LECTURE NOTES ON GENERALIZED HEEGAARD SPLITTINGS


 Tabitha Summers
 3 years ago
 Views:
Transcription
1 LECTURE NOTES ON GENERALIZED HEEGAARD SPLITTINGS TOSHIO SAITO, MARTIN SCHARLEMANN AND JENNIFER SCHULTENS 1. Introduction These notes grew out of a lecture series given at RIMS in the summer of The authors were visiting RIMS in conjunction with the Research Project on Low Dimensional Topology in the TwentyFirst Century. They had been invited by Professor Tsuyoshi Kobayashi. The lecture series was first suggested by Professor Hitoshi Murakami. The lecture series was aimed at a broad audience that included many graduate students. Its purpose lay in familiarizing the audience with the basics of 3 manifold theory and introducing some topics of current research. The first portion of the lecture series was devoted to standard topics in the theory of 3manifolds. The middle portion was devoted to a brief study of Heegaaard splittings and generalized Heegaard splittings. The latter portion touched on a brand new topic: fork complexes. During this time Professor Tsuyoshi Kobayashi had raised some interesting questions about the connectivity properties of generalized Heegaard splittings. The latter portion of the lecture series was motivated by these questions. And fork complexes were invented in an effort to illuminate some of the more subtle issues arising in the study of generalized Heegaard splittings. In the standard schematic diagram for generalized Heegaard splittings, Heegaard splittings are stacked on top of each other in a linear fashion. See Figure 1. This can cause confusion in those cases in which generalized Heegaaard splittings possess interesting connectivity properties. In these cases, some of the topological features of the 3manifold are captured by the connectivity properties of the generalized Heegaard splitting rather than by the Heegaard splittings of submanifolds into which the generalized Heegaard splitting decomposes the 3manifold. See Figure 2. Fork complexes provide a means of description in this context. Figure 1. The standard schematic diagram 1
2 2 TOSHIO SAITO, MARTIN SCHARLEMANN AND JENNIFER SCHULTENS Figure 2. A more informative schematic diagram for a generalized Heegaard splitting for a manifold homeomorphic to (a surface) S 1 The authors would like to express their appreciation of the hospitality extended to them during their stay at RIMS. They would also like to thank the many people that made their stay at RIMS delightful, illuminating and productive, most notably Professor Hitoshi Murakami, Professor Tsuyoshi Kobayashi, Professor Jun Murakami, Professor Tomotada Ohtsuki, Professor Kyoji Saito, Professor Makoto Sakuma, Professor Kouki Taniyama and Dr. Yo av Rieck. Finally, they would like to thank Dr. Ryosuke Yamamoto for drawing the fine pictures in these lecture notes. 2. Preliminaries 2.1. PL 3manifolds. Let M be a PL 3manifold, i.e., M is a union of 3 simplices σi 3 (i = 1, 2,..., t) such that σi 3 σj 3 (i j) is emptyset, a vertex, an edge or a face and that for each vertex v, v σ σ 3 j 3 j is a 3ball (cf. [14]). Then the decomposition {σi 3} 1 i t of M is called a triangulation of M. Example (1) The 3ball B 3 is the simplest PL 3manifold in a sense that B 3 is homeomorphic to a 3simplex. (2) The 3sphere S 3 is a 3manifold obtained from two 3balls by attaching their boundaries. Since S 3 is homeomorphic to the boundary of a 4simplex, we see that S 3 is a union of five 3simplices. It is easy to show that this gives a triangulation of S 3. Exercise Show that the following 3manifolds are PL 3manifolds. (1) The solid torus D 2 S 1. (2) S 2 S 1. (3) The lens spaces. Note that a lens space is obtained from two solid tori by attaching their boundaries. Let K be a three dimensional simplicial complex and X a subcomplex of K, that is, X a union of vertices, edges, faces and 3simplices of K such that X is a simplicial complex. Let K be the second barycentric subdivision of K. A regular neighborhood of X in K, denoted by η(x; K), is a union of the 3simplices of K intersecting X (cf. Figure 3). Proposition If X is a PL 1manifold properly embedded in a PL 3 manifold M (namely, X M = X), then η(x; M) = X B 2, where X is
3 LECTURE NOTES ON GENERALIZED HEEGAARD SPLITTINGS 3 X K = տ ր η(x; K) Figure 3 identified with X {a center of B 2 } and η(x; M) M is identified with X B 2 (cf. Figure 4). M X η(x; M) Figure 4 Proposition Suppose that a PL 3manifold M is orientable. If X is an orientable PL 2manifold properly embedded in M (namely, X M = X), then η(x; M) = X [0, 1], where X is identified with X {1/2} and η(x; M) M is identified with X [0, 1]. Theorem (Moise [10]). Every compact 3manifold is a PL 3manifold. In the remainder of these notes, we work in the PL category unless otherwise specified Fundamental definitions. By the term surface, we will mean a connected compact 2manifold. Let F be a surface. A loop α in F is said to be inessential in F if α bounds a disk in F, otherwise α is said to be essential in F. An arc γ properly embedded in F is said to be inessential in F if γ cuts off a disk from F, otherwise γ is said to be essential in F. Let M be a compact orientable 3manifold. A disk D properly embedded in M is said to be inessential in M if D cuts off a 3ball from M, otherwise D is said to be essential in M. A 2sphere P properly embedded in M is said to be inessential in M if P bounds a 3ball in M, otherwise P is said to be essential
4 4 TOSHIO SAITO, MARTIN SCHARLEMANN AND JENNIFER SCHULTENS in M. Let F be a surface properly embedded in M. We say that F is parallel in M if F cuts off a 3manifold homeomorphic to F [0, 1] from M. We say that F is compressible in M if there is a disk D M such that D F = D and D is an essential loop in F. Such a disk D is called a compressing disk. We say that F is incompressible in M if F is not compressible in M. The surface F is compressible in M if there is a disk δ M such that δ F is an arc which is essential in F, say γ, in F and that δ M is an arc, say γ, with γ γ = δ. Otherwise F is said to be incompressible in M. Suppose that F is homeomorphic neither to a disk nor to a 2sphere. The surface F is said to be essential in M if F is incompressible in M and is not parallel in M. Definition Let M be a connected compact orientable 3manifold. (1) M is said to be reducible if there is a 2sphere in M which does not bound a 3ball in M. Such a 2sphere is called a reducing 2sphere of M. M is said to be irreducible if M is not reducible. (2) M is said to be reducible if there is a disk properly embedded in M whose boundary is essential in M. Such a disk is called a reducing disk. 3. Heegaard splittings 3.1. Definitions and fundamental properties. Definition A 3manifold C is called a compression body if there exists a closed surface F such that C is obtained from F [0, 1] by attaching 2handles along mutually disjoint loops in S {1} and filling in some resulting 2sphere boundary components with 3handles (cf. Figure 5). We denote F {0} by + C and C \ + C by C. A compression body C is called a handlebody if C =. A compression body C is said to be trivial if C = F [0, 1]. Definition For a compression body C, an essential disk in C is called a meridian disk of C. A union of mutually disjoint meridian disks of C is called a complete meridian system if the manifold obtained from C by cutting along are the union of C [0, 1] and (possibly empty) 3balls. A complete meridian system of C is minimal if the number of the components of is minimal among all complete meridian system of C. Remark The following properties are known for compression bodies. (1) A compression body C is reducible if and only if C contains a 2sphere component. (2) A minimal complete meridian system of a compression body C cuts C into C [0, 1] if C, and cuts C into a 3ball if C = (hence C is a handlebody). (3) By extending the cores of the 2handles in the definition of the compression body C vertically to F [0, 1], we obtain a complete meridian system of C such that the manifold obtained by cutting C along is homeomorphic to a union of C [0, 1] and some (possibly empty) 3balls. This gives a dual description of compression bodies. That is, a compression body C is obtained from C [0, 1] and some (possibly empty) 3balls by attaching
5 LECTURE NOTES ON GENERALIZED HEEGAARD SPLITTINGS 5 Dual discription F [0, 1] ց ւ Figure 5 some 1handles to C {1} and the boundary of the 3balls (cf. Figure 5). (4) For any compression body C, C is incompressible in C. (5) Let C and C be compression bodies. Suppose that C is obtained from C and C by identifying a component of C and + C. Then C is a compression body. (6) Let D be a meridian disk of a compression body C. Then there is a complete meridian system of C such that D is a component of. Any component obtained by cutting C along D is a compression body. Exercise Show Remark An annulus A properly embedded in a compression body C is called a spanning annulus if A is incompressible in C and a component of A is contained in + C and the other is contained in C. Lemma Let C be a nontrivial compression body. Let A be a spanning annulus in C. Then there is a meridian disk D of C with D A =. Proof. Since C is nontrivial, there is a meridian disk of C. We choose a meridian disk D of C such that D intersects A transversely and D A is minimal among all such meridian disks. Note that A C is an essential loop in the component of C containing A C. We shall prove that D A =. To this end, we suppose D A.
