Topological Combinatorics

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Topological Combinatorics

ALEXANDER ENGSTRÖM

Doctoral Thesis Stockholm, Sweden 2009

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TRITA-MAT-09-MA-05 ISSN 1401-2278

ISRN KTH/MAT/DA 09/03-SE ISBN 978-91-7415-256-2

KTH Matematik SE-100 44 Stockholm SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i matematik fredagen den 8 maj 2009 klockan 13.00 i sal E2, huvudbyggnaden, Kungl Tekniska högskolan, Lindstedsvägen 3, Stockholm.

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iii

Abstract

This thesis on Topological Combinatorics contains 7 papers. All of them but paper B are published before.

In paper A we prove that!

idim ˜Hi(Ind(G); Q) ≤ |Ind(G[D])| for any graph G and its independence complex Ind(G), under the condition that G\D is a forest. We then use a correspondence between the ground states with i + 1 fermions of a supersymmetric lattice model on G and ˜Hi(Ind(G); Q) to deal with some questions from theoretical physics.

In paper B we generalize the topological Tverberg theorem. Call a graph on the same vertex set as a (d + 1)(q − 1)-simplex a (d, q)-Tverberg graph if for any map from the simplex to R^d there are disjoint faces F1, F2, . . . , Fq whose images intersect and no two adjacent vertices of the graph are in the same face. We prove that if d ≥ 1, q ≥ 2 is a prime power, and G is a graph on (d + 1)(q − 1) + 1 vertices such that its maximal degree Dsatisfy D(D + 1) < q, then G is a (d, q)–Tverberg graph. It was earlier known that the disjoint unions of small complete graphs, paths, and cycles are Tverberg graphs.

In paper C we study the connectivity of independence complexes. If G is a graph on n vertices with maximal degree d, then it is known that its independence complex is (cn/d + !)–connected with c = 1/2. We prove that if G is claw-free then c ≥ 2/3.

In paper D we study when complexes of directed trees are shellable and how one can glue together independence complexes for finding their homotopy type.

In paper E we prove a conjecture by Björner arising in the study of simplicial polytopes.

The face vector and the g–vector are related by a linear transformation. We prove that this matrix is totaly nonnegative. This is joint work with Michael Björklund.

In paper F we introduce a generalization of Hom–complexes, called set partition complexes, and prove a connectivity theorem for them. This generalizes previous results of Babson, Cukic, and Kozlov, and questions from Ramsey theory can be described with it.

In paper G we use combinatorial topology to prove algebraic properties of edge ideals.

The edge ideal of G is the Stanley-Reisner ideal of the independence complex of G. This is joint work with Anton Dochtermann.

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iv

Sammanfattning

Denna avhandling om topologisk kombinatorik innehåller 7 artiklar. Alla utom artikel B är tidigare publicerade.

I artikel A visar vi att !

idim ˜Hi(Ind(G); Q) ≤ |Ind(G[D])| för varje graf G och dess oberoendekomplex Ind(G), under villkoret att G \ D är en skog. Vi använder sen en korrespondans mellan grundtillstånden med i + 1 fermioner hos en supersymmetrisk latticemodell på G med ˜Hi(Ind(G); Q) för att behandla frågor från teoretisk fysik.

I artikel B generaliserar vi den topologiska Tverbergsatsen. Kalla en graf med samma hörnmängd som ett (d+1)(q−1)-simplex en (d, q)-Tverberggraf om det för varje avbildning från simplexen till R^dfinns disjunkta sidor F1, F2, . . . , Fq vars bilder skär varandra samt inga två hörn ligger i samma sida om de har en kant mellan sig i grafen. Vi visar att om d ≥ 1, q ≥ 2 är en primpotens, och G är en graf på (d + 1)(q − 1) + 1 hörn med vars maximalgrad D som uppfyller att D(D + 1) < q, då är G en (d, q)–Tverberggraf. Det var tidigare känt att den disjunkta unionen av små kompletta grafer, cykler samt stigar är Tverberggrafer.

I artikel C studerar vi hur pass sammanhängande oberoendekomplex är. Om G är en graf på n hörn med maximalgrad d, så är det känt att dess oberoendekomplex är (cn/d + !)–sammanhängande med c = 1/2. Vi visar att om G är klofri så är c ≥ 2/3.

I artikel D så studerar vi först komplexet av riktade träd är skalbart och sen hur man kan limma ihop oberoendekomplex fär att bestämma deras homotopityp.

