Filtered order complexes and magnitude homology of finite graded posets
January 01, 1970
Abstract
In this paper, we study the family of subcomplexes of the order complexes of finite graded posets, defined via its rank function. We address three main topics. We describe the general topological properties of these subcomplexes in relation to magnitude homology of graded posets. For posets whose order complexes are simplicial subdivisions of closed manifolds, we show that the homology groups of these subcomplexes agree with that of the undelying manifold except for the top dimension, where it is a nontrivial free abelian group. For shellable graded posets, we prove that each of the subcomplexes are also shellable. Moreover, in the case of geometric semilattices, we show that each subcomplexes are homotopy equivalent to a nontrivial wedge sums of spheres of the same dimension.
Unless otherwise stated, all abstract simplicial complexes and posets considered in this paper are finite.
1 Introduction↩︎
Magnitude of finite metric spaces are introduced by Leinster in [1] as a invariant of finite metric spaces. Hepworth and Willerton in [2] introduced magnitude homology of undirected graphs as a categorification of magnitude of graphs. Leinster and Shulman in [3] expand this notion to generalized metric spaces and enriched categories, which includes finite posets.
In relation with magnitude homology of posets, we deal the family of subcomplexes of the order comlexes of finite posets. For a finite graded poset \(P\) of rank \(n\) with the rank function \(r\), the subcomplex \(\varDelta^{(k)}(P)\) of the order comlex \(\varDelta(P)\) is defined as follows: \[\varDelta^{(k)}(P) =\{(x_0<\cdots <x_l)\mid r(x_l)-r(x_0)\leq k\}.\] The family of subcomplexes \(\{\varDelta^{(k)}(P)\}_{0\leq k \leq n}\) is the special case of filterd-nerve of directed graphs introduced by Di-Ivanov-Mukoseev-Zhang in [4], and through this article we call this family of subcomplexes filtered order complex of the poset \(P\). By the definition, \(\varDelta^{(0)}(P)\) is equal to \(P\) itself, \(\varDelta^{(1)}(P)\) coincides with the Hasse diagram of \(P\) regarded as a 1-dimensional simplicial complex, and \(\varDelta^{(n)}(P)=\varDelta(P)\). Hence, order complex \(\varDelta(P)\) admits the following natural filtration: \[P= \varDelta^{(0)}(P) \subset \mathop{\varDelta^{(1)}(P)}\limits_{\substack{\rotatebox{90}{=}\\ \text{\scriptsize Hasse diagram of P}}} \subset \cdots \subset \varDelta^{(n)}(P)=\varDelta(P).\]
For a graded poset \((P,r)\), its magnitude homology can be expressed as a relative homology of a pair of subcomplexes \((\varDelta^{(k)}(P), \varDelta^{(k-1)}(P))\) ([4]), and this formula plays an important role throughout this paper when considering the homology of subcomplexes \(\varDelta^{(k)}(P)\).
This paper is divided into four sections. In Section 1, we present the necessary background on simplicial complexes and partially ordered sets. In Section 2, we review the magnitude homology of posets and the notion of filtered order complexes, and we discuss their basic topological properties. When the magnitude homology vanishes except for its diagonal components, it is called diagonal (this notion was introduced in [2]), and the notion of diagonality has been one of the central themes in the study of magnitude homology. The following theorem on the filtered order complexes of diagonal posets plays an important role in the later sections of this paper:
Main Theorem 1 (Theorem 28). If the magnitude homology of \(P\) is diagonal, then we have the following isomorphism of the homology groups of the subcomplexes: \[H_{i}(\varDelta^{(k)}(P))\cong H_{i}(\varDelta(P))\quad (i\leq k-1).\] The rank of the top homology group satisfies the following inequality: \[\mathop{\mathrm{rk}}H_{k}(\varDelta^{(k)}(P))\leq \sum_{\substack{x\leq y\\ r(y)-r(x)=k}}|\mu(x,y)|.\]
In Section 3, we study the properties of filtered order complexes of a poset \(P\) under the assumption that \(\varDelta(P)\) gives simplicial subdivision of manifolds. Yoshinaga-Kaneta used the undirected Hasse diagrams of face posets of triangulated manifold with a minimum and a maximum element adjoined, to create examples of undirected graphs whose magnitude homology have torsions in ([5]). Ivanov and Mukoseev investigated the magnitude homology of such posets in [6]. We establish analogous results for posets whose order complexes provide triangulations of homology manifolds, without adjoining a maximum or a minimum element, and show that the magnitude homology of such posets are diagonal and torsion-free. The proof is essentially the same strategy as that of [6]. Using this fact, we prove the following theorem about the filtered order complex of such posets.
Main Theorem 2 (Theorem 35). If \(\varDelta(P)\) provides a simplicial subdivision of an \(n\)-dimensional homology manifold \(X\), it holds that \(H_{i}(\varDelta^{(k)}(P))\cong H_{i}(X) (i\leq k-1)\), and \(H_{k}(\varDelta^{(k)}(P);\mathbb{Z}) (k\leq n-1)\) is a nontrivial free abelian group.
As a corollary of this theorem, we prove that for a poset whose order complex provides a simplicial subdivision of a closed manifold, each subcomplexes are not contractible.
In Section 4, we consider the case where the posets are shellable. We show that when a graded poset \(P\) is shellable, each subcomplex \(\varDelta^{(k)}(P)\) is also shellable (Theorem 45). Moreover, we show the following theorem, which is a generalization of a result about the homotopy types of Falkman complexes of hyperplane arrangements, stated by Quillen in [7].
Main Theorem 3 (Theorem 58). For a geometric semilattice \(P\) with its rank more than \(2\), each subcomplex \(\varDelta^{(k)}(P)\) is homotopy equivalent to a nontrivial wedge of \(k\)-dimensional spheres.
Also, we consider the relationship between the magnitude homology of a central complex arrangement and the cohomology of its complement. We show that the rank of the magnitude homology of the intersection lattice of a complex central arrangement is isomorphic to the sum of the Betti number of the complement of restrictions of the arrangement(Theorem 63).
2 Preliminaries↩︎
In this section we present some background on abstract simplicial complexes and order complexes of posets.
2.1 Abstract simplicial complex↩︎
Definition 1 ([8], Definition 2.1). A finite abstract simplicial complex is a finite set \(A\) together with a collection \(\varDelta\) of subsets of \(A\) such that if \(X\in \varDelta\) and \(Y\subseteq X\), then \(Y\in \varDelta\). The element \(v\in A\) such that \(\{v\}\in \varDelta\) is called the vertex of \(\varDelta\), and each \(X\in \varDelta\) is called the simplex of \(\varDelta\).
Definition 2 ([8], Definition 2.12, Definition 2.13, Definition 2.14). Let \(\varDelta\) be an abstract simplicial complex, and let \(\tau\) be a simplex of \(\varDelta\). The deletion, link, star of \(\tau\) is the abstract simplicial subcomplexes of \(\varDelta\), denoted by \(\mathrm{dl}_{\varDelta}(\tau)\) defined by \[\begin{gather} \mathrm{dl}_{\varDelta}(\tau):=\{\sigma\in \varDelta \mid \sigma \nsupseteq \tau\}\\ \mathrm{lk}_{\varDelta}(\tau):=\{\sigma\in \varDelta \mid \sigma \cap \tau = \emptyset, \sigma\cup \tau \in \varDelta\}\\ \mathrm{star}_{\varDelta}(\tau):=\{\sigma\in \varDelta \mid \sigma\cup \tau \in \varDelta\}. \end{gather}\] Also, for two abstract simplicial complexes \(\varDelta_1\) and \(\varDelta_2\), the join of \(\varDelta_1\) and \(\varDelta_2\) is the abstract simplicial complex \(\varDelta_1 \ast \varDelta_2\) with the set of vertices \(V(\varDelta_1)\cup V(\varDelta_2)\), and the set of simplices \[\varDelta_1 \ast \varDelta_2:=\{\sigma \in V(\varDelta_1)\cup V(\varDelta_2) \mid \sigma \cap V(\varDelta_1) \in \varDelta_1,\;\sigma \cap V(\varDelta_2) \in \varDelta_2\}.\]
Definition 3 ([8]). A matroid \(M\) is an abstract simplicial complex \(\mathcal{I}\) with its set of vertices \(V\) such that the following property is satisfied: if \(\sigma, \tau \in \mathcal{I}\), and \(|\sigma|>|\tau|\), then there exists \(x\in \sigma \setminus \tau\) such that \(\tau\cup \{x\} \in \mathcal{I}\). The elements of \(\mathcal{I}\) are called independent sets.