6 6 TOSHIO SAITO, MARTIN SCHARLEMANN AND JENNIFER SCHULTENS Claim 1. There are no loop components of D A. Proof. Suppose that D A has a loop component which is inessential in A. Let α be a loop component of D A which is innermost in A, that is, α cuts off a disk δ α from A such that the interior of δ α is disjoint from D. Such a disk δ α is called an innermost disk for α. We remark that α is not necessarily innermost in D. Note that α also bounds a disk in D, say δ α. Then we obtain a disk D by applying cut and paste operation on D with using δ α and δ α, i.e., D is obtained from D by removing the interior of δ α and then attaching δ α (cf. Figure 6). Note that D is a meridian disk of C. Moreover, we can isotope the interior of D slightly so that D A < D A, a contradiction. (Such an argument as above is called an innermost disk argument.) α δ α δ α δ α A D Figure 6 D Hence if D A has a loop component, we may assume that the loop is essential in A. Let α be a loop component of D A which is innermost in D, and let δ α be the innermost disk in D with δ α = α. Then α cuts A into two annuli, and let A be the component obtained by cutting A along α such that A is adjacent to C. Set D = A δ α. Then D ( C) is a compressing disk of C, contradicting (4) of Remark Hence we have Claim 1. Claim 2. There are no arc components of D A. Proof. Suppose that there is an arc component of D A. Note that D + C. Hence we may assume that each component of D A is an inessential arc in A whose endpoints are contained in + C. Let γ be an arc component of D A which is outermost in A, that is, γ cuts off a disk δ γ from A such that the interior of δ γ is disjoint from D. Such a disk δ γ is called an outermost disk for γ. Note that γ cuts D into two disks δ γ and δ γ (cf. Figure 7). If both δ γ δ γ and δ γ δ γ are inessential in C, then D is also inessential in C, a contradiction. So we may assume that D = δ γ δ γ is essential in C. Then we can isotope D slightly so that D A < D A, a contradiction. Hence we have Claim 2. (Such an argument as above is called an outermost disk argument.) Hence it follows from Claims 1 and 2 that D A =, and this completes the proof of Lemma Remark Let A be a spanning annulus in a nontrivial compression body C.
7 LECTURE NOTES ON GENERALIZED HEEGAARD SPLITTINGS 7?γ δ γ γ A Figure 7 } {{ } δ γ } {{ } δ γ D (1) By using the arguments of the proof of Lemma 3.1.5, we can show that there is a complete meridian system of C with A =. (2) It follows from (1) above that there is a meridian disk E of C such that E A = and E cuts off a 3manifold which is homeomorphic to (a closed surface) [0, 1] containing A. Exercise Show Remark Let ᾱ = α 1 α p be a union of mutually disjoint arcs in a compression body C. We say that ᾱ is vertical if there is a union of mutually disjoint spanning annuli A 1 A p in C such that α i A j = (i j) and α i is an essential arc properly embedded in A i (i = 1, 2,..., p). Lemma Suppose that ᾱ = α 1 α p is vertical in C. Let D be a meridian disk of C. Then there is a meridian disk D of C with D ᾱ = which is obtained by cutandpaste operation on D. Particularly, if C is irreducible, then D is ambient isotopic such that D ᾱ =. Proof. Let Ā = A 1 A p be a union of annuli for ᾱ as above. By using innermost disk arguments, we see that there is a meridian disk D such that no components of D Ā are loops which are inessential in Ā. We remark that D is ambient isotopic to D if C is irreducible. Note that each component of Ā is incompressible in C. Hence no components of D Ā are loops which are essential in Ā. Hence each component of D Ā is an arc; moreover since D is contained in + C, the endpoints of the arc components of D Ā are contained in +C Ā. Then it is easy to see that there exists an arc β i ( A i ) such that β i is essential in A i and β i D =. Take an ambient isotopy h t (0 t 1) of C such that h 0 (β i ) = β i, h t (Ā) = Ā and h 1(β i ) = α i (i = 1, 2,..., p) (cf. Figure 8). Then the ambient isotopy h t assures that D is isotoped so that D is disjoint from ᾱ. In the remainder of these notes, let M be a connected compact orientable 3manifold. Definition Let ( 1 M, 2 M) be a partition of components of M. A triplet (C 1, C 2 ; S) is called a Heegaard splitting of (M; 1 M, 2 M) if C 1 and C 2 are compression bodies with C 1 C 2 = M, C 1 = 1 M, C 2 = 2 M and C 1 C 2 = + C 1 = + C 2 = S. The surface S is called a Heegaard surface and the genus of a Heegaard splitting is defined by the genus of the Heegaard surface.