I artikel E bevisar vi en förmodan av Björner från studiet av simpliciella polytoper.

Vektorn som beskriver antalet sidor per dimension och g–vektorn är relaterade genom en linjär transformation. Vi bevisar att den matrisen är totalt ickenegativ. Artikeln skrevs med Michael Björklund.

I artikel F introducerar vi en generalisering av Hom–komplex, kallad mängdpartitions- komplex och bevisar en sammanhängandesats om dem. Detta generaliserar tidigare arbe- ten av Babson, Cukic och Kozlov. Vi beskriver också kopplingar till Ramseyteori.

I artikel G använder vi kombinatorisk topologi för att visa algebraiska egenskaper hos kantideal. Kantidealet till G är Stanley-Reisneridealet till obereoendekomplexet av G.

Artikeln skrevs med Anton Dochtermann.

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Acknowledgements

Vad är det mamma?

En skola där man lär sig finurliga saker.

Mamma och barn vid KTH, våren 2008.

Writing a thesis requires hard work, but also time to reflect and have fun. With- out the guidance of my advisor Svante Linusson in mathematics and the academic life in general this thesis would not be in your hands. I have also had many stim- ulating conversations and lots of help from my co-advisor Jakob Jonsson. I deeply appreciate the support of my advisors.

I have great co-authors! Anton Dochtermann and I are working on topological combinatorics in commutative algebra – our first article is paper G. The many dis- cussions on mathematical didactics with Alan Sola turned into a paper on problem solving. Michael Björklund and I solved a conjecture in paper E, and with Seth Sullivant I work on algebraic statistics.

For half a year I visited Günter Ziegler’s group at TU Berlin and worked with Bernd Sturmfels, and earlier on I worked for one year with Dmitry Kozlov in Emo Welzl’s group at ETH Zurich. Being exposed to international research enviroments has been a great source of inspiration and I thank those who made it possible.

Family, friends, and all of you who weave life together: The fun goes on!

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Introduction and Summary

1

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Chapter 1

Introduction

1.1 Topological combinatorics

During the last century the levels of abstraction in mathematics have increased considerably. While we still want to solve easily stated problems whose solutions can be applied in other sciences, we have also developed very abstract tools which are hard to prove useful even within mathematics.

In topological combinatorics we want to build bridges for using topological ma- chinery to answer combinatorial questions, and sometimes the other way around. I have included seven papers on this theme in the thesis. Before the summary of the papers there are short introductions with literature suggestions on discrete Morse theory, equivariant methods, and independence complexes.

There are several excellent introductions to topological combinatorics, among my favourites are the survey on topological combinatorics by Anders Björner [10], Jakob Jonsson’s study on complexes from graphs [33], the book on polytopes by Günter M. Ziegler [45], and Jiří Matoušek’s introduction to the Borsuk-Ulam theorem [38].

1.2 Discrete Morse theory

Morse theory is a technique from differential topology for finding cell structures on manifolds by using differentiable functions on them. Forman adapted this to a discrete setting where the manifolds are replaced by combinatorial cell structure and the differentiable functions are replaced by acyclic orientations of posets. The aim of using discrete Morse theory is to get a new cell structure with the same topology, but with much fewer cells.

Without discrete Morse theory papers A and F would never have been written, and many of the results in papers B, C, D, and F were motivated or first proved with discrete Morse theory.

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4 CHAPTER 1. INTRODUCTION

The transitive reduction of a poset encoded as a directed graph on the same vertex set is its Hasse diagram. That is, there is an edge from u to v in the Hasse diagram if u is less than v but there exists no element between them. A matching on a poset is a set of edges from its Hasse diagram with no poset element incident to more than one edge of the set. The Hasse diagram of a poset is a directed acyclic graph since it is impossible to return to the same vertex by walking around in the same direction as the edges. If we swap the directions of the edges in a matching and the diagram is still acyclic, then the matching is called acyclic.

In combinatorial situations a convenient class of cell structures are regular CW- complexes since their topology can be reconstructed from combinatorial data on how the cells are attached. The face lattice of a cell complex is the poset of cells ordered by inclusion.

The existence of a large acyclic matching on the face lattice of a cell complex tells us that the space can be described with a smaller poset if the directed edges in its Hasse diagram can carry more information than the order. Given an acyclic matching on a poset, we actually can throw away the elements incident to edges of the matching and still keep the topological data. The elements of the poset that are not incident to a directed edge in a matching on the poset are called critical, and they should be thought of in the same way as the critical points in the differential topology setting of Morse theory.

