Angewandte Mathematik und. Informatik. Universitat zu Koln. Oriented Matroids from Wild Spheres. W. Hochstattler

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1 Angewandte Mathematik und Informatik Universitat zu Koln Report No Oriented Matroids from Wild Spheres by W. Hochstattler 1995 W. Hochstattler Universitat zu Koln Weyertal D{50931 Koln Germany Telefon (0221) 470{6012 e{mail

2 1991 Mathematics Subject Classication: 05B35, 06B30, 51D20, 52B30, 52B40, 57N50 Keywords: Oriented Matroids, Wild Spheres, Sphere Systems, Topology, Matroids, Geometry

3 Oriented Matroids from Wild Spheres W. Hochstattler November 21, 1995 Abstract In a recent article [5] we gave a lattice-theoretical characterization of oriented matroids in terms of the zero-map. In this paper we derive from that characterization a generalization of one direction of the Topological Representation Theorem using hyperspheres which may be wildly embedded, thus solving a problem presented by A. Bjorner (see [1] Exercise 5.6.c). 1 Introduction Matroids, introduced in the 30s of this century, were modelled as a combinatorial abstraction of linear dependency in a vector space. Though they, originally, were supposed to reect a geometric situation their axioms are weak enough to allow structures quite dierent from point congurations in Euclidean space, in particular they encode the combinatoric structure of projective geometries over nite elds. As a consequence they are frequently considered to be \just" set systems. When oriented matroids were introduced in [7], [2] the former of these articles proved a close connection to topology: Every oriented matroid can be realized as a cell decomposition of a sphere induced by codimension 1 spheres such that the signed covectors of the oriented matroid correspond bijectively to the cells of the complex. Thus, oriented matroids always encode a geometric situation. On the other hand certain systems of hyperspheres on a sphere were shown to induce oriented matroids (see [7] and [9]). Both of them impose a \niceness" condition on the spheres, namely that their embedding always can be extended to a global homeomorphism of the sphere. As they are nice they induce a regular cell decomposition of the sphere which is the cell complex realizing the poset of the signed covectors of an oriented matroid. 1

4 Basics and Denitions 2 As by Jordan Brouwer Separation Theorem arbitrarily wild spheres have the combinatorial property to separate the sphere, there does not seem to be a combinatorial reason why the topological property of \being nice" should be necessary for a collection of hyperspheres to encode the combinatorial structure of an oriented matroid. In this paper we will prove that it actually is not. We will show that any niceness condition is redundant. As a corollary we derive an even weaker denition of a sphere system. Finally, we will restate the results in a lattice theory language to derive a characterization of oriented matroids in terms of shellability. We will prove that oriented matroid posets are given as matroidal nested systems of spheres satisfying a certain symmetry property. The paper is organized as follows: In Section 2 we will introduce sphere systems and some tools, in Section 3 we will present our main result and its proof. Section 4 proves that the axiom system for sphere systems can be weakened without loosing structure and in Section 5 we will present a lattice theoretical version of that result. We assume some familiarity with matroids and oriented matroids, standard references are [11] and [1]. Furthermore we will need some basic facts in elementary topology on an undergraduate course level (see e.g. [8]). 2 Basics and Denitions Why is it natural to consider sphere systems in the context of oriented matroids? Oriented matroids may be viewed as a combinatorial abstraction of ane hyperplane arrangements. The diculty with boundary eects of parallel ats as usual can be overcome by adding the hyperplane at innity, thus considering the situation projectively. One common denition of the projective space constructs it from the standard sphere by identifying antipodal points. The resulting space is not orientable which conicts intuition. We can avoid this by not doing the identication and considering the sphere as two-sided oriented projective space with the side eect that vector spaces become spheres. That way hyperplane arrangements \become hypersphere arrangements". Denition 1 Let S n := fx 2 IR n+1 j j kxk = 1g denote the n-dimensional standard sphere. If X S n is homeomorphic to S n?1, we call X a hypersphere of S n. The most important property of a hypersphere on a sphere for our purposes is given by the Jordan-Brouwer Separation Theorem (see e.g. [6]).