8 8 TOSHIO SAITO, MARTIN SCHARLEMANN AND JENNIFER SCHULTENS α i A i Figure 8 K 1 K 2 Figure 9 Theorem For any partition ( 1 M, 2 M) of the boundary components of M, there is a Heegaard splitting of (M; 1 M, 2 M). Proof. It follows from Theorem that M is triangulated, that is, there is a finite simplicial complex K which is homeomorphic to M. Let K be a barycentric subdivision of K and K 1 the 1skeleton of K. Here, a 1skeleton of K is a union of the vertices and edges of K. Let K 2 K be the dual 1skeleton (see Figure 9). Then each of K i (i = 1, 2) is a finite graph in M. Case 1. M =. Recall that K 1 consists of 0simplices and 1simplices. Set C 1 = η(k 1 ; M) and C 2 = η(k 2 ; M). Note that a regular neighborhood of a 0simplex corresponds to a 0handle and that a regular neighborhood of a 1simplex corresponds to a 1 handle. Hence C 1 is a handlebody. Similarly, we see that C 2 is also a handlebody. Then we see that C 1 C 2 = M and C 1 C 2 = C 1 = C 2. Hence (C 1, C 2 ; S) is a Heegaard splitting of M with S = C 1 C 2. Case 2. M. In this case, we first take the barycentric subdivision of K and use the same notation K. Recall that K is the barycentric subdivision of K. Note that no 3simplices of K intersect both 1 M and 2 M. Let N( 2 M) be a union of the 3 simplices in K intersecting 2 M. Then N( 2 M) is homeomorphic to 2 M [0, 1], where 2 M {0} is identified with 2 M. Set 2 M = 2M {1}. Let K 1 ( K 2
9 LECTURE NOTES ON GENERALIZED HEEGAARD SPLITTINGS 9 K 1 K 1 2 M K 2 K 2 2 M 1 M Figure 10 resp.) be the maximal subcomplex of K 1 (K 2 resp.) such that K 1 ( K 2 resp.) is disjoint from 2M ( 1 M resp.) (cf. Figure 10). Set C 1 = η( 1 M K 1 ; M). Note that C 1 = η( 1 M; M) η( K 1 ; M). Note again that a regular neighborhood of a 0simplex corresponds to a 0handle and that a regular neighborhood of a 1simplex corresponds to a 1handle. Hence C 1 is obtained from 1 M [0, 1] by attaching 0handles and 1handles and therefore C 1 is a compression body with C 1 = 1 M. Set C 2 = η(n( 2 M) K 2 ; M). By the same argument, we can see that C 2 is a compression body with C 2 = 2 M. Note that C 1 C 2 = M and C 1 C 2 = C 1 = C 2. Hence (C 1, C 2 ; S) is a Heegaard splitting of M with S = C 1 C 2. We now introduce alternative viewpoints to Heegaard splittings as remarks below. Definition Let C be a compression body. A finite graph Σ in C is called a spine of C if C \ ( C Σ) = + C [0, 1) and every vertex of valence one is in C (cf. Figure 11). Figure 11 Remark Let (C 1, C 2 ; S) be a Heegaard splitting of (M; 1 M, 2 M). Let Σ i be a spine of C i, and set Σ i = im Σ i (i = 1, 2). Then M \ (Σ 1 Σ 2 ) = (C 1 \ Σ 1 ) S (C 2 \ Σ 2 ) = S (0, 1). Hence there is a continuous function f : M [0, 1] such that f 1 (0) = Σ 1, f 1 (1) = Σ 2 and f 1 (t) = S (0 < t < 1). This is called a sweepout picture.
10 10 TOSHIO SAITO, MARTIN SCHARLEMANN AND JENNIFER SCHULTENS Remark Let (C 1, C 2 ; S) be a Heegaard splitting of (M; 1 M, 2 M). By a dual description of C 1, we see that C 1 is obtained from 1 M [0, 1] and 0handles H 0 by attaching 1handles H 1. By Definition 3.1.1, C 2 is obtained from S [0, 1] by attaching 2handles H 2 and filling some 2sphere boundary components with 3handles H 3. Hence we obtain the following decomposition of M: M = 1 M [0, 1] H 0 H 1 S [0, 1] H 2 H 3. By collapsing S [0, 1] to S, we have: M = 1 M [0, 1] H 0 H 1 S H 2 H 3. This is called a handle decomposition of M induced from (C 1, C 2 ; S). Definition Let (C 1, C 2 ; S) be a Heegaard splitting of (M; 1 M, 2 M). (1) The splitting (C 1, C 2 ; S) is said to be reducible if there are meridian disks D i (i = 1, 2) of C i with D 1 = D 2. The splitting (C 1, C 2 ; S) is said to be irreducible if (C 1, C 2 ; S) is not reducible. (2) The splitting (C 1, C 2 ; S) is said to be weakly reducible if there are meridian disks D i (i = 1, 2) of C i with D 1 D 2 =. The splitting (C 1, C 2 ; S) is said to be strongly irreducible if (C 1, C 2 ; S) is not weakly reducible. (3) The splitting (C 1, C 2 ; S) is said to be reducible if there is a disk D properly embedded in M such that D S is an essential loop in S. Such a disk D is called a reducing disk for (C 1, C 2 ; S). (4) The splitting (C 1, C 2 ; S) is said to be stabilized if there are meridian disks D i (i = 1, 2) of C i such that D 1 and D 2 intersect transversely in a single point. Such a pair of disks is called a cancelling pair of disks for (C 1, C 2 ; S). Example Let (C 1, C 2 ; S) be a Heegaard splitting such that each of C i (i = 1, 2) consists of two 2spheres and that S is a 2sphere. Note that there does not exist an essential disk in C i. Hence (C 1, C 2 ; S) is strongly irreducible. Suppose that (C 1, C 2 ; S) is stabilized, and let D i (i = 1, 2) be disks as in (4) of Definition Note that since D 1 intersects D 2 transversely in a single point, we see that each of D i (i = 1, 2) is nonseparating in S and hence each of D i (i = 1, 2) is nonseparating in C i. Set C 1 = cl(c 1 \ η(d 1 ; C 1 )) and C 2 = C 2 η(d 1 ; C 1 ). Then each of C i (i = 1, 2) is a compression body with + C 1 = +C 2 (cf. (6) of Remark 3.1.3). Set S = + C 1 (= +C 2 ). Then we obtain the Heegaard splitting (C 1, C 2; S ) of M with genus(s ) = genus(s) 1. Conversely, (C 1, C 2 ; S) is obtained from (C 1, C 2; S ) by adding a trivial handle. We say that (C 1, C 2 ; S) is obtained from (C 1, C 2 ; S ) by stabilization. Observation Every reducible Heegaard splitting is weakly reducible. Lemma Let (C 1, C 2 ; S) be a Heegaard splitting of (M; 1 M, 2 M) with genus(s) 2. If (C 1, C 2 ; S) is stabilized, then (C 1, C 2 ; S) is reducible. Proof. Suppose that (C 1, C 2 ; S) is stabilized, and let D i (i = 1, 2) be meridian disks of C i such that D 1 intersects D 2 transversely in a single point. Then η( D 1 D 2 ; S) bounds a disk D i in C i for each i = 1 and 2. In fact, D 1 (D 2 resp.) is obtained from two parallel copies of D 1 (D 2 resp.) by adding a band
11 LECTURE NOTES ON GENERALIZED HEEGAARD SPLITTINGS 11 along D 2 \ (the product region between the parallel disks) ( D 1 \ (the product region between the parallel disks) resp.) (cf. Figure 12). D 2 D 1 D 1 D 2 C 1 C 2 D 1 D 2 Figure 12 Note that D 1 = D 2 cuts S into a torus with a single hole and the other surface S. Since genus(s) 2, we see that genus(s ) 1. Hence D 1 = D 2 is essential in S and therefore (C 1, C 2 ; S) is reducible. Definition Let (C 1, C 2 ; S) be a Heegaard splitting of (M; 1 M, 2 M). (1) Suppose that M = S 3. We call (C 1, C 2 ; S) a trivial splitting if both C 1 and C 2 are 3balls. (2) Suppose that M = S 3. We call (C 1, C 2 ; S) a trivial splitting if C i is a trivial handlebody for i = 1 or 2. Remark Suppose that M = S 3. If (M; 1 M, 2 M) admits a trivial splitting (C 1, C 2 ; S), then it is easy to see that M is a compression body. Particularly, if C 2 (C 1 resp.) is trivial, then M = 1 M and + M = 2 M ( M = 2 M and + M = 1 M resp.). Lemma Let (C 1, C 2 ; S) be a nontrivial Heegaard splitting of (M; 1 M, 2 M). If (C 1, C 2 ; S) is reducible, then (C 1, C 2 ; S) is weakly reducible. Proof. Let D be a reducing disk for (C 1, C 2 ; S). (Hence D S is an essential loop in S.) Set D 1 = D C 1 and A 2 = D C 2. By exchanging subscripts, if necessary, we may suppose that D 1 is a meridian disk of C 1 and A 2 is a spanning annulus in C 2. Note that A 2 C 2 is an essential loop in the component of C 2 containing A 2 C 2. Since C 2 is nontrivial, there is a meridian disk of C 2. It follows from Lemma that we can choose a meridian disk D 2 of C 2 with D 2 A 2 =. This implies that D 1 D 2 =. Hence (C 1, C 2 ; S) is weakly reducible Haken s theorem. In this subsection, we prove the following. Theorem Let (C 1, C 2 ; S) be a Heegaard splitting of (M; 1 M, 2 M).
