Shellable Complexes and Topology of Diagonal Arrangements
Discrete Comput Geom
Shellable Complexes and Topology of Diagonal Arrangements
Sangwook Kim 0 1
0 Present address: S. Kim Department of Mathematical Sciences, George Mason University , Fairfax, VA 22030 , USA
1 S. Kim ( ) School of Mathematics, University of Minnesota , Minneapolis, MN 55455 , USA
We prove that if a simplicial complex is shellable, then the intersection lattice L for the corresponding diagonal arrangement A is homotopy equivalent to a wedge of spheres. Furthermore, we describe precisely the spheres in the wedge, based on the data of shelling. Also, we give some examples of diagonal arrangements A where the complement MA is K (π, 1), coming from rank3 matroids. Consider Rn with coordinates u1, . . . , un. A diagonal subspace Ui1...ir is a linear subspace of the form ui1 = · · · = uir . A diagonal arrangement (or a hypergraph arrangement) A is a finite set of diagonal subspaces of Rn. Throughout this paper, we assume that for any H1, H2 ∈ A, H1 is not included in H2. For a simplicial complex on a vertex set [n] := {1, 2, . . . , n} such that dim ≤ n − 3, one can associate a diagonal arrangement A in Rn as follows. For a facet F of , let UF be the diagonal subspace ui1 = · · · = uir where F = [n] − F = {i1, . . . , ir }. Define
Shellable simplicial complexes; Diagonal arrangements; K (π; 1)

1 Introduction
A
= {UF  F is a facet of
}.
This work forms part of the author’s doctoral dissertation at the University of Minnesota, supervised
by Vic Reiner, and partially supported by NSF grant DMS0245379.
are
If every subspace in a diagonal arrangement A in Rn has the form Ui1...ir with r ≥ 2,
one can find a simplicial complex on [n] satisfying A = A .
Two important spaces associated with an arrangement A of linear subspaces in Rn
MA = Rn −
H
and
VA◦ = Sn−1 ∩
H,
H ∈A H ∈A
called the complement and the singularity link.
Diagonal arrangements arise in connection waitnhdimVAportant questions in many
differ
We are interested in the topology of MA ◦ for a diagonal arrangement A.
ent fields. In computer science, Björner, Lovász and Yao [
5
] found lower bounds on
the complexity of kequal problems using the topology of diagonal arrangements (see
also [
4
]). In group cohomology, it is well known that MBn for the braid arrangement
Bn in Cn is a K(π, 1) space with the fundamental group isomorphic to the pure braid
group [
10
]. Khovanov [
13
] showed that MAn,3 for the 3equal arrangement An,3 in
Rn is also a K(π, 1) space.
Note that MA and VA◦ are related by Alexander duality as follows:
H i (MA; F) = Hn−2−i (VA◦; F) (F is any field).
(1.1)
In the mid1980s, Goresky and MacPherson [
11
] found a formula for the Betti
numbers of MA, while the homotopy type of VA◦ was computed by Ziegler and
Živaljevic´ [
18
] (see Sect. 5). The answers are phrased in terms of the lower intervals in
the intersection lattice LA of the subspace arrangement A, that is, the collection of all
nonempty intersections of subspaces of A ordered by reverse inclusion. For general
subspace arrangements, these lower intervals in LA can have arbitrary homotopy type
(see [18, Corollary 3.1]).
Our goal is to find a general sufficient condition for the intersection lattice LA
of a diagonal arrangement A to be well behaved. Björner and Welker [
8
] showed
that LAn,k has the homotopy type of a wedge of spheres, where An,k is the kequal
arrangement consisting of all Ui1...ik for all 1 ≤ i1 < · · · < ik ≤ n (see Sect. 3), and
Björner and Wachs [6] showed that LAn,k is shellable. More generally, Kozlov [15]
showed that LA is shellable if A satisfies certain technical conditions (see Sect. 3).
Suggested by a homological calculation (Theorem 5.4), we prove the following main
result, capturing the homotopy type assertion from [
15
] (see Sect. 4).
Theorem 1.1 Let be a shellable simplicial complex. Then the intersection lattice
L for the diagonal arrangement A is homotopy equivalent to a wedge of spheres.
Furthermore, one can describe precisely the spheres in the wedge, based on the
shelling data. Let be a simplicial complex on [n] with a shelling order F1, . . . , Fq
on its facets. Let σ be the intersection of all facets, and σ¯ its complement. Let
G1 = F1 and for each i ≥ 2, let Gi be the face of Fi obtained by intersecting the
walls of Fi that lie in the subcomplex generated by F1, . . . , Fi−1, where a wall of Fi
is a codimension 1 face of Fi . An (unordered) shellingtrapped decomposition (of σ¯
over ) is defined to be a family {(σ¯1, Fi1 ), . . . , (σ¯p, Fip )} such that {σ¯1, . . . , σ¯p} is a
decomposition of σ¯ as a disjoint union
σ¯ =
p
j=1
σ¯j
p
j=1
and Fi1 , . . . , Fip are facets of such that Gij ⊆ σj ⊆ Fij for all j = 1, . . . , p. Then
the wedge of spheres in Theorem 1.1 consists of (p − 1)! copies of spheres of
dimension
p(2 − n) +
for each shellingtrapped decomposition D = {(σ¯1, Fi1 ), . . . , (σ¯p, Fip )} of σ¯ .
Moreover, for each shellingtrapped decomposition D of σ¯ and a permutation w of [p − 1],
there exists a saturated chain CD,w (see Sect. 4.1) such that removing the simplices
corresponding to these chains leaves a contractible simplicial complex.
The following example shows that the intersection lattice in Theorem 1.1 is not
shellable in general, even though it has the homotopy type of a wedge of spheres.