5 Basics and Denitions 3 Figure 1: The third step on the way to Alexander's horned sphere Theorem 1 (Jordan-Brouwer Separation Theorem) Any hypersphere H e of S n separates S n into exactly two disjoint domains of which it is the common boundary. Although combinatorially any hypersphere is, therefore, well-behaved, the components may have non-trivial topology (such spheres are called wild). Example 1 A rst example of a sphere wildly embedded into IR 3 was given by Alexander (see e.g [10]). The construction is done by the following iterative process. S 0 is a standard sphere, this sphere grows two horns which almost link to get S 1, each of these horn splits into two ngers which almost link and the pairs of ngers mesh thus forming S 2. Iterating this process we get uniform convergence towards a homeomorphic image S 1 of the sphere. The unbounded component of IR 3 ns 1 has non-trivial topology, it is not simply connected. A sketch of S 3 is given in Figure 1. For notational reasons it will be convenient given a hypersphere of S n to mark one of the two components of its complement. Denition 2 Let H e be a hypersphere of S n. Marking one of the two components of S n nh e to be the positive side H + e we get an oriented hypersphere. Let H? := e Sn n (H e [ H + ). We will use e H0 and H e e synonymously. Let H T = (H e ) e2e be a family of oriented hyperspheres of S n. If A E, we call T f2a H f the at of A. Furthermore we denote by cl (A) = fe 2 E j H e f2a H f g the closure of A. H is called a sphere system if

6 Basics and Denitions 4 (S1) All ats are topological spheres. (S2) The intersection of a at F with any oriented hypersphere H e 6 F, is a hypersphere H e \F of F, separating F into F + H + e and F? H? e. If s : E! f+;?; 0g is some map then we call a pseudo cell. \ e2e H s(e) e Remark 1 Note, that a pseudo cell is not necessarily homeomorphic to ball. The author does not even know whether it is always connected. Considering only hyperspheres which are tamely embedded, i.e. the two components of their complements are simply connected, the above axioms are known to dene a regular CW-complex the poset of which is isomorphic to the big face lattice of an oriented matroid (see [1]). Take a S hypersphere H f of such a tame sphere system and consider a cell Z of H f n e2enf H e. This cell will be in the boundary of two maximal cells T 1 and T 2. Now observe that given a cell decomposition of a standard S n -sphere embedded in IR n the modications of the sphere to get Alexander's horned sphere can be restricted to any maximal cell of that decomposition. Therefore, it is clear that we can have H f grow horns such that say T 2 is not simply connected anymore but neither the topology of any other cell changes nor any boundary relation changes. Nevertheless, it is not very surprising that we get a \pseudo cell complex" with the same poset as before. This will be made more precise later (see Denition 4). In other words it is immediate that there are oriented matroids \from" wild spheres. What we are going to prove in the following is that the above axiom system is strong enough to always guarantee an oriented matroid. Two minors of sphere systems are immediate. Lemma 1 Let H be a sphere system on a nite set E; A E; G = T e2a H e. Then 1. H ja := fh e 2 H j e 2 Ag, 2. H=A := ff(h e \ G) j e 2 E n Ag, where f : G! S d is a suitable homeomorphism are sphere systems. 2

7 Basics and Denitions 5 To prove that the pseudo cells of any sphere system give rise to the signed covectors of an oriented matroid we are going to use our result from [5]. There we showed that a map from a lattice to a geometric lattice is the zero map of an oriented matroid if and only if it is a symmetric antitonic folding. This result is quite technical. Therefore we cannot avoid making the following denitions. Denition 3 Let P be a nite partially ordered set. We say P is connected, if for any two elements a; b 2 P there is a path in the Hasse-Diagram from a to b. We say P is openly connected, if all open intervals with at least two comparable elements are connected. If I = [x; y] is an interval in a poset, the length of I is given by the number of elements in a maximum chain minus one. P has the diamond property, if each interval of length 2 consists of exactly four elements. A nite lattice R is a combinatorial d-manifold[3], if R is openly connected, R has length d + 2, R has the diamond property. If P is a poset and a < b 2 P, then b covers a, if for all c 2 P we have a c b only if a = c or b = c. We denote this by a < b or b > a. If P und R are JD-lattices of nite length and r(1i P ) = r(1i R ) + 1 then an antitonism is a map ' : P n f1i P g! R satisfying (a < b 2 P and b 6= 1I P ) =) '(a) > '(b), (a; b 2 P and a _ b 6= 1I P ) =) '(a _ b) = '(a) ^ '(b). Let P be a combinatorial d-manifold and L be a geometric lattice of rank d + 1. An antitonic folding ' is a surjective antitonism ' : P n f1i P g! L satisfying folding property: If [x; y] L is an interval of length 1 and b 2 '?1 (y), then there are exactly two elements a; a 0 2 P in the preimage of x covering b. In this situation we call the triple (a; b; a 0 ) a fold. local atness: If [x; y] L is an interval of length 2 and b 2 '?1 (y), then the Hasse-Diagram of the set fa 2 P j a > b und '(a) xg is connected. We say an antitonic folding (L; P; ') is symmetric, if for all folds a; 0; a 0 and any path a = a 0 ; a 1 ; : : : ; a n = a 0 in P n f1i P g we have: 80 < p 2 L 90 i n : p '(a i ).