12 12 TOSHIO SAITO, MARTIN SCHARLEMANN AND JENNIFER SCHULTENS (1) If M is reducible, then (C 1, C 2 ; S) is reducible or C i is reducible for i = 1 or 2. (2) If M is reducible, then (C 1, C 2 ; S) is reducible. Note that the statement (1) of Theorem is called Haken s theorem and proved by Haken [4], and the statement (2) of Theorem is proved by Casson and Gordon [1]. We first prove the following proposition, whose statement is weaker than that of Theorem 3.2.1, after showing some lemmas. Proposition If M is reducible or reducible, then (C 1, C 2 ; S) is reducible, reducible, or C i is reducible for i = 1 or 2. We give a proof of Proposition by using Otal s idea (cf. [11]) of viewing the Heegaard splittings as a graph in the three dimensional space. Edge slides of graphs. Let Γ be a finite graph in a 3manifold M. Choose an edge σ of Γ. Let p 1 and p 2 be the vertices of Γ incident to σ. Set Γ = Γ \σ. Here, we may suppose that σ η( Γ; M) consists of two points, say p 1 and p 2, and that cl(σ \ (p 1 p 2 )) consists of α 0, α 1 and α 2 with α 0 = p 1 p 2, α 1 = p 1 p 1 and α 2 = p 2 p 2 (cf. Figure 13). σ p 1 p 2 p 1 p 2 Figure 13 Take a path γ on η( Γ; M) with γ p 1. Let σ be an arc obtained from γ α 0 α 2 by adding a straight short arc in η( Γ; M) connecting the endpoint of γ other than p 1 and a point p 1 in the interior of an edge of Γ (cf. Figure 14). Let Γ be a graph obtained from Γ σ by adding p 1 as a vertex. Then we say that Γ is obtained from Γ by an edge slide on σ. γ α 0 { }} { p 1 p 2 p 1 Figure 14 If p 1 is a trivalent vertex, then it is natural for us not to regard p 1 as a vertex of Γ. Particularly, the deformation of Γ which is depicted as in Figure 15 is realized by an edge slide and an isotopy. This deformation is called a Whitehead move.
13 LECTURE NOTES ON GENERALIZED HEEGAARD SPLITTINGS 13 Figure 15 A Proof of Proposition Let Σ be a spine of C 1. Note that η( C 1 Σ;M) is obtained from regular neighborhoods of C 1 and the vertices of Σ by attaching 1handles corresponding to the edges of Σ. Set Σ η = η(σ;m). The notation h 0 v, called a vertex of Σ η, means a regular neighborhood of a vertex v of Σ. Also, the notation h 1 σ, called an edge of Σ η, means a 1handle corresponding to an edge σ of Σ. Let = D 1 D k be a minimal complete meridian system of C 2. Let P be a reducing 2sphere or a reducing disk of M. If P is a reducing disk, we may assume that P C 2 by changing subscripts. We may assume that P intersects Σ and transversely. Set Γ = P (Σ η ). We note that Γ is a union of disks P Σ η and a union of arcs and loops P in P. We choose P, Σ and so that the pair ( P Σ, P ) is minimal with respect to lexicographic order. Lemma Each component of P is an arc. Proof. For some disk component, say D 1, of, suppose that P D 1 has a loop component. Let α be a loop component of P D 1 which is innermost in D 1, and let δ α be an innermost disk for α. Let δ α be a disk in P with δ α = α. Set P = (P \δ α ) δ α if P is a reducing disk, or set P = (P \δ α ) δ α and P = δ α δ α if P is a reducing 2sphere. If P is a reducing disk, then P is also a reducing disk. If P is a reducing 2sphere, then either P or P, say P, is a reducing 2 sphere. Moreover, we can isotope P so that ( P Σ, P ) < ( P Σ, P ). This contradicts the minimality of ( P Σ, P ). By Lemma 3.2.3, we can regard Γ as a graph in P which consists of fatvertices P Σ η and edges P. An edge of the graph Γ is called a loop if the edge joins a fatvertex of Γ to itself, and a loop is said to be inessential if the loop cuts off a disk from cl(p \ Σ η ) whose interior is disjoint from Γ Σ η. Lemma Γ does not contain an inessential loop. Proof. Suppose that Γ contains an inessential loop µ. Then µ cuts off a disk δ µ from cl(p \Σ η ) such that the interior of δ µ is disjoint from Γ Σ η (cf. Figure 16). We may assume that δ µ = δ µ D 1. Then µ cuts D 1 into two disks D 1 and D 1 (cf. Figure 17).
14 14 TOSHIO SAITO, MARTIN SCHARLEMANN AND JENNIFER SCHULTENS an inessential monogon Figure 16 D 1 P P δ µ D 1 Figure 17 Let C 2 be the component, which is obtained by cutting C 2 along, such that C 2 contains δ µ. Let D 1 + be the copy of D 1 in C 2 with D+ 1 δ µ and D1 the other copy of D 1. Note that C 2 is a 3ball or a (a component of C 2 ) [0, 1]. This shows that there is a disk δ µ in +C 2 such that δ µ = δ µ and δ µ δ µ bounds a 3ball in C 2. Note that δ µ D 1 +. By changing superscripts, if necessary, we may assume that δ µ D 1 (cf. Figure 18). D + 1 { }} { D 1 D 1 + C 2 } {{ } δ µ δ µ Figure 18
15 LECTURE NOTES ON GENERALIZED HEEGAARD SPLITTINGS 15 Set D 0 = δ µ D 1 if δ µ D 1, and D 0 = δ µ D 1 if δ µ D 1 =. We may regard D 0 as a disk properly embedded in C 2. Set = D 0 D 2 D k. Then we see that is a minimal complete meridian system of C 2. We can further isotope D 0 slightly so that P < P. This contradicts the minimality of ( P Σ, P ). A fatvertex of Γ is said to be isolated if there are no edges of Γ adjacent to the fatvertex (cf. Figure 19). an isolated fatvertex Figure 19 Lemma If Γ has an isolated fatvertex, then (C 1, C 2 ; S) is reducible or reducible. Proof. Suppose that there is an isolated fatvertex D v of Γ. Recall that D v is a component of P Σ η which is a meridian disk of C 1. Note that D v is disjoint from (cf. Figure 20). P D v Figure 20 Let C 2 be the component obtained by cutting C 2 along such that C 2 contains D v. If D v bounds a disk D v in C 2, then D v and D v indicates the reducibility of (C 1, C 2 ; S). Otherwise, C 2 is a (a closed orientable surface) [0, 1], and D v is a boundary component of a spanning annulus in C 2 (and hence C 2 ). Hence we see that (C 1, C 2 ; S) is reducible. Lemma Suppose that no fatvertices of Γ are isolated. Then each fatvertex of Γ is a base of a loop.
