Department of Mathematical Sciences, University of Copenhagen. Kandidat projekt i matematik. Jens Jakob Kjær. Golod Complexes

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1 F A C U L T Y O F S C I E N C E U N I V E R S I T Y O F C O P E N H A G E N Department of Mathematical Sciences, University of Copenhagen Kaniat projekt i matematik Jens Jakob Kjær Golo Complexes Avisor: Alexaner Berglun Seconary-Avisor: Jesper Michael Möller April 13th, 2012

2 Abstract We will in this paper iscuss the concept of a Golo complex, that is for the Stanley-Reisner ring of a simplicial complex, k[ ], the torsion (k, k is as large as it coul possibly be, i.e. the Poincaré series of k[ ] viewe as a local ring to be coefficient wise as large as possible. We will see a theorem which gives a purely combinatorial requirement for this otherwise algebraic notion. We will then spen some time stuying the concept of shellability both as a shellable complex an of an EL-shellable poset. We will use these notions to be able to calculate the imensions of homology Tor k[ ] i groups of orer complexes of certain interval posets. Now let 2 be a simplicial complex on 2 vertices, with all faces of carinality less than an no other faces. We will use the combinatorial requirement for Goloness of a complex an our iscussion of shellability to show that 2 is a Golo complex for all > 1.

3 Contents 1 Preliminaries Simplicial complexes Partially orere sets Golo complexes Helpful results Introuction to shellability The h integer array Lexicographic shelling The curious case of Further problems

4 Chapter 1 Preliminaries 1.1 Simplicial complexes We will start with some rather basic efinitions of a simplicial complex which is the objects we will stuy throughout the paper; these efinitions can be foun in e.g. [St] Definition 1.1 (Simplicial complexes. We write n for the set {1, 2,..., n} A simplicial complex on the vertex set V, where V is finite, is a subset of the power set of V, such that if τ σ then τ. For a simplicial complex, we call elements of for faces. If σ is an inclusionwise maximal face of we say that σ is a facet of. We say that is a ( 1-complex if max{ F F is a facet of } =. For a simplicial complex the minimal non-faces of, M(, is the set of inclusionwise least elements of P(V We write n i for the simplicial complex on the vertex set n where n i := {σ σ n, σ < i}. Given a simplicial complex on the vertex set V = {v i,..., v n } we call R n the geometric realization of if for each {v i0,..., v ir } the convex hull of e i0,..., e ir, where e 1,..., e n R n with e 2 e 1,..., e n e 1 being linear inepenent. We equip R n with the subspace topology. It is trivial that any two geometric realizations of a simplicial complex are homeomorphic an there is thus no nee to istinguish between them. Given a complex we can always rename its vertices so the vertex set becomes n for some n N. We will also nee the homology of simplicial 3

5 CHAPTER 1. PRELIMINARIES complexes, which we will efine in the usual sense 1. The reason for the efinition of the complex n i is that these complexes are rather simple an we will be able to prove that for a certain class of these 2, with > 1 the complex is Golo over any fiel k. We will for our main theorem nee to be able to make rings out of our complexes, this we will o in the following way Definition 1.2. Let be a simplicial complex on the vertex set V = {v 1,..., v n }, then the Stanley-Reisner ring of, k[ ] for a fiel k, is the ring k[x 1,..., x n ]/I, being the polynomial ring moe with the ieal which contains x i1 x ir if {v i1,..., v ir } / This allows us to talk about algebraic notions of our simplicial complexes in much the same way as the geometric realization allows us to talk about topological properties of our otherwise combinatorial complexes. 1.2 Partially orere sets We will also nee another class of combinatorial objects closely relate to the simplicial complexes above, see for instance the efinition of the orer complex. These efinitions are likewise taken from [St] Definition 1.3 (Partially orere sets. A set P is calle a poset, if it is partially orere by some relation which is reflexive, anti-symmetric an transitive. We call m = (c 1, c 2,... a chain in the poset P if c i P for all i, an c 1 < c 2 <..., we partially orer the chains of a poset by for m = (c 1, c 2,... then m m c i = c n i for all i an some sequence n i. For a finite poset P we efine the orer-complex of P, (P, to be the simplicial complex on the vertex set P, where the faces of (P are the chains of P. For a poset P the set M(P is the set of all maximal chains of P. If P is finite this is the facets of (P The eges of P, E(P, is the set of elements of the form (p i < p j where p i, p j P with p i < p j. If P is finite this is exactly the eges (faces of carinality 2 of (P. A poset P is calle boune if it has a unique minimal an maximal element calle respectively ˆ0 an ˆ1. The proper part of a boune poset, 1 Again see [St] 4

