IIIVERTISN’G AND MONOTONE PROPERTIES OF COMPLEXES Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY SHASHICHAND FATEHCHAND KAPOOR 196,7 This is to certify that the thesis entitled Inverting and Monotone PrOperties of Complexes presented by Shashichand Fatehchand Kapoor has been accepted towards fulfillment of the requirements for 211.12 .___ degree inmatics DMeSQDtGMber 6. 19é7 0-169 Mnjor professor P. H. Doyle 07 . I W ‘WB ,5 ‘I'FBRARV I ‘ “' .1“, "' QT" "1 J .LIIIDau t s.-- ljnrve*' \ ‘. . .5!" 1' ‘d' ABSTRACT INVERTING AND MONOTONE PROPERTIES OF COMPLEXES by Shashichand Fatehchand Kapoor Doyle and Hocking introduced the concept of inverti- bility for topological spaces and then applied this idea to finite geometric simplicial complexes. Characterizations of l- and 2-complexes with a single invert point were given by Doyle and Klassen respectively. Hocking proved that if K is a complex with O s dim 1(K) s dim K, then K is a multiple suspension. In Chapter II we show that if a complex K has dim {I(K)} 2 1, then CI(K) = I(K). For a complex K with I(K) f C or SO and p e I(K) we show that there exists an inverting homeomorphism which fixes p and that K-p Ill-3 . w 18 an open monotone union l IZLi where each 1L1 i=1 Lkr>x El. For products of complexes it is shown that if K1 and K2 are non-degenerate connected complexes and I(K13 = SO. For a l-complex, we prove that this is true, and for 2- and 3—complexes we get the invert set as a O-sphere if the complex has two orbits under isotopy. The uniqueness of the open cone neighborhood is used to show that local homology groups are invariant under trian- gulations of any complex. For any complex K with I(K) = {p}, we prove that if I(Af(K)) % SO, then dim {I(QI(K)) } 2 2. A result of Doyle on suspension rings in a double suspension is generalized to show that for any complex K, I(4fk(Kl) 2,8k-l for k = 1,2,3, In the last chapter we introduce the concept of an expanding n-star graph E(n) as a monotone union of star graphs and show that all such graphs can be embedded in a plane. This concept suggests a possible generalization of the self—avoiding walks discussed by Kesten and generalizes a result of Doyle on complexes which are monotone unions of l—cells. INVERTING AND MONOTONE PROPERTIES OF COMPLEXES By Shashichand Fatehchand Kapoor A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1967 a wage-'3‘» '3«fi's€ ACKNOWLEDGMENTS I am very grateful to Professor P. H. Doyle for his helpful guidance throughout the preparation of this thesis. His comments, suggestions, the long discussions we had at hours which must have been surely inconvenient to him and his eagerness to lend a helpful and understanding ear to my questions are most sincerely appreciated. By their kind words and sincere concern, many friends and colleagues have made my stay in this country and at this school a most enjoyable, satisfying and meaningful experience. I take this unique opportunity of humbly ex— pressing my gratitude to all of them and conveying my sin- cere thanks. ii To Bibiji and Bowji iii CONTENTS Chapter I Page I. INTRODUCTION. . . . . . . . . . 1 II. GENERAL RESULTS. . . . . . . . . 9 III. ORBITS AND BROUWER PROPERTY. . . . . 23 IV. SUSPENSIONS . . . . . . . . . . 31 V. AN APPLICATION TO GRAPHS. . . . . . H7 BIBLIOGRAPHY. . . . . . . . . . . . . . 5% iv LIST OF FIGURES Figure Page 2.1 . . . . . . . . . . . . 9 2.2 . . . . . . . . . . . . 1H 4.1 . . . . . . . . . . . . 3H 5.1 . . . . . . . . . . . . 1+8 5.2 .‘ . . . . . . . . . . . H9 CHAPTER I INTRODUCTION The concept of invertible spaces was introduced by Doyle and Hocking in [:3]. This lead to the investigation of such concepts as continuous invertibility, dimensional invertibility and local invertibility in [u l, [ 5] and [(5]. These papers discussed the above concepts with ref- erence to general topological spaces. For manifolds they gave rise to some very interesting and useful results. We give below the relevant basic definitions and results. Let 'X be a topological space. The symbol ‘3*(X) denotes the group of all homeomorphisms of X onto itself and ’5}(X) denotes the subgroup of '1¥(X) consisting of all maps in 23%(X) which are isotopic to the identity map on X. D i i Let p E X. Then p is an 1ng3:§_pgin£ of X if and only if for each open neighborhood U of p there exists h e MX) such that h(X-U) c: U. Here h is an inygrting_map for U. The collec- tion of all invert points of X is called the in1e11_1et of X and is denoted by 1(X). X is called inygztible if and only if I(X) = X. Wm Let p e X. Then p is a W- xext_ng1nt of X if and only if for each open neighborhood U of p there exists g e f;(X) such that g(X-U) c U. The set of all continuous invert points of X is called the continuous Luger; set of X and is denoted by CI(X). Clearly, CI(XJ c I(X). X is said to be QQQLlflr uggsly igyeztible if and only if CI(X) = X. In [ll] Doyle discussed the invert set in a finite geometric simplicial complex. He proved that if K is a complex, then I(K) carries subcomplexes of each triangu- lation of K. In other words, I(K) is invariant under triangulations of K in this sense. He also showed that if K is a complex, then I(K) is null, a point, or a finite simplicial sphere. The next two theorems give char- acterizations of l- and 2-complexes with a single invert point. Ekugugijgl_12gylel Let K be a l-complex. Then I(K) = {p} if and only if K is a set of r (22) simple closed curves meeting in p but otherwise disjoint in pairs (an r-leafed rose). Theorem 1,2 SKIEESEDZ Let K be a 2-complex. Then I(K) = {p} if and only if .. m 2 n 1 K-(Iglci)U(jL=Jisj)’ 3 where (1) CE is a 2-cell, a 2-sphere, a pinched annulus or a pinched torus for l s i s m such that C: n CE = {p} or a union of l-spheres containing p for all s f t and l l s s s m, l s t s m; and (ii) 83 is a l-sphere which is disjoint from for l s j s n. In [12] Klassen gave the characterization of a l-complex with a O-sphere as its invert set. For the pur- pose of simplicity, Sn will denote an n-sphere for n = 0,1,2, --- and .45k(K) will denote the k-fold suspen- sion of K with 451(K) written as 400. The cardinal— ity of a set A will be written as |A|. T m . K se Let K be a connected l-com- plex. Then I(K) = S0 if and only if K =jAI(F) where F is a set of finite number of points with IFI f 2. In [$9] Hocking generalized a result of Klassen and proved the following: Theprem_lifi_iflcskingl A complex K is a suspension if and only if I(K) contains a O-sphere. The next result discusses complexes K with dim I(K) = dim K. I+ Th em . Let K be an n-complex where n 2 1. Then I(K) = Sn if and only-if K 2 Sn. Proof. Klassen proved the result for n = l in [12]. Moreover, if K = Sn, then I(K) = Sn (See [22] and I 3]). So let K be an n-complex with I(K) = Sn. Then Sn is a subcomplex of K. Let p 6 Sn such that p 6 Int on where on is a principal n-simplex in Sn. Let U be an open neighborhood of p in on and h the corresponding