2.2 Partially ordered sets (posets)↩︎
Definition 4 ([8], Definition 2.18). A partially ordered set, or simply poset, \(P\) is a set together with the relation \(\leq\) that satisfies the following three axioms:
idenpotency: for any \(x\in P\), \(x\leq x\);
antisymmetry: for any \(x,y\in P\), if \(x\leq y\) and \(y\leq x\), then \(x=y\);
transivity: for any \(x,y,z\in P\), if \(x\leq y\) and \(y \leq z\), then \(x\leq z\);
If \(x<y\in P\) and no element \(u\in P\) satisfies \(x<u<y\), then we say that \(y\) covers \(x\), denoted \(x \prec y\).
Definition 5 ([8], Definition 9.3). Let \(P\) be a finite poset. The order complex of \(P\) is an abstract simplicial complex \(\varDelta(P)\) whose vertices are all elements of \(P\) and whose simplices are all chains of \(P\).
Remark 6. If a poset \(P\) has either minimum element \(\hat{0}\) or maximam element \(\hat{1}\), then \(\varDelta(P)\) is a cone with the apex \(\hat{0}\)(or \(\hat{1}\)), hence it is contractible. In this case, we often consider the order complex of the proper32part of \(P\), namely \(\bar{P}:=P-\{\hat{0},\hat{1}\}\).
Definition 7 ([8], Definition 10.8). Let \(P\) and \(Q\) be posets. Ordinal sum of \(P\) and \(Q\) is the poset \(P\oplus Q\) whose set of vertices is \(P \sqcup Q\) and whose order relation is given by \[x\leq_{P\oplus Q} y \Longleftrightarrow \left\{ \begin{array}{lll} \text{either} & x,y\in P, & x\leq_{P} y;\\ \text{or} & x,y\in Q, & x\leq_{Q} y;\\ \text{or} & x\in P, & y\in Q. \end{array} \right.\]
Proposition 8 ([8], p.157). For arbitrary two posets \(P\) and \(Q\), we have the following isomorphism of abstract simplicial complexes; \[\varDelta(P\oplus Q)=\varDelta(P)\ast \varDelta(Q).\]
Definition 9 ([9]). Let \(P\) be a finite poset. If every maximal chain of \(P\) has the same length \(n\), then we say that \(P\) is pure, or graded of rank \(n\). In this case there is a unique rank function \(r:P\rightarrow \mathbb{Z}_{\geq 0}\) such that \(r(x)=0\) if \(x\) is a minimal element of \(P\), and \(r(y)=r(x)+1\) if \(x\prec y\). If \(r(x)=i\), then we say that \(x\) has rank\(i\).
Definition 10 ([9]). Let \(P\) be a finite graded poset of rank \(n\), with a rank function \(r:P\rightarrow \{0,\ldots, n\}\). If \(S\subseteq \{0,\ldots, n\}\) then define the subposet \[P_{S}=\{x\in P \mid r(x)\in S\},\] called the subposet \(S\)-rank-selected subposet of \(P\).
Definition 11 ([8]). For an arbitrary poset \(P\), define the function \(\mu:P \times P \rightarrow \mathbb{Z}\) as follows:
\(\mu(x,x)=1\),for all \(x\in P\);
\(\mu(x,y)=-\sum_{x\leq z < y}\mu(x,z)\), for all \(x<y \in P\);
\(\mu(x,y)=0\), (otherwise).
The function \(\mu\) is called the Möbius function of \(P\).
Proposition 12 ([8], Theorem 10.24). For any finite poset \(P\) with a maximal and minimal elements, we have \[\mu(\hat{0},\hat{1})=\tilde{\chi}(\varDelta(\bar{P})),\] where \(\tilde{\chi}\) denotes the reduced Euler characteristic (Euler characteristic minus one).
Corollary 13. Let \(\mu\) be the Möbius function of \(P\). For elements \(x\leq y\), the following equation holds: \[\mu(x,y)=\chi(\varDelta(x,y)),\] where \(\varDelta(x,y)\) denotes the order complex of the open interval \((x,y):=\{z\mid x<z<y\}\).
3 Magnitude homology and filtered order complexes↩︎
3.1 Magnitude homology of quasi-metric spaces↩︎
Definition 14 ([10]). A Generalized metric space \((X,d)\) is a set \(X\) with a map \(d:X\times X\rightarrow \mathbb{R}_{\geq 0}\cup \{\infty\}\) that satisfy the following conditions;
\(d(x,x)=0\;(\forall x\in X)\),
\(d(x,y)+d(y,z) \geq d(x,z)\;(\forall x,y,z\in X)\).
A generalized metric space \((X,d)\) is symmetric if it satisfies that \(d(x,y)=d(y,x)\) for any \(x,y\in X\), and is non-degenerate if \(d(x,y)=d(y,x)=0\) implies that \(x=y\) for any \(x,y\in X\). A generalized metric space \((X,d)\) is called quasi-metric space if it is symmetric and non-degenerate.
Example 15. Let \((P,r)\) be a finite graded poset with a rank function \(r:P\rightarrow \mathbb{Z}_{\geq 0}\). We metrize \((P,r)\) by setting the quasi-metric function \(d:P\times P\rightarrow \mathbb{R}_{\geq 0}\cup \{\infty\}\) as follows: \[\begin{align} \label{q-metric} d(x,y)=\begin{cases} r(y)-r(x) & (x\leq y),\\ \infty & (\mathrm{otherwise}). \end{cases} \end{align}\tag{1}\]
Definition 16 (magnitude homology of quasi-metric spaces). Let \((X,d)\) be a quasi-metric space. For a tuple \((x_0,\ldots,x_n)\in X^{n+1}\), its length \(L(x_0,\ldots,x_n)\) is defined by \[L(x_0,\ldots,x_n):=\sum_{i=0}^{n-1}d(x_i,x_{i+1}).\] For \((X,d)\) and \(k\in \mathbb{R}_{\geq 0}\), the chain complex \((MC^{k}_{\ast}(X),\partial_{\ast})\) is defined as follows: \[\mathop{\mathrm{MC}}_{n}^{k}(X):=\mathbb{Z}\langle (x_0,x_1,\ldots,x_n)\in X^{n+1} \mid x_i\neq x_{i+1},\;L(x_0,\ldots,x_n)=k \rangle.\] The boundary map \(\partial^{k,n}:\mathop{\mathrm{MC}}_{n}^{k}(X)\rightarrow \mathop{\mathrm{MC}}_{n-1}^{k}(X)\) is defined by \(\partial^{k,n}:=\sum_{i=0}^{n}(-1)^{i}\partial^{k,n}_{i}\), where \[\partial^{k,n}_{i}(x_0,\ldots,x_n):=\begin{cases} (x_0,\ldots,x_{i-1},x_{i+1},\ldots,x_n) & (L(x_0,\ldots,x_{i-1},x_{i+1},\ldots,x_n)=k)\\ 0 & (\mathrm{otherwise}). \end{cases}\] The chain complex \((\mathop{\mathrm{MC}}^{k}_{\ast}(X),\partial^{k}_{\ast})\) is called the magnitude chain complex of \((X,d)\) in grading \(k\), and the magnitude homology group of \((X,d)\) in grading \(k\) is the homology \(\mathop{\mathrm{MH}}_{n}^{k}(X;\mathbb{Z}):=H_{n}(\mathop{\mathrm{MC}}_{\ast}^{k}(X;\mathbb{Z}),\partial)\) of the magnitude chain complex \((\mathop{\mathrm{MC}}_{\ast}^{k}(X), \partial^{k}_{\ast})\) in grading \(k\).