Example 1.2 Let be a simplicial complex on [8] with a shelling 123456 (short for
{1, 2, 3, 4, 5, 6}), 127, 237, 137, 458, 568, 468. Then the order complex of the upper
interval (U78, 1ˆ) is a disjoint union of two circles, hence is not shellable. Therefore,
the intersection lattice L for the diagonal arrangement A is also not shellable.
The intersection lattice L is shown in Fig. 1 (thick lines represent the open interval
(U78, 1ˆ)). In Fig. 1, the subspace Ui1...ir is labeled by i1 . . . ir . Also note that a facet
F of corresponds to the subspace U[n]−F . For example, the facet 127 corresponds
The next example shows that there are nonshellable simplicial complexes whose
intersection lattices are shellable.
Example 1.3 Let be a simplicial complex on [4] whose facets are 12 and 34. Then
is not shellable. But the order complex of L consists of two vertices, hence is
shellable.
In Sect. 3, we give Kozlov’s result and show that its homotopy type consequence
is a special case of Theorem 1.1. Also, we give a new proof of Björner and Welker’s
result using Theorem 1.1. In Sect. 4, we prove Theorem 1.1. In Sect. 5, we deduce
from it the homotopy type and the homology of the singularity link (and hence the
homology of the complement) of a diagonal arrangement A for a shellable
simplicial complex . In Sect. 6, we give some examples in which MA are K(π, 1),
coming from matroids.
2 Basic Notions and Definitions
In this section, we give a few definitions used throughout this paper.
First, we start with the definition of (nonpure) shellability of simplicial complexes;
see [
6, 7
] for further background on this notion.
is called a shelling order or shelling.
Definition 2.1 A simplicial complex is shellable if its facets can be arranged in linear
dimensional for all k = 2, . . . , q, where 2F =ik=−{11G2FGi )⊆∩ F2F}k. Sisucphuraenaonrdde(rdiinmg Fofk
f−ac1e)tsorder F1, F2, . . . , Fq in such a way that (
There are several equivalent definitions of shellability. The following restatement
of shellability is often useful.
Lemma 2.2 [6, Lemma 2.3] A linear order F1, F2, . . . , Fq of facets of a simplicial
complex is a shelling if and only if for every Fi and Fk with Fi < Fk , there is a
facet Fj < Fk such that Fi ∩ Fk ⊆ Fj ∩ Fk Fk , where G F means that G has
codimension 1 in F .
It is well known that a pure ddimensional shellable simplicial complex has the
homotopy type of a wedge of dspheres. Björner and Wachs [
6
] generalized this result
to the nonpure case, i.e., a nonpure shellable simplicial complex has the homotopy
type of a wedge of spheres. However, these spheres need not be equidimensional.
The link of a face σ of a simplicial complex is
link σ = {τ ∈
 τ ∩ σ = ∅ and τ ∪ σ ∈
}.
Björner and Wachs [
7
] showed that shellability is inherited by all links of faces in a
simplicial complex.
Proposition 2.3 If is shellable, then so is link σ for all faces σ ∈
induced order on facets of link σ .
, using the
Now we give the definition of the order complex of a poset which we will use
frequently.
Definition 2.4 The order complex (P ) of a poset P is the simplicial complex
whose vertices are the elements of P and whose faces are the chains of P .
For the order complex ((x, y)) of an open interval (x, y), we will use the
notation (x, y). When we say that a finite lattice L with bottom element 0ˆ and top
element 1ˆ has some topological properties, such as purity, shellability and homotopy
type, it means the order complex of L := L − {0ˆ, 1ˆ} has those properties.
3 Special Cases That Were Known
In this section, we give Kozlov’s theorem and show how its consequence for
homotopy type follows from Theorem 1.1. Also, we give a new proof of Björner and
Welker’s theorem about the intersection lattice of the kequal arrangements using
Theorem 1.1.
Kozlov [15] showed that A has shellable intersection lattice if satisfies some
conditions. This class includes kequal arrangements and all other diagonal
arrangements for which the intersection lattice was proved shellable.
Theorem 3.1 [15, Corollary 3.2] Consider a partition of [n] = E1 · · ·
that max Ei < min Ei+1 for i = 1, . . . , r − 1. Let
Er such
f : {1, 2, . . . , r} → {2, 3, . . .}
be a simplicial complex on [n] such that F is a facet
be a nondecreasing map. Let
of if and only if
(1) Ei − F  ≤ 1 for i = 1, . . . , r
(2) if min F ∈ Ei , then F  = n − f (i)
Then the intersection lattice for A
type of a wedge of spheres.
Proposition 3.2
in Theorem 3.1 is shellable.
Proof We claim that a shelling order is F1, F2, . . . , Fq such that the words w1, w2, . . . ,
wq are in lexicographic order, where wi is the increasing array of elements in F i . Let
Fs , Ft be two facets of with 1 ≤ s < t ≤ q. Then ws ≺lex wt . Let m be the first
number in [r] such that Em − Fs = Em − Ft . Construct the word w as follows:
is shellable. In particular, L
has the homotopy
w
(1) w ∩ Ei = ws ∩ Ei for i = 1, . . . , m
(2) for i = m + 1, . . . , q,
w ∩ Ei =
⎧
⎨ wt ∩ Ei if w ∩
⎩ ∅
otherwise,
i
j=1 Ej
≤ f (l),
where min ws ∈ El .
Note that the length of w is f (l) and w ≺lex wt . Let F be the set of all elements
which do not appear in w. Since F satisfies the two conditions from Theorem 3.1,
F is a facet of . Since F ∩ Ft = Ft − (Em − Fs ) and Em − Fs is a subset of Ft of
size 1, F ∩ Ft has codimension 1 in Ft . Also Fs ∩ Ft ⊆ F ∩ Ft . Hence F1, F2, . . . , Fq
is a shelling by Lemma 2.2.