inverting map such that h(K-Ul c U. Now U 2 ED and it can be so arranged that K-U 2 En. Then K is an n-manifold. Using the characterizations in [:2] and [:3], we get K 2 Sn. In a recent unpublished work, Hocking proved the following result which shows that all complexes K with O s dim I(K) s dim K are multiple suspensions. First we state a lemma whose proof is omitted. Lemma 1.6 Let Ak A; be a k-simplex in the barycentric subdivision of Ak. be a k—simplex in a complex K and let T Then Lk(A§,K') = Lk(Ak,K). T . H in If the n-complex Kn has T I(K) = Sk, O s k s n, then Kn = 4fk+l(L)- P1991. Let Ak be a principal k-simplex in 1(Kn). k Choose p 5 Int A such that p lies interior to a k-sim- plex in each barycentric subdivision of Ak. This is pos- 5 sible since a barycentric subdivision introduces finitely many points of the subdivision leading to a countable set of points by successive barycentric subdivisions. Let UO be a closed neighborhood of p in Int Ak such that the boundary of U0 relative to Ak is Bd UO = k-l~ k—l k-l s and U0 = poBd UO = poS . Now poS °Lk(Ak,Kn) is a closed neighborhood of p in Kn. Choose q 6 I(Kn)-Ak. Then there exists hO e 3*(Kn) such that ho(q) = p and hO ( Kn - Int (posk‘leLk(Ak,Kn))) c Int ( pcSk—loLk(Ak,Kn)) This may be rewritten as h;1(p) = q and Kn-Int < poSk-luLkIAk,Kn;) hgl (Int ( pesk‘1.Lk(Ak,Kn))) k Passing to the barycentric subdivision, let p 6 Al 0 where A? is a k-simplex in K'. Keeping p and Bd Ak pointwise fixed, shrink 81‘"1 to lie in Int A? and then Ul = posi'l is a closed neighborhood of p in A; and poSE-IULK(AE,K') is a closed neighborhood of p in Kn. By Lemma 1.6, puSE—lULk(A§,K') g peSk'loLk(Ak,Kn). Also, there exists hl e 31(Kn) such that hl(q) = p and k-l ' k hl (Kn-Int (p051 oLk(Al,K'))) c Int (p.s11<-1.Lk(A1{,K'>) . or hil(p) = q and 6 Kn-Int ( poS§-loLk(A§,K')) -l k-l k Let gl 6 3*(Kn) such that g1 < pesk'loLk(Ak,Kn)) poSE-loLk(A§,K'). Consider hil gl ( Int ( poSk-loLk(Ak,Knll) as an open cone neighborhood of q. By construction, we get Q n _ -l k-l k n K _p — lIzIO hi gi (Int (p08 oLk(A ,K D) . By uniqueness of the open cone neighborhood (see [13]) we get T _ Kn-p = Int ('poSk loLk(Ak,KnZ) . Then Kn is the 1-point compactification of k'loLk(Ak,Kn), or Kn is homeomorphic to k—l poS SOoS oLk(Ak,Kn). The induction on k is now obvious, and we get r K = SkoLk(Ak,Kn) =Jk+l(L), where L = Lk(Ak,Kn) is an (n-k—l)—complex. Also, Theorem l.% and Theorem 1.5 correspond to k = O and n respec- tively. This completes the proof. The next theorem is due to Klassen (Theorem H.l of [12]). We present a simplified version of the proof. Th . K Let K be an n-complex with I(K) = {p}. Then p e CI(K) and consequently CI(K)==I(K). 7 mg. Let V = StL p be the open star of p. Let h E 3¥(K) be an inverting map for V. Then h(K-V) c V' and h(p) = p since I(K) = {p}. Also, K- ’c h(V) implies that h'l(K-V) c V. By uniform con- tinuity and h(p) = p, there exists a neighborhood U of p such that U c V' and h(U) : V. Let r E 3*(K) such that r is the identity out- side IV and rh'l(K«-V) c U. Then hrh'l(K-V) c h(U)<: V. Now r can be accomplished by an isotopy rt, 0 s t s l, -1 such that = r. Then gt = hrth is an inverting map r1 for V with gO = id and gl(K-V) c V. Thus gl e‘§;(K) and p e CI(K). Since CI(K) c I(K), we get CI(K) = I(K). Remark. Let K be any complex with I(K) = {p,q}. We assert that CI(K) = C. If not, let p e CI(K) and U be any open neighborhood of p which excludes q. There exists an inverting map g E €;(K) such that g(q) 6 U. But every point of the arc gt(q), O s t s l, is an in- vert point of K. This is a contradiction. However, Hocking proved the following theorem in [$9]. Thggzgm 1,9 QHnging2 Let K be a complex such that dim I(K) 2 l and CI(K) f O. Then CI(K) = I(K). Hocking conjectured in [‘9] that dim I(K) 2 l is enough to imply CI(K) = I(K). We prove this in the next chapter. The next theorem is also due to Hocking. T O H ‘ For complexes. P and Q, let P =,4JIQ). If dim I(P) 2 1, then (i) I(Q) c Q n I(P) and (ii) CI(Q) c Q n CI(P). It was mentioned in.[ 9] that if equality could be proved in Theorem 1.10, other well known results may then be used with this to prove the Poincare Conjecture in die mension four. In other words, the concept of an invert set and some current problems in combinatorial topology are re- lated. Unless otherwise specified, we will follow the standard terminology of [10]. CHAPTER II GENERAL RESULTS Th§92§m_2&1 Let K be a complex with dim I(K) 2 1. Then CI(K) = I(K). Egggf. By Theorem l.% we can write K = 45(L) with p and q as the vertices of suspension. Let s 6 L n I(K) and h €-3¥(K) such that h(p) = 5. Let gt (0 s t s 1) be an isotOpy such that gth(p) moves away from s. This is possible since there is a product neigh- borhood of s in K. Let ft = h’1 gth. We use this to move p. Since fo = h’l(id)h = id and ft is a homeo- morphism, ft is isotopic to the identity map. Also h'lgth(p) f p. Figure 2.1 9 10 Let U be an open set in K containing p. For our purpose it will be enough to take U'c Stla. Also, Lk1> is bicollared in K. Choose a collar C of LkI) in Sty) such that U n C = O. Let ft 6 j;(K) such that itlis fixed outside St;>- C and moves p to ft(p) = p' with p' E U. Since K is a suspension, there exists at E :éiK) such that al(K-Stp)c V and at|Stp= id, where V is a sufficiently small open neighborhood of q such that V c: Stq. Let St E €(K) such that Btl (K-St q) = id and is such that it slides V away from q. This is done by arguments similar to that used above to construct ft’ Let Yt e ’9’“) such that y tI (Stp-C) = id and slides B1(V) inside C. Finally, let 5t 6 é;(K) be such that 6t|(K-Stpfl = id and 61(p') = p. Now define fit = bt‘vt St at ft’ 0 s t s 1. Then ¢o = id and ¢l(K-Stp)= blYl 51 cl fl (K—Stp) c olvl BI :11 (K-Stp) c bl'Yl 51 (V) c 61 (C) o c Stph This shows that Cl is the required inverting map for 9 Sty) and is isotopic to the identity map on K. Hence p e CI(K). By Theorem 1.9, CI(K) = I(K). This completes the proof. ll Tnggpgm 2,2 Let K be a complex with I(K) f'fl or SO. If p e I(K), then there is an inverting homeomorphism f E 3*(K) such that f(p) = p. 2199:. If I(K) = {p}, then every inverting homeo- morphism fixes p. So let dim I(K) = k 2 1. Without any loss of generality we may assume that p 6 Int ck, where 0k is a principal k-simplex in I(K). Let U be any k k open set containing p such that U n a = V c Int 0 . Choose q E V and p # q. Let W<: V, p z W, q E W such that an open set A in K containing q has A n ck = W and A.c U. Since q 6 I(K), there exists h €.§+(K) such that h(K- A) c A and q ,1! h(p) 6 W. Choose 8 6 3*(K) such that (i) gh(p) = p and (ii) g(A) c U. Define f = goh. Then f e §4(K). Also, f(K-U) = gh(K-U) c: gh(K-A) c g(A) c U and f(p) = gh(p) = p. CQIQJJQII 2,3 Let K be a complex with I(K) % U or SO. Let p e I(K) and U be an open set containing p. Then there exists an open set V": U and p 6 V such that some inverting homeomorphism fixes V pointwise. 