The main theorem of discrete Morse theory states that any regular CW-complex with an acyclic matching on its face lattice is homotopy equivalent with a cell complex whose cells are the critical cells, but they are (almost always) glued together another way.

There is a geometrical interpretation of the deformation between the cell complexes in the main theorem. For any finite cone there is a deformation retraction sending all points on rays crossing the interior of the cone towards its apex. One can think of a simplex as a cone with one of its vertices as the apex, and the deformation retraction would then remove the interior of the simplex and the face opposite to the apex. The removal of the interior and the opposite face correspond to changing the orientation of an edge in the Hasse diagram of the face lattice and turning the two cells non-critical.

Usually one wants small cell complexes, for example bouquets of spheres if possible, and finding large acyclic matchings is then the name of the game. It used to be common to very explicitly describe matchings and then prove them acyclic. In paper A a more poset theoretic idea first introduced by Björner, Hersh and Jonsson is employed. A poset map is a set map between posets respecting the order relation.

The fibers of a poset map are also posets and the trick is that acyclic matchings on the fibers put together gives an acyclic matching. One should think of this trick together with the main theorem as a combinatorial Quillen fiber theorem.

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1.3. EQUIVARIANT METHODS 5

Literature

· Morse Theory.

[14, 41]

· Discrete Morse theory, basic theory.

[17, 25, 27]

· Discrete Morse theory, extensions and used as a tool.

[5, 6, 8, 9, 15, 21, 22, 26, 30, 33, 34, 36, 35]

· On cell complexes, poset maps, and the Quillen fiber theorem.

[10, 37, 42]

1.3 Equivariant methods

A common misconception by new topology students is that if there is a map from one topological space into another one, and certain homology groups of the first space vanish, then the corresponding homology groups of the second space also vanish. This is not true - for example it is impossible to relate the dimensions of two spheres just because of a map between them.

But if we have a group acting on both of the spaces and the map commutes with the action (it is equivariant), then, under some conditions, it is true. Returning to the spheres, but now adapting them with an antipodal action and an equivariant map, we can never map into a sphere of lower dimension. Comparing spheres can be replaced by connectivity conditions, calculating characteristic classes, and using more advanced index theorems. Even group cohomology can be used.

In paper B we prove a generalized topological Tverberg theorem and employ an equivariant method called the configuration space/test map scheme. Here is a toy example of the scheme.

We want to prove that any absolutely continuous probability measure on the plane can be dissected by a line through the origin. To any point on the unit circle there is a unique line through the origin perpendicular to the line through that point and the origin. The half plane on the other side of the unique line is associated to the point. The equivariant map is from the unit circle to the unit interval; it sends any point on the circle to the measure of the half plane we associated to it. The antipodal action on the circle and flipping the interval turns the map equivariant.

Assuming that no line dissects the measure we can remove the middle point of the interval since it is outside the circle image. The configuration space of the lines is parameterized by the unit circle and is mapped equivariantly into the test space.

The configuration space is a homology one sphere and the test space a homology zero sphere. There can be no such equivariant map and our assumption about the non-existence of the dissection is wrong.

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6 CHAPTER 1. INTRODUCTION

The topological spaces in paper G are constructed to allow nice group actions, and we intend to return to those spaces and prove Ramsey theory results by using the equivariant method.

Literature

· Equivariant methods, basic theory.

[16, 19, 38, 46, 47]

· Equivariant methods, more advanced and its applications.

[4, 7, 12, 13, 44]

1.4 Independence complexes

If all minimal simplices missing from a simplicial complex are one-dimensional, then it is called a flag complex, a clique complex, or an independence complex. We stick to the terminology of “independence complexes” since many of our questions are motivated by graph theory. An independent set of a graph is a subset of vertices without edges between them, and the independence complex is the collection of those sets.

Many natural problems in topological combinatorics concerns independence complexes or close relatives to them. As for many other combinatorially constructed cell complexes, a first step towards understanding them is to determine their topological connectivity. There is a large literature on using well-known local and global properties of graphs to understand their topological connectivity. In papers B,C, and D we prove new connectivity theorems.

Let us assume that for a certain class of graphs the independence complex of an n vertex graph is n/k–connected. Then any partition of the vertex set of a graph from the class into pieces with at least k vertices will give rise to an independent set with one vertex in every part. It is conjectured that one can find complete covering with such independent sets, and that results would prove several difficult conjectures on the strong chromatic number.