Example 3.3 Consider the partition of
[
7
] = {1} {2, 3} {4} {5, 6, 7}
and the function f given by f (1) = 2, f (2) = 3, f (3) = 4, and f (4) = 5. Then the
facets of the simplicial complex that satisfy the conditions from Theorem 3.1 and
the corresponding words can be found in Table 1. Thus the ordering 34567, 24567,
23567, 23467, 23457, 23456, 1367, 1357, 1356, 1267, 1257 and 1256 is a shelling
for .
One can also use Theorem 1.1 to recover the following theorem of Björner and
Welker [
8
].
Theorem 3.4 The order complex of the intersection lattice LAn,k has the homotopy
type of a wedge of spheres consisting of
(p − 1)!
0=i0≤i1≤···≤ip=n−pk j=0
p−1
copies of (n − 3 − p(k − 2))dimensional spheres for 1 ≤ p ≤ nk .
Proof It is clear that An,k = A n,n−k , where n,n−k is a simplicial complex on
[n] whose facets are all n − k subsets of [n]. Here, σ = ∅, and so σ¯ = [n].
By ordering the elements of each facet in increasing order, the lexicographic
order of facets of n,n−k gives a shelling. Also, one can see that the facets of the
form Fi = {1, 2, . . . , m, am+1, . . . , an−k}, where m + 1 < am+1 < · · · < an−k , have
Gi = {1, . . . , m}. Thus, Gi ⊆ σi ⊆ Fi implies F i ⊆ σ¯i ⊆ Gi = {m + 1, . . . , n}. Note
that min σ¯ = min F = mp + 1 and F has k elements. Thus, in any shellingtrapped
decomposition [n] = j=1 σ¯j , one has p ≤ nk .
Let 1 ≤ p ≤ nk and 0 = i0 ≤ i1 ≤ · · · ≤ ip = n − pk. We will construct a
shellingtrapped family {(σ¯1, Fi1 ), . . . , (σ¯p, Fip )} as in Theorem 1.1. Since Fi1 <
· · · < Fip , we have min σ¯1 > · · · > min σ¯p. In particular, 1 ∈ F ip ⊆ σ¯p. Thus there
are kn−−11 ways to pick F ip (equivalently, Fip ). Now suppose that we have chosen
Fip , . . . , Fip−j+1 . We pick Fip−j so that min F ip−j = min σ¯ip−j is the ij + 1st element
in [n] − (Fip ∪ · · · ∪ Fip−j+1 ). Then we have n−jkk−−1ij −1 ways to choose Fip−j . For
each j = 1, . . . , p, there are ij − ij−1 elements in [n] − (Fip ∪ · · · ∪ Fip−j+1 ) which
are strictly between min F ip−j+1 and min F ip−j and they must be contained in one of
σ¯p, . . . , σ¯p−j+1 (i.e., there are j ij −ij−1 choices). Therefore there are
p−1
j=0
n − j k − ij − 1
k − 1
p
j=1
j ij −ij−1 =
p−1
j=0
shellingtrapped families. By Theorem 1.1, each of those families contributes (p − 1)!
copies of spheres of dimension
Theorem 1.1 will be deduced from a more general statement about homotopy types
of lower intervals in L , Theorem 4.1. Throughout this section, we assume that is
a simplicial complex on [n] with dim ≤ n − 3.
Theorem 4.1 Let F1, . . . , Fq be a shelling of and Uσ¯ a subspace in L for some
subset σ¯ of [n]. Then (0ˆ, Uσ¯ ) is homotopy equivalent to a wedge of spheres,
consisting of (p − 1)! copies of spheres of dimension
δ(D) := p(2 − n) +
for each shellingtrapped decomposition D = {(σ¯1, Fi1 ), . . . , (σ¯p, Fip )} of σ¯ .
Moreover, for each such shellingtrapped decomposition D and each permutation
w of [p − 1], one can construct a saturated chain CD,w (see Sect. 4.1), such that if
one removes the corresponding δ(D)dimensional simplices for all pairs (D, w), the
remaining simplicial complex (0ˆ, Uσ¯ ) is contractible.
To prove this result, we begin with some preparatory lemmas.
First of all, one can characterize exactly which subspaces lie in L when
is shellable. Recall that for σ¯ = {i1, . . . , ir } ⊆ [n], we denote by Uσ¯ the diagonal
subspace of the form ui1 = · · · = uir . We also use the notation Uσ¯1/···/σ¯p to denote
Uσ¯1 ∩ · · · ∩ Uσ¯p for pairwise disjoint subsets σ¯1, . . . , σ¯p of [n].
A simplicial complex is called galleryconnected if any pair F, F of facets are
connected by a path
F = F0, F1, . . . , Fr−1, Fr = F
of facets such that Fi ∩ Fi−1 has codimension 1 in Fi for i = 1, . . . , r . Since it is
known that Cohen–Macaulay simplicial complexes are galleryconnected, shellable
simplicial complexes are galleryconnected.
Lemma 4.2
(1) Given any simplicial complex
on [n], every subspace H in L
has the form
for pairwise disjoint subsets σ¯1, . . . , σ¯p of [n] such that σi can be expressed as
an intersection of facets of for i = 1, 2, . . . , p.
(2) Conversely, when is galleryconnected, every subspace H of Rn that has the
above form lies in L .
Proof To see (1), note that every subspace H in L
has the form
for pairwise disjoint subsets σ¯1, . . . , σ¯p of [n]. Since H =
F of facets of ,
F ∈F UF for some family
for some subfamily Fj of F for all j = 1, . . . , p. Therefore
for j = 1, . . . , p.
For (2), suppose that H has the form
H = Uσ¯1/···/σ¯p
for pairwise disjoint subsets σ¯1, . . . , σ¯p of [n] such that σi can be expressed as an
intersection of facets of for i = 1, 2, . . . , p. It is enough to show the case when
H = Uσ¯ . Since galleryconnectedness is inherited by all links of faces in a simplicial
complex, we may assume σ = F ∈F F , where F is the set of all facets of , without
loss of generality. Then σ¯ = F ∈F F .