2199:. As in the proof of Theorem 2.2, assume that p 6 Int ck. Since there is an inverting homeomorphism f which fixes p, by uniform continuity there exists a symmetric ball V in U, with p as center and such that f|V = id. l2 TDQQIQQ 2.h Let K be a complex with dim I(K) 2 2. Then I(K) is continuously w-homogeneously embedded in K. Proof. By Theorem 2.1, CI(K) = I(K). Let An:{al,a2’ coo, a and B n {bl’b ..., b 2’ n} be any two sets of distinct points in I(K). We can choose a triangulation T of K fine enough to ensure that there is a principal simplex o in I(K) such that (An u Bn) n Ste: 91. Let p 6 Int 0 c Sta. Since p e I(K), there exists f e.3$(K) such that f(K-Sto): St 0. Thus f(An) and f(Bn) are contained in Int 0. If n = l, we use Lemma 0 of [1.] to obtain g E j§(K) such that g|(K-—Int a) = id and gh(al)==f(bl). Define h = f'lgf. Then h 6 éiiK) and h(Al) = Bl' As induction hypothesis, assume that for all i such that l < i < n, there exists h e fi;(K) such that h(Ai) = Bi’ Let A.n = n-1 U an and En = n-l U bn' Then there exists h e jaéK) such that h(An-l) = Bn-l’ If h(an) = b we are done. Otherwise, let D be a n’ closed set containing Bn-l and if needed, attach a col- lar C to D. Now there exists an isotopy 9t which moves an to bn in o-(C u D) leaving D fixed. Then at = et-h is the required isotopy. 13 Remark. When dim I(K) = l, a similar result may be proved if we disregard the order of points in An and Bn. Theorem 2.5 Let K be a complex with I(K) f U or SO and p E I(K). Then K-p is an open T monotone union Qiruq, where 211 = LkIDX El. - i=1 Proof. Let U be an open cone neighborhood of p and C be any compact set in K-p. Then there exists an inverting homeomorphism h E 3*(K) such that h(C) c.U. Consider U = Lkgax [0,1) with kaax 0 identified with p. Then every compact set in K-p is contained in a pro- duct space Lklax E1. Thus K-p is the monotone union LNJUi, where u, g Lkpx El. i=1 We observe that if_ K is a l-complex with I(K) f U or SO and p e I(K), then K-p E FJcEl, where F is a finite set of points such that |F| = l for I(K) = S1 and |F| 2 2 for I(K) = {p}. If K is a 2-complex with I(K) f U or SO and p e I(K), then E1 x E1 if I(K) 32 K-p = l s B xEl if I(K) where B is a one point union of b (23) semi-open inter- vals. If I(K) = {p}, the cases are more complicating in view of Theorem 1.2. It may be possible to show that K-p 2 leE1 where G is a graph. 1 1% Let G be a graph and consider U = LE)U1 where ‘U% E GJCEl. The l-point compactification of U gives only one spaCe K which is invertible at a point p. Thus mono- tone union property gives rise to a unique space in this sense and the failure of this property may not yield unique— ness. The following example is illustrative of the first part and serves as a counter example for many intuitive con- jectures for complexes with a single point of invertibility. Example. Let G be a graph which is a one point union of a l-sphere and an open interval. Then the one point com- pactification of L§)ui, where 1(1 3 anEl and L is a union of two l-spheris joined by an arc, is a pinched torus with a spanning disk. If we call this complex K then I(K) = {p}. We note that K-p g G)(El and K is not a pinched suspension. G: r—O 1gp; GxEl Figure 2.2 15 If G is a connected graph, consider 211 2 GJcEl. We note that G can be embedded in E3. Let v be a vertex in G of maximum degree d. We claim that 211 can be embedded in a d-book if G is a tree. In order to see this, note that vertices in G of degree 2 present no problem. The same is true for the open end of a l-simplex. Since G is connected, let a vertex x of degree a be joined to a vertex y of degree B. Construct a- and B-books at x and y respectiVely. Let B s a. Since these books have one page in common, they can be embedded in an a-book. A repeated application of the same argument yields the result. Since GJcEl contains a copy of @211, we conclude that the monotone union @ui can be embedded in a d-book. If G is not a tree,1the number of pages in the book may have to be increased. In [12] Klassen proved that if K is a 2—complex with I(K) = {p}, then K g Lx[0,l] where LxO U Lxl U Mx [0,1] is identified with p and M is a finite set of points in a l-complex L. This leads to the following result. Exgpgsijjgn_2‘§ Let K be a l-complex with I(K) f'fl. Let F be a finite set of points with |F| = f. Then (a) I(K) = {p} if and only if K E FJ{[0,1] where f 2 2 and FxO U Fxl is identified with p, (b) I(K) = S0 if and only if K 2 F3c[O,l] where f 7! 2 and FxO and in are identified with 16 p and q respectively T 1 if and only if K = F]([O,l] where and (c) I(K) = S f = 1 and FxO U Fxl is identified with p or f = 2 and F)(O and F](l are identified with p and q respectively. Proof. Obvious from Theorems 1.1, 1.3 and 1.5. Similarly it is possible to write down the corre- sponding result for a 2-complex in view of the earlier theorems. So far it has not been possible to factor higher dimensional complexes with a non-empty invert set in this fashion. The aim of the last proposition is to exhibit a factorization with [0,1] as one of its factors, as com- pared to Theorem 2.5 in which K-p can be written as a monotone union UEHXi where 111 has a factor (0,1). i=1 Proposition 2,2 Let K be a l-complex such that I(K) = so. If I(K/I(K)) # {p}, then K = D1 (l-cell). Ezoofi. I(K) = so implies that K =,A((F) where F is a finite set of points with |F| # 2. Then K/I(K) is a rose with |F| leaves. [F| # 1 implies that I (K/l(K)) = {p}. Hence |F| = l and K = D1. Expansiiign_21§ Let K be a 2-complex such that I(K) = S31. If I(K/I(K)) 7" Ipl’ then K = D2 (2-cell). Proof, we have K = 452(F) where F is a finite set of points with |F| # 2. Clearly, |F| 2 3 implies that K/I(K) is a one point union of IFI 2-spheres. 17 Thus I (K/I(K)) = {p}. For [F] = 1, K = 120?) = D2 and I (K/I(K)) = I(SZ) = s2. Next we quote from [1.] a general theorem in this context. T Do Let K be an n-complex which is not a point and let I(K) flfl. If K/I(K) is invertible, then K is a sphere or a cell if K is a manifold or has a free (n-1)-face. In.