16 16 TOSHIO SAITO, MARTIN SCHARLEMANN AND JENNIFER SCHULTENS the following three cases.?? D w D w D w D w γ D w δ γ D w D D w D w 1 Figure 21 Proof. Suppose that there is a fatvertex D w of Γ which is not a base of a loop. Since no fatvartices of Γ are isolated, there is an edge of Γ adjacent to D w. Let σ be the edge of Σ with h 1 σ D w. (Recall that h 1 σ is a 1handle of Σ η corresponding to σ.) Let D be a component of with D h 1 σ. Let C w be a union of the arc components of D P which are adjacent to D w. Let γ be an arc component of C w which is outermost among the components of C w. We call such an arc γ an outermost edge for D w of Γ. Let δ γ D be a disk obtained by cutting D along γ whose interior is disjoint from the edges incident to D w. We call such a disk δ γ an outermost disk for (D w, γ). (Note that δ γ may intersects P transversely (cf. Figure 21).) Let D w ( D w ) be the fatvertex of Γ attached to γ. Then we have Case 1. ( δ γ \ γ) (h 1 σ D). In this case, we can isotope σ along δ γ to reduce P Σ (cf. Figure 22). δ γ D w σ Dw γ δ γ P D w D w σ γ Figure 22 Case 2. ( δ γ \ γ) (h 1 σ D) and D w (h 1 σ D). Let p be the vertex of Σ such that p σ and h 0 p δ γ. Let β be the component of cl(σ \ D w ) which satisfies β p. Then we can slide β along δ γ so that β contains γ (cf. Figure 23). We can further isotope β slightly to reduce P Σ, a contradiction. Case 3. ( δ γ \ γ) (h 1 σ D) and D w (h1 σ D). Let p and p be the endpoints of σ. Let β and β be the components of cl(σ \ (D w D w )) which satisfy p β and p β. Suppose first that p p. Then we can slide β along δ γ so that β contains γ (cf. Figure 24). We can further isotope β slightly to reduce P Σ, a contradiction.
17 LECTURE NOTES ON GENERALIZED HEEGAARD SPLITTINGS 17 σ D w γ p δ γ δ γ D w D w P σ γ Figure 23 δ γ β D w p p β D w σ γ δ γ σ P p p β β D w D w σ γ Figure 24 Suppose next that p = p. In this case, we perform the following operation which is called a broken edge slide. p β β P D w D w w Figure 25 We first add w = D w Σ as a vertex of Σ. Then w cuts σ into two edges β and cl(σ \ β ). Since γ is an outermost edge for D w of Γ, we see that β β (cf. Figure 25). Hence we can slide cl(β \ β ) along δ γ so that cl(β \ β ) contains γ. We now remove the verterx w of Σ, that is, we regard a union of β and cl(σ \β ) as an edge of Σ again. Then we can isotope cl(σ \ β ) slightly to reduce P Σ, a contradiction (cf. Figure 26).
18 18 TOSHIO SAITO, MARTIN SCHARLEMANN AND JENNIFER SCHULTENS p w Figure 26 Proof of Proposition By Lemma 3.2.5, if there is an isolated fatvertex of Γ, then we have the conclusion of Proposition Hence we suppose that no fatvertices of Γ are isolated. Then it follows from Lemma that each fatvertex of Γ is a base of a loop. Let µ be a loop which is innermost in P. Then µ cuts a disk δ µ from cl(p \ Σ η ). Since µ is essential (cf. Lemma 3.2.4), we see that δ µ contains a fatvertex of Γ. But since µ is innermost, such a fatvertex is not a base of any loop. Hence such a fatvertex is isolated, a contradiction. This completes the proof of Proposition Proof of (1) in Theorem Suppose that M is reducible. Then by Proposition 3.2.2, we see that (C 1, C 2 ; S) is reducible or reducible, or C i is reducible for i = 1 or 2. If (C 1, C 2 ; S) is reducible or C i is reducible for i = 1 or 2, then we are done. So we may assume that C 1 and C 2 are irreducible and that (C 1, C 2 ; S) is reducible. By induction on the genus of the Heegaard surface S, we prove that (C 1, C 2 ; S) is reducible. Suppose that genus(s) = 0. Since C i (i = 1, 2) are irreducible, we see that each of C i (i = 1, 2) is a 3ball. Hence M is the 3sphere and therefore M is irreducible, a contradiction. So we may assume that genus(s) > 0. Let P be a reducing disk of M with P S = 1. By changing subs cripts, if necessary, we may assume that P C 1 = D is a disk and P C 2 = A is a spanning annulus. Suppose that genus(s) = 1. Since C i (i = 1, 2) are irreducible, we see that C i contain no 2sphere components. Since C 1 contains an essential disk D, we see that C 1 = D 2 S 1. Since C 2 contains a spanning annulus A, we see that C 2 = T 2 [0, 1]. It follows that M = D 2 S 1 and hence M is irreducible, a contradiction. Suppose that genus(s) > 1. Let C 1 (C 2 resp.) be the manifold obtained from C 1 (C 2 resp.) by cutting along D (A resp.), and let A + and A be copies of A in C 2. Then we see that C 1 consists of either a compression body or a union of two compression bodies (cf. (6) of Remark 3.1.3). Let C 2 be the manifold obtained from C 2 by attaching 2handles along A+ and A. It follows from Remark that C 2 consists of either a compression body or a union of two compression bodies. Suppose that C 1 consists of a compression body. This implies that C 2 consists of a compression body (cf. Figure 27). We can naturally obtain a homeomorphism + C 1 + C 2 from the homeomorphism + C 1 + C 2. Set + C 1 = + C 2 = S. Then (C 1, C 2 ; S ) is a Heegaard splitting of the 3manifolds M obtained by
19 LECTURE NOTES ON GENERALIZED HEEGAARD SPLITTINGS 19 cutting M along P. Note that genus(s ) = genus(s) 1. Moreover, by using innermost disk arguments, we see that M is also reducible. A D S C 1 C 2 S C 1 C 2 Figure 27 Claim. (1) If C 1 is reducible, then C 1 is reducible. (2) If C 2 is reducible, then one of the following holds. (a) C 2 is reducible. (b) The component of C 2 intersecting A is a torus, say T. Proof. Exercise Recall that we assume that C i (i = 1, 2) are irreducible. Hence it follows from (1) of the claim that C 1 is irreducible. Also it follows from (2) of the claim that either (I) C 2 is irreducible or (II) C 2 is reducible and the condition (b) of (2) in Claim 1 holds. Suppose that the condition (I) holds. Then by induction on the genus of a Heegaard surface, (C 1, C 2 ; S ) is reducible, i.e., there are meridian disks D 1 and D 2 of C 1 and C 2 respectively with D 1 = D 2. Note that this implies that C i (i = 1, 2) are nontrivial. Let α + and α be the cocores of the 2handles attached to C 2. Then we see that α+ α is vertical in C 2. It follows from Lemma that we may assume that D 2 (α + α ) =, i.e., D 2 is disjoint from the 2 handles. Hence the pair of disks D 1 and D 2 survives when we restore C 1 and C 2 from C 1 and C 2 respectively. This implies that (C 1, C 2 ; S) is reducible and hence we obtain the conclusion (1) of Theorem Suppose that the condition (II) holds. Then it follows from (2) of Remark that there is a separating disk E 2 in C 2 such that E 2 is disjoint from A and that E 2 cuts off T 2 [0, 1] from C 2 with T 2 {0} = T. Let l be a loop in S (T 2 {1}) which intersects A S(= A S = D) in a single point. Let E 1 be a disk properly embedded in C 1 which is obtained from two parallel copies of D by adding a band along l \ (the product region between the parallel disks). Since genus(s) > 1, we see that E 1 is a separating meridian disk of C 1. Since E 1 is isotopic to E 2, we see that (C 1, C 2 ; S) is reducible. Hence we obtain the conclusion (1) of Theorem The case that C 1 is a union of two compression bodies is treated analogously, and we leave the proof for this case to the reader (Exercise 3.2.8).