We claim that the simplicial complex whose facets are {F  F ∈ F } is connected.
Since dim ≤ n − 3, every facet of has at least two elements. Let F , F be two
facets of with F < F . Since is galleryconnected, there is F = F1, F2, . . . , Fk =
F such that Fi ∩ Fi−1 has codimension 1 in Fi for all i = 2, . . . , k. Thus F i and F i−1
share at least one vertex for all i = 2, . . . , k. This implies that F and F are connected.
Hence is connected.
Therefore Uσ¯ = F ∈F UF .
The next example shows that the conclusion of Lemma 4.2(2) can fail when
not assumed to be galleryconnected.
is
Example 4.3 Let be a simplicial complex with two facets 123 and 345. Then is
not galleryconnected. Since L has only four subspaces R5, U12, U45 and U12/45, it
does not have the subspace U1245, even though 1245 = 3 is an intersection of facets
123 and 345 of . Thus the conclusion of Lemma 4.2(2) fails for .
[0ˆ, Uσ¯ ].
The following easy lemma, whose obvious proof is omitted, shows that every
lower interval [0ˆ, H ] can be written as a product of lower intervals of the form
Lemma 4.4 Let H ∈ L
be a subspace of the form
for pairwise disjoint subsets σ¯1, . . . , σ¯p of [n]. Then
In particular,
where “∗” denotes join of spaces.
Note that the join of a space X with an empty set equals X.
The next lemma, whose proof is completely straightforward and omitted, shows
that the lower interval [0ˆ, Uσ¯ ] is isomorphic to the intersection lattice for the diagonal
arrangement corresponding to link σ .
Lemma 4.5 Let Uσ¯ be a subspace in L for some face σ of . Then the lower
interval [0ˆ, Uσ¯ ] is isomorphic to the intersection lattice of the diagonal arrangement
Alink (σ ) corresponding to link (σ ) on the vertex set σ¯ .
The following lemma shows that upper intervals in L are at least still homotopy
equivalent to the intersection lattice of a diagonal arrangement.
Lemma 4.6 Let Uσ¯ be a subspace in L for some face σ of . Then the upper
interval [Uσ¯ , 1ˆ ] is homotopy equivalent to the intersection lattice of the diagonal
arrangement A σ corresponding to the simplicial complex σ on the vertex set
σ ∪ {v} whose facets are obtained in the following ways:
(A) If F ∩ σ is maximal among
F ∩ σ  F is a facet of
with σ
F and F ∪ σ = [n] ,
then F = F ∩ σ is a facet of σ .
(B) If a facet F of satisfies F ∪ σ = [n], then F = (F ∩ σ ) ∪ {v} is a facet of
σ .
Proof We apply a standard crosscut/closure lemma [3, Theorem 10.8] saying that
a finite lattice L is homotopy equivalent to the sublattice consisting of the joins of
subsets of its atoms. By the closure relation ψ1 on [Uσ¯ , 1ˆ] which sends a subspace
to the intersection of all subspaces that lie weakly below it and cover Uσ in [Uσ , 1ˆ],
¯ ¯
one can see that [Uσ , 1ˆ] is homotopy equivalent to the sublattice Lσ generated by the
¯
subspaces of [Uσ¯ , 1ˆ] that cover Uσ . Using the map ψ2 defined by
¯
ψ2(Uτ¯ ) =
U(τ¯−σ¯ )∪{v}
Uτ
¯
if σ¯ ∩ τ¯ = ∅,
otherwise,
one can see that Lσ is isomorphic to the intersection lattice L σ for a simplicial
complex σ on the vertex set σ ∪ {v}. The facets of σ correspond to the subspaces
that cover Uσ¯ , giving the claimed characterization of facets of σ . Therefore the map
ψ := ψ2 ◦ ψ1 gives the homotopy equivalence between [Uσ¯ , 1ˆ] and L σ .
Facets of σ that do not contain v are called facets of type (A), and facets of
containing v are called facets of type (B).
σ
Example 4.7 Let be a simplicial complex on [5] with facets 123, 234, 35, 45 and
let σ = {1, 2, 3}. Then σ is a simplicial complex on {1, 2, 3, v} and its facets are 23
and v. The intersection lattices L and L σ are shown in Fig. 2 and it is easy to see
that the order complex for L σ is homotopy equivalent to the order complex for the
open interval (U45, 1ˆ) in L . Note that the thick lines in Fig. 2a represent the closed
interval [U45, 1ˆ] in L .
In general, the simplicial complex σ from Lemma 4.6 is not shellable, even
though is shellable (see Example 1.2). However, the next lemma shows that σ is
shellable if σ is the last facet in the shelling order.
Lemma 4.8 Let be a shellable simplicial complex. If F is the last facet in a
shelling order of , then F is also shellable. Moreover, if Fi is a facet of F of
type (B), then Gi = Gi ∩ F .
Proof One can check that the following gives a shelling order on the facets of F .
First list the facets of type (A) according to the order of their earliest corresponding
facet of , followed by the facets of type (B) according to the order of their
corresponding facet of .
To see the second assertion, let Fi be a facet of F of type (B), i.e.,
Fi = (Fi ∩ F ) ∪ {v}
for some facet Fi of such that F ∪ Fi = [n]. Then Gi = Gi ∩ F follows from the
observations that Fi ∩ F is an old wall of Fi , and all other old walls of Fi are Fk ∩ Fi
for some facets Fk of F of type (B) such that Fk ∩ Fi is an old wall of Fi .
