a (P+1)-simplex of st(vK, L), where the product meansjoin [2:p. 35]. Thus. Bc1le(Af.-,',L) and BG!k(vK,L). Therefore, B c 1le(A?Knt)', L) A 1le(vK, L)･. PRooF of (b). If C is a q-simplex of 1le(A?Km)i,L)A1le(vK, L),then C is con-. tained in a (q+r+1)-simplex vKA?K")iC of st(A', L). Thus C is contained in 1fe(Ar, L). That is 1le(ArK-)t, L) A1le(v., L) c lk(AT, L). Repeating this operation on 1le(ATK-)`,L),. sn-r+i = sr+i A Ss+i A ･･･ A S ge+'i･.

(4) 38 H. ToKuDA PRooF of Lemma 2.. PRooF of (i). Since S?Ks'"2 is a (n-r+2)-manifold by the definition of manifold, E?-r+2 is in (n-r+2)-element. ES"r"2 is (n-r+2)-element by J.W. ALExANDER's lemma [5: p. 316, lemma [14: 26]]: The residue of each (n:O)-cluster of the n-sPhere is n-element. Proof of (ii). From the constructions of E,n-r+2 and E,n-r+2, intE?-r+2c. intst(vK, L) and OE?'r+2=OEK-7+2cSk+i.. From J.W. ALExANDER's lemma, above mentioned, RVoo is divided into Ei""2= st(vK, L) and E2n+2=cl(RVoo-E?+2) by Sr'i. Since ArK-,i is in Sxt+i= Ost(vK, L), ES"2 contains the interior point P of Es'r"2. If the point q of intEs-r+2 is intEr+2 or OE?"2, any polygonal arc Pq in S?K-)"2 has at least a common point r of intEs"+2 with S"K'`. Then r is in both S?i)r"2 and Sn.+i which contradict lemma 1. Thus all theinternal points of E,"-r'2 are in E,n+2.. as. .. And all this will bring out the following proof :-int E℃'r+2 c int E?+2, int Es-r+2. cint Es+2 and OE?･-r+2 =OEc'r+2cSnK+i. i sn-r+l. L. E2-r+2. E ",+i. ve' Ein-r+2 s?K-)"2 A2-,-:x Ar Nxv, E:'i sn,:i. Fig. 1. Lemmas 1-2. PRooF of Lemma 2. Let F?-r-i==ist(vK, Tn-"i)l. Fc-r-i == cl(Tlt--r-imF?-r-i). Then intF:-r'icintE#･-r+i (i=1, 2) and OF7･-rwicaE:･z-r+i by the both En,'r+i. L. 9. and Fy.-r-i (i=1,2). And the above-mentioned proof of lemma 2 (ii) will lead to the conclusion that T?i)' is divided into two (n-r)-elements-F7-r and F℃-r by SnK+i.. This will be the complete proof of lemma 3, I. 3. Proof of Theorems PRooF of Theorem 1. If the singularity of M imbedded at ArK')i is O, the (n-r+1)-element Bn-'+i such that OB"-r+i=T7W will be found in Slk-,r+2 by definition 1. We may regard that Bn'r"' is the polyhedron of a subcomplex' of L, because R is combinatorial. We may choose the following:-. j. :. '. l. Li:this is.the subdivision of L so that it is a subcomplex common to 1. l i i. I.

(5) l. L. Singularities of n-rpanifold in (n+2)-space 39. both the partitions of R and Bn-r+i [2: p. '34, 2.21].. K/:'this is the subdivision of K so that it is a subcomplex of Li. And let' St?K-)r+2=I1fe(A?k)i,Lf)1, Stn-r+i==i1fe(Ar,Li)1, Tilk-,r==1lk(Alk-,i,Ki)I, Tm-r-i='11fe(Ar,Ki)l.. Since B"-r+i is (n-r+1)-element, it is the (n-r+1)-manifold with boundary. Since vK is in the boundary T7i,' of Bn-'+i, llle(vK,B"-r+iAL')i is (n-r)-ele-. ment Bn-r by [2: p. 31]. From Bn"r+i and S'n-r+i, Bn-rcSm-r÷i and 6Bn-r == 11le(vK, T/n-r)1. By lemma 3, l1le(vK, Ti?.-)r)1== Tin-r-i.. The knot type of (lk(Ar, K/), lh(Ar,Lt)) will be denoted by (1le(Ar,Ki), 1le(Ar, L,))... Judging from the above mentioned, we conclude that the (n-r)-element Bn'r such'that OBn-r =Tm=r-i exists in S/nmr+i. Then (1le(vK, Ti?K-)r), 1le(vK, Si?K-,r+2)). is O,. and (1fe(vK, Ti?i)r), 1le(vK, S/?K-)r+2))* =(1fe(vK, T?jsr), lk(vK, S?K-)r+2))*. [3:pp.134-135]. , .. (1le(vK,T?i)'),1le(vK,S?K-)r+2))=(lk(Ar,K), lk(A',L)) by lemma 1. Thus. (1le(Ar,K),lk(Ar,L))=(Tn-r-i,Sn-r+i)istrivial. '･ All this has proved Theorem 1 completely.. ,n-r+2 SCK) '. ,n-r+l '. 's ･ ....,r 1'') ?. .Aw. Bts B n-r+1. l. T 'n-r-Fl. //. Tln-r (K)i. i Fig. 2.. Theorem 1.. PRooF of Theorem This 2. is analogus to that of lemma 1, except that BnLr+i is completely in H. (i. e. the outside) or H- (i. e. the inside) of S?K--Tt)2 in S?K=ri+,k), where Sn(K-!+i,3m i's llk(ATkai,io =viv2 ･･･ vK-iDK ･･･ v.+i, L)l -. .. By the common subdivision of Bn-r÷i and L, and by the construction of. Bn-r, , ..