8 Oriented Matroids from Wild Spheres 6 Theorem 2 ([5]) Let (L; P; ') be a symmetric antitonic folding. Then P is the big face lattice of an oriented matroid with ' for its zero-map. 3 Oriented Matroids from Wild Spheres While Theorem 2 actually characterizes reorientation classes of oriented matroids we could actually examine sphere systems without specied orientations. But since we provided each sphere in our system with an orientation the Separation Theorem of Jordan Brouwer immediately yields a unique signed vector X 2 2 E for each point p 2 S n und thus for each pseudo cell, rendering notation easier. Denition 4 A signed vector U on a nite set E is a map U : E! f0; +;?g. Instead of U(e) we will simply write U e. If U and V are signed vectors sep (U; V ) := fe 2 E j U e =?V e 6= 0g and supp (U) := U?1 (f+;?g). On signed vectors we have the partial order P:. X Y :, 8e 2 E : (X e 6= Y e ) X e = 0) We have a canonical right hand side for an antitonism: Lemma 2 The map cl satises the axioms of a matroid closure. Proof. By set inclusion we get an atomic nite lattice on the image of cl : 2 E! 2 E with the lattice operations supfa; Bg = cl (A[B) und inffa; Bg = A \ B = cl (A \ B) for two sets A; B in the image of cl. The rst half of Axiom (S2) implies that in this lattice L the following holds: If e 2 L is atomic and x 2 L such that e 6 x, then e _ x > x. Hence Theorem 3 implies that L is geometric. The lemma follows by the 1-1-correspondence between geometric lattices and simple matroids (see [11] 3.4.1). 2 Theorem 3 (see e.g. [11] Exercise 3.8) A nite, atomic lattice is geometric if and only if 8a 2 L 8 0 < p 2 L : (p 6 a ) a < a _ p): Thus cl provides us with the possibility to construct a map onto a geometric lattice. Before we give its denition we mention the following fact the easy verication of which is left to the reader.

9 Oriented Matroids from Wild Spheres 7 Proposition 1 Let H be a sphere system and X one of its signed vectors. Then cl (E n supp (X)) = E n supp (X). 2 Denition 5 For all X 2 P let '(X) := E n supp (X). By the above proposition ' is a map from some poset onto a geometric lattice. With the next lemma we prove that P is pure dimensional. Lemma 3 Let P be the poset of a sphere system. Then: 1. Each point of the pseudo cell 0 P is in the boundary of every component of S n n S e2e H e. 2. If x 2 S n with signed vector X, then for all Y 2 P: X Y, x is on the boundary of Y: 3. P satises the Jordan-Dedekind chain condition. 4. P is pure dimensional. Proof. 1: We proceed by induction on n = jej. The case n = 1 is clear by Jordan Brouwer Separation Theorem. Let n 2, and p 2 T e2e H e, let be a component of S n n S e2e H e and f 2 E. In the sphere system (H e ) e2enf there is a component 0 of S n n S e2enf H e such that 0. If 0 = we are done. Otherwise the intersection of the boundary of with H f is non-empty and thus contains a component of H f n S e2enf H e. By inductive assumption p is in the boundary of. Since is contained all in the boundary of, the claim follows. 2:and 3: The backward implication of 2) is obvious. Considering the statement of 3) we rst remark that each pseudo cell dierent from 0 P is a relatively open proper subset of the minimal at containing it. Hence the dimension of a pseudo cell is well dened. Instead of 3) we may thus prove that X Y and Y covers X then dim(x) = dim(y )? 1. To verify suciency of 2) and to prove 3) we proceed by induction on n = jsupp (X)j. The case n = 0 is clear from the rst claim of this lemma. Thus let f 2 supp (X) 6= ;. Hence, let X 0 ; Y 0 denote the