6 CHAPTER 1. PRELIMINARIES P, is the poset P {ˆ0, ˆ1}, an we efine P = P {ˆ0, ˆ1} with ˆ0 an ˆ1 being the respectively unique minimal an unique maximal elements of P shoul they exists, otherwise they are elements ae such that they become the unique minimal respectively maximal elements. For a poset P we can have intervals, so for a, b P we efine both the open an close (an semi-open intervals in the usual sense. We will note in which poset the interval is, if oubt coul arise, so we woul write [a, b] P for the close interval from a to b in P. For a poset P we efine the n th reuce homology group of P with coefficients in some group G, H n (P, G, to be the n th reuce homology group of the simplicial complex (P with coefficients in G. We will also nee some rather specific posets, i.e. the coorinate an iagonal lattices efine below. These play a very important role in our main theorem. But we will start with a simple efinition of a lattice Definition 1.4. A poset P is calle a lattice if a, b P there is are unique elements u, v P such that u a, u b with no c P such that u > c > a an u > c > b, an v a, v b with no c P such that v < c < a an v < c < b. Meaning a poset is lattices if there for all two elements are a unique supremum an infimum. Definition 1.5. For σ n, we efine C σ := {(x 1,..., x n C n i σ : x i = 0} For the simplicial complex, on the vertex set n, the coorinate arrangement lattice, L C(, is the lattice consisting of all intersections of C σ for σ M( orere by reverse inclusion, where the empty intersection is viewe as C = C n. For σ n, we efine D σ := {(x 1,..., x n C n i, j σ : x i = x j } For σ 1,..., σ r n isjoint sets, we write D σ1... σ r r i=1 D σ i for the intersection For the simplicial complex, on the vertex set n, the iagonal arrangement lattice, L D(, is the lattice consisting of all intersections of D σ for σ M( orere by reverse inclusion, where the empty intersection is viewe as D = C n. For each simplicial complex on the vertex set n we have a map ɛ : L D( L C( given by for D σ1... σ r L D( then ɛ(d σ1... σ r = C σ1... σ r 5

7 CHAPTER 1. PRELIMINARIES We have a map c : L D( N 0 given by for D σ1... σ r L D( then c(d σ1... σ r = r. In other wors, this function counts the number of isjoint subsets in the inex of the element. We can now see that for all C in L C( there is a unique σ n such that C = C σ. Likewise for all D in L D( there are unique σ 1,..., σ r n such that D = D σ1... σ r. It is trivially clear that C σ C τ = C σ τ, for σ, τ n, an likewise if σ τ then D σ D τ = D σ τ. It is clear that ɛ is a surjective map, since for C L C( then there is σ 1,..., σ r M such that r i=1 C σ i = C, but then trivially r i=1 D σ i L D( with ɛ( r i=1 D σ i = C. So let s give an example as to how these posets might look. So we will raw the Hasse-iagrams, that is a graph with the vertices being the elements of the poset, P an E(P being the eges, of both the coorinate- an the iagonal-intersection lattice for 4 2. We will enote C {1,2} with the simpler C 12 to ease notation an the same for the iagonal elements. So first the poset L C( 4 2, clearly M( 4 2 = {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}} an hence we get C 1234 (1.1 C 123 C 124 C 134 C 234 C 12 C 13 C 14 C 23 C 24 C 34 Now lets raw the Hasse-iagram of L D( 4 2, I will color the eges an vertices not mirrore in L C( 4 2 re, to high light the ifferences between the two posets. C D 1234 D D 123 D 124 D 134 D 234 D D D 12 D 13 D 14 D 23 D 24 D 34 D (1.2 As we can see these posets rather quickly grow rather large. 6