Proposition 17 ([10] Definition 4.6, [11] p.47). Let \(a,b\) be points of a quasi-metric space \((X,d)\) and \(\mathop{\mathrm{MC}}^{\ast}_{\ast}(a,b)\) be a submodule of \(\mathop{\mathrm{MC}}^{\ast}_{\ast}(X)\) generated by the tuples \((a,x_1\ldots,x_{n-1},b)\in X^{n+1}\). Then, we have decompositions as follows: \[\begin{align} \mathop{\mathrm{MC}}^{k}_{\ast}(X;\mathbb{Z}) &\cong \bigoplus_{a,b\in X}\mathop{\mathrm{MC}}_{\ast}^{k}(a,b),\\ \mathop{\mathrm{MH}}^{k}_{\ast}(X;\mathbb{Z}) &\cong \bigoplus_{a,b\in X}\mathop{\mathrm{MH}}_{\ast}^{k}(a,b). \end{align}\]
Definition 18. The magnitude homology of \(\mathop{\mathrm{MH}}_{*}^{*}(X)\) is diagonal if \(\mathop{\mathrm{MH}}_{n}^{k}(P)\) is trivial whenever \(n\neq k\).
Definition 19 (Magnitude homology of graded posets). Let \((P,r)\) be a graded poset. We define the magnitude homology \(\mathop{\mathrm{MH}}^{\ast}_{\ast}(P)\) of \((P,r)\) by the magnitude homology of the quasi-metric space induced by the rank function \(r\).
Proposition 20 (cf. [5] Corollary 5.12, [12] Proposition 7.2). The magnitude homology of a graded poset \((P,r)\) decomposes as follows: \[\label{m32decomposition} \mathop{\mathrm{MH}}_{n}^{k}(P)\simeq \begin{dcases} \bigoplus_{\substack{x\leq y\\r(y)-r(x)=k}}\tilde{H}_{n-2}(\varDelta(x,y)) & (k\geq 1),\\ \mathbb{Z}^{\#P} & (n=k=0). \end{dcases}\tag{2}\]
Proof. A path \((x_0,\ldots,x_n)\) of finite length in \(P\) forms a chain in this order, and the length of this path is \(r(x_n)-r(x_0)\). If \(n \geq 2\), then we have the following isomorphism: \[\mathop{\mathrm{MC}}_{n}^{k}(x_0,x_n) \cong \mathbb{Z}\langle (x_1<\cdots<x_{n-1})\mid x_0<x_1<x_{n-1}<x_n\rangle \cong \tilde{C}_{n-2}(\varDelta(x_0,x_n)).\] If \(n=1\), we have \(\mathop{\mathrm{MC}}_{n}^{k}(x,y) \cong \mathbb{Z}\langle (x,y)\rangle\). Also for the boundary operators of magnitude chain complex, we have \[\partial(x_0< \cdots <x_n)=\sum^{n-1}_{i=1}(-1)^{i}(x_0< \cdots < x_{i-1} < x_{i+1}< \ldots < x_n).\] Therefore, these isomorphisms are compatible with the boundary operators, and we obtain \(\mathop{\mathrm{MH}}_{n}^{k}(x,y)\cong H_{n-2}(\varDelta(x,y))\;(n\geq 1)\). ◻
Proposition 21. For a ranked poset \((P,r)\), we have \[\sum_{n=0}^{k}(-1)^{n}\mathop{\mathrm{rk}}\mathop{\mathrm{MH}}^{k}_{n}(P)=\sum_{\substack{x,y\in P\\ r(y)-r(x)=k}}\mu(x,y).\]
Corollary 22. If \((P,r)\) is diagonal, we have \[\mathop{\mathrm{rk}}\mathop{\mathrm{MH}}^{k}_{k}(P)=\sum_{\substack{x,y\in P\\ r(y)-r(x)=k}}|\mu(x,y)|.\]
3.2 Filtered order complex of posets↩︎
The following definition is the special case of "filtered nerve of digraphs" defined in [4].
Definition 23 (Filtration of the order complex of a graded poset, cf. [4], Section 1.3). Let \((P,r)\) be a graded poset of rank \(n\). For each \(0\leq k \leq n\), we define the subcomplex \(\varDelta^{(k)}(P)\) of the order complex \(\varDelta(P)\) of \(P\) by \[\varDelta^{(k)}(P):=\{(x_0<\cdots<x_l)\mid r(x_l)-r(x_0)\leq k\}.\] The family of subcomplexes \(\{\varDelta^{(k)}(P)\}_{0\leq k \leq n}\) will be reffered to as the filtered order complexes of \(P\).
Example 24. Let \(P\) be a chain \([n]:=\{0<1<2<\cdots<n\}\) of length \(n\), and define its rank function by \(r(i)=i\;(0\leq i \leq n)\). Hasse diagram of \(P\) is a line graph consists of \((n+1)\) vertices, and the order complex \(\varDelta(P)\) is a \(n\)-simplex. \(\varDelta^{(k)}(P)\) is contractible.
Figure 1: 
.
Remark 25. It is easy to see that when \(\varDelta^{(1)}(P)\) is contractible, subcomplexes \(\varDelta^{(k)}(P)\;(k\geq 2)\) are also contractible(sequence of collapsing \(k\)-simplices having vertices corresponding the leaf of \(\varDelta^{(1)}(P))\). It is an open problem whether \(\varDelta^{(k)}(P)\) is contractible whenever \(\varDelta^{(k-1)}(P)\) is contractible.
When considering the topological properties of filtered order complexes, sometimes it is useful to represent these subcomplexes as the union of the order complexes of rank selections glued together.
Proposition 26. For a graded poset \((P,r)\) of rank \(n\), we have \[\label{sum} \varDelta^{(k)}(P)=\bigcup_{i=0}^{n-k}\varDelta(P_{[i,i+k]}).\tag{3}\] Also, we have the injection \(H_{k}(\varDelta(P_{[i,i+k]}))\hookrightarrow H_k(\varDelta^{(k)}(P))\).
Proof. We have 3 by definitions. Injectivity follows from Mayer-Vietoris sequence. ◻
Proposition 27. For a graded poset \((P,r)\), we have the following isomorphism: \[\mathop{\mathrm{MH}}^{k}_{n}(P)\cong \begin{cases} H_{n}(\varDelta^{(k)}(P),\varDelta^{(k-1)}(P)) & (n \geq 1)\\ \mathbb{Z}^{\#P} & (k=n=0),\\ 0 & (\mathrm{otherwise}). \end{cases}\]
Proof. If \(n\geq 1\), we have the following isomorphism regarding the magnitude chain complexes of \(P\): \[\begin{align} \mathop{\mathrm{MC}}^{k}_{n}(P) & = \mathbb{Z}\langle (x_0 < \cdots < x_n) \mid r(x_n)-r(x_0)=k\rangle \\ & = C_{n}(\varDelta^{(k)}(P),\varDelta^{(k-1)}(P)) \end{align}\] It is easy to see that the boundary operators are compatible with the isomorphism. Therefore, we obtain \(\mathop{\mathrm{MH}}^{k}_{n}(P)\cong H_{n}(\varDelta^{(k)}(P),\varDelta^{(k-1)}(P))\). ◻
Theorem 28. If \(P\) is diagonal, then we have the following isomorphism of the homology groups of the subcomplexes: \[\label{homology32of32Delta94k} H_{i}(\varDelta^{(k)}(P))\cong H_{i}(\varDelta(P))\quad (0\leq i\leq k-1).\tag{4}\] The rank of the top homology group satisfies the following inequality: \[\label{rank} \beta_{k}(\varDelta^{(k)}(P))\leq \sum_{\substack{x\leq y\\ r(y)-r(x)=k}}|\mu(x,y)|.\tag{5}\] Here, \(\beta_{i}\) denotes the \(i\)-th Betti number.