10 Oriented Matroids from Wild Spheres 8 pseudo cells of (H e ) e2enf, containing X resp. Y. Necessarily we must have dim(x 0 ) = dim(x) and dim(y 0 ) = dim(y ). If Y 0 still covers X 0 we are done since 2) follows from the fact for all x 2 X there is some suciently small neighborhood intersecting Y 0 only in points of Y. Therefore, assume there is some Z 0 such that X 0 Z 0 Y 0. Since Y covers X no point of Z 0 must be on the same side of H f than any point of X, contradicting the fact that each point of X must be on the boundary of Z 0. Note, that in fact we proved that the rank function of P satises 8X 2 P : r(x) = dim(x) + dim(0 P ). 4: This is immediate from the above and the fact each point which is not an interior point of a full dimensional pseudo cell is in the boundary of some full dimensional pseudo cell. The following is a direct consequence of the denition of a sphere system. Lemma 4 Let H = (H e ) e2e be a sphere system, X a signed vector of the corresponding partial order P and '(X) = A. The partial order of H=A is isomorphic to the order ideal generated by '?1 (A). 2 2 The other minor is also compatible with the partial order. Lemma 5 Let H = (H e ) e2e be a sphere system, X one of its signed vectors and A = '(X). The partial order of H ja is isomorphic to the lter generated by X in P. Proof. It suces to verify that V is a signed vector of H ja if and only if (X; V ) is a signed vector of P (where (X; V ) denotes the signed vector satisfying (X; V ) jsupp (X) = X jsupp (X) and (X; V ) ja = V ). The non-trivial implication is suciency. This is done by induction on n = jej. The case n = 0 is clear, thus T let n > 0. If '(V ja ) 6= ;, then V as well as X are contained in the sphere e2'(v ja ) H e, and the claim follows from inductive hypothesis and Lemma 4. So we may assume '(V ja ) = ; and V 6= 0 and choose a signed vector W of H ja such that W V and V covers W. Let f 2 supp (V ) n supp (W ): By inductive hypothesis applied to H=f and Lemma 4, (X; W ) is a signed vector of P. By Jordan Brouwer Separation Theorem S n is separated into two components by H f. The point set corresponding to (X; W ) is a full dimensional open subset of the point

11 Oriented Matroids from Wild Spheres 9 set with vector W in H ja, which is in the boundary of V (Lemma 4). In any neighborhood of a point in (X; W ) there have to be points of V with coordinates (X; V ) with respect to the original system. 2 Recall that the denition of an antitonism requires the domain of the map to be a lattice where the top element has been removed. Therefore we have to verify that P [ 1I P is a lattice. First we show the existence of non-trivial joins, i.e. those which are not equal to 1I P. Lemma 6 Assume X; Y are pseudo cells of a sphere system H such that sep (X; Y ) = ;. Then there is a pseudo cell Z in the sphere system satisfying Z jsupp (X) = X jsupp (X), Z jsupp (Y ) = Y jsupp (Y ) and supp (Z) = supp (X) [ supp (Y ). Proof. We proceed by induction on n = jej the case n = 1 being trivial. Thus let n > 1. If supp (X) [ supp (Y ) E the claim follows immediately from inductive assumption and Lemma 4. We may assume that there is an f 2 E : X f 6= 0 = Y f. In the sphere system H n f all points of Y belong to the same pseudo cell Y 0. By inductive assumtion applied to a H n f there is a pseudo cell Z ~ which is as desired with respect to X n f and Y 0. The claim now again follows from the fact that any point in the pseudo cell X has suciently small neighborhood not intersecting H f. 2 Now we are prepared to prove Lemma 7 ^P is a lattice and a combinatorial manifold. Proof. According to Lemma 3 P is pure dimensional. To verify the remaining conditions we proceed by induction on the dimension d of ^P. d = 1 The only possible partial order is the diamond satisfying all conditions. d = 2 The poset of any such sphere system must be the pseudo cell complex of an even cycle. d > 2 Since T e2e H e separates S n only if it is a hypersphere, Lemmas 4 and 5 and inductive hypothesis imply that open connectivity and the diamond property are satised. We are left to verify that P is a lattice. Thus let U be a lower bound for X 1 ; X 2 2 P. Let A = cl (sep (X 1 ; X 2 )[('(X 1 )['(X 2 T )). The subset M S n with signed vector U has to be part of F = e2a H e : Thus Lemma 6 guarantees the existence of a unique maximal lower bound.