8 CHAPTER 1. PRELIMINARIES 1.3 Golo complexes To efine what we mean with a Golo complex we will nee some efinitions an results on the Poincaré series. But first we will nee something as elementary as a local ring. Definition 1.6. A ring R is sai to be local if it has a unique maximal left ieal I. We call the fiel R/I the resiue fiel. Now trivially we can view R/I as an R-moule. We can therefore give the following efinition of the Poincaré series from [GrTh] Definition 1.7. Given a local ring, R, with resiue fiel, k, we efine the Poincaré series of R, P (R, to be the formal power series b p Z p for b p = Tor R p (k, k Now we woul like to stuy the Poincaré series of the Stanley-Reisner ring of a simplicial complex. So for a simplicial complex on a vertex set V an a fiel k, we can easily see that k[ ] is local since the ieal generate by V is maximal. We are now able to give the following theorem from [Av] Theorem 1.8. For a simplicial complex on a vertex set with n vertices an a fiel k, the Poincaré series of k[ ] is always coefficient wise less than or equal to (1 + Z n 1 Z (H (K k[ ] (Z 1 (1.3 where H (K k[ ] (Z is the polynomial with the n th coefficient im k H n (K k[ ] with K k[ ] being the Koszul chain complex of k[ ]. In other wors for any simplicial complex, the Poincaré series of its Stanley-Reisner ring is boune. We will now efine a complex to be Golo over a fiel, if it is in some sense maximal with respect to the bounary. Notice this is not the usual efinition of a Golo complex, but the usual efinition woul require to much algebraic theory for this project an hence we will instea use the following equivalent efinition. Definition 1.9. A simplicial complex is sai to be a Golo complex over k if P (k[ ] = (1 + Z n 1 Z (H (K k[ ] (Z 1 (1.4 with the same naming as in theorem 1.8. Normally when giving the efinition of a Golo complex will rely on a efinition of a Golo ring, which is a ring with the largest possible Poincaré 7

9 CHAPTER 1. PRELIMINARIES series, an then a complex is calle Golo if k[ ] is a Golo ring. I have here avoie that efinition as to spare myself some algebraic consierations. For more information on Golo rings the reaer is referre to [Av]. Now this efinition of a Golo ring is rather bulky since P (k[ ] is rather ifficult to calculate since it relies on fining a resolution of k as a k[ ]-moule, which might be of infinite length. We will therefore nee another characterization of a Golo complex which only relies on pure combinatorics. The following theorem follows from [Be], an it is to this theorem we will eicate the bulk of this paper Theorem The following are equivalent for a simplicial complex on a vertex set n, where i n : {i} an a fiel k (1 is a Golo complex over k. (2 for every C L C( there is an equality of polynomials im k H ((C n, C LC( ; k(z = D ɛ 1 (C ( z c(d 1 im k H ((C n, D LD( ; k(z where im k H ((C n, C LC( ; k(z is the polynomial with variable z an the n th coefficient im k Hn ((C n, C LC( ; k, an likewise im k H ((C n, D LD( ; k(z is the polynomial with variable z where the n th coefficient is im k Hn ((C n, D LD( ; k. This theorem gives a purely combinatorial requirement for the Goloness of a ring. So let s see an example of how a use of this theorem might go. Lets take 4 2 from above. Now I will only check the combinatorial requirement for C 1234 since the other cases are trivial. Now we can see that the geometric realization of the orer complex of (C, C 1234 is homotopic equivalent to S 1 S 1 S 1 an hence im k H ((C n, C 1234 LC( 4 ; k(z = 3z. 2 We now that ɛ 1 (C 1234 = D 1234, D 12 34, D 13 24, D Again we see that the realization of the orer complex of (D, D 1234 is homotopic equivalent to S 1 S 1 S 1 S 1 S 1 S 1 an hence im k H ((C n, D 1234 ; k(z = LD( 4 2 6z. Lastly we see that for σ = {12}, {13}, {14} that the realization of the orer complex of (D, D σ σ c is homotopic equivalent to S 0, an hence that im k H ((C n, D σ σ c LD( 4 ; k(z = 1. So we get the equation (2 to be 2 3z = 6z z 1 z 1 z 1 (1.5 which is true, an hence 4 2 is a Golo complex. We will later see that this is true for all complexes on the form 2. But for now we will use the theorem for a more trivial case. 8