Example 4.9 Let be a shellable simplicial complex on [7] with a shelling
12367, 12346, 13467, 34567, 13457, 14567, 12345 and let F = 12345. Then
F is a simplicial complex on {1, 2, 3, 4, 5, v} and its facets are 123v, 1234,
134v, 345v, 1345, 145v. Since 1234, 1345 are facets of F of type (A) and 123v,
134v, 345v, 145v are facets of F of type (B), the ordering 1234, 1345, 123v,
134v, 345v, 145v is a shelling of F .
We next construct the saturated chains appearing in the statement of Theorem 4.1.
4.1 Constructing the Chains CD,w
Let be a simplicial complex on [n] with a shelling F1, F2, . . . , Fq and let Uσ¯ be a
subspace in L . Let
D = {(σ¯1, Fi1 ), . . . , (σ¯p, Fip )}
be a shellingtrapped decomposition of σ¯ with i1 < i2 < · · · < ip, and let w be a
permutation on [p − 1].
It is well known that the lattice p of partitions of the set [n] ordered by refine
ment is homotopy equivalent to a wedge of (p − 1)! spheres of dimension p − 3
and there is a saturated chain Cw in p for each permutation w of [p − 1] such
that removing {Cw = Cw − {0ˆ, 1ˆ}w ∈ Sp−1} from the order complex of p gives a
contractible subcomplex (see [1, Example 2.9]).
We construct a chain CD,w in [0ˆ, Uσ¯ ] as follows:
(1) Since Gij ⊆ σj ⊆ Fij for all j = 1, 2, . . . , p, Lemma 4.2 shows that the interval
[UF i1 /···/F ip , Uσ¯ ] contains the subspaces UB1/···/Br , where Bm = k∈Km τ¯k such
that
• (K1/K2/ · · · /Kr ) is a partition of [p], and
• F ik ⊆ τ¯k ⊆ σ¯k for all k = 1, 2, . . . , p
Choose a saturated chain Cw in [UF i1 /···/F ip , Uσ¯ ] whose covering relations are
one of the following types (see [
15
]):
(a) UB1/B2/···/Br UB1∪B2/B3/···/Br ,
(b) UB1/···/Br UB1∪{a}/B2/···/Br , where a ∈ σ¯k − F ik such that F ik ⊆ B1,
where the covering relations of type (a) correspond to the saturated chain Cw
in p, i.e., the covering relation UB1/B2/···/Br UB1∪B2/B3/···/Br appears
exactly where (K1/K2/ · · · /Kr ) (K1 ∪ K2/K3/ · · · /Kr ) appears in p (where
means the covering relation).
(2) Define a saturated chain CD,w by
0
ˆ
UF ip
UF ip−1 /F ip
· · ·
UF i1 /···/F ip
followed by the chain Cw.
Note that the length of the chain CD,w = CD,w − {0ˆ, Uσ¯ } is
l CD,w = p + (p − 1) +
σ¯j  − F ij
− 2
p
j=1
p
j=1
= p(2 − n) +
Fij  + σ¯  − 3.
Example 4.10 Let
one can see that
be the shellable simplicial complex from Example 4.9. Then
D = {(45, F1 = 12367), (123, F6 = 14567), (67, F7 = 12345)}
is a shellingtrapped decomposition of [
7
]. Let w be the permutation in S2 with
w(1) = 2 and w(2) = 1. Then the maximal chain Cw in 3 corresponding to w is
(1/2/3) (1/23) (123). One can choose
where the covering U45/123/67 U45/12367 corresponds to (1/2/3) (1/23), and
U45/12367 U1234567 corresponds to (1/23) (123). Thus CD,w is the chain
0
ˆ
U67
The upper interval (U67, 1ˆ) is shown in Fig. 3 and the chain CD,w is represented by
thick lines.
The following lemma gives the relationship between the shellingtrapped
decompositions of [n] containing F and the shellingtrapped decompositions of F ∪ {v}.
Lemma 4.11 Let be a shellable simplicial complex such that the intersection of
all facets is empty. If F is the last facet in the shelling order of , then there is a
onetoone correspondence between
• Pairs (D, w) of shellingtrapped decompositions D of [n] over
and w ∈ SD−1, and
• Pairs (D, w) of shellingtrapped decompositions D of F ∪ {v} over
containing F
F and w ∈
Moreover, one can choose CD,w and CD,w in such a way that the homotopy
equivalence ψ appearing in the proof of Lemma 4.6 maps the chain CD,w − UF to the chain
Proof Let D = {(σ¯1, Fi1 ), . . . , (σ¯p, Fip )} be a shellingtrapped decomposition of [n]
over with Fip = F and let w be a permutation in Sp−1. Then Fij = (Fij ∩ F ) ∪ {v}
are facets of F of type (B) for j = 1, . . . , p − 1 and Fi1 < · · · < Fip−1 . By
Lemma 4.8, Gij = Gij ∩ F for all j = 1, . . . , p − 1.
There are two cases to consider:
(σj ∩ F ) ∪ {v} for j = 1, . . . , p − 1,
σj for j = p.
For j = 1, . . . , p − 1, Gij ⊆ σ˜j ⊆ Fij since Gij ⊆ σj ⊆ Fij .
Since Gip ⊆ σp, it must be that σp is an intersection of some old walls of F .
Thus one can find a family G of facets of such that σp = F ∈G (F ∩ F ) and
F ∩ F F for all F ∈ G. Since F ∪ F  = F  + 1 < n, one knows that F ∩ F
is a facet of F of type (A) for all F ∈ G. Let F = Fk ∩ F be the last facet in the
family {F ∩ F F ∈ G} (pick k as small as possible). Since all facets of F occurring
earlier than F have the form F ∩ Fi such that Fi < Fk and Fi ∩ F F , one can see
G ⊆ σ˜p ⊆ F . Thus D is a shellingtrapped decomposition of F ∪ {v} over F . Also
one can define w ∈ Sp−1 by
In this case, we claim that
Then
and
is a shellingtrapped decomposition of F ∪ {v}.
