Geometric approach to graph magnitude homology
GGEOMETRIC APPROACH TOGRAPH MAGNITUDE HOMOLOGY
Yasuhiko Asao (cid:63) and Kengo Izumihara † (cid:63) Graduate School of Mathematical Sciences, University of Tokyo,Tokyo, Japan ([email protected]) † Graduate School of Mathematics, Kyushu University, Fukuoka,Japan ([email protected])
Abstract
In this paper, we introduce a new method to compute magnitudehomology of general graphs. To each direct sum component of magni-tude chain complexes, we assign a pair of simplicial complexes whosesimplicial chain complex is isomorphic to it. First we states our maintheorem specialized to trees, which gives another proof for the knownfact that trees are diagonal. After that, we consider general graphs,which may have cycles. We also demonstrate some computation as anapplication.
Leinster ([6]) introduced magnitude of finite metric spaces which measures“ the number of efficient points”. Magnitude homology has been invented asa categoryfication of magnitude of a graph which is equipped with a graphmetric, by Hepworth-Willerton ([3]). Magnitude homology
M H k,(cid:96) ( G ) of agraph G is defined by the k -th homology group of a chain complex M C ∗ ,(cid:96) ( G ),whose chain groups are generated by tuples of vertices of length (cid:96) .Several tools for computing magnitude homology of a graph have beenstudied so far. For examples, Hepworth-Willerton ([3]) proves a Mayer-Vietoris type exact sequence and a K¨unneth type formula, and Gu ([2])uses algebraic Morse theory for computation for some graphs. Although, ingeneral, computation of magnitude homology remains a difficult problem.In this paper, we introduce another method to compute magnitude ho-mology of general graphs. Our strategy is to replace the computation of mag-nitude chain complex M H k,(cid:96) ( G ) by that of simpicial homology. A similarmethod using order complex is studied by Kaneta-Yoshinaga ([4]), whereaswe assign simplicial complexes in another way. A subtle difference from1 a r X i v : . [ m a t h . A T ] M a r aneta-Yoshinaga’s method which restricts us to work within a range withno 4-cuts, is that our method can be applied to general graphs.For a magnitude chain complex M C ∗ ,(cid:96) ( G ), we denote by M C ∗ ,(cid:96) ( a, b ) adirect sum component of it, which consists of tuples with ends a and b . Ourmain result is the following which appears as Theorem 4.3 in this paper. Weassume that graphs are connected and contain no loops. Theorem .
Let a, b be vertices of a graph G , and fix an integer (cid:96) ≥ . Thenwe can construct a pair of simplicial complexes ( K (cid:96) ( a, b ) , K (cid:48) (cid:96) ( a, b )) whichsatisfies C ∗ ( K (cid:96) ( a, b ) , K (cid:48) (cid:96) ( a, b )) ∼ = M C ∗ +2 ,(cid:96) ( a, b ) . In particular, we have
M H k,(cid:96) ( a, b ) ∼ = H k − ( K (cid:96) ( a, b ) , K (cid:48) (cid:96) ( a, b )) for k, (cid:96) ≥ . Moreover, for k = 2 , we also have M H ,(cid:96) ( a, b ) ∼ = (cid:40) H ( K (cid:96) ( a, b ) , K (cid:48) (cid:96) ( a, b )) if d ( a, b ) < (cid:96), ˜ H ( K (cid:96) ( a, b )) if d ( a, b ) = (cid:96), where ˜ H ∗ denotes the reduced homology group. Our theorem yields an interpretation of magnitude homology groups asa homology group of a simplicial complex. Therefore, our method allowsus to apply sophisticated tools of homotopy theory. In the special cases of2 ≤ (cid:96) ≤ Sq introduced in [7] as an application. In this section, we recall some basic definitions for graphs and their magni-tude homology together with related notation. Main definitions are takenfrom [3]. 2 .1 Simplicial complexes
Definition 2.1.
Let V be a set, and let P ( V ) be its power set. A subset S ⊂ P ( V ) \ {∅} is called a simplicial complex if it satisfies that B ∈ S for every ∅ (cid:54) = B ⊂ A ∈ S. A subset S (cid:48) of a simplicial complex S is called a subcomplex of S if S (cid:48) itselfis a simplicial complex. Definition 2.2.
For a simplicial complex S ⊂ P ( X ) \ {∅} , we associate achain complex ( C ∗ ( S ) , ∂ ∗ ) defined as follows: (cid:40) C n ( S ) = Z (cid:104) σ ∈ S | σ = n + 1 (cid:105) ,∂ n { s , . . . s n } := (cid:80) ni =0 ( − i { s , . . . , ˆ s i , . . . , s n } , where the index s i is a fixed total order on X , the notation ˆ s i means theremoval of the vertex s i , and S (cid:48) of S , the associated chain complex C ∗ ( S (cid:48) ) is obviously a sub-complex of C ∗ ( S ). We define C ∗ ( S, S (cid:48) ) := C ∗ ( S ) /C ∗ ( S (cid:48) ) . We suppose that any chain complex C ∗ has no negative component, thatis, C i = 0 for i <
0. For a chain complex ( C ∗ , ∂ ∗ ), we denote by C ∗ + N achain complex ( D ∗ , ∂ (cid:48)∗ ) defined as follows: D i = (cid:40) C i + N ( i ≥ , i < ,∂ (cid:48) i = (cid:40) ∂ i + N ( i ≥ , i < . Definition 2.3.