[ 6] it was proved that if I(SlcT) % C, then I(S) xI(T) ; I(SxT), and that the product of two invert- ible spaces is either invertible or has empty invert set. This, when applied to complexes gives some interesting re- sults. For instance, if K1 and K2 are complexes such that I(Kl) # a, I(K2) y! p and I(leKZ) ;£ 93, then C 7! I(Kl)xI(K2) ; I(KIXK2)' If I(leK2) = (p,q), I(Kl) #,U, and I(K2) fyfl, then I(Kl) = {p} and I(K2) ={q}. We note that this may be vaccuously satisfied. Moreover, if I(Kl) 2,80 and I(KQ) 2 SO, then I(leKZ) ;! p implies that dim I(leKZ) 2 1. k1 k2 Let Kl = S , K2 = S where kl 2 1 and k2 2 1. kl+k2 S . Then we assert that and I(K2) = K Assume that Kl xK2 ,11 I(leKZ) = 325. Note that I(Kl) = K1 2. If I(leK2) 7525, then leK2 5; I(KIXK2)’ But I(leKg) ; leK2 implies that I(KIXK2) = leK2. By - kl+k2 Theorem 1.5 this implies that Klch2 = S This is a 18 contradiction. Tgoorgg 2,]0 {Doyloz Let K be an n-complex with dim I(K) = k 2 O and n > k. Then Hi(K) and ni(K) are trivial for O < i s k. Eroof, For the groups under consideration it is enough to consider continuous cycles or singular spheres that lie in the k-skeleton of, K. First note that their homotopy classes are all represented by maps having I(K) as carrier. Let 0k be a k—simplex of I(K). Since n > k, k k+1 ok+l 0 lies on the face of a (k+l)—simp1ex o If has more than one k-simplex in I(K)~ then use its bary- center to ensure that ok+l n I(K) = ok. This means that each map with I(K) as carrier can be homotoped away from I(K) leaving Int 0k uncovered. This completes the proof. Erooooition_2oll Let K1 and K2 be kl- and k2-complex— es respectively with dim I(KlJcKZ) = k 2 0. If k<K such that Ostsl and hl(lep2) =q. This shows that K1 and K2 are contractible. Since there is a retraction g of K onto Kllcp2, 'htog gives the required contraction of K into a point. For example, let Kl = fake 3-cell and K2 = 2-disk. Then leK2 is a 5-cell with I(KlJcKZ) = H—sphere. By the last theorem, K is contractible. The result of the 1 theorem is more effective in a negative sense. Thus,if P is a non-contractible complex such that it is a factor of another complex K. Then I(K) = U or SO. Bomork. Let Kn be an n-complex with dim I(Kn) = k. If k = n, then Kn E Agn(F) with |F| = 2; and if k = n-1, then Kn E jn(F) with |F| # 2, where F is a finite set of points. This follows from Theorems 1.5 and 1.7. When k = O, we can prove the following. EIQDQ§2L12n_Zill Let K be an n-complex and I(K) = SO. Then K =j(L) where I(L) = p or {p}. Proof, Under the hypothesis, K = 4f(L) by Theorem l.h. Assume that I(L) 2 So. Again by Theorem 1.H, 20 we get L =j(M). But then K = 1201) and I(K) con- tains at least a l-sphere, namely the suspension ring (Theorem h of [l.]). This is a contradiction. The next two results complete the discussion of 2- complexes in view of Theorems 1.2 and 1.5. A part of the result is obtainable from Theorem 1.7; but we present an intuitive and independent proof. Proposition 2.14 Let K be a 2-complex. Then I(K) = S1 if and only if K = A32(F), where F is a finite set of points with |F| % 2. m. If |F| = 2, then 1207‘) = 52 and we get an immediate contradiction by using Theorem 1.5. So let F be a finite set of points with |F| # 2. Since K = A¥2(F) is a double suspension, by Theorem M of [1 ], l S ; I(K) where S1 is the suspension ring. Clearly 2 I(K) # S2, otherwise K = S and we can write K = 4f2(F) where |F| = 2. Hence I(K) = 81. Let K be a 2—complex with I(K) = 81. By Theorem 1.H, K =.4/(L) where L is a l-complex and the vertices of suspension belong to I(K). Since I(K) is a l-sphere, there exist x and y in L such that {x,y} c I(K). A small product neighborhood of x in K is an n-book for some n. Since all points on the back of this n-book are in I(K), it follows that K is locally euclidean except at points of I(K) and hence L is locally euclidean except at x and y. This shows that {x,y} ;_I(L). We cannot 21 have I(L) as a l-sphere otherwise L, becomes a l-sphere by Theorem 1.5 and consequently K is a 2-sphere. Then I(L) = {x,y} implies that L =,2V(F), where F is a finite set of pOints with |F| # 2 (Theorem 1.3). This completes the proof. Proposition 2.15 Let K be a connected 2-complex. Then I(K) = S0 if and only if K = ‘f(L), where L is a l-com- ples which is not a suspension of a finite number of points. Eroof. If I(K) = S0 we can write K ="V(L) where L is a l-complex. Assume that I.=g‘f(FU where F is a finite set of points. If |F| = 2, then L = s1 and K =I‘f(L) = 82, whence I(K) = S2. If |F| # 2, by Theorem 2.1% we get I(K) = 81. This proves the necessary part of the theorem. For the sufficiency we note that if K =’4((L) where 1.2/fol), then I(K) #51 or 32. But K =/ k. Eroof. The proof is by induction on k. When k = —1, I(K) = p and dim {01(p)} 2 o. For k = o, I(K) is a point or a O-sphere. But p l I(K) implies that p 23 2% is not a singularity of K and dim {OI(p)} 2 1. Assume that the result is true for all k < m. Let K be a connected n-complex with dim {I(K)} = m, where m 2 1. Let p 2 I(K) and dim {01(p)} s m. Under some triangulation T of K, let CI(p) be written as a union of open simplices. Then L = OEIET' is a subcomplex of K under T and dim L s m. Now I(K) = Sm and each simplex of I(K) is principal. Also, Sm n L fi C. Let M = SIn U L be a subcomplex of K under T. Then dim M = m and Sm‘c:I(M). This implies that M 2 3m and L = E. This is a contradiction. Hence dim {01(p)} > m and the proof is complete. Proposition 3.3 Let K be a l-complex. Then (i) I(K) = Ip} implies that NOH(K) = NOI(K) = r-tl, where r 2 2 is the number of leaves in K. (ii) I(K) = 80 implies that NOI(K)=() and NOH(K) = f3-2, where f is the number of points over which K is a suspension and f # 2. (iii) I(K) = 81 implies that NOI(K) = NOH(K) = l. Proofi. Follows from the definitions of orbits and earlier theorems. Efigungflgfirurgigi Let K be a 2-complex with I(K) = {p}. Then (i) NOI(K) = 2 implies that K is a pinched torus and (ii) NOI(K) = 3 implies that K is one of the fol- lowing: one point union of two pinched tori, 25 two 2-spheres, a pinched torus and a 2-sphere, a 2-sphere and a l-sphere or a pinched torus and a l-sphere. Proof, For (i) we note that one orbit is necessary for p since I(K) = {p}. This implies that we cannot have free l—simplices in K. .80 K = 3?] C? as in Theorem i = {p} for i # jl—lotherwise it is a l-sphere and needs one orbit. Then clearly m < 2, other- 1.2. Also, Ci n c wise NOI(K) 2 3. This shows that m = l and a pinched torus is the only possibility for K. m n When NO (K) = 3, we write K = U C? U U St I i=1 1 j=l J as in Theorem 1.2. If CE is a 2-cell, we get K = c? and I(K) f {p}. If C? is a pinched annulus, then NOI(K) 2 A. This leaves C? as a 2-sphere or a pinched torus. Clearly we must have m = l or 2, n = O or 1 2 j orbits exceed 3. With m = 2 we get the first three pos- and C? n C = {p} for i f j otherwise the number of sibilities. If m = l, we must have r = 1 otherwise I(K) # {p}. This yields the remaining possibilities. For higher dimensional complexes with a single in- vert point, the restriction on the number of orbits under isotopy does not simplify the problem to any significant degree. It is useful to impose some extra restriction on the complex. For a 3-complex K with I(K) = {p}, the imposition of Brouwer Property and the restriction of NOI(K) leads to the following results. First we need the 26 definition of Brouwer PrOperty. Dofioitioo 3.3 A topological space X has Brourer prop- orty_if and only if homeomorphic images in X of open sub- sets of X are also open subsets of X. This definition follows G.T. Whyburn in [1A]. The following results by Duda appear in [‘7]. By Brouwer's Theorem on the Invariance of Domain, Euclidean Spaces and manifolds have the Brouwer Property, whereas manifolds with non-empty boundary