Let k = w(1). Define
D =
σ¯1, Fi1 , . . . , σ¯k ∪ {v}, Fik , . . . , σ¯p−1, Fip−1
σ˜j =
σj ∩ F
for j = k,
(σj ∩ F ) ∪ {v} for j = 1, . . . , kˆ, . . . , p − 1.
Gik ⊆ σ˜k = σk ∩ F ⊆ Fik ,
Gij ⊆ σ˜j = (σj ∩ F ) ∪ {v} ⊆ Fij ,
for j = 1, . . . , kˆ, . . . , p − 1. Thus D is a shellingtrapped decomposition of F ∪ {v}.
Also one can define w ∈ Sp−2 as follows:
be a shellingtrapped decomposition of F ∪ {v}, where Fi1 < · · · < Fip are facets of
F , and let w be a permutation in Sp−1. There is at most one facet of F of type
(A) because Fij ∪ Fik = F ∪ {v} for all j = k. Since Fi1 < · · · < Fip and the facets of
type (A) appear earlier than the ones of type (B), there are two possible cases.
Case 1. v ∈/ Fi1 and v ∈ Fij for j = 2, . . . , p.
In this case, v ∈/ σ˜p, i.e., v ∈ [F ∪ {v}] − σ˜p. One can show that a family
D =
[F ∪ {v}] − σ˜2, Fi2 , . . . , [F ∪ {v}] − σ˜p, Fip , [n] − σ˜1, F
,
where Fij = (Fij ∪ F ) − {v} for j = 2, . . . , p, is a shellingtrapped decomposition of
[n] and w ∈ Sp−1 is defined by
Case 2. v ∈ Fij for j = 1, . . . , p.
In this case, there is a k such that v ∈ [F ∪ {v}] − σ˜k. One can show that the family
D = (F − σ˜1, Fi1 ), . . . , (F − σ˜p, Fip ), ([n] − F , F ) ,
where Fij = (Fij ∪ F ) − {v} for j = 2, . . . , p, is a shellingtrapped decomposition of
[n] and w ∈ Sp can be defined by
⎧ w(j − 1) if 1 < j and w(j − 1) < k,
w(j ) = ⎨ w(j − 1) + 1 if 1 < j and w(j − 1) ≥ k,
⎩ k
if j = 1.
For the second assertion, one can show that each subspace in CD,w is not changed
under the map ψ1 in the proof of Lemma 4.6 since it is the intersection of all
subspaces that lie weakly below it and cover UF . We will show that ψ (CD,w − UF ) =
ψ2(CD,w − UF ) is a chain satisfying all conditions for CD,w. If τ¯j is a set satisfying
F ij ⊆ τ j ⊆ σ¯j , then
⎧ Uτ¯j
⎪
ψ (Uτ¯j ) = ⎨ U[τ¯p∩F ]∪{v}
⎪⎩ RF +1
if j = 1, 2, . . . , p − 1,
if j = p and τ¯p = F ,
if j = p and τ¯p = F ,
where the coordinates of RF +1 are indexed by F ∪ {v}.
There are two cases:
since F ij = [F ∪ {v}] − Fij for j = 1, 2, . . . , p − 1 and [F ∪ {v}] − F =
{v, x}. Moreover, the image of Cw − UF i1 /···/F ip under ψ is a saturated chain in
[U[F ∪{v}]−F /[F ∪{v}]−Fi1 /···/[F ∪{v}]−Fip−1 , UF ∪{v}] whose covering relations of type
(a) (in Step (1) of Sect. 4.1) correspond to the covering relations in the chain Cw
in p. Therefore, the image of CD,w under ψ can be chosen as CD,w.
In this case, it is not hard to see that the image of Cw under ψ is the saturated
chain in [U[F ∪{v}]−Fi1 /···/[F ∪{v}]−Fip−1 , UF ∪{v}] whose covering relations of type (a)
(in Step (1) of Sect. 4.1) correspond to the covering relations in the chain Cw in
p−1. Thus the image of CD,w under the map ψ can be chosen as CD,w.
Example 4.12 Let be the shellable simplicial complex from Example 4.9. In
Example 4.10, CD,w is the chain
0
ˆ
U67
for the shellingtrapped decomposition
D = (45, F1 = 12367), (123, F6 = 14567), (67, F7 = 12345)
of [
7
] and the permutation w in S2 with w(1) = 2 and w(2) = 1.
Since 67 = F 7, the corresponding shellingtrapped decomposition D of the set
{1, 2, 3, 4, 5, v} is
D =
45, F1 = 123v , 123v, F6 = 145v ,
and the corresponding permutation w ∈ S1 is the identity.
The map ψ from the proof of Lemma 4.6 sends the chain
and this chain satisfies the conditions for CD,w.
The intersection lattice for F is shown in Fig. 4 and the chain CD,w is repre
sented by thick lines.
Proof of Theorem 4.1 By Lemma 4.5, it is enough to show the assertion for the
case when σ¯ = iq=1 F i = [n]. Since every chain CD,w is saturated, it is enough to
show that (L ), the simplicial complex obtained after removing the corresponding
simplices for all pairs (D, w), is contractible. We use induction on the number q of
facets of .
Base case: q = 2. If has only two facets F1 and F2 and F 1 ∪ F 2 = [n], then F2
has only one element and G2 = ∅. It is easy to see that the order complex (L ) is
homotopy equivalent to S0 and D = {([n], F2)} is the only shellingtrapped
decomposition of [n], while CD,∅ = (UF 2 ) is the corresponding saturated chain. Therefore,
(L ) is contractible when q = 2.