20 20 TOSHIO SAITO, MARTIN SCHARLEMANN AND JENNIFER SCHULTENS Exercise Show the claim in the proof of Theorem Exercise Prove that the conclusion (1) of Theorem holds in case that C 1 consists of two compression bodies. Proof of (2) in Theorem Suppose that M is reducible. If (C 1, C 2 ; S) is reducible, then we are done. Let Ĉi be the compression body obtained by attaching 3balls to the 2sphere boundary components of C i (i = 1, 2). Set M = Ĉ 1 Ĉ2. Then M is also reducible. Then it follows from (1) of Remark and Proposition that (Ĉ1, Ĉ2; S) is reducible or reducible. If (Ĉ1, Ĉ2; S) is reducible, then we see that (C 1, C 2 ; S) is also reducible. Hence we may assume that (Ĉ1, Ĉ2; S) is reducible. By induction on the genus of a Heegaard surface, we prove that (C 1, C 2 ; S) is reducible. Let P be a reducing 2sphere of M with P S = 1. For each i = 1 and 2, set D i = P Ĉi, and let Ĉ i be the manifold obtained by cutting Ĉi along D i, and let D + i and D i be copies of D i in Ĉ i. Then each of Ĉ i (i = 1, 2) is either (1) a compression body if D i is nonseparating in Ĉi or (2) a union of two compression bodies if D i is separating in Ĉi. Note that we can naturally obtain a homeomorphism + Ĉ 1 +Ĉ 2 from the homeomorphism + Ĉ 1 + Ĉ 2. Set M = Ĉ 1 Ĉ 2 and +Ĉ 1 = +Ĉ 2 = S. Then (Ĉ 1, Ĉ 2 ; S ) is either (1) a Heegaard splitting or (2) a union of two Heegaard splittings (cf. Figure 28). D 1 S Ĉ 1 Ĉ 2 D 2 S Ĉ 1 Ĉ 2 Figure 28 By innermost disk arguments, we see that there is a reducing disk of M disjoint from P. This implies that a component of M is reducible and hence one of the Heegaard splittings of (Ĉ 1, Ĉ 2; S ) is reducible. By induction on the genus of a Heegaard surface, we see that (Ĉ1, Ĉ2; S) is reducible. Therefore (C 1, C 2 ; S) is also reducible and hence we have (2) of Theorem Waldhausen s theorem. We devote this subsection to a simplified proof of the following theorem originally due to Waldhausen [21]. To prove the theorem, we exploit Gabai s idea of thin position (cf. [3]), Johannson s technique (cf. [6]) and Otal s idea (cf. [11]) of viewing the Heegaard splittings as a graph in the three dimensional space. Theorem (Waldhausen). Any Heegaard splitting of S 3 is standard, i.e., is obtained from the trivial Heegaard splitting by stabilization.
TOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS
TOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS OSAMU SAEKI Dedicated to Professor Yukio Matsumoto on the occasion of his 60th birthday Abstract. We classify singular fibers of C stable maps of orientable
More informationSurface bundles over S 1, the Thurston norm, and the Whitehead link
Surface bundles over S 1, the Thurston norm, and the Whitehead link Michael Landry August 16, 2014 The Thurston norm is a powerful tool for studying the ways a 3manifold can fiber over the circle. In
More informationA Short Proof that Compact 2Manifolds Can Be Triangulated
Inventiones math. 5, 160162 (1968) A Short Proof that Compact 2Manifolds Can Be Triangulated P. H. DOYLE and D. A. MORAN* (East Lansing, Michigan) The result mentioned in the title of this paper was
More informationFinite covers of a hyperbolic 3manifold and virtual fibers.
Claire Renard Institut de Mathématiques de Toulouse November 2nd 2011 Some conjectures. Let M be a hyperbolic 3manifold, connected, closed and oriented. Theorem (Kahn, Markovic) The fundamental group
More informationCS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992
CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992 Original Lecture #6: 28 January 1991 Topics: Triangulating Simple Polygons
More informationA program to search for homotopy 3 spheres
A program to search for homotopy 3 spheres Michael Greene Colin Rourke Mathematics Institute University of Warwick Coventry CV4 7AL, UK Email: mtg@maths.warwick.ac.uk and cpr@maths.warwick.ac.uk Abstract
More informationFIXED POINT SETS OF FIBERPRESERVING MAPS
FIXED POINT SETS OF FIBERPRESERVING MAPS Robert F. Brown Department of Mathematics University of California Los Angeles, CA 90095 email: rfb@math.ucla.edu Christina L. Soderlund Department of Mathematics
More informationCHAPTER 1 BASIC TOPOLOGY
CHAPTER 1 BASIC TOPOLOGY Topology, sometimes referred to as the mathematics of continuity, or rubber sheet geometry, or the theory of abstract topological spaces, is all of these, but, above all, it is
More informationFiber sums of genus 2 Lefschetz fibrations
Proceedings of 9 th Gökova GeometryTopology Conference, pp, 1 10 Fiber sums of genus 2 Lefschetz fibrations Denis Auroux Abstract. Using the recent results of Siebert and Tian about the holomorphicity
More informationPSEUDOARCS, PSEUDOCIRCLES, LAKES OF WADA AND GENERIC MAPS ON S 2
PSEUDOARCS, PSEUDOCIRCLES, LAKES OF WADA AND GENERIC MAPS ON S 2 Abstract. We prove a BrucknerGarg type theorem for the fiber structure of a generic map from a continuum X into the unit interval I. We
More information1 Local Brouwer degree
1 Local Brouwer degree Let D R n be an open set and f : S R n be continuous, D S and c R n. Suppose that the set f 1 (c) D is compact. (1) Then the local Brouwer degree of f at c in the set D is defined.
More informationSingular fibers of stable maps and signatures of 4 manifolds
359 399 359 arxiv version: fonts, pagination and layout may vary from GT published version Singular fibers of stable maps and signatures of 4 manifolds OSAMU SAEKI TAKAHIRO YAMAMOTO We show that for a
More informationGraph Theory Notes. Vadim Lozin. Institute of Mathematics University of Warwick
Graph Theory Notes Vadim Lozin Institute of Mathematics University of Warwick 1 Introduction A graph G = (V, E) consists of two sets V and E. The elements of V are called the vertices and the elements
More informationComplete graphs whose topological symmetry groups are polyhedral
Algebraic & Geometric Topology 11 (2011) 1405 1433 1405 Complete graphs whose topological symmetry groups are polyhedral ERICA FLAPAN BLAKE MELLOR RAMIN NAIMI We determine for which m the complete graph
More informationMonotone Partitioning. Polygon Partitioning. Monotone polygons. Monotone polygons. Monotone Partitioning. ! Define monotonicity
Monotone Partitioning! Define monotonicity Polygon Partitioning Monotone Partitioning! Triangulate monotone polygons in linear time! Partition a polygon into monotone pieces Monotone polygons! Definition
More informationOn end degrees and infinite cycles in locally finite graphs
On end degrees and infinite cycles in locally finite graphs Henning Bruhn Maya Stein Abstract We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel
More informationPOLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS
POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS RUSS WOODROOFE 1. Unique Factorization Domains Throughout the following, we think of R as sitting inside R[x] as the constant polynomials (of degree 0).