Proof. We have the following long exact sequence of homology groups for a pair of spaces \((\varDelta^{(k+1)}(P),\varDelta^{(k)}(P))\): \[\label{LES}
\dots \rightarrow H_{i}(\varDelta^{(k)}(P))\rightarrow H_{i}(\varDelta^{(k+1)}(P)) \rightarrow H_{i}(\varDelta^{(k+1)}(P),\varDelta^{(k)}(P))\rightarrow \dots.\tag{6}\] By proposition 27 and diagonality, we obtain \(H_i(\varDelta^{(k+1)}(P))\cong H_{i}(\varDelta^{(k)}(P))\). Applying this repeatedly, we get 4 .
By 6 , we have the inclusion \(H_{k}(\varDelta^{(k)}(P))\hookrightarrow \mathop{\mathrm{MH}}^{k}_{k}(P)\); hence Proposition 22 yields 5 . ◻
Example 29. The following is an example of a poset which is diagonal, non-contractible and the equality of 5 holds. Let \(P\) be a poset of rank \(2\) with the Hasse diagram shown in Figure 2. The magnitude homology of \(P\) and the homotopy types of \(\varDelta^{(k)}(P)\;(0\leq k \leq 2)\) are as follows: \[\begin{gather} \mathop{\mathrm{MH}}^{2}_{i}(P)\cong 0\;(i=0,1,2), \quad \mathop{\mathrm{MH}}^{1}_{i}(P)\cong \begin{cases} \mathbb{Z}^{8} & (i=1)\\ 0 & (\mathrm{otherwise}), \end{cases} \quad \mathop{\mathrm{MH}}^{0}_{0}(P)\cong \mathbb{Z}^{8},\\ \varDelta^{(2)}(P)\simeq \mathbb{S}^{1}, \quad \varDelta^{(1)}(P) \simeq \mathbb{S}^1, \quad \varDelta^{(0)}(P) \simeq \bigvee^{7}\mathbb{S}^0. \end{gather}\] The equality of 5 holds in the case \(k=2\).
4 Magnitude homology and filtered order complexes associated with manifolds↩︎
4.1 Magnitude homology of posets associated with manifolds↩︎
First, we review some of the combinatorial aspects of manifolds. Throughout this section, we deal with triangulable spaces.
Definition 30. A topological space \(X\) is called a homology \(n\)-manifold (without boundaries) if for each point \(x\) of \(X\), the local homology group \(H_{i}(X,X-x)\) vanishes if \(i\neq n\) and is isomorphic to \(\mathbb{Z}\) if \(i=n\).
Usual topological \(n\)-manifolds (without boundaries) are homology \(n\)-manifolds(by excisions and long exact sequence of a pair of spaces).
Proposition 31 ([13] Theorem 63.2). Let \(\varDelta\) be a triangulated homology \(n\)-manifold. Let \(\sigma\in \varDelta\) be a \(k\)-simplex of \(\varDelta\). If \(\mathrm{lk}\;\sigma\) is non-empty, then it is a homology \(n-k-1\) sphere.
Corollary 32 ([9] p.310). Let \(P\) be a finite poset. If \(\varDelta(P)\) is a homology \(n\)-manifold, then any non-empty open interval is a homology sphere.
Proof. For a non-empty open interval \((x,y)\) in \(P\), take a maximal chain of \(P_{\leq x}\) by \(x_0<\cdots<x_i<x\), and a maxial chain in \(P_{\geq y}\) by \(y<y_{1}<\cdots < y_{j}\). Let \(\sigma\) be a simplex of \(\varDelta(P)\) represented by a chain \(x_0<\cdots < x_i < y_1 < \cdots <y_j\), then the link of \(\sigma\) is equal to \(\varDelta(x,y)\); hence Proposition 31 yields that \(\varDelta(x,y)\) is a homology \(n-k-1\) sphere. ◻
If \(\varDelta(P)\) is a homology manifold, then the following proposition tells us that the magnitude homology of \(P\) is diagonal and torsion-free.
Proposition 33 (cf. [12], Theorem 7.8). Let \((P,r)\) be a finite graded poset. If \(\varDelta(P)\) is a homology \(n\)-manifold, then the magnitude homology of \(P\) is diagonal and torsion free. In particular, \[\label{mfd32magnitude} \mathop{\mathrm{MH}}_{i}^{k}(P) \cong \begin{cases} \mathbb{Z}^{\#\{(x,y)\mid r(y)-r(x)=k\}} & (i=k),\\ 0 & (i \neq k). \end{cases}\tag{7}\]
The following proposition implies that the diagonality and torsion-freeness of magnitude homology can be collapsed simply by adding a maximum and a minimum elements. Though it is a straightforward extension of [6] and proof is essentially identical to that of [6]; we include it for completeness.
Proposition 34 (cf.[5] Corollary 5.12, [12] Theorem 7.8). Under the same assumption as above, let \(\hat{P}\) be \(P\cup \{\hat{0},\hat{1}\}\). Then the magnitude homology of \(\hat{P}\) is \[\label{mfd32magnitude322} \mathop{\mathrm{MH}}^{k}_{i}(\hat{P})\cong \begin{cases} \tilde{H}_{i-2}(\varDelta(P)) & (k=r(P) + 2),\\ \mathbb{Z}^{\#\{(x,y)\mid x\leq y, r(y)-r(x)=k\}} & (k\neq r(P) +2,\;i=k),\\ 0 & (\mathrm{otherwise}). \end{cases}\tag{8}\]
Proof. If \(x,y\in \hat{P}\) are neither \(\hat{0}\) nor \(\hat{1}\), it is the same as Proposition 7 .
For an open interval \((\hat{0},\hat{1})=P\), \(\mathop{\mathrm{MH}}^{r(P)}_{i}(\hat{P})\cong \tilde{H}_{i-2}(\varDelta(\hat{0},\hat{1}))=\tilde{H}_{i-2}(\varDelta(P))\).
For any \(x\in \hat{P}\setminus \{\hat{1}\}\), an open interval \((\hat{0},x)\) is isomorphic to \(P_{< x}\). Let \(\sigma\in
\varDelta(P)\) be a simplex indexed by a maximal chain \(x<x_1<\cdots<x_j\) in \(P_{\geq x}\), then \(\varDelta(P_{<x}) =
\mathrm{lk}_{\varDelta(P)}(\sigma)\) and Proposition 31 implies that \(\varDelta(x,y)\) is a \(r(y)-r(x)-2\) dimensional homology sphere. The same applies for an open interval of type \((y,\hat{1})\). Now Proposition 20 yields 8 . ◻
4.2 Filtered order complexes of posets associated with homology manifolds↩︎
Theorem 35. Let \((P,r)\) be a finite graded poset, and suppose that \(\varDelta(P)\) is a triangulation of homology \(n\)-manifold \(X\). Then for \(0\leq k\leq n\), we have the following isomorphisms of the homology groups of \(\varDelta^{(k)}(P)\), and the inequality for the \(k\)-th Betti number of \(\varDelta^{(k)}(P)\): \[\begin{gather} H_{i}(\varDelta^{(k)}(P))\cong H_{i}(X) \quad (i\leq k-1),\\ n-k\leq \beta_{k}(\varDelta^{(k)}(P)) \leq \#\{(x,y)\mid x\leq y,\;r(y)-r(x)=k\} \label{mfd32rank}. \end{gather}\tag{9}\]
Proof. \(P\) is diagonal by Proposition 7 , and Corollary 28 implies that \(H_{i}(\varDelta^{(k)}(P))\cong H_{i}(X)\;(i\leq k-1)\). Also, Proposition 33 implies the right-hand side of the inequality 9 . Take \(\sigma=(x_i<x_{i+1}<\cdots<x_{i+k-1})\in \varDelta(P)\) so that the rank of an element \(x_i\) is \(i\), then \(\mathrm{lk}\;\sigma\in \varDelta(P_{[i,i+k]})\) and Proposition 31 implies that \(\mathrm{lk}\;\sigma\) is a nontrivial cycle of \(H_{k}(\varDelta(P_{[i,i+k]}))\). Proposition 26 implies that \(\mathrm{lk}\;\sigma\) is also a nontrivial cycle of \(H_{k}(\varDelta^{(k)}(P))\). Given that \(\varDelta(P_{[i,i+k]})\cap\varDelta(P_{[i+1,i+k+1]})=\varDelta(P_{[i+1,i+k]})\), if we take one \(k\)-dimensional cycle from each of \(\varDelta(P_{[0,k]}),\ldots,\varDelta(P_{[n-k,n]})\), then these cycles are independent. Thus, we obtain the left-hand side of the inequality 9 . ◻
Corollary 36. Let \((P,r)\) be a graded poset, and suppose \(\varDelta(P)\) gives a triangulation of a homology \(n\)-manifold \(X\) without boundary. Then for \(0\leq k \leq n\), \(\varDelta^{(k)}(P)\) is not contractible.