do not. If K is an n-complex with Brouwer Property, then every r-simplex or, r s n-1, is the face of an n-simplex, every‘ on-1 is the face of exactly two n-simplices, and if as belongs to St(or) then St(or)--oS cannot contain the homeomorphic image of an open n—cell intersecting or. If K is an n-complex with n < 3, then K has Brouwer Property if and only if K is an n-manifold. Also, there exist non-manifolds with Brouwer Property in all dimensions greater than 2. lgmmmrriri (a) Let K be an n-complex with I(K) = {p}. If K has Brouwer Property then so does Llcp. (b) Let K = ‘f(L) have Brouwer Property. Then L has Brouwer Property. Proot. If not, consider LlcthEl. Romark. Let K be an n-complex with I(K) = {p}. Let «£(K) be the subcomplex of K determined by the closed (n-l)-simp1ices which are faces of none, one, three or 27 more n-simplices of K. Klassen proved that if’ JéKK) = fl, then p e I(,{(K)) (Theorem ’+.ll of [12]). Moreover, &C(K) was used to effect a separation of K useful for characterizing a 2-complex with a single invert point. Clearly £(K) = U if and only if I (9((K)) = D'. Also, Jfi(K) = U is a necessary condition for a complex K with I(K) = {p} to have Brouwer Property. Thoorom 3,6 Let K be a 3—complex with Brouwer Property and dim {I(K)} 2 1. Then K 2 S3. Eroof. Since dim {I(K)} 2 l, we can write K = A!(L) where L is a 2-complex which has Brouwer Prop- erty since K has the same. Then L is a 2-manifold. Also, there exist x and y in L such that {x,y} : L n I(K). Since L is a manifold, L«;_I(K). Thus K = 1(L) ; I(K). Consequently K = I(K) and by Theorem 1.5 we get K 3 S3. Romark. Let K be a 3-comp1ex with I(K) = S0 and having Brouwer Property. Then K = 4((L) where L is a 2-complex with Brouwer Property by Lemma 3.5 and hence it is a 2-man- ifold M2. It is possible that M2 may be a disjoint union of m (21) 2-manifolds. From such a complex it is easy to obtain another with a single point invert set by identifying the two suspension points of ,JF(L) as is the case in the next result. W Let K be a 3-complex with 28 Brouwer Property and I(K) = {p}. If NOI(K) = 2, then K is a suspension of a closed 2-manifold M2 with the sus- pension points identified at p. I Proof. We note that Llcp has Brouwer Property by Lemma 3.5. Since dim {Llfp} = 2, Llcp is a closed 2—man- ifold. Out of the two orbits under isotopy, one orbit is necessary for ‘p. This shows that K does not contain any simplex of dimension less than or equal to (i-l) which is not a face of an i-simplex in K ‘for O s i s 3. Moreover Llcp must have precisely two components, for if it has one, then I(K) ;_{p}. Then K is a suspension over one of the components of Llcp with the suspension points identified at p. Ccrcllarx 3,§ Let K be a 3-complex with Brouwer Property and I(K) = {p}. Suppose that NOI(K) = 3. Then (i) K = Kl U K2 where Kl n K2 = {p} and for i = 1, 2 K1 is a suspension of a 2-manifold with the suspension points identified at p or a cone over a 2—manifold from p, or (ii) K is a suspension over a 2-manifold with the suspension points identified at p. Proof. The proof proceeds as in the last theorem. However, since we have NOI(K) = 3, it is possible to have two 3-complexes K1 and K2 with Kl n K2 = {p} and each K. behaving as in Theorem 3.7. This gives the first part 1 of (i). But it is possible that Llcp n Ki may be connect— 29 ed, in which case we get a cone over a 2—manifold from p. This completes case (i). The proof of (ii) is similar to that of the last theorem. Let K be an n-complex with I(K) = {p}. Let x e K — p such that dim {01(x)} = k is minimal. If 0 = CI(x), then ’6 = o u p and p 6 01(6). Also, '6—p is a k-manifold Mk with Ed Mk = C. By an earlier re— mark, Mk has Brouwer Property and consequently Lk(p,O) has Brouwer Property. F3 If k = 1, then 6: 81. If k = 2, then Lk(p,O) has dimension one and Brouwer Property. Thus it is a l- manifold without boundary and so it is a collection of disjoint l-spheres. If Lk(p,O) is a l-sphere then 6': S2. If Lk(p,65 is a collection of two disjoint l— spheres, then IO 2 a pinched torus. If k = , then Lk(p,O) has dimension two and Brouwer Property, and is a 2-manifold without boundary. All this leads to the next result. PIQDQEIELQD 3,9 Let K be an n-complex with I(K) = {p}. Let x e K-p such that dim {01(x)} = k is minimal. Then (i) k = 1 implies that W2 s1. (ii) k = 2 implies that Lk(p{5;7;7) is a col- lection of disjoint l-spheres and (iii) k = 3 implies that Lk(p,5ET§€I) is a 2- manifold without boundary. 30 In particular, the last proposition is useful for a 3—complex where the possible values of k are l, 2 and 3, from which the nature of Lk(p,OI(x)) is available. CHAPTER IV SUSPENSIONS W Let K be a complex with I(K) = {p}. Then p e I (I(K)) if and only if dim {I (A3(K)) } 2 1. Prgmf. Let u and v be suspension points for $00. Then {u,v} g I (I(K)) . If p e I (I(K)) , then [I (I(K)) | 2 3. This means that dim {I(j(K)) } 21. On the other hand, if dim {I (I(K)) } 2 1, then I(K) ; K n I (j (K)) (by Theorem 1.10). This shows that pe I(j(K)) . Corollary 4,2 Let K be a complex with I(K) = {p}. Then p e I “(K0 if and only if CI (,J (K)) = I( (K)) . Proof, If p e I(j(K)) , then dim {I(j(K)) } 21 by the last result. Using Theorem 2.1 we get CI(X(K)) = I(1(K)) . If CI(j(K)) = I(j(K)) ,then |I(X(K)) | 2 2 since f(K) is a suspension. But |I([(K)) | = 2 implies that 016/ (K)) = O. This gives dim {I(AYCK)) } 2 1. Now use Theorem h.l. Proooottion H,3 Let K be an n-complex with dim {I(K)} = k 2 1. Then dim {I(L)} s k-—1 where K =X. Proof. We use Theorems 1.h and 1.10 to write K = [(L) with I(L) g; L n I(K). This shows that 31 32 I(IJ :,I(K). It is evident that equality is not possible as the vertices of suspension lie in I(K) but not in I(L). The result is now obvious. Thoorom H.h Let K be an n-complex with I(K) = {p}. If I([(K)) ,f 3°, then dim {I(j(K)) } 2 2. Proof, Let Iw(K)) a S0 and assume that dim {I(}{(K)) } = 1. Let u and v be the vertices of suspension used in obtaining ‘AfKK) from K. By Theorem h.l we note that p e I(Af(K)) . Also, there exists q gK such that q s I(J(K)) and p 7! q. Let U be an open neighborhood of p in k. Then there exists h e.§F(K) such that h(K-U) g,U. In particular, h(q) e U. Now we can construct a sequence [hi(q)} con— verging to pI in K and hi(q) e U for i = 1,2, "° By suspending each hi’ we can show that hi(q) e I(j(K)) . By compactness and uniform continuity, this cannot happen unless dim {164((K)) } 2 2. CQIQJJEIX h,§ Let K be an n-complex. If dim [I(K)}==l, then K =X(L) where I(L) is empty or a O-sphere. Proof. We use Theorem 1.4 to write K =‘4((L). By Proposition H.3, dim I(L) s 0. Again, I(L) {p} is not possible by the last theorem. Remark. We may compare the last result with that of Propo- sition 2.13. 