More informationAssignment 7; Due Friday, November 11
Assignment 7; Due Friday, November 9.8 a The set Q is not connected because we can write it as a union of two nonempty disjoint open sets, for instance U = (, 2) and V = ( 2, ). The connected subsets are
More informationSOLUTIONS TO ASSIGNMENT 1 MATH 576
SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts
More informationGeometry and Topology from Point Cloud Data
Geometry and Topology from Point Cloud Data Tamal K. Dey Department of Computer Science and Engineering The Ohio State University Dey (2011) Geometry and Topology from Point Cloud Data WALCOM 11 1 / 51
More informationPlanar Tree Transformation: Results and Counterexample
Planar Tree Transformation: Results and Counterexample Selim G Akl, Kamrul Islam, and Henk Meijer School of Computing, Queen s University Kingston, Ontario, Canada K7L 3N6 Abstract We consider the problem
More informationTwo fixed points can force positive entropy of a homeomorphism on a hyperbolic surface (Not for publication)
Two fixed points can force positive entropy of a homeomorphism on a hyperbolic surface (Not for publication) Pablo Lessa May 29, 2012 1 Introduction A common theme in lowdimensional dynamical systems
More informationThe Goldberg Rao Algorithm for the Maximum Flow Problem
The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }
More informationON SELFINTERSECTIONS OF IMMERSED SURFACES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 12, December 1998, Pages 3721 3726 S 00029939(98)044566 ON SELFINTERSECTIONS OF IMMERSED SURFACES GUISONG LI (Communicated by Ronald
More informationTHE 0/1BORSUK CONJECTURE IS GENERICALLY TRUE FOR EACH FIXED DIAMETER
THE 0/1BORSUK CONJECTURE IS GENERICALLY TRUE FOR EACH FIXED DIAMETER JONATHAN P. MCCAMMOND AND GÜNTER ZIEGLER Abstract. In 1933 Karol Borsuk asked whether every compact subset of R d can be decomposed
More informationChapter 1. SigmaAlgebras. 1.1 Definition
Chapter 1 SigmaAlgebras 1.1 Definition Consider a set X. A σ algebra F of subsets of X is a collection F of subsets of X satisfying the following conditions: (a) F (b) if B F then its complement B c is
More informationA REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries
Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do
More informationHomework Exam 1, Geometric Algorithms, 2016
Homework Exam 1, Geometric Algorithms, 2016 1. (3 points) Let P be a convex polyhedron in 3dimensional space. The boundary of P is represented as a DCEL, storing the incidence relationships between the
More informationThe Ideal Class Group
Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned
More informationChapter 6 Planarity. Section 6.1 Euler s Formula
Chapter 6 Planarity Section 6.1 Euler s Formula In Chapter 1 we introduced the puzzle of the three houses and the three utilities. The problem was to determine if we could connect each of the three utilities
More informationISOPERIMETRIC INEQUALITIES FOR THE HANDLEBODY GROUPS
ISOPERIMETRIC INEQUALITIES FOR THE HANDLEBODY GROUPS URSULA HAMENSTÄDT AND SEBASTIAN HENSEL Abstract. We show that the mapping class group of a handlebody V of genus at least 2 has a Dehn function of at
More informationOdd induced subgraphs in graphs of maximum degree three
Odd induced subgraphs in graphs of maximum degree three David M. Berman, Hong Wang, and Larry Wargo Department of Mathematics University of New Orleans New Orleans, Louisiana, USA 70148 Abstract A longstanding
More informationA 2factor in which each cycle has long length in clawfree graphs
A factor in which each cycle has long length in clawfree graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science
More informationCabling and transverse simplicity
Annals of Mathematics, 162 (2005), 1305 1333 Cabling and transverse simplicity By John B. Etnyre and Ko Honda Abstract We study Legendrian knots in a cabled knot type. Specifically, given a topological
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More information8.1 Examples, definitions, and basic properties
8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A kform ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)form σ Ω k 1 (M) such that dσ = ω.
More informationDO NOT REDISTRIBUTE THIS SOLUTION FILE
Professor Kindred Math 04 Graph Theory Homework 7 Solutions April 3, 03 Introduction to Graph Theory, West Section 5. 0, variation of 5, 39 Section 5. 9 Section 5.3 3, 8, 3 Section 7. Problems you should
More informationTreerepresentation of set families and applications to combinatorial decompositions
Treerepresentation of set families and applications to combinatorial decompositions BinhMinh BuiXuan a, Michel Habib b Michaël Rao c a Department of Informatics, University of Bergen, Norway. buixuan@ii.uib.no
More informationPartitioning edgecoloured complete graphs into monochromatic cycles and paths
arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edgecoloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political
More informationA CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE
A CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE DANIEL A. RAMRAS In these notes we present a construction of the universal cover of a path connected, locally path connected, and semilocally simply
More informationLabeling outerplanar graphs with maximum degree three
Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationORIENTATIONS. Contents
ORIENTATIONS Contents 1. Generators for H n R n, R n p 1 1. Generators for H n R n, R n p We ended last time by constructing explicit generators for H n D n, S n 1 by using an explicit nsimplex which
More informationLattice Point Geometry: Pick s Theorem and Minkowski s Theorem. Senior Exercise in Mathematics. Jennifer Garbett Kenyon College
Lattice Point Geometry: Pick s Theorem and Minkowski s Theorem Senior Exercise in Mathematics Jennifer Garbett Kenyon College November 18, 010 Contents 1 Introduction 1 Primitive Lattice Triangles 5.1
More information5. GRAPHS ON SURFACES
. GRPHS ON SURCS.. Graphs graph, by itself, is a combinatorial object rather than a topological one. But when we relate a graph to a surface through the process of embedding we move into the realm of topology.
More informationLecture 17 : Equivalence and Order Relations DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion
More informationCharacterizations of Arboricity of Graphs
Characterizations of Arboricity of Graphs Ruth Haas Smith College Northampton, MA USA Abstract The aim of this paper is to give several characterizations for the following two classes of graphs: (i) graphs
More informationA simple existence criterion for normal spanning trees in infinite graphs
1 A simple existence criterion for normal spanning trees in infinite graphs Reinhard Diestel Halin proved in 1978 that there exists a normal spanning tree in every connected graph G that satisfies the
More informationOn planar regular graphs degree three without Hamiltonian cycles 1
On planar regular graphs degree three without Hamiltonian cycles 1 E. Grinbergs Computing Centre of Latvian State University Abstract. Necessary condition to have Hamiltonian cycle in planar graph is given.
More informationMathematical Research Letters 1, 249 255 (1994) MAPPING CLASS GROUPS ARE AUTOMATIC. Lee Mosher
Mathematical Research Letters 1, 249 255 (1994) MAPPING CLASS GROUPS ARE AUTOMATIC Lee Mosher Let S be a compact surface, possibly with the extra structure of an orientation or a finite set of distinguished
More informationA Turán Type Problem Concerning the Powers of the Degrees of a Graph
A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of HaifaORANIM, Tivon 36006, Israel. AMS Subject Classification:
More information136 CHAPTER 4. INDUCTION, GRAPHS AND TREES
136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics
More informationCycles in a Graph Whose Lengths Differ by One or Two
Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDEBERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS
More information6. GRAPH AND MAP COLOURING
6. GRPH ND MP COLOURING 6.1. Graph Colouring Imagine the task of designing a school timetable. If we ignore the complications of having to find rooms and teachers for the classes we could propose the following
More informationMath 317 HW #7 Solutions
Math 17 HW #7 Solutions 1. Exercise..5. Decide which of the following sets are compact. For those that are not compact, show how Definition..1 breaks down. In other words, give an example of a sequence
More informationARC AND CURVE GRAPHS FOR INFINITETYPE SURFACES
ARC AND CURVE GRAPHS FOR INFINITETYPE SURFACES JAVIER ARAMAYONA, ARIADNA FOSSAS & HUGO PARLIER Abstract. We study arc graphs and curve graphs for surfaces of infinite topological type. First, we define
More informationWhen is a graph planar?