Inductive step. Now, assume that (L ) is contractible for all shellable simplicial
complexes with less than q facets. For simplicity, denote L = L . Let F = Fq be
the last facet in the shelling order of and H = UF . Let L be the intersection lattice
for , where is the simplicial complex generated by the facets F1, . . . , Fq−1.
Let L≥H denote the subposet of elements in L which lie weakly above H .
Consider the decomposition of (L) = X ∪ Y , where X is the simplicial complex
obtained by removing all simplices corresponding to chains CD,w and CD,w − H
from (L≥H ) for all CD,w containing H , and Y is the simplicial complex obtained
by removing all simplices corresponding to chains CD,w not containing H from
(L − {H }). Our goal will be to show that X, Y and X ∩ Y are all contractible,
and hence so is X ∪ Y (= (L)).
Step 1. Contractibility of X.
Since X has a cone point H , it is contractible.
Step 2. Contractibility of Y.
Define the closure relation π on L which sends a subspace to the join of the
elements covering 0ˆ which lie below it except H . Then the closed sets form a sublattice
of L, which is the intersection lattice L for the diagonal arrangement
corresponding to . It is wellknown that the inclusion of closed sets L ∩ L → L − {H } is a
homotopy equivalence (see [6, Lemma 7.6]). We have to consider the following two
cases:
Case 1. S :=
iq=−11 F i = [n].
Case 2. S :=
iq=−11 F i = [n].
Then L ∩ L = L − {0ˆ} since US ∈ L. Since L ∩ L has a cone point US , it is
contractible. Since S = [n], there is no shellingtrapped decomposition of [n] for .
Thus Y is contractible.
In this case, L ∩ L = L . Moreover, (L ) is homotopy equivalent to Y since
every element in a chain CD,w in (L − {H }) is fixed under π . Since has q − 1
facets, the induction hypothesis implies that (L ) is contractible and hence so is Y .
Step 3. Contractibility of X ∩ Y.
Note that X ∩ Y is obtained by removing simplices corresponding to CD,w − H for
all CD,w containing H from L>H . By Lemma 4.8, (L)>H is isomorphic to the proper
part of the intersection lattice LF for the diagonal arrangement corresponding to F
on F ∪ {v}. Also, Lemma 4.11 implies that X ∩ Y is isomorphic to (LF ), where
(LF ) is obtained by removing simplices corresponding to CD,w for all
shellingtrapped decomposition D of F ∪ {v} and w ∈ SD−1 from LF . Since F has fewer
facets than , the induction hypothesis implies (LF ) is contractible and hence
X ∩ Y is also contractible.
be a simplicial complex from Example 4.7. Then one can see
Example 4.13 Let
that
is a shelling, and
Let σ¯ = 12345. Then there are two possible (unordered) shellingtrapped
decompositions of σ¯ (see Table 2).
Thus, Theorem 4.1 implies (0ˆ, U12345) is homotopy equivalent to a wedge of
two circles. The intersection lattice L and the order complex for its proper part are
shown in Fig. 5. Note that the chains CD,w and the simplices corresponding to each
shellingtrapped decomposition are represented by thick lines.
In this section, we give the corollary about the homotopy type of the singularity link
of A when is shellable. We also give the homology version of the corollary.
Ziegler and Živaljevic´ [
18
] showed the following theorem about the homotopy
type of VA◦.
Theorem 5.1 For every subspace arrangement A in Rn,
◦
VA
x∈LA−{0ˆ}
(0ˆ, x) ∗ Sdim(x)−1 .
From this and our results in Sect. 4, one can deduce the following.
Corollary 5.2 Let be a shellable simplicial complex on [n]. The singularity link
of A has the homotopy type of a wedge of spheres, consisting of p! spheres of
dimension
n + p(2 − n) +
p
j=1
Fij  − 2
for each shellingtrapped decomposition {(σ¯1, Fi1 ), . . . , (σ¯p, Fip )} of some subset
of [n].
Proof sketch This is a straightforward, but tedious, calculation. By Theorem 5.1, one
needs to understand homotopy types of (0ˆ, H ) for H ∈ L . Lemmas 4.2 and 4.4
reduce this to the case of (0ˆ, Uσ¯ ), which is described fully by Theorem 4.1. The
rest is some bookkeeping about shellingtrapped decompositions.
Table 2 shows shellingtrapped decompositions {(σ¯1, Fi1 ), . . . , (σ¯p, Fip )} of subsets
of [
5
] with corresponding dimensions
Therefore Corollary 5.2 shows that the singularity link of A is homotopy equivalent
to a wedge of three threedimensional spheres and eight twodimensional spheres.
The following theorem is a homology version of Corollary 5.2.
Theorem 5.4 Let be a shellable simplicial complex and F1, . . . , Fq be a shelling
of . Then dimF Hi (VA◦ ; F) is the number of ordered shellingtrapped
decompositions ((σ¯1, Fi1 ), . . . , (σ¯p, Fip )) with
i = n + p(2 − n) +
Fij  − 2.
This can be proven without Theorem 4.1 by combining a result of Peeva, Reiner
and Welker [16, Theorem 1.3] with results of Herzog, Reiner and Welker [12,
Theorems 4, 9] along with the theory of Golod rings. It is what motivated us to prove the
stronger Corollary 5.2 and eventually Theorem 1.1.
6 K(π, 1) Examples from Matroids
In this section, we give some examples of diagonal arrangements A where the com
plement MA is K(π, 1), coming from rank3 matroids.
One should note that an arrangement having any subspace of real codimension 1
(hyperplane) will have MA disconnected. So one may assume without loss of
generality that all subspaces have real codimension at least 2. Furthermore, if any maximal
subspace U in A has codimension at least 3, then it is not hard to see that MA is not
K(π, 1). Hence we may assume without loss of generality that all maximal subspaces
have real codimension 2.
A hyperplane arrangement H in Rn is simplicial if every chamber in MH is a
simplicial cone. Davis, Januszkiewicz and Scott [
9
] showed the following theorem.