33 From the standard results in topology, we note that if a complex K is a suspension of a complex L, then K is simply connected if and only if L is connected. Now let K be a complex such that dim {I(K)} 2 2. Then we assert that nl(K) = 1. This is obvious in view of Theo- rem 1.7. For if dim {I(K)} = k, then K =jk+l(L) where k 2 2. Hence K is the suspension of a connected complex. In other words, if K is an n-complex such that nl(K) f 1, then dim {I(K)} s 1. Moreover, if K is an n-complex , i-w’i'th dim {I(K)} 2 2, then rl(K/I(K)) = 1. Let K be an n-complex with I(K) = S1 and nl(K) # 1. Then K =.X?(L) and L is not connected. Let S1 =/A?(x U y) where x,y e L. Then L has just two com— ponents. If u is a suspension vertex for K, u is a local cut point of K. Since I(K) consists of local cut points of continuous invertibility (see Theorem 2.1), we must have dim K = l and K = $1. This result can also be stated as follows. If K is an n-complex such that I(K) = S1 and K # 81, then wl(K) = 1. Alternately, if K is an n-complex such that K # S1 and nl(K) # 1, then I(K) is empty, a single point, or a O-sphere. We collect these results in the following proposition. Proposition H.o (a) K =,A?(L) is simply connected if and only if L is connected. (b) dim {I(K)} 2 2 implies that Trl(K) =1 and Trl(K/I(K)) = l. 3% (c) I(K) = S1 and nl(K) % 1 implies that K = $1, or I(K) = s1 and K # Sl implies that nl(K) = 1, or K f s1 and nl(K) # 1 implies that I(K) = O, {p} or SO. Next we discuss a few results on double suspensions. Let K be an n-complex with I(K) = {p}. Let a1, bl be the vertices of suspension for ‘((K) and let a2, b2 be the vertices of suspension fer ‘¥2(K). We will write the double suspension of K as D(K). The suspension ring is b U b b2 U b2a 1 and is written as R = . a182 U 82 1 1 Doyle proved in [1] that R 4; CI (D(K)) . Figure H.l 35 Proposition thz (i) dim {I (D(K)) } 2 1 (ii) I (D(K)) = c1(p) a I(,J(K)) = {al,bl} and CI (D(K)) = I (mm) with dimension 2 or more. If p g I(J(K)) and R = I ( D(K)) , then $2! = CI(1(K)) g I(j(K)) = {al,bl} g R = CI(D(K))= I (D(K)) . We assert that it is impossible to have p e I([(K)) and R = I (D(K)) . Assume to the contrary. Then CI(1(K)) = I(j(K)) with dimension 1 or more by Theorems 2.1 and 4.1. Using Proposition H.7 we get R = Clam) = I([(K)) = CI (D(K)) = I (D(K)) . But the disk spanned by R U are (alpbl) U arc (a2pb2) must be contained in I (D(K))I . This is a contradiction. This leads to the next result. P 'ti H. Let K be an n—complex with I(K) = {p}. Then (i) p 6 I(/(K)) implies that R 55 I (D(K)) and 36 then dim’II (I(K))l 2 l and dim[1(D(K))I 2 2, (ii) p E I(X(K)) and R S, I (D(K)) implies that I(D(K)) : k22’ o = CI(j(K)) g I(jm) = so 3 CI (D(K)) and (iii) p g I([(K)) and R = I (D(K)) implies that a = CI(J(K)) s, I(j(K)) = so 5 CI (D(K)) I (D(K)) = sl. In connection with double suspensions, we quote a theorem due to Doyle which provides a scheme for constructing complexes with precisely one invert point. For example, if K is a non-simply connected compact n-manifold, then D(K)/R is precisely this type of complex. T e D Let K be a triangulated compact n-manifold. Then. I.(LKKJ) = R, unless D(K) is a sphere. Further, if I (D(K)) = R, then D(K)/I ( D(K)) is locally an (n+2)-manifold except at one point. IDSQI§m_E12 Let K be an n-complex with I(K) = {p} and NOI(K) = 2. If dim {I(jm» } 2 1, then I(K) Z Sn+l. Proof. Since NOI(K) = 2, K-p is locally euclid- ean of dimension n. Also, dim {I(Q!(K)) } 2 1 implies that there exists x e K-p such that x 6 I(j(K)) . By homogeneity, K - p c: IM(K)) and K c Iw(K)) since 37 p e ICAV(K)) in view of Theorem 4.1. Since dim K = n, we get dim [TC/(KD } = dim {I(K)} = n+1. Now by Theorem 1.5 we get ’JV(K) = Sn+l. Coro]lo 1 b.10 Let K be an n-complex with I(K) = {p} and NOI(K) = 2. If p e I(Af(K)) , then Lk(p,K) has Brouwer Property. Proof, By Theorem H.l, dim {I(AF(K)) } 2 l. The last theorem gives [4((K) = Sn+1 and this has Brouwer Pro- perty since it is a manifold without boundary. By Lemma 3.5, both K and Lk(p,K) have Brouwer Property. Let K be an n-complex with I(K) % O. Assume that p e I(K) and St}: embeds in En. Now let ,27(K) have Brouwer Property and dim {I(,J(K)) } 2 1. As noted Sn and earlier, K has Brouwer Property. Then K ,4F(K) Z Sn+l. Consider the case when I(K) = {p}. If AfKK) has Brouwer Property, then we must have dim {I(j(K)) } < 1 or I(j(K)) = SO. We have the fol- lowing: ETQDQEALAQE_E111 Let K be an n-complex such that n 2 l, p E I(K) and St;) embeds in En. Then (i) I(K) has Brouwer Property and dim [Iw(K)) } 2 1 imply that K 2 Sn, (ii)/JQ(K) has Brouwer Property and I(K) = C, {p} or SO imply that I(‘f(K)) = 30: and (iii) "I(K) = So" and ”I(K) has Brouwer Property" 38 are mutually exclusive. Proof. We need show only (iii). Assume that I(K) = S0 and ”XKK) has Brouwer Property. Then K =Af(L) by Theorem 1.4, and ‘AICK) = A!2(L)' By Doyle‘s theorem, we get dim {I(”F(K)) } 2 1. Using (i) we get K 3 Sn and this contradicts I(K) = $0. Propooitioo b.12 Let K be a 2-comp1ex with I(K) = {p} _ _ O and NOI(K) - 2. Then I(j(K)) — s . Proof. Assume that dim {I(j(K)) } 2 1. By T Theorem H.9, AfIK) = S3. Now we use Theorem 1.2 to get the result. It may be useful to remark that NOI(K) = 2 does not imply that NOIw(K)) = 2. We can only say that l s Nol(j(K)) 5 LI. Let n > 1 and identify two antipodal points of Sn in a nice way to obtain an n-complex K. This may be called a generalized pinched torus. It is evident that I(K) = {p} and NOI(K) = 2. Moreover, IM(K)) = S0 since 147(K) # Sn+l. This suggests the next set of results. EIQDQ§1319D_5213 Let K be an n-complex with I(K) = {p} and NOI(K) = 2. If K is not a homotopy n-sphere, then I(jmv = so. 21:99.2. Let dim {I(j(K)) } 2 1, Then j=Sn+1 and K is a homotopy n-sphere. This proves the result. 39 Let K be an n-complex (n2:2) such that ,2FCK) = Sn+l. Then K is a homotopy n—sphere. Let v be any vertex of K in the given triangulation. Then K and Lk(v,K) have the Brouwer Property. Since Lk(v,j(K)) =j(Lk(v,K)) and I(K) = so”, ,‘F(IMKV,K)) has the integral homology groups of an n-sphere. Moreover, o : Hi-1( Lk(v,k)) —> Hi (JLk(v,K)) is an onto isomorphism for 2 s i s n with H0(Lk(v,K)) = Z. Thus 0 for l s k s n—2 Hk (Lk(v,K)) = z for k = o, n-l Pomork, The fact that the local homology groups are in- variant under all triangulations of K can be justified by using the uniqueness of the open cone neighborhood (see [13]). Let v be any vertex of K under any triangulation. Consider Stir-— v. There exists a deformation of this onto Lk\A Now Stxl is an open cone neighborhood of v. By Kwun's theorem, we get the result that the links of v are homeomorphic under all triangulations of K. This proves the desired result. PEQDQEILiQQ b.1h Let K be an n-complex with n 2 2, I(K) = {p} and N0I(K) = 2. Let v be any vertex of K to under the given triangulation such that either (i) Hk (Lk(v,K)) ; o for some k such that 1 s k s n-2 or (ii) Bk <1k(v,K)) f Z for k = 0 or n-l. Then I(j(K)) = so. Proof. If we deny the assertion, then “f(K) = Sn+1 by Theorem H.9. This contradicts preceeding remarks. Thoorom_&rlfi Let K be a 