When is a graph planar? Theorem(Euler, 1758) If a plane multigraph G with k components has n vertices, e edges, and f faces, then n e + f = 1 + k. Corollary If G is a simple, planar graph with n(g) 3,
More informationGraphs without proper subgraphs of minimum degree 3 and short cycles
Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract
More information1 Plane and Planar Graphs. Definition 1 A graph G(V,E) is called plane if
Plane and Planar Graphs Definition A graph G(V,E) is called plane if V is a set of points in the plane; E is a set of curves in the plane such that. every curve contains at most two vertices and these
More information1. Relevant standard graph theory
Color identical pairs in 4chromatic graphs Asbjørn Brændeland I argue that, given a 4chromatic graph G and a pair of vertices {u, v} in G, if the color of u equals the color of v in every 4coloring
More informationRemoving Even Crossings
EuroComb 2005 DMTCS proc. AE, 2005, 105 110 Removing Even Crossings Michael J. Pelsmajer 1, Marcus Schaefer 2 and Daniel Štefankovič 2 1 Department of Applied Mathematics, Illinois Institute of Technology,
More informationSets of Fibre Homotopy Classes and Twisted Order Parameter Spaces
Communications in Mathematical Physics ManuscriptNr. (will be inserted by hand later) Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces Stefan BechtluftSachs, Marco Hien Naturwissenschaftliche
More informationOn Integer Additive SetIndexers of Graphs
On Integer Additive SetIndexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A setindexer of a graph G is an injective setvalued function f : V (G) 2 X such that
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationMincost flow problems and network simplex algorithm
Mincost flow problems and network simplex algorithm The particular structure of some LP problems can be sometimes used for the design of solution techniques more efficient than the simplex algorithm.
More informationBest Monotone Degree Bounds for Various Graph Parameters
Best Monotone Degree Bounds for Various Graph Parameters D. Bauer Department of Mathematical Sciences Stevens Institute of Technology Hoboken, NJ 07030 S. L. Hakimi Department of Electrical and Computer
More informationNOTES ON MEASURE THEORY. M. Papadimitrakis Department of Mathematics University of Crete. Autumn of 2004
NOTES ON MEASURE THEORY M. Papadimitrakis Department of Mathematics University of Crete Autumn of 2004 2 Contents 1 σalgebras 7 1.1 σalgebras............................... 7 1.2 Generated σalgebras.........................
More informationIntroduction to Topology
Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................
More informationHandout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs
MCS236: Graph Theory Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010 Chapter 7: Digraphs Strong Digraphs Definitions. A digraph is an ordered pair (V, E), where V is the set
More informationCombinatorics: The Fine Art of Counting
Combinatorics: The Fine Art of Counting Week 9 Lecture Notes Graph Theory For completeness I have included the definitions from last week s lecture which we will be using in today s lecture along with
More informationCOMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH. 1. Introduction
COMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the HigmanSims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact
More information(a) (b) (c) Figure 1 : Graphs, multigraphs and digraphs. If the vertices of the leftmost figure are labelled {1, 2, 3, 4} in clockwise order from
4 Graph Theory Throughout these notes, a graph G is a pair (V, E) where V is a set and E is a set of unordered pairs of elements of V. The elements of V are called vertices and the elements of E are called
More informationarxiv:1203.1525v1 [math.co] 7 Mar 2012
Constructing subset partition graphs with strong adjacency and endpoint count properties Nicolai Hähnle haehnle@math.tuberlin.de arxiv:1203.1525v1 [math.co] 7 Mar 2012 March 8, 2012 Abstract Kim defined
More information1 Definitions. Supplementary Material for: Digraphs. Concept graphs
Supplementary Material for: van Rooij, I., Evans, P., Müller, M., Gedge, J. & Wareham, T. (2008). Identifying Sources of Intractability in Cognitive Models: An Illustration using Analogical Structure Mapping.
More informationAll trees contain a large induced subgraph having all degrees 1 (mod k)
All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New
More informationGROUPS ACTING ON A SET
GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for
More information3. Linear Programming and Polyhedral Combinatorics
Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the
More informationRemoving even crossings
Removing even crossings Michael J. Pelsmajer a, Marcus Schaefer b, Daniel Štefankovič c a Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA b Department of Computer
More informationeach college c i C has a capacity q i  the maximum number of students it will admit
n colleges in a set C, m applicants in a set A, where m is much larger than n. each college c i C has a capacity q i  the maximum number of students it will admit each college c i has a strict order i
More informationAn Introduction to Braid Theory
An Introduction to Braid Theory Maurice Chiodo November 4, 2005 Abstract In this paper we present an introduction to the theory of braids. We lay down some clear definitions of a braid, and proceed to
More informationClassification of Cartan matrices
Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms
More informationUnicyclic Graphs with Given Number of Cut Vertices and the Maximal Merrifield  Simmons Index
Filomat 28:3 (2014), 451 461 DOI 10.2298/FIL1403451H Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Unicyclic Graphs with Given Number
More informationLarge induced subgraphs with all degrees odd
Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order
More informationCourse 221: Analysis Academic year , First Semester
Course 221: Analysis Academic year 200708, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................
More informationSubsets of Euclidean domains possessing a unique division algorithm
Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationCOLORED GRAPHS AND THEIR PROPERTIES
COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloring
More informationColoring Eulerian triangulations of the projective plane
Coloring Eulerian triangulations of the projective plane Bojan Mohar 1 Department of Mathematics, University of Ljubljana, 1111 Ljubljana, Slovenia bojan.mohar@unilj.si Abstract A simple characterization
More informationIntroduction to Graph Theory
Introduction to Graph Theory Allen Dickson October 2006 1 The Königsberg Bridge Problem The city of Königsberg was located on the Pregel river in Prussia. The river divided the city into four separate
More information1 Polyhedra and Linear Programming
CS 598CSC: Combinatorial Optimization Lecture date: January 21, 2009 Instructor: Chandra Chekuri Scribe: Sungjin Im 1 Polyhedra and Linear Programming In this lecture, we will cover some basic material
More informationLecture 7: Approximation via Randomized Rounding
Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining
More information4 Basics of Trees. Petr Hliněný, FI MU Brno 1 FI: MA010: Trees and Forests
4 Basics of Trees Trees, actually acyclic connected simple graphs, are among the simplest graph classes. Despite their simplicity, they still have rich structure and many useful application, such as in
More informationFIBRATION SEQUENCES AND PULLBACK SQUARES. Contents. 2. Connectivity and fiber sequences. 3
FIRTION SEQUENES ND PULLK SQURES RY MLKIEWIH bstract. We lay out some foundational facts about fibration sequences and pullback squares of topological spaces. We pay careful attention to connectivity ranges
More informationZachary Monaco Georgia College Olympic Coloring: Go For The Gold
Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various
More informationWeek 2 Polygon Triangulation
Week 2 Polygon Triangulation What is a polygon? Last week A polygonal chain is a connected series of line segments A closed polygonal chain is a polygonal chain, such that there is also a line segment
More informationIntersection Dimension and Maximum Degree
Intersection Dimension and Maximum Degree N.R. Aravind and C.R. Subramanian The Institute of Mathematical Sciences, Taramani, Chennai  600 113, India. email: {nraravind,crs}@imsc.res.in Abstract We show
More informationMidterm Practice Problems
6.042/8.062J Mathematics for Computer Science October 2, 200 Tom Leighton, Marten van Dijk, and Brooke Cowan Midterm Practice Problems Problem. [0 points] In problem set you showed that the nand operator
More informationOberwolfach Preprints
Oberwolfach Preprints OWP 201001 There is a Unique Real Tight Contact 3Ball Mathematisches Forschungsinstitut Oberwolfach ggmbh Oberwolfach Preprints (OWP) ISSN 18647596 Oberwolfach Preprints (OWP)
More informationDisjoint Compatible Geometric Matchings
Disjoint Compatible Geometric Matchings Mashhood Ishaque Diane L. Souvaine Csaba D. Tóth Abstract We prove that for every even set of n pairwise disjoint line segments in the plane in general position,
More informationINCIDENCEBETWEENNESS GEOMETRY
INCIDENCEBETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full
More information