(6) H. ToKuDA. 40. Bn-r = Bn-r+1 A SInK+1 c Sln-r÷2.. On the other hand, OBn-r ,,. T/n-r-1 = Tn-r A SinN . Therefore, (Tin-r-i, S/n-r÷i)* =(Tn-r-i, Sn-r+i)* is the slice. type.. All this wi!1 be the complete proof of Theorem 2.. n-r+3. SCK--LK). l)sxlll Bn-r+I. sc. l}ii,IIEinE', illZII.Ili.I,: .. T'n-r+I. × VK-l s n-r+2 {K- l). //. Fig. 3. Theorem 2.. PRooF of Theorem 3. This theorem will emerge when we construct the product of a certain pair of knots [4: pp. 28-29, 3.1].. (A) With regard to the fact that thesingularity is O, we shall construct. an n-sphere S" in euclidean (n+2)-space Rn'2 marked by K and L as the partitions of S" and Rn+2, respectively, andKbeinga subcomplex of L. And Sn satisfies the following conditions*:. (1) In a euclidean (n+1)-space R"+icRn+2, SrHi is a locally fiat combinatorial (n-1)-sphere, which, however, is non-trivial in R"+i. Such a knot may be constructed in this way: Let Ei be a 1-element, in euclidean half 3-space H3, with two points in R2(cOH3) as its boundary and let (Ei, H3) be a non-trivialpunctured knot. If we rotate H3 around R2, a locally fiat non-trivial (2, 4)-knot will be brought out. Repeat this way, and the required (n-1, n+1)-knot will be established.. ". ". (2) Choose two points vi in R""2-R""i, such that they lie Qpposite to each other with R""i between them. (3) Let S?'i joint with vi, and S"=S?-'(v,Vv,) is an n-sphere with K ==(the partition of S?'i)(v,Vv,) as its partition. It is clear, therefore, that the non-trivial singularities of K in L are only vi and v2.. (4) Choose a vertex v, of K in lk(v,, K) and L which, as a partition of * In this case, let i=1, 2.. :. l.

(7) Singularities of n-manifold in -(n+2)-space. 41. Rn+2, contains K, and the singularity of S" in Rn+2 at viv, is O by All the above-mentioned-(1), (2), (3) and (4)-will amply prove with reference to case (A).. '. Theorem L Theorem 3 '. Vl Rn{-2. x. n-i X. S3 'X Sn. /. / t'. --- S-x. -f. Rn+l. ×. g. ve N × xs n-2. ×. x. ,-i g'. NSi S. X .s. / /. /. /. /. ×. 7. V2 Fig. 4. Theorem 3. (B) The same may be said of the slice type, so long as we can bring it out in some way other than rotation, for, different from what was made in case (A), the slice (n-1, n+1)-knot, which is non-trivial, cannot be produced by. means of rotation.. In this way Theorem 3 has been completely proved.. References 1) R.H. Fox and J.W. MiLNoR: Singularities of 2-spheres in 4-space and cobordism of knots. Unpublished. ･2) V.K.A.M. GuGENHEiM: Piecewise linear isotopy and embedding of elements and spheres (1). Proc. London Math. Soc., 3 (1951).. 3) (11): ibid.3 (1951).. 4) H.ToKuDA: Singularities of n-spheres in (n+2)-space. The Yokohama Math.. Jour. 11 (1963). -. 5) J.W. ALExANDER: Thecombinatorialtheoryofcomplex. Ann.ofMath. 31 (1930).. t.

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