Proof. \(\varDelta^{(n)}(P)=\varDelta(P)\) is not contractible because if \(X\) is orientable, \(H_{n}(X)\cong \mathbb{Z}\) and if \(X\) is not orientable, then \(H_{n-1}(X)\) has a torsion subgroup of order \(2\) (see, [13]). If \(k\leq n-1\), Theorem 35 implies that \(\varDelta^{(k)}(P)\) is not contractible. ◻
Remark 37. If \(\varDelta(P)\) is a triangulation of a manifold with boundary, \(\varDelta^{(k)}(P)\) can be contractible. For example, let \(P\) be a chain of length \(n\), then \(\varDelta(P)\) is a triangulation of a \(n\)-dimensional ball, and for \(1\leq k \leq n\), \(\varDelta^{(k)}(P)\) are all contractible by Example 24.
The following proposition implies that even if the diagonality of the magnitude homology collapses by adding the maximam and minimum elements, the homology of subcomplexes have similar behaviors.
Proposition 38. Let \((P,r)\) be a graded poset such that \(\varDelta(P)\) gives a triangulation of a homology \(n\)-manifold. For \(1\leq k\leq n+1\), the homology groups of subcomplexes \(\varDelta^{(k)}(\hat{P})\) are as follows: \[\begin{align} \label{hat} H_{i}(\varDelta^{(k)}(\hat{P})) & \cong \tilde{H}_{i-1}(X) & (1\leq i \leq k-1),\\ H_{0}(\varDelta^{(k)}(\hat{P}))& \cong \mathbb{Z} & (i=0). \end{align}\tag{10}\]
Proof. By Proposition 27 and Proposition 8 , the long exact sequence of the homology of a pair of subcomplexes \((\varDelta^{(n+2)}(\hat{P}),\varDelta^{(n+1)}(\hat{P}))\), we have the following exact sequence of homology groups: \[\cdots \rightarrow H_{i}(\varDelta^{(n+2)}(\hat{P}))\rightarrow \tilde{H}_{i-2}(X)\rightarrow H_{i-1}(\varDelta^{(n+1)}(\hat{P}))\rightarrow H_{i-1}(\varDelta^{(n+2)}(\hat{P}))\rightarrow \cdots\] Thus we obtain the isomorphism \(\tilde{H}_{i-1}(X) \cong H_{i}(\varDelta^{(n+1)}(\hat{P}))\), since \(\varDelta^{(n+2)}(\hat{P})=\varDelta(\hat{P})\) is contractible. Also, we get \(H_{i}(\varDelta^{(k)}(\hat{P}))\cong H_{i}(\varDelta^{(k+1)}(\hat{P}))\;(1\leq i \leq k-1)\) for \(k\leq n\) by the long exact sequence of a pair \((\varDelta^{(k+1)}(\hat{P}),\varDelta^{(k)}(\hat{P}))\), and we have \(H_{i}(\varDelta^{(k)}(\hat{P}))\cong H_{i-1}(X)\), hence 10 holds. ◻
Example 39. Let \(P_n=\{a^{1}_{1},a^{1}_{2},\ldots,a^{1}_{n},a^{2}_{1},a^{2}_{2},\ldots,a^{2}_{n}\}\) with the partial order generated by \(a^{p}_{i}<a^{q}_{i+1} \;(p,q\in \{1,2\})\). \(\varDelta(P_n)\) is a \(n\)-joins of \(0\)-sphere \(\mathbb{S}^0\), which is homeomorphic to a \(n\)-sphere. For \(1\leq k \leq n\), \(\varDelta^{(k)}(P_n)\) is homotopy equivalent to a wedge of \((2n-2k-1)\) copies of \(\mathbb{S}^k\).
Figure 3: 
.
Example 40. Let \(K\) be a regular CW decomposition of a real projective space \(\mathbb{R}P^2\) taken as Figure 4. Let \(P\) be a face poset of \(K\), then \(\varDelta^{(2)}(P)\) is a triagulation of \(\mathbb{R}P^2\), \(\varDelta^{(1)}(P)\) is homotopy equivalent to a wedge of \(12\) copies of \(\mathbb{S}^1\).
5 Magnitude homology and filtered order complexes of shellable graded posets.↩︎
5.1 Shellablity of simplicial complexes and posets↩︎
A simplicial complex \(\varDelta\) is pure or pure dimensional of \(n\) if all its maximal simplices have the same dimension \(n\).
Definition 41 ([8], Definition 12.1). A simplicial complex \(\varDelta\) is called shellable if its maximal simplices can be arranged in linear order \(F_1,\ldots,F_t\) in such a way that the subcomplex \((\bigcup_{i=1}^{k-1}F_i)\cap F_{k}\) is pure and \((\mathrm{dim}F_k-1)\)-dimensional for all \(k=2,\ldots,t\). An ordering of maximal simplices satisfying the above condition is called a shelling order. The maximal faces whose entire boundary is contained in the union of the earlier maximal simplices are called spanning simplices.
The next theorem is one the most important topological properties of a shellable complex (we refer the reader to [8]).
Theorem 42. Let \(\varDelta\) be a shellable simplicial complex, with \(F_1,\ldots,F_t\) being the corresponding shelling order of the maximal simplices, and \(\varSigma\) being the set of spanning simplices. Then \(\varDelta\) is homotopy equivalent to \(\bigvee_{\sigma\in \varSigma}\mathbb{S}^{\mathrm{dim}\;\sigma}\).
Remark 43. A poset \(P\) is called a shellable poset if its order complex \(\varDelta(P)\) is shellable. When \(P\) has a maximal element and a minimal element, \(P\) is shellable if and only if \(\bar{P}=P\setminus \{\hat{0},\hat{1}\}\) is shellable.
The next result is important when considering the topology of the rank selection of a graded poset.
Proposition 44 ([8], Proposition 12.6). Let \((P,r)\) be a shellable graded poset of rank \(n\). Then for any \(S\subseteq \{0,\ldots,n\}\), the rank selection \(P_S\) is also shellable and the shelling order for \(\varDelta(P_{S})\) can be obtained from any shelling order for \(\varDelta(P)\) by taking the restriction and then omittimg repetitions.
5.2 Filtered order complexes of shellable posets↩︎
The next theorem states that the shellability of the order complex of a graded poset inherites to the subcomplexes.
Theorem 45. When a graded poset \((P,r)\) is shellable, \(\varDelta^{(k)}(P)\) is also shellable.
Proof. Let \(F_1,\ldots,F_t\) be a shelling order for \(\varDelta(P)\), and take \(D^{0}_1,\ldots, D^{0}_{m_0}\) to be the sequence obtained by restricting \(F_1,\ldots,F_t\) to \(\varDelta(P_{[0,k]})\) and removing repetitions. By Proposition 44, \(D^{0}_1,\ldots, D^{0}_{m_0}\) is a shelling order for \(\varDelta(P_{[0,k]})\). Let \(D^{1}_1, \ldots, D^{1}_{m_1}\) be the sequence obtained by restricting \(F_1,\ldots,F_t\) to \(\varDelta(P_{[1,k+1]})\) and removing repetitions, and \(i_j\) be the smallest number so that \(D^{1}_{j}={F}_{i_j}\cap \varDelta(P_{[1,k+1]})\). \((\bigcup_{i=1}^{m_0}D^{0}_i)\cap D^{1}_{1}=\varDelta(P_{[1,k]})\cap F_{i_1}\) is a pure \((k-1)=(\mathrm{dim}\;D^{1}_1 -1)\)-dimensional subcomplex of \(F_{i_1}\). We have \[\label{shelling32condition} ((\bigcup_{i=1}^{m_0}D^{0}_i)\cup (\bigcup_{i=1}^{j-1}D^{1}_i))\cap D^{1}_{j} =((\varDelta(P_{[0,k]}))\cap D^{1}_{j})\cup ((\bigcup_{i=1}^{j-1}D^{1}_j)\cap D^{1}_{j}).\tag{11}\] Both \(\varDelta(P_{[0,k]})\cap D^{1}_{j}=\varDelta(P_{[1,k]})\cap F_{i_j}\) and \((\bigcup_{i=1}^{j-1}D^{1}_j)\cap D^{1}_{j}\) are pure \((k-1)=(\mathrm{dim}\;D^{1}_{j}-1)\)-dimensional subcomplexes of \(D^{1}_{j}\), so that 11 is also a pure \((k-1)=(\mathrm{dim}\;D^{1}_{j}-1)\)-dimensional subcomplex of \(D^{1}_{j}\). This implies that \(D^{0}_1,\ldots, D^{0}_{m_0}, D^{1}_1, \ldots, D^{1}_{m_1}\) is a shelling order for \(\varDelta(P_{[0,k]})\cup \varDelta(P_{[1,k+1]})\). By applying the same method to \(\varDelta^{(k)}(P)=\varDelta(P_{[0,k]})\cup \varDelta(P_{[1,k+1]}) \cup \cdots \cup \varDelta(P_{[n-k,n]})\), we obtain a shelling order for \(\varDelta^{(k)}(P)\). ◻
Corollary 46. For a shellable graded poset \((P,r)\) and \(0\leq k \leq r(P)\), \(\varDelta^{(k)}(P)\) is homotopy equivalent to a wedge of \(k\)-spheres, and the magnitude homology of \(P\) is diagonal.
Proof. Since all maximal simplices of \(\varDelta^{(k)}(P)\) have dimension \(k\), homotopy type of \(\varDelta^{(k)}(P)\) follows by Theorem 42 and Theorem 45. By Proposition 27 and the long exact sequence of homology groups of a pair \((\varDelta^{(k)}(P),\varDelta^{(k-1)}(P))\), we have the following long exact sequence; \[0\rightarrow H_k(\varDelta^{(k)}(P)) \rightarrow \mathop{\mathrm{MH}}^{k}_{k}(P) \rightarrow H_{k-1}(\varDelta^{(k-1)}(P)) \rightarrow \dots.\] Therefore, diagonality of the magnitude homology of \(P\) holds by the homotopy type of \(\varDelta^{(k)}(P)\). ◻
Example 47. Let \(W\) be a finite Coxeter group with generators \(S\), and \(T:=\{\alpha s \alpha^{-1}\mid s\in S, \alpha\in W\}\). For \(\sigma\in W\), the length \(l(\sigma)\) of \(\sigma\) is defined to be the minimum number \(k\) such that there exist the expression \[\sigma=s_1\ldots s_k,\] where \(s_i\in S\). Bruhat order on \(W\) is a partial order relation on \(W\) that is defined via the covering relation, \(\sigma \prec \tau\) if
\(\tau=t\sigma\), for some \(t\in T\),
\(l(\tau)=l(\sigma)+1\).
It is known that Bruhat order on \(W\) is pure shellable, and that every open interval \((\sigma,\tau)\) of \(W\) is homeomorphic to a \((l(\tau)-l(\sigma)-2)\) sphere (see, [14]). Thus, by Corollary 46, \(\varDelta^{(k)}(W)\) is homotopy equivalent to a wedge of \(k\)-spheres (including contractible case), and analogous to the closed manifold cases, the magnitude homology of \(W\) is as follows: \[\mathop{\mathrm{MH}}^{k}_{i}(W)\cong \begin{cases} \mathbb{Z}^{\#\{(x,y)\mid l(y)-l(x)=k\}} & (k=i),\\ 0 & (\mathrm{otherwise}). \end{cases}\]
Remark 48. Even if a graded poset \(P\) has an order complex homotopy equivalent to a wedge of spheres, \(\varDelta^{(k)}(P)\) is not always homotopy equivalent to a wedge of \(k\)-spheres, and also the magnitude homology of \(P\) is not necessarily diagonal. For example, let \(P\) be a poset having a Hasse diagram obtained by adding the midpoints of edges of the Hasse diagram of Boolean algebra \(\mathcal{B}_4\). \(\varDelta(P)\) is homotopy equivalent to a \(2\)-sphere, but \(\varDelta^{(2)}(P)\) is not. The magnitude homology of \(P\) is as follows: \[\mathop{\mathrm{MH}}^{4}_{i}(P)\cong \begin{cases} \mathbb{Z}^{12} & (i=2)\\ 0 & (\mathrm{otherwise}), \end{cases} \mathop{\mathrm{MH}}^{3}_{i}(P)=0,\;\mathop{\mathrm{MH}}^{2}_{i}(P)=0, \mathop{\mathrm{MH}}^{1}_{1}(P)=\mathbb{Z}^{48}.\] The homology groups of \(\varDelta^{(k)}(P)\) is as follows: \[\begin{align} H_{i}(\varDelta^{(4)}(P))\cong \begin{cases} \mathbb{Z} & (i=2)\\ 0 & (\mathrm{otherwise}), \end{cases} & \quad H_{i}(\varDelta^{(3)}(P))\cong \begin{cases} \mathbb{Z}^{12} & (i=1)\\ 0 & (\mathrm{otherwise}), \end{cases}\\ H_{i}(\varDelta^{(2)}(P))\cong \begin{cases} \mathbb{Z}^{12} & (i=1)\\ 0 & (\mathrm{otherwise}), \end{cases}& \quad H_{i}(\varDelta^{(1)}(P))\cong \begin{cases} \mathbb{Z}^{12} & (i=1)\\ 0 & (\mathrm{otherwise}). \end{cases} \end{align}\]
5.3 Filtered order complexes of geometric semilattices↩︎
For a poset \(P\) and its elements \(x,y\), the upper bound of \(x\) and \(y\) is an element \(u\in P\) such that \(u\geq x,y\), and if there exists a minimal element of the upperbound of \(x\) and \(y\), we call the element the join of \(x\) and \(y\), which we denote \(x\vee y\). Dually, one can define the meet of \(x\) and \(y\), which we denote \(x\wedge y\). If every pair of elements has a meet(respectively, join), then \(P\) is called meet-semilattice(or respectively, join-semilattice). If every pair of elements has both meet and join, then \(P\) is called lattice. An element of a finite lattice that covers \(\hat{0}\) is called atom.
Definition 49 ([15], pp369). A lattice is called geometric if its semimodular and atomistic(every element is a join of atoms). A ranked meet-semilattice \(L\) is called geometric32semilattice if \(L\) satisfy the following conditions:
every closed interval of \(L\) is a geometric lattice.
for all \(x \in L\) and subset \(A\) of atoms of \(L\) whose join exists, if \(r(x)<r(\vee A)=|A|\), there is an atom \(a\in A\) such that \(a\nleq x\) and \(a\vee x\) exists.
Proposition 50 ([15], Theorem 4.1). Let \(L\) be a geometric semilattice, and \(x\in L\). Then \(L_{\geq x}:=\{y\in \mid x\leq y\}\) is a geometric semilattice.
Proposition 51 ([15], Proposition 4.2). Any truncation of the top row ranks of a geometric semilattice is a geometric semilattice.
Proposition 52. For any two elements \(x,x'\in L\) with a element \(u\in L\) such that \(x\geq u\) and \(x'\geq u\), their join \(x\vee x'\) exists.
Proof. By [15], there exists a geometric lattice \(M\) and its atom \(a\in M\) such that \(L=M\setminus [a,\hat{1}]\). Since \(M\) is a lattice, the join \(x\vee x'\) exists in \(M\). If \(u\geq x,x'\), then \(u \geq x\vee x'\). As \(u\ngeq a\), we must have \(x\vee x'\ngeq a\), hence \(x\vee x'\in L\). ◻
Definition 53. Let \(\mathcal{A}=\{H_1,\ldots,H_n\}\) be an arrangement of hyperplanes in \(\mathbb{K}^{d}\;(\mathbb{K}=\mathbb{R},\mathbb{C})\). The intersection poset of \(\mathcal{A}\) is a poset whose set of element is \[\{K\subset \mathbb{K}^{d}\mid \exists I\subseteq \{1,\ldots,n\},\;\text{such that } \bigcap_{i\in I}H_i=K,\;K\neq \emptyset\}\cup \{\mathbb{K}^{d}\}\] with its order relation given by the reverse inclusion.
Proposition 54 ([15] Proposition 3.1, [16] Lemma 2.3). The intersection poset \({L}(\mathcal{A})\) of a hyperplane arrangement \(\mathcal{A}\) is a geometric semilattice.
Proof. See [16] for the fact that every interval is a geometric semilattice. Let \(A=\{H_{i_1},\ldots,H_{i_k}\}\) be a set of atoms whose join exists, and set \(H_{i_j}=\{(x_1,\ldots,x_d)\in \mathbb{K}^d\mid a_{j1}x_1+\cdots +a_{jd}x_d=b_j,\;a_{j1},\ldots,a_{jd},b_{j}\in \mathbb{K}\}\). then \[x\in H_{i_1}\vee \cdots \vee H_{i_k} \Leftrightarrow \begin{pmatrix} a_{11} & a_{12} & \dots & a_{1d}\\ a_{21} & a_{22} & \dots & a_{2d}\\ \vdots & \vdots & \ddots & \vdots\\ a_{k1} & a_{k2} & \dots & a_{kd} \end{pmatrix} \begin{pmatrix} x_1\\ x_2\\ \vdots\\ x_d \end{pmatrix} = \begin{pmatrix} b_1\\ b_2\\ \vdots\\ b_d \end{pmatrix}.\] This implies that a set of atoms is independent if and only if their normal vectors are linearly independent. Therefore, the second condition of the geometric semilattice follows by the exchange axiom of linear algebra. ◻
Theorem 55 ([15] Theorem 7.2, Corollary 7.3). Geometric semilattices are pure shellable.
The above theorem combined with the following proposition yields that the order complex of the proper part of a geometric semilattice is homotopy equivalent to a nontrivial wedge of spheres.
Proposition 56 ([17], Theorem 3.10). Let \(L\) be a geometric semilattice with the rank function \(r\), and \(\mu\) be Möbius function of \(L\). For any \(x\leq y \in L\), the following inequality follows: \[\label{nonzero} (-1)^{r(y)-r(x)}\mu(x,y)>0.\tag{12}\]
Proposition 57. Let \(L\) be a geometric semilattice, and \(x,y\) be elements of \(L\) such that \(r(y)-r(x)=k\). Then, \(\varDelta(x,y)\) is homotopy equivalent to a nontrivial wedge of \(k\)-spheres.
Proof. The open interval \((x,y)\) is a proper part of the closed interval \([x,y]\), which is a geometric lattice by the definition of a geometric semilattice. By Theorem 55, \(\varDelta(x,y)\) is a wedge of \(k\)-spheres and Proposition 56 guarentees the non-triviality since Möbius function \(\mu(x,y)\) equals Euler characteristic. ◻
Theorem 58. Let \(L\) be a geometric semilattice of rank \(n\). For \(0\leq k \leq n-1\), \(\varDelta^{(k)}(L)\) is homotopy equivalent to a nontrivial wedge of \(k\)-spheres, and the following inequality holds for the top Betti number of \(\varDelta^{(k)}(L)\): \[\begin{align} \beta_{0}(\varDelta^{(0)}(L))=|L|,\\ \beta_{n-1}(\varDelta^{(n-1)}(L))=\mathop{\mathrm{rk}}\mathop{\mathrm{MH}}^{n}_{n}(L), \tag{13}\\\sum_{i=0}^{n-k-1} \max_{r(x)=i} \{\mathop{\mathrm{rk}}(\mathop{\mathrm{MH}}(L^{\ge x}_{[i,i+k+1]}))\} +\sum_{i=1}^{n-k} \max_{r(x)=i-1,\;r(y)=i+k+1} \{|\mu(x,y)|\} \\ \le \beta_k(\varDelta^{(k)}(L)) \le \mathop{\mathrm{rk}}\mathop{\mathrm{MH}}^{k}_{k}(L) \qquad (1\le k\le n-2)\tag{14} \end{align}\]
Proof. By Theorem 45 and Theorem 55, \(\varDelta^{(k)}(L)\) is homotopy equivalent to a wedge of \(k\)-spheres, and 28
implies the righthand side of the inequality 14 .
Now we prove the lefthand side of the inequality 14 and the equation 13 .
When \(k=n-1\), let \(\{a_1,\ldots,a_m\}\) be a set of elements of rank \(n\). For a closed interval \([\hat{0},a_i]\) in
\(L\), \(\varDelta^{(n-1)}([\hat{0},a_i])\subseteq \varDelta^{(n-1)}(L)\) is a suspension of \(\varDelta(\hat{0},a_i)\), and the subcomplexes \(\varDelta^{(n-1)}([\hat{0},a_1]),\ldots, \varDelta^{(n-1)}([\hat{0},a_m])\) of \(\varDelta^{(n-1)}(L)\) is glued together over a cone with the apex \(\hat{0}\);
hence \[\begin{align}
\beta_{n-1}(\varDelta^{(n-1)}(L)) & =\sum_{x\in L,\;r(x)=n}\beta_{n-1}(\varDelta^{(n-1)}([\hat{0},x]))\\ & = \sum_{i=1}^{m}\beta_{n-2}(\varDelta(\hat{0},a_i))\\ & =\sum_{i=1}^{m}|\mu(\hat{0},a_i)|\\ & = \mathop{\mathrm{MH}}^{n}_{n}(L),
\end{align}\] and we have the equation 13 .