In paper B we use the connectivity of some independence complexes to prove a generalized topological Tverberg theorem using the equivariant method.

The harmonic representatives of the cohomology of an independence complex can be interpreted as the ground states of a certain supersymmetric lattice model where the fermion sites are the vertices of a graph. In paper A we use discrete Morse theory on such independence complexes to enumerate or identify the ground states.

There is a natural correspondence between simplicial complexes and commutative algebra. Given a complex, one first constructs a polynomial ring whose variables are indexed by the vertices of the complex. Then one generates an ideal of the ring by including all monomials whose vertices do not span a face of the simplicial complex. This ideal is called the Stanley-Reisner ideal, and if the complex

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1.4. INDEPENDENCE COMPLEXES 7

is an independence complex, it is called an edge ideal. In paper G we use topology to give stronger and shorter proofs of algebraic theorems on edge ideals.

One nice question, unfortunatly not answered in this thesis, is if all independence complexes of triangle-free graphs are torsion-free.

Literature

· Connectivity of independence complexes and graph colorings.

[1, 2, 3, 11, 39, 40]

· Supersymmetric lattice models and independence complexes.

[15, 20, 23, 24, 31, 32]

· Independence complexes in commutative algebra.

[18, 28, 29, 43]

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Chapter 2

Summary

2.1 Overview of Paper A – Upper bounds on the Witten index for supersymmetric lattice models by discrete Morse theory

The cohomology classes of the independence complex of a graph G correspond to the ground states of a supersymmetric lattice model with fermions on sites described by the vertices of G. The graphs investigated by theoretical physicists are planar lattices, for example square, hexagonal, or triangular ones. These kinds of lattices usually have large induced subtrees, and the idea behind paper A is to use that the independence complex of a tree is either contractible or it is a cross polytope. If all discrete Morse theory fibers are contractible or spheres, then one can relate the number of fibers and the number of cohomology classes. The main result of paper A is that

"

i

dim ˜Hi(Ind(G); Q) ≤ min

∅"=D⊆V (G) G\Dis a forest

|Ind(G[D])|.

Using supercomputers the actual number of ground states have been calculated for lattices of different types with up to about 200 vertices. It is belived that for some of these lattices the model is superfrustrated and the number of ground states grows exponentially with the number of vertices. We calculated general upper bounds on the contribution by each vertex for lattices with arbitrary number of vertices and then compared it to the calculations for small lattices.

The contribution per vertex.

Lattice type For small lattice General upper bound

Hexagonal 1.2 ± 0.1 1.19

Hexagonal dimer 1.25 ± 0.1 1.26 Triangular 1.14 ± 0.01 1.27 Triangular dimer 1.36 ± 0.01 1.38 Square dimer 1.15 ± 0.01 1.27

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10 CHAPTER 2. SUMMARY

We also generalize and formalize an idea called the “3-rule” on how one can get independence complexes which are bouquets of spheres of the same dimension; or in the context of supersymmetric lattice models, that all ground states have equally many fermions.

2.2 Overview of Paper B – Tverberg graphs

The Tverberg theorem states that any subset of R^d on (d + 1)(q − 1) + 1 points can be partitioned into q disjoint subsets F1, F2, . . . , Fq such that their convex hulls intersect,

conv(F1) ∩ conv(F2) ∩ · · · ∩ conv(Fq) $= ∅.

The topological version of the statement is that for any continuous map f from the (d + 1)(q − 1)-simplex to R^d there are q disjoint faces F1, F2, . . . , Fq of the simplex such that their images intersect,

f (F1) ∩ f(F2) ∩ · · · ∩ f(Fn) $= ∅.

The topological Tverberg theorem is only proved for prime powers q = p^k, and it is done with the equivariant method using a Z^kp group action.

The number of ways to choose the disjoint faces to get their images to intersect is larger than what one first guess, and it make sense to study other conditions one could put on the disjoint faces. We call a graph on the same vertex set as the (d+1)(q −1)-simplex a (d, q)-Tverberg graph if for any map from the simplex to R^d there are disjoint faces F1, F2, . . . , Fq whose images intersect and no two adjacent vertices of the graph are in the same face.

Hell proved that disjoint unions of small complete graphs, paths, and cycles are Tverberg graphs by employing connectivity results on chessboard complexes.

Our main theorem gives a local condition for Tverberg graphs. Stated using the maximal degree we get the following corollary: If d ≥ 1, q ≥ 2 is a prime power, and G is a graph on (d + 1)(q − 1) + 1 vertices such that its maximal degree D satisfy D(D + 1) < q, then G is a (d, q)–Tverberg graph.