3-complex with _ _ _ o I(K) - {p} and NOI(K) — 2. Then Iw(K)) — s . Proof. Assume that dim {I(A((K)) } 2 1. Then by T Theorem n.9,“f(K) = 8”. Also from earlier remarks, we get for i 1 Hi ( Lk(p,K)) = Z for 1 0,2 Moreover, Lk(p,K) has Brouwer Property by Corollary H.10. Then it is a 2-manifold without boundary with the prescribed 2 and K = p-Lk(p,K) is homology groups. Thus Lk(p,K) = S a 3-cell with a 2-sphere of invert points. This contradicts I(K) = {p}. If Lk(p,K) is connected and simply connected, we get an immediate contradiction. Theorom_&rlo Let K be a l-complex with I(K) = {p}. Then Twat» = so. Proof. By Theorem 1.1, K is an r-leafed rose with r 2 2. Assume that I(A((K)> # SO. Then by Theorem h.h, dim {I(4((K)) } = k 2 2. So there exists at least a l-sphere of invert points of',da(K) in K. Since K is a l-complex, it can contain only a l-sphere in it. Thus k = 2. Let M1 I(jflfi) =/(Sl) where 81 c: K. Now K is an r-leafed rose. Hence Sl must be one of the petals of this rose. Then p e I(‘f(K)) and the symmetry of the figure shows that every point of K 'is an invert point of /dV(K). This is easily seen to be impossible. Pomork. A more direct proof of the last theorem can also be given. Let q 6 K-p. Let U be an open neighborhood of q in ,AF(K). We can take U to be an open 2-cell. Clearly there does not exist any h.e.§$(4((K)) such that h CJKK)-U) c U. In particular, we cannot have h(p) e U. Hence q 6 K-p implies that q z I(j(K)) . This means that [K n I(j(K)) | s 1. But dim [I(j(K)) }2 2 implies that [K n I (I(K)) | 2 2. This shows that K n I(‘f(K)) = O. By Theorem H.l we get 0 s dim [I(1(K)) } < l, and the result is now obvious. Let R be an r-leafed rose, r 2 2. We now inves— tigate I(Jk(R)) for k 2 l. The result for k = l is given in the last theorem. For k = 2, we note that ’dV(R) is topologically the union of r 2-spheres with an are com- mon to all of them. Then “f2(R) is the union of r 3- spheres with a common 2-disk D2. It is also easy to see that IU2(R)) = Ed D2 = 81. j3(R) may be considered as a double suspension of ‘ofKR). Let Iai’bi} denote the Set of vertices of suspension for obtaining ‘fi(R) from ‘fi'l(R) with the obvious restrictions on i. Then I 13“”) contains the suspension ring . #2 Also, I. Moreover, ’4F3(R) is a union of r H-spheres with a common 3-disk D3. We claim that Iw3(R)) = Ed D3 = S2, The suspension of . In order to see this, we observe that 2 _ 3 CI(j (R)) — and ; Clj (31)) . Then points of are equivalently embedded in 4f3(R) and we are finished. The induction on k is now obvious. We have the following: Prooooitioo H.1Z Let R be an r-leafed rose (r2:2) and let K =jk(R). Then I(K) = $1"1 for k 2 l. The last result suggests a generalization of a re- sult of Doyle in [1.]. Proposition H,1o Let K be an n-complex. Then I (jk(K)) 2 SI"1 for k = 1,2,3, Proof. The result is true for k = l and 2 by Theorem 1.H and Theorem H of [1.] respectively. So assume that the result is true for k 2 3. Let ak+1’ bk+l be the vertices of suspension for getting jk+l(K) from jk(K). By induction hypothesis, Sk-l ; I(jk(K)) . Clearly, dim {I(A{k+l(K)) } 2 1 and by Theorem 1.10 we 8913 IUkIKU C. lab-I(K)) . By homogeneity, the sus- . k—l . . pen51on of S from vertices ak+l and bk+l lles 1n Iwk+l(K)) . Then Sk ; law-I(K)) , and the proof is complete. ’+3 Remark. In view of the earlier theorems, we note that if k 2 2 and K =l4fk(L), then CI(K) = I(K). This result, together with the last proposition, gives a simplified proof of a similar result in [S9]. It is also clear that by taking successive suspensions, the dimension of the in- vert set is raised by at least one. By Proposition H.17, we note that the dimension is raised precisely by one when we take the successive suspensions of a rose R, and in this sense the result is the best possible. Let K be an n-complex. Then . k k—ls dlm I(AI (K)) sn+k, where k 2 l; and K = R, K = Sn respectively give the equality at the extremes. Let F be a finite set of points. Then dim {1km} = k for k 2 1. By Proposition 4.18, Sk'l t; I(jk(F)) . Now either (i) Sk-l $- Iwk(F)) or (ii) Sk-l = lwk(F)) . In case of (i) we get I(jkm) = Sk since jk(F) has dimension k. By T Theorem 1.5 this means that Jk(F) = SR and this is im- possible unless |F| = 2. The following remark is now obvious and appears to be converse of the remark preceding Proposition 2.13. Romork. Let F be a finite set of points and k 2 l. _ k _ k _ k Then (i) |F| — 2 implies that I(‘f (F) -,4( (F) - S and (ii) |F| 7! 2 implies that WINE?) = sk‘l. AA Let K be a complex such that dim {I(K)} 2 1. Then write K =‘A((L) with p and q as the vertices of suspension. Clearly there exists v e L n I(K). Now I(‘St(v,L)) gives a proper suspension neighborhood N of v. Without any loss of generality we may assume that I(K) ¢'N, otherwise a smaller open set may be chosen in St(v,L) to get the desired result. Choose w e (K-N) n I(K). Let Ul be an open set containing w. There exists an inverting map hl such that hl(K"Ul) c U In particular, choose diameter of 1' U1 < 1. We get h1(N) c U1. Since I(K) is continuously homogeneous, we get hl(v) = w. By uniform continuity, w 6 Int hl(N). Let U2 be an open set containing w such that U2 c Int hl(N) and diameter of U2 <-% . Then there exists h2 e WK) such that h2(K-U2) c U2. Again,we get h2(N) c U2 and h2(v) = w 6 Int h2(N). Proceeding inductively we get a sequence of inverting maps {hi}I=l with the property that w = 1E1 hi(N) and dia- meter of hi(N) < %-. Then w e I(K) and has arbitrarily small suspension neighborhoods, and this shows that every point in I(K) has this property. In particular, if C is an open cone neighborhood of v in L, then ,47(C) embeds in C x I . We can obtain the same result by the following argument. Let w e (K-N) n I(K). Let Nl be an open cone neighborhood of w such that Nl n N = U and ”5 diameter of N1 < 1. Then there exists hl €.?¥(K) such that hl(N) c: Int N1. Now hl(v) e I(K). In Int hl(N), select an open cone neighborhood N2 of hl(v) with dia- meter <'%. Inversion about hl(V) yields an inverting map h2 such that h2(N) c Int-N2. Proceeding inductiVely we get a sequence {hi}I=l of inverting maps such that Z = n hi(N) 6 I(K) and has arbitrarily small suspension i=1 neighborhoods. We now state the next result. Theorom h,19 Let K be an n-complex with dim {I(K)} 2 1. Let p e I(K). Then p has arbitrarily small suspension neighborhoods. Pomork, Following the arguments leading to the last theo- rem and using Theorem 1.7, it is evident that if dim {I(K)} = k 2 1, then every invert point has arbitrarily small k—fold suspension neighborhoods. Let K be an n-complex with I(K) = {p}. It was remarked earlier that if equality could be established in Theorem 1.10, the Poincare Conjecture could be proved in dimension 1+. Thus, the equality I(K) = K n I(I(K)) for dim { I(Af(K)) } 2 l is stronger than the Poincare Conjecture. If we use this for the complex K with a single invert point, the following result is obtained. Let dim {I(X(K)) ] 2 1, Then {p} = I(K) = K n'I(,f(K)) . 