12 The Crossing Property 10 Since the meet of two upper bounds always is an upper bound again, we are done. 2 Next we are going to verify the properties of '. Lemma 8 The map ' : P! L is an antitonic folding. Proof. We have already proven that ' is an order reversing, cover preserving surjection from a combinatorial manifold onto a geometric lattice. Consider a pseudo cell X being the join of X 1 and X 2. Lemma 6 guarantees that if X 1 _ X 2 6= 1I P we have '(X 1 _ X 2 ) = '(X 1 ) ^ '(X 2 ). The validity of the folding property and local atness follows from Lemmas 4 and 5 and the observations in the proof of the last lemma. 2 Theorem 4 The set of signed vectors given by a sphere system is the set of signed covectors of an an oriented matroid. Proof. The symmetry of the antitonic folding is immediate from the Jordan Brouwer Separation Theorem since the two pseudo cells of any minimal nonzero signed vector must be separated by each sphere not containing them. 2 4 The Crossing Property Considering the results of the last section in more detail we nd that axiom (S2) splits into two conditions. (S2a) If F 6 H 0 e is a at then F \ H0 e is a hypersphere of F. (S2b) Under the assumptions of (S2a) the components of F n H 0 e dierent sides of H 0. e are on To prove that the map ' is an antitonic folding condition (S2b) (which Edmonds called the crossing property) was not used at all. It was needed only to prove the symmetry. As it is known that, given a matroid, a cocircuit signature is an oriented matroid cocircuit signature i the signed cocircuit elimination axiom is satised for modular pairs of cocircuits one might ask whether it is possible

13 Shellable Spheres and Lattices 11 to restrict condition (S2b) to the cases where F is 1-dimensional. It is immediately clear that this way we always will get a proper oriented matroid cocircuit sign pattern on the 0-dimensional cells. But our above examinations actually yield the following stronger result. Theorem 5 Let H = (H e ) e2e be a nite family of hyperspheres of S n. Then H is a sphere system if and only if (S1) For all A E T e2a H e is a topological sphere. (S2 0 a) The map cl : 2 E! 2 E, given by cl (A) = fe 2 E j H e T f2a H f g is a matroid closure and its rank function satises: r(a) = n?dim( T e2a H e ). (S2 0 b): If X; Y are copoints of M(E; cl ) such that X ^ Y is a coline then the two components of \ e2x H e lie in dierent components of \ \ H e n e2x^y e2y H e Proof. In Lemma 2 we proved that a sphere system satises (S2 0 a). Since (S2 0 b) is a special case of (S2b) it is clear that any sphere system satises the above axioms. On the other hand the proofs in the last section showed that a nite family of hyperspheres of S n satisfying (S1) and (S2 0 a) canonically induces an antitonic folding. Assume this folding were not symmetric and there were some copoint X 2 L and an atom e 2 L such that there exists a path in P n 0 P between the two preimages of X missing e. Clearly we must have X 6= 0 L and, therefore we may choose some coline G 2 L which is covered by X. Now X and Y := G _ e contradict (S2 0 b). 2 5 Shellable Spheres and Lattices To get a lattice-theoretical characterization of oriented matroids similar to Theorem 2 from the above it is necessary to review some facts from combinatorial topology. The required material is completely covered by the Appendix 4.7 from [1]. The following denition is a lattice-theoretical counterpart of shellability of regular cell complexes.

14 Shellable Spheres and Lattices 12 Denition 6 (cf. [1] ) Let P be a nite lattice with rank funktion. A linear ordering x 1 ; : : : ; x t of its coatoms is a recursive coatom ordering if either r(1 P ) 2, or if r(1 P ) > 2 and for all 1 j t 1. [x j ], the ideal generated by x j has a recursive coatom ordering y 1 ; : : : ; y s and 2. there is an 1 k s such that [x j ] \ Si<j[x i ] = S ik[y i ]: The notion of recursive coatom ordering is important for us since the big face lattice of an oriented matroid admits a recursive coatom ordering and the following Theorem therefore allows to construct, given an oriented matroid O, a sphere systems which is a topological model for O. Theorem 6 (cf. [1] ) Let P be a nite lattice with rank function r and r(1 L ) = d + 2. Then P is isomorphic to the face poset of some shellable regular cell decomposition of the d-sphere if and only if it has the diamond property and admits a recursive coatom ordering. Furthermore is uniquely determined by P up to cellular homeomorphism. Denition 7 A matroidal sphere lattice is a tripel (L; P; ), where L is a geometric lattice, P is a poset and is a surjective map : P! L satisfying 1. 8p; q 2 P : (p > q =) (q) > (p)), 2. 8x 2 L :?1 ([x; 1I L ]) with a top element 1?1 ([x;1i L ] admits a recursive coatom ordering and has the diamond property. It is well known that given the big face lattice P of an oriented matroid, L the geometric lattice of the underlying matroid and ' the zero map then the tripel (L; P; ) satises the above conditions. In the following we will prove that vice versa any matroidal sphere lattice arises from an oriented matroid this way. By the above theorem, given a matroidal sphere lattice (L; P; ) P has a topological realization ^F() as a regular cell decomposition of the sphere. Let isom denote the isomorphism from to P and ' = isom. By Theorem 6 we have that for all atoms of e 2 L the space k'?1 ([e; 1 L ])k is a hypersphere H e of kk. Denote by E the set of all atoms of L and let A E. Then \ \ H e = k'?1 ([e; 1])k (1) e2e e2a

15 Shellable Spheres and Lattices 13 = kisom?1 ( \ e2a?1 ([e; 1]))k (2) = kisom?1 (fz 2 P j 8e 2 A : _ (Z) eg)k (3) = kisom?1 (fz 2 P j (Z) eg)k (4) _ e2a = k'?1 ( e)gk: (5) e2a Therefore, the collection of spheres (H e ) e2e satises (S1). A similar computation veries that (S2 0 a) holds as well. Obviously (S2 0 b) will not be valid without further assumptions. This is easily seen on a rank 2 example. But (S2 0 b) is easily translated into our lattice theoretical notation. Denition 8 A matroidal sphere lattice (L; P; ) is symmetric, if for all copoints X 2 L and all colines G 2 L such that X covers G we have?1 ([G; 1 L ]) n?1 ([X; 1 L ]) consists of two connected components of the same cardinality. Theorem 7 Let (L; P; ') be a symmetric matroidal sphere lattice. Then P is the partial order of an oriented matroid and ' is its zero map. Proof. By the above remarks we are left to verify that (S2 0 b) holds. Assume to the contrary that X; Y are copoints of M(E; cl ) such that X ^Y is a coline and the two components of \ e2x H e were in the same component of \ \ H e n e2x^y e2y H e : Consider the two components A 1 ; A 2 of?1 ([X ^ Y; 1 L ]) n?1 ([X; 1 L ]). By symmetry they have the same cardinality. Now let B 1 and B 2 denote the two components of?1 ([X ^ Y; 1 L ]) n?1 ([Y; 1 L ]). By assumption wlog. A 1 B 1 and thus B 2 A 2 and we get ja 1 j < jb 1 j = jb 2 j < ja 2 j contradicting symmetry. 2

16 REFERENCES 14 References [1] A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White, G. Ziegler: Oriented Matroids. Cambridge, [2] R. G. Bland and M. Las Vergnas: Orientability of Matroids Journal of Combinatorial Theory, Series B (1978) [3] R. Cordovil and K. Fukuda: Oriented Matroids and Combinatorial Manifolds Working Paper CNRS Paris (1987) [4] W. Hochstattler: Seitenachenverbande orientierter Matroide Dissertation Universitat zu Koln, [5] W. Hochstattler: A lattice-theoretical characterization of oriented matroids submitted to: European Journal of Combinatorics. [6] J.G. Hocking and G.S. Young: Topology, Addison Wesley, London [7] J. Folkman and J. Lawrence: Oriented Matroids Journal of Combinatorial Theory, Series B (1978) [8] L.C. Kinsey: Topology of Surfaces, Springer Verlag, New York [9] A. Mandel: Topology of Oriented Matroids Ph.D. Thesis University of Waterloo (1982) [10] T. B. Rushing: Topological Embeddings Academic Press New York and London [11] N. White (ed.): Theory of Matroids. Cambridge, 1986.