Together with Bertrand’s postulate we get affine special cases for which no other proofs are known.

2.3 Overview of Paper C – Independence complexes of claw-free graphs

A graph is claw-free if any three vertices adjacent to some other vertex also have two vertices adjacent. It is one of the most important classes of graphs since claw-free is the right condition in many theorems.

It is known that the independence complex of an n vertex graph with maximal degree d is (cn/d + !)–connected with c = 1/2. There are many different conditions to get c larger than 1/2, for example from the spectrum of the Laplacian of G.

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2.4. OVERVIEW OF PAPER D – COMPLEXES OF DIRECTED TREES AND

INDEPENDENCE COMPLEXES 11

In our paper we prove for the first time that c > 1/2 for any large natural class of graphs. The main result states that c ≥ 2/3 for claw-free graphs.

The proof of c ≥ 2/3 follows from a technical type of nerve lemma which is created for this application. We also calculate new connectivity bounds for some complexes in the literature.

2.4 Overview of Paper D – Complexes of directed trees and independence complexes

This paper is the oldest one included in the thesis. It was posted on the arxiv¹ some years ago and this overview reflects how results from it are used today.

In the second part of the paper we prove several theorems on how independence complexes can be glued together and how this can be used to calculate their homotopy type.

There is an elementary but earlier overlooked lemma in the paper: If u and v are distinct vertices of a graph G and any vertex adjacent to u is also adjacent to v, then Ind(G) deformation retracts to Ind(G \ u), and in particular they are homotopy equivalent.

For some classes of graphs, for example forests and complements of chordal graphs, there are always such vertices u and v or their homotopy type is swiftly determined.

After papers A and D were published a new connection between them was found. In Paper A a supersymmetric model was studied were fermions are on the vertices of a graph. If we replace the graph by a finite number of sites on R and do not allow fermions to be within a fixed distance r, then the results on anti- Rips complexes (which are independence complexes) in Proposition 4.2 of paper D completely determines the groundstates of that model.

2.5 Overview of Paper E – The g–theorem matrices are totally nonnegative

This is a joint paper with Michael Björklund.

The number of faces of a polytope tabulated by dimension in a vector is called the f–vector of the polytope. If the polytope is simplicial, then by a famous theo- rem by Billera, Lee, and Stanley, the f–vector is given by a linear transformation of a vector g with about half as many elements. The vector g is the g–vector of a sim- plicial polytope if and only if it satisfy a combinatorial condition from commutative algebra.

1A collection of 1/2 million mathematics, physics, and computer science preprints on the Internet. It was the largest preprint server in 2009.

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12 CHAPTER 2. SUMMARY

Björner proved that the 2 × 2-minors of the matrix sending g–vectors to f- vectors are nonnegative. He conjectured that all minors are nonnegative, and that is the result of the paper.

We proved the conjecture using Lindström’s path counting lemma.

2.6 Overview of Paper F – Set partition complexes

Lovász introduced Hom–complexes to find obstructions to graph colorings. A graph coloring is simply a partition of the vertex set into independent sets, and in this paper we study other partitions than those from graph colorings.

The polyhedral complex Hom(G, Kn) has all n–colorings of G as vertex set. Let Σ be a simplicial complex. We generalize Hom(G, Kn) to the polyhedral complex Part(Σ, [n]) whose vertices are all partitions of the vertex set of Σ into n sets such that each part is a cell of Σ. One gets Hom back by setting Σ = Ind(G).

We generalize the theorem that Hom(G, Kn) is (n − d − 2)–connected to set partition complexes: The complex Part(Σ, [n]) is (n − gr(Σ) − 1)–connected where gr(Σ) is the largest number of faces a greedy algorithm would use to cover Σ.

We also discuss connections to Ramsey Theory.

2.7 Overview of Paper G – Algebraic properties of edge ideals via combinatorial topology

This is a joint paper with Anton Dochtermann.

For any graph G on {1, 2, . . . , n} the edge ideal IG ⊆ k[x¹, x2, . . . , xn] is gen- erated by all the monomials xixj whenever ij is an edge of G. Another way to state it is that the edge ideal of G is the Stanley-Reisner ideal of the independence complex of G.

We tried to prove as many algebraic theorems on edge ideals as possible by only using combinatorial topology. We managed to find new proofs, strengthen several theorems, and also prove new types of theorems.

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