Theorem 6.1 Let H be a simplicial real hyperplane arrangement in Rn. Let A be
any arrangement of codimension2 subspaces in H which intersects every chamber
in a codimension2 subcomplex. Then MA is K(π, 1).
Remark 6.2 In order to apply this to diagonal arrangements, we need to consider
hyperplane arrangements H which are subarrangements of the real braid arrangement
Bn in Rn and also simplicial. It turns out (and we omit the straightforward proof) that
all such arrangements H are direct sums of smaller braid arrangements. So we only
consider H = Bn itself here.
Corollary 6.3 Let A be a subarrangement of the 3equal arrangement of Rn so that
A =
Uijk  {i, j, k} ∈ TA ,
for some collection TA of 3element subsets of [n]. Then A satisfies the hypothesis
of Theorem 6.1 (and hence MA is K(π, 1)) if and only if every permutation w =
w1w2 · · · wn in Sn has at least one triple in TA consecutive, i.e., there exists j such
that {wj , wj+1, wj+2} ∈ TA.
Proof It is easy to see that there is a bijection between chambers of the real Braid
arrangement Bn in Rn and permutations w = w1 · · · wn in Sn. Moreover, each
chamber has the form xw1 > · · · > xwn with bounding hyperplanes
xw1 = xw2 ,
xw2 = xw3 , . . . ,
xwn−1 = xwn
and intersects the 3equal subspaces of the form xwi = xwi+1 = xwi+2 for i =
1, 2, . . . , n − 2.
We seek shellable simplicial complexes for which A satisfies this condition.
If is the independent set complex I(M) for some matroid M (see [17] for the
definition of independent sets and further background on matroids), then facets of
are bases of M . Simplicial complexes of this kind are called matroid complexes, and
they are known to be shellable [
2
]. For a rank3 matroid M on [n], consider
AI(M⊥) =
Uijk  {i, j, k} = [n] − B for some B ∈ B(M⊥)
= Uijk  {i, j, k} ∈ B(M) ,
where M⊥ is the dual matroid of M . Note that adding a loop to M does not change
the structure of the intersection lattice for AI(M⊥). Thus, if B(M) on the set of all
nonloop elements satisfies the condition of Corollary 6.3, then the diagonal
arrangement corresponding to the matroid in which all loops have been deleted has K(π, 1)
complement, and hence AI(M⊥) has K(π, 1) complement.
Definition 6.4 Let M be a rank3 matroid on [n]. Say M is DJS if B(M) on the set
of all nonloop elements satisfies the condition of Corollary 6.3.
The following example shows that matroid complexes are not DJS in general.
Thus we look for some subclasses of matroid complexes which are DJS, and hence
whose corresponding diagonal arrangements have K(π, 1) complements; for these,
Theorem 5.4 gives us the group cohomology H •(π, Z).
Example 6.5 Let M be a matroid on [6] which has three distinct parallel classes
{1, 6}, {2, 4} and {3, 5}. Then M is selfdual and I(M⊥) is a simplicial complex on
[
6
] whose facets are 123, 134, 145, 125, 236, 256, 346 and 456. But w = 124356 is a
permutation that does not satisfy the condition of Corollary 6.3.
Recall that a matroid is simple if it has no loops nor parallel elements. The
following proposition shows that rank3 simple matroids are DJS.
Proposition 6.6 Let M be a matroid of rank3 on [n]. If M does not have parallel
elements, then M is DJS. In particular, rank3 simple matroids are DJS.
Proof Without loss of generality, we may assume that M is simple. M is not DJS if
and only if there is a permutation w ∈ Sn such that every consecutive triple is not in
B(M). Since M is simple, the latter statement is true if and only if each consecutive
triple in w forms a circuit, i.e., all elements lie on a rank2 flat. But this is impossible
since M has rank 3.
The following two propositions give some subclasses of matroids with parallel
elements which are DJS.
Proposition 6.7 Let M be a rank3 matroid on [n] with no circuits of size 3. Let
P1, . . . , Pk be the distinct parallel classes which have more than one element, and let
N be the set of all nonloop elements which are not parallel with anything else. Then,
M is DJS if and only if
P1
2
+ · · · +
Pk
2
− k < N  − 2.
Proof We may assume that M does not have loops. Since M does not have loops
nor circuits of size 3, M is not DJS if and only if one can construct a permutation
w ∈ Sn such that for each consecutive triple in w there are at least two elements
which are parallel. This means if wi ∈ N , then wi−2, wi−1, wi+1 and wi+2 (if they
exist) must be in the same parallel class. Such a w can be constructed if and only if
P21 + · · · + P2k − k ≥ N  − 2.
A simplicial complex on [n] is shifted if, for any face σ of , replacing any
vertex i ∈ σ by a vertex j < i with j ∈/ σ gives another face in . A matroid M is
shifted if its independent set complex is shifted. Klivans [
14
] showed that a rank3
shifted matroid on the ground set [n] is indexed by some set {a, b, c} with 1 ≤ a <
b < c ≤ n as follows:
B(M ) = (a , b , c ) : 1 ≤ a < b < c ≤ n, a ≤ a, b ≤ b, c ≤ c .
It is not hard to check the following.
Proposition 6.8 Let M be the shifted rank3 matroid on the ground set [n] indexed
by {a, b, c}. Then, M is DJS if and only if c−2 b < a.
We have not yet been able to characterize all rank3 matroids which are DJS.
Acknowledgements The author thanks his advisor, Vic Reiner, for introducing the topic of subspace
arrangements and encouraging him to work on this problem. I also thank Ezra Miller, Dennis Stanton,
Michelle Wachs and Volkmar Welker for valuable suggestions. Finally, the author thanks the referee for a
careful reading of the manuscript, and valuable suggestions for improvement.
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