10 CHAPTER 1. PRELIMINARIES Corollary A simplicial complex on the vertex set n with the property that σ M( : σ n 2 + 1, then is a Golo complex over any fiel k. Proof. Now let be a complex fulfilling the requirement of the corollary. Now all the minimal faces of the complex consist of over half the elements of vertex set. That means for all σ, τ M( : σ τ. So there are no elements in L D( on the form D σ1... σ n. Now this means that if for D, D L D( ɛ(d = ɛ(d, then we have D = D so since we alreay knew that ɛ was surjective, it is therefore bijective. So we nee to check that for all C L C( we have im k H ((C n, C LC( ; k(z = im k H ((C n, ɛ 1 (C LD( ; k(z since for all D L D( n i c(d = 1. Now take D σ, D τ L D( n i such that D σ < D τ, this means that σ τ an hence that ɛ(d σ = C σ < C τ = ɛ(d τ, an hence the poset L D( an L C( are the same poset up to a renaming of the elements. An hence the requirement is trivial. 9

11 Chapter 2 Helpful results To prove that 2 is a Golo complex that is, fulfills the theorem 1.10 (2 requirement we will nee a way of computing the imensions of the homology groups of certain intervals. Now in general I know of no metho to o this, but if we coul recognize the orer complexes of the intervals as something nice, as for example a wege of spheres, an fin some way to count the number of spheres with imension, then we woul be able to check the requirement quite easily. So we will in this chapter state some efinitions an theorems about shellability of complexes, in the hope that this will ai us to o exactly as we want. This is entire chapter is full of results taken from [BjWa], an you are aske to go there for proofs an further iscussions of the results. 2.1 Introuction to shellability Before we can begin stating what we mean with shellability we will nee this rather basic efinition, of intervals with respect to the inclusion orer. Definition 2.1. For a set F, we write F for the set {σ σ F } i.e. the power set of F. The efinition might seem reunant, but this will lea the following to confirm with [BjWa]. Definition 2.2. A simplicial complex is calle shellable if its facets can be arrange in a linear orer F 1, F 2,..., F t in such a way that the subcomplex ( k 1 F i=1 i F k is pure, that is all its facets have equal carinality, an a ( F k 2-complex for all k = 2, 3,..., t. Such an orering is calle a shelling. So the efinition means that we are able to buil our complexes by aing facets in the shelling orer, such that they intersect our part complex in a face of carinality one less. Notice this is the unpure version of the efinition which allows for greater flexibility later on. Since we o not nee to require the facets have the same carinality. 10

12 CHAPTER 2. HELPFUL RESULTS 2.2 The h integer array Now to state more firmly what we want of this new concept of shellability is two-fol. First we woul like that geometric realizations of a shellable complex i homotopic equivalent to a wege of spheres since we rather trivially know how homology interacts with both wege-sums an spheres. But knowing just that, will o us a small help if we have no way of counting both the number of spheres an their respective imensions in the wege-sum. We therefore introuce the following: Definition 2.3. For a ( 1complex, let (i f i,j = number of faces σ of carinality j an with i = δ(σ := max{ τ σ τ } (ii h i,j = j ( i k ( 1 j k j k f i,k (iii The triangular integer arrays an f = (f i,j 0 j i h = (h i,j 0 j i are calle the f-triangle an the h-triangle of, respectively. Notice again that this is a generalization of the normal f- an h-vectors which we usually see with simplicial complexes. We woul hope that these to new integer arrays somehow woul be able to tell us how many spheres there are in our wege-sum, an luckily they are: Theorem 2.4. Let be shellable ( 1-complex. Then the geometric realization of has the homotopy type of a wege of spheres, consisting of h j,j copies of the (j 1-sphere for 1 j Corollary 2.5. Let be shellable complex, then im k Hj 1 ( ; k = h j,j The corollary follows from the well known facts that H (S, k = k an that H ( v V T v = H v V (T v, where S is the -sphere, with > 0 an for T v some pointe topological space. We also know that H 0 (S 0, k = k, an that a -sphere oesn t have homology in other than the th imension. Furthermore it is well known that homology is stable uner homotopy equivalences an that a simplicial complex an its geometric realization have the same homology. 11