46 But dim {I([(K)) } 2 1 implies that 1KnI(,J(K)) I 2 2. This is a contradiction. So we must have I(A{(K)) = SO. Moreover, Theorem 4.16 for n = 1, Proposition 4.12 for n = 2 and Theorem 4.15 for n = 3, the last two with the additional hypothesis of two orbits under isotopy, indicate that the following result may be true. We end this chapter with this conjecture. Coojooturo. Let K be an n-complex with I(K) 3 IPI° Then I(j(K)) = SO. CHAPTER V AN APPLICATION TO GRAPHS In a recent study, Doyle extended to a class of spaces called monotonic complexes, the result that every open connected set in En is a monotone union of closed n-cells. The relevant definition and the statement of his unpublished result are as follows. Dofioitioo 5,1 A simplicial complex Kn is monotonic if P and only if Kn = I.) Ki where each K1 is a subcomplex i=1 of Kn, K1 is an n-simplex, and for l s i s p-l, Ki+l is obtained from Ki by adding just one n-simplex L1 to Ki such that L1 and K1 have an (n-l)—simp1ex in common. EZEEHEELJUTDLURL If Kn is a monotonic complex T a of dimension n (2 2), then Kn = U Ci where C1 is a 1:1 closed n-cell and C1 c Ci+l for i = 1,2,3: For n s 1, it was mentioned that a monotonic 0-complex is a point and that every connected l-complex is monotonic. Pomork, Doyle proved that if K is a monotone union of l-cells, then K is homeomorphic to one of the six figures 47 48 listed below. ‘I =0 (5.. Q o—o Figure 5.1 It was also remarked that five of these configura- tions represent the termination of a self-avoiding walk discussed by Kesten in [11]. These considerations lead to the following definition. Dofinitioo 5.2 If E(n) is a graph such that it can be 1 no written as A.) Si(n) where Si(n) is a closed star graph i=1 ‘ of order n (2 2) and Si(n) c: Si+l(n) for i = 1,2,3, then E(n) is said to be an oroooofog_n;otor_grooh, Then a monotone union of l—cells may be written as E(2) and has the non-homeomorphic forms given in Figure 5.1. Also, the generalization of self-avoiding walks of Kesten is immediate. By a direct counting process, it was possible to obtain the 30 configurations of E(3) as given in Figure 5.2. 50 Remark. The collection of all expanding n-star graphs for n 2 2 contains examples of all l-complexes with a nonempty invert set. Moreover, we also obtain many examples of com- plexes with an empty invert set. D i i ' . Let G be any graph. Then D(k,G) will represent the number of vertices of G whose degree is greater than or equal to k. Theorom 5.; Let E(n) be an expanding n- star graph. Then D(k,E(n)) s 1 + [E{}§j for 3 s k s 2n and n 2 3. Proof. Let E(n) = (:I Si(n) where each Si(n) i=1 is a star graph with a vertex p such that n s p(p) 2 Zn. The maximum number of vertices in the graph with degree 2 3 is obtained if every end point of an arc meets the interior of that arc. Thus D(3,E(n)) s l + [%3. For obtaining the maximum number of vertices of degree 2 4, the n ends of the arcs from p can be paired in such a way that every pair meets on the interior of an arc to produce a vertex of degree 4. Then D(4,E(n)) s 1 + [23. In general, k-2 ends have to meet on the interior of an arc to produce a vertex of degree k. But the n ends can be paired to produce at most [E{?§] vertices with degree k. This completes the proof. 51 Corollary 5.2 Let E(n) be an expanding n-star graph and n 2 3. (a) If n4-3 s k 5 2n, then D(k,E(n)) s l. (b) If k = ni-l or ni-2, then D(k,E(n)) s 2. (C) If n 2 5 and k = n, then D(k,E(n)) s 2. Proof, Parts (a) and (b) follow from the last theorem. For part (c) we note that n 2 5 implies ._1L_ : [n-2] 1’ Pomorko. The preceding results show that we cannot have too many vertices of high degree in an expanding n-star graph. In fact, an expanding n-star graph is locally euclidean everywhere except at (n+1) points at most. Moreover, if E(n) is an expanding nestar graph and x is any vertex of E(n) then 1 s 9(x) s 2n, and if P(x) > ni-2 then x must be the center of Sl(n) where o E(n) = 32% Si(n). Theorem_523 Let E(n) be an expanding n-star graph and n 2 3. Let p = max {p(x)}. Then er(n) (k-—2) (D(k,E(n))-1) s n s p for 3 s k s 2n. Proof. By Theorem 5.1 we get D(k,E(n)) s 1 + [E%}Efl s l + k-2 . This gives (k-2) (D(k,E(n))-l) s n. Obviously p 2 n. Using the standard terminology of graph theory, let K denote the complete graph on n vertices and let K n m,n 52 denote the complete bipartite graph on m and n vertices. From previous remarks and results we note that K1’ K2, K3, K4’ K1,n and K2,n are expanding r-star graphs for a suitable r. This naturally leads to the investigation of K5 and K3,3. Thoorom_5gi I K5 and K3,3 are not expand- ing n-star graphs. Proof, For K5 we note that p = 4. If k = 4 then D(4,K5) = 5. These values give (k-2) (D(k,K5)-l> = and there is no n which can satisfy the inequality in Theorem 5.3. Then K5 cannot be an expanding n-star graph. For K3,3 we have p = 3. Taking k = 3, we get 3.3) conclude as before that K3 3 is not an expanding n-star ’ D(3,K 6. Now (k-2) E(n) be a mapping which is one to one except on the end points of Sl(n). Let 'Sl(n) be a star graph of order (n-—l) obtained from Sl(n) by deleting a semi—open branch of Sl(n). Then (E(n-l) = f(Sl(n)) is an expanding (n-l)-star graph. By the mini- mality of n, (E(n-l) is planar. This shows that every proper subset of E(n) is planar. By Kuratowski's theorem, a graph is planar if and only if it has no subgraph homeomorphic with K5 or K3,3. Consequently, E(n) must be K5 or K3,3. But this is impossible in view of Theorem 5.4 and the proof is complete. IO. ll. 12. 13. 14. BIBLIOGRAPHY Doyle, P. H., "Symmetry in Geometric Complexes," Amer- ican Math. Monthly, 73, 625-628 (1966). Doyle, P. H. and Hocking, J. G., "A Characterization pf gutlidean n-Space," Michigan Math. J., 7, 199-200 19 O . Doyle, P. H. and Hocking, J. G., "Invertible Spaces," American Math. Monthly, 68, 959—965 (1961). Doyle, P. H. and Hocking, J. G., "Continuously Invert- ible Spaces," Pacific J. Math., 12, 499-503 (1962). Doyle, P. H. and Hocking, J. G., "Dimensional Inverti- bility," Pacific J. Math., 12, 1235-1240 (1962). Doyle, P. H., Hocking, J. G. and Osborne, R. P., "Local Invertibility," Fund. Math., LIV, 15-25 (1964). Duda, E., "Brouwer Property Spaces," Duke Math. J., 30, 6h7-660 (1963). Edwards, C. H., Jn, "Products of Pseudo Cells," Bull. Amer. Math. Soc., 68, 583-584 (1962). Hocking, J. G., "Invert Sets in Polyhedra," American Math. Monthly (to appear). Hocking, J. G. and Young, G. S., Tooojogy, Addison- Wesley, 1961. Kesten, H., "0n the Number of Self-avoiding Walks," J. Math. Phys., A, 960-969 (1963). Klassen, V. M., "Complexes with Invert Points," Ph.D. Thesis, Virginia Polytech. Inst. (1964). Kwun, K. W., "Uniqueness of the Open Cone Neighborhood," Proc. Amer. Math. Soc., 15, 476-479 (1964). Whyburn, G. T., "Decomposition Spaces," opology of -m d goo Rolotoo Poofoo, Prentice—Hall, 2 19 2 .