Path complexes and their homologies Yong Lin Renmin University of China July 16, 2014
This is a joined work with: Alexander Grigor’yan, Yuri Muranov, Shing-Tung Yau 1
Paths on A Finite Set 2
Let V be an arbitrary non-empty finite set whose elements will be called vertices. For any non-negative integer p , an elementary p - path on a set V is any sequence { i k } p k = 0 of p + 1 vertices of V (a priori the vertices in the path do not have to be distinct). For p = − 1, an elementary p -path is an empty set ∅ . The p -path { i k } p k = 0 will also be denoted simply by i 0 ... i p , without delimiters between the vertices. Fix a field K and consider a K -linear space Λ p = Λ p ( V ) that consists of all formal linear combinations of all elementary p -paths with the coefficients from K . The elements of Λ p are called p - paths on V . An elementary p -path i 0 ... i p as an element of Λ p will be denoted by e i 0 ... i p . The empty set as an element of Λ − 1 will be denoted by e . 3
� � By definition, the family e i 0 ... i p : i 0 , ..., i p ∈ V is a basis in Λ p . Each p -path v has a unique representation in the form � v i 0 ... ip e i 0 ... i p , v = (1) i 0 ,..., i p ∈ V where v i 0 ... ip ∈ K . For example, Λ 0 consists of all linear combinations of elements e i that are the vertices of V , Λ 1 consists of all linear combinations of the elements e ij that are pairs of vertices, etc. Note that , Λ − 1 consists of all multiples of e so that Λ − 1 ∼ = K . 4
For any p ≥ 0, define the boundary operator ∂ : Λ p → Λ p − 1 is a linear operator that acts on elementary paths by p � ( − 1 ) q e i 0 ... � ∂ e i 0 ... i p = i q ... i p , (2) q = 0 where the hat � i q means omission of the index i q . For example, we have ∂ e i = e , ∂ e ij = e j − e i , ∂ e ijk = e jk − e ik + e ij . (3) It follows that, for any v ∈ Λ p , p � � ( − 1 ) q v j 0 ... j q − 1 k j q ... j p − 1 . ( ∂ v ) j 0 ... j p − 1 = (4) k ∈ V q = 0 5
For example, for any u ∈ Λ 0 and v ∈ Λ 1 we have � v ki − v ik � � � u k and ( ∂ v ) i = ∂ u = . k ∈ V k ∈ V Set also Λ − 2 = { 0 } and define ∂ : Λ − 1 → Λ − 2 to be zero. 6
Lemma We have ∂ 2 = 0 . 7
For all p , q ≥ − 1 and for any two paths u ∈ Λ p and v ∈ Λ q define their join uv ∈ Λ p + q + 1 as follows: ( uv ) i 0 ... i p j 0 ... j q = u i 0 ... i p v j 0 ... j q . (5) Clearly, join of paths is a bilinear operation that satisfies the associative law (but is not commutative). It follows from (5) that e i 0 ... i p e j 0 ... j q = e i 0 ... i p j 0 ... j q . (6) If p = − 2 and q ≥ − 1 then set uv = 0 ∈ Λ q − 1 . A similar rule applies if q = − 2 and p ≥ − 1 . 8
Lemma (Product rule) For all p , q ≥ − 1 and u ∈ Λ p , v ∈ Λ q we have ∂ ( uv ) = ( ∂ u ) v + ( − 1 ) p + 1 u ∂ v . (7) 9
We say that an elementary path i 0 ... i p is non - regular if i k − 1 = i k for some k = 1 , ..., p , and regular otherwise. For example, a 1-path ii is non-regular, while a 2-path iji is regular provided i � = j . For any p ≥ − 1, consider the following subspace of Λ p spanned by the regular elementary paths: � � R p = R p ( V ) := span e i 0 ... i p : i 0 ... i p is regular . Note that R p = Λ p for p ≤ 0 but R p is strictly smaller than Λ p for p ≥ 1. The elements of R p are called regular p -paths. 10
We would like to consider the operator ∂ on the spaces R p . However, ∂ is not invariant on spaces of regular paths. For example, e iji ∈ R 2 for i � = j while its boundary ∂ e iji = e ji − e ii + e ij is not in R 1 as it has a non-regular component e ii . The same applies to the notion of join of paths: the join of two regular path does not have to be regular, for example, e i e i = e ii . 11
However, it is easy to define a regular boundary operator ∂ and a regular join that are invariant on the spaces R p : if after applying ∂ or join the outcome contains non-regular terms then all these terms should be discarded. A careful definition requires taking quotient over a space of non-regular paths, but we omit the obvious details. 12
For example, we have for the non-regular operator ∂ ∂ e iji = e ji − e ii + e ij , whereas for the regular operator ∂ ∂ e iji = e ji + e ij since e ii is non-regular and, hence, is replaced by 0. For non-regular join we have e ij e ji = e ijji whereas for the regular join e ij e ji = 0 since e ijji is non-regular. One can show that the regular versions of ∂ and join also satisfy ∂ 2 = 0 and the product rule (7), for all u ∈ R p and v ∈ R q . 13
Path Complexes 14
Definition A path complex over a set V is a non-empty collection P of elementary paths on V with the following property: for any n ≥ 0, if i 0 ... i n ∈ P then also the truncated paths i 0 ... i n − 1 and i 1 ... i n belong to P . [ 1 ] The set of all n -paths from P is denoted by P n . When a path complex P is fixed, all the paths from P are called allowed , whereas all the elementary paths that are not in P are called non-allowed . 15
The set P − 1 consists of a single empty path e . The elements of P 0 (that is, allowed 0-paths) are called the vertices of P . Clearly, P 0 is a subset of V . By the property [1], if i 0 ... i n ∈ P then all i k are vertices. Hence, we can (and will) remove from the set V all non-vertices so that V = P 0 . The elements of P 1 (that is, allowed 1-paths) are called edges of P . By [1], if i 0 ... i n ∈ P then all 1-paths i k − 1 i k are edges. 16
Example By definition, an abstract finite simplicial complex S is a collection of subsets of a finite vertex set V that satisfies the following property: if σ ∈ S then any subset of σ is also in S . 17
Let us enumerate the elements of V by distinct reals and identify any subset s of V with the elementary path that consists of the elements of s put in the (strictly) increasing order. Hence, we can regard S as a collection of elementary paths on V . Then the defining property of a simplex can be restated the following: if an elementary path belongs to S then its any subsequence also belongs to S . [ 2 ] Consequently, the family S satisfies the property [1]so that S is a path complex. The allowed n -paths in S are exactly the n -simplexes. 18
For example, a simplicial complex on Fig. 1(left) has the following path complex: 0-paths: 0 , 1 , ..., 8 1-paths: 01 , 02 , 03 , 04 , 05 , 06 , 07 , 08 , 12 , 34 , 35 , 45 , 67 , 68 , 78 2-paths: 012 , 678 , 034 , 035 , 045 , 678 3-paths: 0345 . 19
5 5 8 8 4 3 4 3 6 6 7 7 0 0 1 2 2 1 Figure: A simplicial complex (left) and a digraph (right) 20
Example Let G = ( V , E ) be a finite digraph, where V is a finite set of vertices and E is the set of directed edges, that is, E ⊂ V × V . The fact that ( i , j ) ∈ E will also be denoted by i → j . An elementary n -path i 0 ... i n on V is called allowed if i k − 1 → i k for any k = 1 , ..., n . Denote by P n = P n ( G ) the set of all allowed n -paths. In particular, we have P 0 = V and P 1 = E . Clearly, the collection { P n } of all allowed paths satisfies the condition [1] so that { P n } is a path complex. This path complex is naturally associated with the digraph G and will be denoted by P ( G ) . 21
For example, a digraph on Fig. 1(right) has the following path complex: 0-paths: 0 , 1 , ..., 8 1-paths: 01 , 02 , 03 , 04 , 05 , 06 , 07 , 08 , 12 , 34 , 35 , 45 , 67 , 68 , 78 2-paths: 012 , 678 , 034 , 035 , 045 , 067 , 068 , 678 3-paths: 0345 , 0678. 22
It is easy to see that a path complex arises from a digraph if and only if it satisfies the following additional condition: if in a path i 0 ... i n all pairs i k − 1 i k are allowed then the whole path i 0 ... i n is allowed. 23
We say that a path complex P is perfect , if any subsequence of any allowed elementary path of P is also an allowed path. We say that a path complex P is monotone , if there is an injective real-valued function on the vertex set of P that is strictly monotone increasing along any path from P . It is easy to show that a path complex P arises from a simplicial complex if and only if P is perfect and monotone. 24
Given an arbitrary path complex P = { P n } ∞ n = 0 over a finite set V , consider for any integer n ≥ − 1 the K -linear space A n that is spanned by all the elementary n -paths from P , that is � v i 0 ... i n e i 0 ... i n : i 0 ... i n ∈ P n , v i 0 ... i n ∈ K A n = A n ( P ) = . i 0 ,..., i n ∈ V The elements of A n are called allowed n -paths. By construction, A n is a subspace of Λ n . For example, A p = Λ p for p ≤ 0, while A 1 is spanned by all edges of P and can be smaller than Λ 1 . 25
We would like to restrict the boundary operator ∂ on the spaces Λ n to the spaces A n . For some path complexes it can happen that ∂ A n ⊂ A n − 1 , so that the restriction is straightforward. If it is not the case then an additional construction is needed as will be explained below. The inclusion ∂ A n ⊂ A n − 1 takes place, for example, for perfect path complexes. In this case we obtain a chain complex 0 ← K ← A 0 ← ... ← A n − 1 ← A n ← ... (8) whose homology groups are denoted by � H n ( P ) , n ≥ − 1 , and are referred to as the reduced path homologies of P . 26
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