13 CHAPTER 2. HELPFUL RESULTS 2.3 Lexicographic shelling Now all this new theory will not help us out unless we are able to recognize a complex as shellable. Now remember that in our case we want to recognize an interval of a posets orer complex as shellable, so let s look at some theory for how this is one. So we will efine a map from a poset with certain properties, an if they are fulfille the poset will be shellable. Definition 2.6. For a poset P, then (i an ege labeling of P is a map λ : E(P Λ, for some poset Λ (ii for a maximal chain m = (c 1, c 2,... in P, an an ege labeling λ we write λ(m = (λ((c 1 < c 2, λ((c 2 < c 3,.... (iii an ege labeling λ is calle an ege rising labeling (ER-labeling if for every interval [x, y], with x, y P, there is an unique maximal chain m = (c 1, c 2,... with a rising label, that is λ((c 1 < c 2 < λ((c 2 < c 3 <... in Λ. (iv an ER-labeling is calle an ege lexicographic labeling (EL-labeling if for every interval [x, y] in P the unique maximal chain with rising label is lexicographic first among the maximal chains in the interval. (v a boune poset P is calle EL-shellable if it amits an EL-labeling. We will efine what we mean with lexicographic in (iv of the efinition above. Definition 2.7. For a = (a 1, a 2,... an a = (a 1, a with i N : a i, a i P for some poset P. We say that a is lexicographic before a if j such that i < j a i = a i an a j < a j in the poset P. Now if a an a are of ifferent length in the sense that a = (a 1, a 2,..., a k an a = (a 1, a 2,..., a l then we say that a is lexicographic before a if j such that i < j a i = a i an a j < a j in the poset P or if i k a i = a i an k < l. Theorem 2.8. If a boune poset P is EL-shellable, then ( P is shellable So that means if we want to check if some orer complex is shellable it is enough to check that if we a minimal an maximal elements that there is some EL-shelling. We will see later that this is quite hany since, it for some posets, are a rather easy thing to construct. 12

14 Chapter 3 The curious case of 2 We will here show using theorem 1.10 (2 that 2 is a Golo complex. We will o this by stuying the iagonal- an coorinate arrangement lattices, showing that the interesting intervals are shellable with the help of ELshellability an then using the h array to rewrite the equality of polynomials. Proposistion is a Golo complex over any fiel k, for > 1. To prove this we will nee some further results, so let s start with two lemmas. Here we will use the theory state in chapter 2 to show that the relevant intervals for our complex are shellable. Lemma 3.2. [C n, C 2 ] LC( n k is EL-shellable for k > 1 Proof. We see that this is a boune poset an ientify C n with C. We efine an ege labeling λ by λ(c σ < C τ = max(τ σ. Now clearly for all σ n with k σ, C σ [C n, C n ] LC( n k Now take arbitrary C σ, C τ (C n, C n ] LC( n, now all chain labels of maximal chains in the interval [C σ, C τ ] will be permutations of the elements k τ σ, an all these permutations are also the chain labels, so trivially they can only be arrange in a rising way in one way, an this way will also be lexicographic first. Now take the interval [C, C τ ], take i 1 <... < i k to be the first k elements of τ. Since σ = {i 1,..., i k } consists of k elements an is a subset of n we know that C σ [C, C τ ], now from above we know that there is a chain m which is uniquely rising maximal an lexicographic first in the interval [C σ, C τ ], lets call λ(m = (j 1,..., j r. Now efine m to be the maximal chain from C to C τ by first going to C σ an then follow m. Now since σ {j 1,..., j r } = τ by the way we have efine λ we know that i k < j 1, an hence λ(m = (i k, j 1,..., j k is rising. Since i 1 <... < i k was the first k elements of τ, then all other elements of k elements γ τ we will get that λ(c < C γ > λ(c < C σ, an hence m will be lexicographic first among the maximal chains. 13