For a simplicial complex S , an element A ∈ S with A = 1is called a vertex of S , and an element A ∈ S with A = 2 is called an edge of S . Definition 2.4. A graph G is a simplicial complex with A ≤ A ∈ S . We denote by V ( G ) the set of vertices of G , and denote by E ( G ) theset of edges of G , which are called a vertex set and an edge set respectively. Definition 2.5.
1. A graph G is finite if its vertex set V ( G ) is a finiteset. 3. A graph G is connected if any two vertices a, b ∈ V ( G ) are connected,that is, there exists a finite sequence of edges e , . . . , e n ∈ E ( G ) , with (cid:40) a ∈ e , b ∈ e n ,e i ∩ e j (cid:54) = ∅ .
3. A cycle in a graph G is a finite sequence of edges e , . . . , e n ∈ E ( G ) , with (cid:40) e ∩ e n (cid:54) = ∅ ,e i (cid:54) = e j ( i (cid:54) = j ) .
4. A graph is a tree if it is finite, connected, and contains no cycles.Throughout the paper, we assume that graphs are connected.
Definition 2.6.
For a graph G , we define a distance function d : V ( G ) × V ( G ) −→ Z ≥ as follows: • In case a (cid:54) = b , we define d ( a, b ) = n where n is the smallest integersuch that there exist a sequence of edges e , . . . , e n ∈ E ( G ) , with (cid:40) a ∈ e , b ∈ e n ,e i ∩ e j (cid:54) = ∅ . • In case a = b , we define d ( a, b ) = 0. Definition 2.7.
For a graph G , we call a tuple ( x , . . . , x k ) ∈ V ( G ) k +1 a( k + 1)- sequence if it satisfies x j (cid:54) = x j +1 for every 0 ≤ j ≤ k − Definition 2.8.
Let G be a graph and fix an integer (cid:96) ≥ Magnitude chaincomplex of length (cid:96) of G is defined as follows. We denote it by M C ∗ ,(cid:96) ( G ).The graded module M C ∗ ,(cid:96) ( G ) is defined as the family of free Z -modules { M C k,(cid:96) ( G ) } k ≥ generated by all ( k + 1)-sequences x = ( x , . . . , x k ) ∈ V ( G ) k +1 k − (cid:88) i =0 d ( x i , x i +1 ) = (cid:96). The boundary map is defined by ∂ = (cid:80) k − i =1 ( − i ∂ i with ∂ i ( x ) := (cid:40) ( x , . . . , ˆ x i , . . . , x k ) if d ( x i − , x i +1 ) = d ( x i − , x i ) + d ( x i , x i +1 ) , x i means the removal of the vertex x i .It has been proved that ∂ = 0 in [3, Lemma 11]. Magnitude homology
M H k,(cid:96) ( G ) of a graph G is defined as the homology group H k ( M C ∗ ,(cid:96) ). Forconvenience, we define the length function L byL( x ) := (cid:80) k − i =0 d ( x i , x i +1 )for x ∈ V ( G ) k +1 , and we call it the length of x . The condition d ( x i − , x i +1 ) = d ( x i − , x i ) + d ( x i , x i +1 ) in Definition 2.8 is equivalent to the conditionL( x , . . . , ˆ x i , . . . , x k ) = L( x , . . . , x k ) . By definition, we have the following proposition.
Proposition 2.9.
For (cid:96) ≥ , we have the direct sum decomposition of amagnitude chain complex M C ∗ ,(cid:96) ( G ) = (cid:77) a,b ∈ V ( G ) M C ∗ ,(cid:96) ( a, b ) , where M C ∗ ,(cid:96) ( a, b ) is the subcomplex of M C ∗ ,(cid:96) ( G ) generated by sequenceswhich start at a and end at b . Hence the computation of magnitude homology of a graph G reduces tothe computation of each ( a, b )-component. We define a subsequence of asequence as follows. Definition 2.10.
Let x = ( x , . . . , x k ) be a sequence, and y = ( y , . . . , y k (cid:48) )be a tuple. We call the tuple y a subsequence of x if there exists integers0 = i < · · · < i k (cid:48) = k such that x i j = y j for each 0 ≤ j ≤ k (cid:48) . When y is asubsequence of x , we denote it by y ≺ x .Note that a subsequence need not to be a sequence. Definition 2.11.
For a subsequence y = ( x , x j , . . . , x j k (cid:48) , x k ) ≺ x , we calla set { j , . . . , j k (cid:48) } the indices of y ≺ x .5 Computation for trees
In this section, we compute magnitude homology of a tree which is knownin [3, Corollary 31] by using simplicial homology.
Definition 3.1.
Let k ≥
1. We call a sequence ( x , x , . . . , x k ) ∈ V ( G ) k +1 a path in a graph G if it satisfies d ( x i , x i +1 ) = 1 , for every 0 ≤ i ≤ k −
1. For vertices a, b ∈ V ( G ), we denote by P ≤ (cid:96) ( a, b ) theset of all paths ( x , . . . , x k ) in G satisfying (cid:40) x = a, x k = b L( x , . . . , x k ) = k ