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MATROID RELATIONSHIPS: MATROIDS FOR ALGEBRAIC TOPOLOGY DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Charles Estill, BS/MS Graduate Program in Mathematics The Ohio State University 2013 Dissertation Committee: Sergei Chmutov, Advisor Matthew Kahle Thomas Kerler Azita Manouchehri c Copyright by Charles Estill 2013 ABSTRACT In [ACE+13] we found a relationship between two polynomials cellularly em- bedded in a surface, the Krushkal polynomial, based on the Tutte polynomial of a graph and using data from the algebraic topology of the graph and the surface, and the Las Vergnas polynomial for the matroid perspective from the bond ma- troid of the dual graph to the circuit matroid of the graph, B(G∗) !C(G). With Vyacheslav Krushkal having (with D. Renardy) expanded his polynomial to the nth dimension of a simplicial or CW decomposition of a 2n-dimensional mani- fold, a matroid perspective was found whose Las Vergnas polynomial would play a similar role to that in the 2-dimensional case. We hope that these matroids and the perspective will prove useful in the study of complexes. ii This is dedicated to my family, whose trust in me has finally been justified. iii ACKNOWLEDGMENTS Thanks are due especially to my two thesis advisors: Ian Leary, who helped me learn so much, even if our wonderful possible result got snatched out from under us by a genius; and Sergei Chmutov, who helped me to the finish line. In addition, my gratitude to everyone associated with the Mathematics department of The Ohio State University is limitless. Finally, wothout the work of Ross Askanazi, Jonathan Michel, and Patrick Stollenwork on the paper we wrote with Dr. Chmutov, this result may not have ever existed. iv VITA 1972 . Year of birth 2004 . B.Sc. in Mathematics 2008 . MS in Mathematics 2004-Present . Graduate Teaching Associate, The Ohio State University PUBLICATIONS Askanazi, Ross; Chmutov, Sergei; Estill, Charles; Michel, Jonathan; Stollen- werk, Patrick Polynomial invariants of graphs on surfaces FIELDS OF STUDY Major Field: Mathematics Specialization: Algebraic Topology v TABLE OF CONTENTS Abstract . ii Dedication . ii Acknowledgments . iv Vita......................................v List of Figures . viii List of Tables . ix CHAPTER PAGE 1 Introduction . .1 2 Matroids . .3 2.1 Axioms of independence . .3 2.2 Bases . .6 2.3 The circuit axioms and graphical matroids . 10 2.4 The rank function . 18 2.5 Closure, hyperplanes, and spanners . 24 2.6 Bond matroids and more general dual matroids . 32 2.7 Minors . 39 2.8 Matroid perspectives . 41 3 Topology . 45 3.1 Cell complexes . 45 3.2 Cohomology and the cup product . 50 3.3 Manifolds and Poincarè duality . 51 vi 3.4 Krushkal’s polynomial . 53 3.5 Matroid perspectives for chain complexes . 55 4 Graphs on Surfaces . 57 4.1 The Las Vergnas polynomial . 57 4.2 Krushkal’s polynomial for graphs in surfaces . 60 4.3 Relationship . 61 4.4 An example . 66 4.5 Duality . 67 5 Higher Dimensions . 70 5.1 Main result . 70 5.2 Another simple example . 74 6 Further Directions . 76 6.1 More than just the middle, more than just one structure . 76 6.2 Orientation . 77 6.3 Infinite structures . 78 6.4 Simple-homotopy . 80 Bibliography . 81 vii LIST OF FIGURES FIGURE PAGE 2.1 A simple four vertex graph . 15 4.1 A cellular graph (with loops and parallel edges) on a 2-holed torus. 58 4.2 The dual graph of the graph in figure 4.1 . 59 4.3 A graph cellularly embedded in the two-holed torus on the right, and its dual, represented as a ribbon graph, on the left. 67 viii LIST OF TABLES TABLE PAGE 4.1 Calculations by subset of the necessary data for our polynomials. 68 ix CHAPTER 1 INTRODUCTION In the summer of 2010, in a working group on knot theory funded by VIGRE we considered a possible relation between a polynomial defined by Vyacheslav Krushkal in [Kru11] defined for graphs embedded in a surface and the Tutte polynomial for the matroid perspective between the bond matroid of the dual of a ribbon graph to the circuit matroid of the graph. This exploration led to our paper [ACE+13]. Subsequently, Krushkal, together with David Renardy, gave us [KR10], which expanded his polynomial to one defined on the nth level of a triangulation of a 2n-manifold. In chapter 2, I introduce and explain many of the basic concepts concerning matroids. Most of this follows the work in [Oxl11] and [Wel10], which are the main reference works for matroids. There are many axiom systems, all equivalent, for defining matroids. We will need to know several of them. In addition, we will need to know about dual matroids and matroid perspectives, both covered, along with the useful notion of a minor of a matroid, in the later sections of chapter 2. In chapter 3 I cover some of the basics of algebraic topology needed. Most of 1 what I cover is well known. I used [Hat02] and [Mun84] as my main references. It is in this chapter that I also share the definition of Krushkal’s polynomial from [KR10]. I also define here the matroid perspective that will fulfill the role that B(G∗) !C(G) does in the 2-dimensional case. To help our geometric understanding of the final result, I recapitulate [ACE+13] in chapter 4. This is followed by my main theorem in chapter 5. And we finish in chapter 6 with some associated topics and ideas that might be worth exploring in the future. 2 CHAPTER 2 MATROIDS Matroids were first conceived by H. Whitney as an abstract generalization of matrices, with a focus on questions of independence of subsets of the set of column vectors. Having previously defined an independent subgraph as one containing no cycles, he was able to simultaneously generalize graphs. Follow- ing his lead we define a matroid as giving some information on subsets of a fixed set. There are several different ways of indicating this information, from the vector algebra influenced notion of independence to the nearly topological closure operator. 2.1 Axioms of independence A matroid, M, is a finite set E and a collection of subsets I ⊆ P(E) satisfying the following axioms, (i1)-(i3). (i1) ; 2 I. (i2) If X 2 I and Y ⊆ X then Y 2 I. 3 (i3) If U and V are in I with jV j < jUj, then there exists an element of E, x 2 U r V , such that V [ fxg 2 I. The subsets in I are called independent and those not in I are called dependent. Further, we’ll call the set E the ground set. Two matroids M & N are called isomorphic if there is a bijection between their ground sets and the structure is the same. That is to say, for example, that independent sets are mapped to independent sets in both directions. The following proposition and many others in this chapter follow the path well-trod by James Oxley in [Oxl11]. Those proofs not guided by Oxley were guided by D. J. A. Welsh’s [Wel10]. Proposition 2.1.1. Let A be an m × n matrix over a field F. Set E as the set of column vectors of A and let I consist of those subsets X ⊆ E which are m linearly independent in F , the m-dimensional vector field over F. (E; I) is a matroid. Proof. Trivially ; 2 I, as the empty set of vectors is trivially linearly indepen- dent, so (i1) is satisfied. And if X is an independent set of column vectors, so is any subset, which means that (i2) is satisfied. To show that (i3) is satisfied, let U and V be two independent sets of column m vectors of A with jV j < jUj. Let W be the subspace of F spanned by U [ V . Note that the dimension of W , dim W is at least jUj. If (i3) weren’t true and 4 hence V [ fxg were linearly dependent for every x 2 U r V , then W would be contained in the span of V and thus jUj ≤ dim W ≤ jV j < jUj; (2.1.1) a contradiction. Hence (i3) is satisfied. We denote the matroid thus obtained M[A] and call it the vector matroid of A. Any matroid, M, for which we can find a field F and a matrix A such that M is isomorphic to M[A] is called representable over the field F or just representable. Example 2.1.2. Consider the vector matroid of the following matrix over R. 2 3 0 0 1 0 1 0 0 6 7 6 7 A = 60 1 1 0 0 1 17 6 7 4 5 1 0 0 0 0 1 1 If we label the column vectors as one to seven from left to right, so that E = f1; 2; 3; 4; 5; 6; 7g, then the set of independent subsets, is n I = ;; f1g; f2g; f3g; f5g; f6g; f7g f1; 2g; f1; 3g; f1; 5g; f1; 6g; f1; 7g; f2; 3g; f2; 5g; f2; 6g; f2; 7g; f3; 5g; f3; 6g; f3; 7g; f5; 6g; f5; 7g; f1; 2; 3g; f1; 2; 5g; f1; 3; 5g; f1; 3; 6g; f1; 3; 7g; f1; 5; 6g; o f1; 5; 7g; f2; 3; 6g; f2; 3; 7g; f2; 5; 6g; f2; 5; 7g; f3; 5; 6g; f3; 5; 7g : Theorem 2.1.3. If U and V are independent sets such that jV j < jUj then there is a set W ⊆ U r V such that jV [ W j = jUj and V [ W 2 I.
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  • The Fixed-Point Partition Lattices

    The Fixed-Point Partition Lattices

    Pacific Journal of Mathematics THE FIXED-POINT PARTITION LATTICES PHILIP HANLON Vol. 96, No. 2 December 1981 PACIFIC JOURNAL OF MATHEMATICS Vol. 96, No. 2, 1981 THE FIXED-POINT PARTITION LATTICES PHIL HANLON Let σ be a permutation of the set {1,2, — -, n} and let Π(N) denote the lattice of partitions of {1,2, •••,%}. There is an obvious induced action of σ on Π(N); let Π(N)σ — L denote the lattice of partitions fixed by σ. The structure of L is analyzed with particular attention paid to ^f, the meet sublattice of L consisting of 1 together with all elements of L which are meets of coatoms of L. It is shown that -^ is supersolvable, and that there exists a pregeometry on the set of atoms of ~^ whose lattice of flats G is a meet sublattice of ^C It is shown that G is super- solvable and results of Stanley are used to show that the Birkhoff polynomials B (λ) and BG{λ) are BG{λ) = W - 1)U - i) U - (m - and Here m is the number of cycles of σ, j is square-free part of the greatest common divisor of the lengths of σ and r is the number of prime divisors of j. ^ coincides with G exactly when j is prime. !• Preliminaries* Let (P, ^) be a finite partially ordered set. An automorphism σ of (P, 50 is a permutation of P satisfying x ^ y iff xσ 5Ξ yσ for all x, y eP. The group of all automorphisms of P is denoted Γ(P).