15 CHAPTER 3. THE CURIOUS CASE OF 2D D Now take a rising maximal chain m from C to C τ, lets call the secon element of the chain C σ, then clearly σ = k, now the maximal chain m from C σ to C τ, which is just m just skipping C must be rising as well, lets assume λ(m = (j 1,..., j r then we know that σ {j 1,..., j r } = τ, but since the σ {j 1,..., j r } =, an since m was rising we know that for i k = max σ i k < j 1, an hence σ must consist of the first k elements of τ, we therefore have uniqueness of the rising chain an hence λ is an EL-shelling. An again much in the same way as before we will se that also L D( 2 is EL-shellable. Even our shelling map will be taken as before an expane to a new shelling map. Lemma 3.3. The poset L D( 2 is EL-shellable. Proof. We see that this is a boune poset an ientify C n with D. We efine an ege labeling λ by λ(d σ < D τ = max τ σ, an for any σ 2 with carinality, we efine λ(d σ < D σ σ c = λ(d σ σ c < D 2 = 2. Now since this is almost the same labeling as in lemma 3.2, for interval which oes not contain elements of the form D σ σ c with the same argument as before λ fulfills the EL-requirement. Trivially the EL-requirement is fulfille for the intervals [D σ, D σ σ c] an [D σ σ c, D 2 ], for all σ 2 with elements, since they both only contain on ege. Now take the interval [D, D σ σ c] for some σ 2 with elements. The Hasse-iagram of it will look like this 2 D σ σ c 2 (3.1 D σ D σ c max σ max σ c D with the ege labeling written on. We clearly see that 2 is an element in either σ or σ c. So assume without loss of generality that 2 / σ, then max σ < 2 an hence the left chain in the iagram above will be rising an lexicographic first. Since pr assumption 2 σ c we can see that the left chain has the label (2, 2 an is therefore not rising, an hence the EL-criteria is fulfille in this case. Now take D τ L D( 2, such that there is some σ 2 with elements such that D σ σ c [D τ, D 2 ], then either D τ = D or τ = σ or τ = σ c. So lets cover the case were τ = σ or τ = σ c. Now we can assume without loss of generality that τ = σ, an take m to be the maximal chain which goes (D σ, D σ σ c, D 2, then the label of m is λ(m = (2, 2, an hence m is 14

16 CHAPTER 3. THE CURIOUS CASE OF 2D D clearly not a rising chain. Now we know since λ is almost the same as in lemma 3.2 that there is a chain m which is rising, oesn t go through D σ σ c an is the lexicographic first among the maximal chains which on t. Now assume that D δ is the secon element of the chain m, now if max(δ σ = 2, then since the label of m is rising then δ = 2, but this can t be since > 1, an since δ is the secon element it must contain + 1 elements. Hence m will be lexicographic before m. Now take the interval [D, D 2 ]. All maximal chains through D σ σ c will have the label (i, 2, 2 for some i =, + 1,..., 2 an are therefore not rising. Again we can by the same arguments as in the coorinate lattice case state that there is a unique rising maximal chain, m which oesn t go through D σ σ c which is lexicographic before all other maximal chains not going through D σ σ c. So let m be a maximal chain through D σ σ c, with λ(m = (i, 2, 2 an let λ(m = (j 1, j 2,..., j +1. Now if j 1 < i we are one, an m will be before m in the lexicographic orer. Now assume j 1 > i, but we saw in lemma 3.2 that j 1 woul be since the only rising maximal chain form D to D 2 starte with D σ where σ consiste of the first elements, an since is the lowest label on any ege from D we have a contraiction. So assume that i = j 1. Now the chain m consists of + 2 elements, so if the j 2 = 2 then since m is rising there will be no j k, k > 2, but that woul mean that the thir element of m was the last, now since > 1 this cannot be true, an hence j 2 < 2, an therefore m is before m in the lexicographic orer. We have now covere all possible intervals an can therefore conclue that λ is an EL-shelling An we are now reay to prove our proposition. We will start by arguing in the various trivial cases first, an then see that we can use the two lemmas an the results we know about shellability to come to the conclusion. Proof of proposition 3.1. Now take any fiel k. I will use theorem 1.10 (2 for this, an will therefore nee to verify, that for all C L C( 2 that H ((C n, C LC( 2 ; k(z = D ɛ 1 (H ( z c(d 1 H ((C n, D ; k(z LD( 2 Now assume that C C 2, trivially there is some σ 2, such that C = C σ, but then since σ 2 we see that ɛ 1 (C σ = {D σ }, since that trivially ɛ(d σ = C σ an assume there was an element of the form D τ1... τ n such that ɛ(d τ1... τ n = C σ, but this woul imply that n i=1 τ i = σ, but all the τ i are isjoint an contain at least elements, an hence σ n, which cannot be if n 1. Furthermore we see that (C n, C LC( is the same poset up to 2 renaming as (C n, ɛ 1 (C LD( 2, an hence the requirement is trivial. 15

17 CHAPTER 3. THE CURIOUS CASE OF 2D D So now take C 2 L C( 2, we see that ɛ 1 (C 2 = {D 2 } {D σ σ c σ 2, σ = }. Now we know from lemma 3.2 an theorem 2.8 that the orer complex of (C, C 2 LC( 2 is shellable. Likewise we know from lemma 3.3 an theorem 2.8 that the orer complex of (D, D 2 LD( 2 is shellable. Now from (3.1 we can see that the chain complex of (D, D σ σ c, for LD( 2 σ 2 with elements, is two isjoint points an hence trivially shellable, since any orering of the points fulfills efinition 2.2. Now this means that we can use corollary 2.5 to restate the requirement of theorem 1.10 (2 in the following way: For all j N h C 2 j,j = D ɛ 1 (C 2 ( 1 c(d 1 h D j c(d+1,j c(d+1 (3.2 where h D i,j is h i,j for the orer complex of (D, D an likewise LD( 2 hc 2 j,j is h j,j integer for the orer complex of (C, C 2. LC( Now with same 2 naming convention for the f integers we can use efinition 2.3 (ii to rewrite this with the left han sie of (3.2 being j ( j k ( 1 j k f C 2 j k j,k = an the right han sie of (3.2 being D ɛ 1 (C 2 j c(d+1 ( 1 c(d 1 = = D ɛ 1 (C 2 D ɛ 1 (C 2 j ( 1 j k f C 2 j,k (3.3 ( j c(d + 1 k ( 1 j c(d+1 k j c(d + 1 k j c(d+1 ( 1 c(d 1 j c(d+1 ( 1 j c(d+1 k f D j c(d+1,k f D j c(d+1,k ( 1 j k f D j c(d+1,k (3.4 Now fix j 2 an lets stuy f D 2 j c(d 2 +1,k = f D 2 j,k, now all maximal chains in the interval (D, D 2 LD( which goes through elements on the form 2 D σ σ c for σ having elements, are of length 2, an hence since the rest of poset is like that of (C, C 2, the only ifference between f D 2 LC( 2 j,k an f C 2 j,k is in the case of j = 2. Now f D 2 2,k counts the number of faces which are at most a face of a one-simplex. Now assume that we have such an element, then it is a chain in the interval (D, D 2 LD(. If it oes not go through 2 D σ σ c, then there will be a mirror chain in (C, C 2, so f D 2 LC( 2 2,k is equal 16

18 CHAPTER 3. THE CURIOUS CASE OF 2D D to f C 2 2,k plus the number of elements of carinality k containing D σ σ c which are not also in a facet not containing D σ σ c. Of these there are only the following faces of carinality one: {D σ σ c}, with σ 2 σ =, an of 2 these there are (2 2, i.e. one for every other subset of carinality of 2. Of carinality two there are only the faces {D σ, D σ σ c} of which there are ( 2, one for each subset of carinality of 2. So all in all we see that f D 2 j,k = f D σ σ c j 1,k = f C 2 j,k when j 2 k 1, 2 f C 2 2,1 + (2 2 f C 2 2,2 + ( when j = 2 k = 1 when j = 2 k = 2 (3.5 So let us now turn our attention to f D σ σ c j c(d 2 +1,k = f D σ σ c j 1,k. Now this complex consists of two isjoint points, an therefore rather trivial we see that 1 when j = 2 k = 0 2 when j = 2 k = 1 0 else (3.6 Now since there are (2 2 elements on the form D 2, we get the right sie of the equality (3.2 by way of the rewrite of (3.4 as for j 2 j ( 1 j k f C 2 j,k + 0 (3.7 i.e. the left han sie of (3.2 by way of (3.3, now in the case of j = 2 we get D=D 2 D=D σ σ c 2 c(d+1 2 c(d 2 +1 = = = = ( 1 j k f D j c(d+1,k (3.8 ( 2 ( 1 j k f D 2 j c(d 2 +1,k + 2 ( ( 1 j k f D 2 j,k + ( 1 j k f D σ σ c 2 j 1,k ( 2 2 ( ( 2 ( 1 j k f C 2 2 j,k ( c(d σ σ c +1 2 ( 1 2 k f C 2 j,k + 0 (3.9 i.e. exactly the left han sie of (3.2 by way of (3.3. We have now shown that the complex 2 fulfills (2 in theorem 1.10, an hence it is a Golo complex over k. ( 1 j k f D σ σ c j c(d σ σ c +1,k 17

19 CHAPTER 3. THE CURIOUS CASE OF 2D D 3.1 Further problems I will en this project with some further problems which have occurre to me. Several other complexes which have stuie to some egree seems to have intervals of their coorinate- an iagonal intersection lattices which are homotopic equivalent to a wege of spheres, an hence easy to calculate the homology of. Is there some characterization of a complex with these types of coorinate- an iagonal intersection lattices? When working with simplicial complexes with 4 vertices, all of them faces of the complex; I foun no immeiate non-golo complex. The examples I i both the coorinate- an iagonal intersection lattices were weges of spheres, with the iagonal intersection lattice having one extra sphere for each element of the form D σ σ c, which, ue to the nature of (D, D σ σ c being the zero sphere, mae the polynomials in theorem 1.10 (2 match. So one coul woner if there is no non-golo complex on 4 vertices, or if not what the minimum number of vertices require are before non-golo complexes start showing. We know ue to corollary 1.11 that complexes on 1, 2 an 3 vertices were all the vertices are faces of the complex always are Golo, since if they have non-faces they are of rather large carinality. The ege labeling we use in lemma 3.2 worke in more cases that the one we use in lemma 3.3, so one coul woner if you coul expan the EL-labeling from 3.3 to at least n i, an thusly maybe be able to prove that n i is a Golo complex, which we know is true from [GrTh], using theorem 1.10, an hence expaning our main result. 18

20 Bibliography [Av] Avramov, Luchezar L.: Infinite free resolutions. Six lectures on commutative algebra (Bellaterra, In: Progressions in Mathemathics, vol. 166, Birkhäuser, Basel, pp [Be] Berglun, Alexaner: Poincaré series of monomial rings. In: Journal of Algebra, vol. 295 (2006, pp [BeJö] Berglun, Alexaner; Jöllenbeck, Michael. On the Golo property of Stanley-Reisner rings. In: Journal of Algebra, vol 315 (2007, pp [BjWa] Björner, Aners; Wachs, Michelle L.: Shellable nonpure complexes an posets. I. In: Transactions of the American mathematical society, vol 348 (1996, pp [GrTh] Grbić, Jelena; Theriault, Stephen: The homotopy type of the complement of coorinate subspace arrangement. In Topology, vol 46 (2007, pp [St] Stanley, Richar P.: Enumerative Combinatorics - Secon eition version of 9 May rstan/ec/ec1 visite 10th of may