When \(k \leq n-2\), for \(0< i < n-k\), let \(x\) be an element of rank \(i\) and \(y\) be an element of rank \(i+k+1\). Then by Proposition 56, \(\varDelta(x,y)\) has
nontrivial \(k\)-dimensional cycles contained in \(\varDelta(L_{[i,i+k]})\), hence it is also a nontrivial \(k\)-dimensional cycle of \(H_{k}(\varDelta^{(k)}(L))\) by Proposition 27. Also, let \(x\) be a element of rank \(i\), and set \(L^{\geq x}_{[i,i+k+1]}:=\{y\in L \mid x\leq y,\;r(y)\in [i,i+k+1]\}\). By Proposition 50 and
Proposition 51, \(L^{\geq x}_{[i,i+k+1]}\) is a geometric semilattice, and \(\varDelta^{(k)}(L^{\geq
x}_{[i,i+k+1]})\) is a nontrivial \(k\)-cycle in \(\varDelta^{(k)}(L)\). The \(k\)-the Betti number of \(\varDelta^{(k)}(L^{\geq x}_{[i,i+k+1]})\) is \(\mathop{\mathrm{rk}}(\mathop{\mathrm{MH}}(L^{\geq x}_{[i,i+k+1]}))\) by 13 . If \(x',y'\) are elements of \(L\) such that \(r(x')=i,\;r(y')=i+k+2\), then \(\varDelta^{(k)}(L^{\geq
x}_{[i,i+k+1]})\) and \(\varDelta(x',y')\) are both nontrivial \(k\)-cycles contained in \(\varDelta(L_{[i,i+k]})\cup\varDelta(L_{[i+1,i+k+1]})\), and their intersection \(\varDelta^{(k)}(L^{\geq x}_{[i,i+k+1]})\cap \varDelta(x',y')\) is empty or a cone with an apex \(x\vee x'\) by Proposition 52. Also, \((\varDelta(L_{[0,k]})\cup \cdots \cup \varDelta(L_{[i,i+k]}))\cap (\varDelta^{(k)}(L^{\geq
x}_{[i,i+k+1]})\cup \varDelta(x',y'))=\varDelta(L_{i,i+k})\cap(\varDelta^{(k)}(L^{\geq x}_{[i,i+k+1]})\cup \varDelta(x',y'))\) is a union of a cone and a \((k-1)\)-dimensional cycle, therefore the
lefthand side of 14 follows. ◻
Corollary 59. Let \(L\) be a geometric semilattice of rank \(r(L) \geq 2\), and \(\bar{L}\) be its proper part. For \(k\leq r(L)-2\), \(\varDelta^{(k)}(\bar{L})\) is homotpy equivalent to a nontrivial wedge of \(k\)-spheres.
Proof. For \(k=0\), it is trivial since \(L\) has at leasy 2 atoms.
Suppose \(r(L)>2\). For \(1\leq k \leq r(L)-2\), choose a pair of elements \(x,y \in L\) such that \(x\leq y\) and \(r(y)-r(x)=k+2\), then \(\varDelta(x,y)\) is a nontrivial \(k\)-dimensional cycle in \(\varDelta^{(k)}(\bar{L})\) by the same
argument as Theorem 58. ◻
The next corollary is a generalization of a result such that the Falkman complex of a rank \(n\) hyperplane arrangement is homotopy equivalent to a nontrivial wedge of \((n-2)\) spheres (see, [16]).
Corollary 60. Let \(\mathcal{A}\) be a hyperplane arrangement of rank \(n\), and \(L(\mathcal{A})\) be its intersection poset. Then \(\varDelta^{(k)}(\bar{L}(\mathcal{A}))\) is homotopy equivalent to a nontrivial wedge of \(k\)-spheres.
Proof. Since \(L(\mathcal{A})\) is a geometric semilattice by Proposition 54, the statement follows from Corollary 59. ◻
Example 61. Let \(\mathcal{B}_n\) be a Boolean lattice on \(\{0,\ldots,n\}\), and \(\varPi_n\) be a partition lattice on \(\{1,\ldots,n\}\). For \(\bar{\mathcal{B}_5}\) and \(\varPi_4\), the homotopy type of the filtered order complexes of these posets are as follows: \[\begin{align} \varDelta^{(3)}(\bar{\mathcal{B}}_5) &\simeq \mathbb{S}^3, &\quad \varDelta^{(2)}(\bar{\mathcal{B}}_5) &\simeq \bigvee^{19} \mathbb{S}^{2}, &\quad \varDelta^{(1)}(\bar{\mathcal{B}}_5) &\simeq \bigvee^{41} \mathbb{S}^{1},\\ \varDelta^{(3)}(\varPi_4) &\simeq \ast, &\quad \varDelta^{(2)}(\varPi_4) &\simeq \bigvee^{6} \mathbb{S}^2, &\quad H_{1}(\varDelta^{(1)}(\varPi_4)) &\simeq \bigvee^{17} \mathbb{S}^1. \end{align}\] Magnitude homologies of these posets are as follows: \[MH^{k}_{i}(\bar{\mathcal{B}_5}) \cong \begin{cases} \mathbb{Z}^{20} & (k=i=3)\\ \mathbb{Z}^{60} & (k=i=2)\\ \mathbb{Z}^{70} & (k=i=1)\\ \mathbb{Z}^{30} & (k=i=0)\\ 0 & (\mathrm{otherwise}), \end{cases} \qquad MH^{k}_{i}(\varPi_4) \cong \begin{cases} \mathbb{Z}^{6} & (k=i=3)\\ \mathbb{Z}^{23} & (k=i=2)\\ \mathbb{Z}^{31} & (k=i=1)\\ \mathbb{Z}^{15} & (k=i=0)\\ 0 & (\mathrm{otherwise}). \end{cases}\]
Figure 5: 
.
5.4 Magnitude homologies of the intersection semilattices of complex hyperplane arrangements↩︎
The following theorem is the well-known fact about the relation between the topology of the complement of a complex hyperplane arrangement and its intersection semilattice.
Theorem 62 ([16], Theorem 5.93). Let \(\mathcal{A}\) be a hyperplane arrangement on \(\mathbb{C}^n\). The integral homology of the complement \(M_{\mathcal{A}}:=\mathbb{C}^n\setminus \bigcup\mathcal{A}\) of \(\mathcal{A}\) is torsion-free, and its betti number has a formula as follows: \[\label{betti} \beta_{k}(M_{\mathcal{A}})=\sum_{x\in L(\mathcal{A}),\;r(x)=k}|\mu(\hat{0},x)|.\tag{15}\]
The following theorem holds for the magnitude homology of the intersection lattice of a complex arrangement and the topology of the complement of restricted arrangements.
Theorem 63. In the same settings above, the next equation holds for the degree of the diagonal components of the magnitude homology of \(L(\mathcal{A})\): \[\label{betti32sum} \mathop{\mathrm{rk}}\mathop{\mathrm{MH}}^{k}_{k}(L(\mathcal{A}))=\sum_{X\in L(\mathcal{A})}\beta_{k}(M(\mathcal{A}^{X})).\tag{16}\] Here, \(\mathcal{A}^{X}:=\{X\cap H \mid X\nsubseteq H,\;X\cap H=\emptyset \}\).
Proof. By 15 , the betti numbers of the complement of \(\mathcal{A}^{X}\) in \(\mathbb{C}^{\mathrm{dim}X}\) are as follows: \[b_k(M_{\mathcal{A}^X})=\sum_{\substack{Y\in L(\mathcal{A}),\;Y\geq X\\ r(Y)-r(X)=k}}|\mu(X,Y)|\] Therefore, by the diagonality of the magnitude homology of \(L(\mathcal{A})\), Corollary 56 yields 16 . ◻
Remark 64. Let \(L\) be a geometric lattice, and \(\mathop{\mathrm{OS}}^{\ast}(L)\) be its Orlik-Solomon algebra. It is known that the degree of \(\mathop{\mathrm{OS}}^{k}(L)\) is equal to \(\sum_{x\in L,\;r(x)=k}|\mu(\hat{0},x)|\) (see [18]) hence by the same argument as Theorem 63, we have \[\mathop{\mathrm{rk}}\mathop{\mathrm{MH}}^{k}_{k}(L)=\sum_{X\in L}\mathop{\mathrm{rk}}\mathop{\mathrm{OS}}^{k}(L_{\leq X}).\] Combining with Theorem 58, we get \(\sum_{X\in L}\mathop{\mathrm{rk}}\mathop{\mathrm{OS}}^{n-1}(L_{\leq X})\leq \sum_{X\in L}\mathop{\mathrm{rk}}\mathop{\mathrm{OS}}^{n}(L_{\leq X})\). It would be interesting to understand whether this inequality is related to the unimodality of the coefficients of the characteristic polynomial of a matroid (see, [19]).
Acknowledgements↩︎
I would like to thank Yoshinaga Masahiko for his constant guidance and discussions. I also thank Yosuke Kusano and Ye Liu for their valuable advice.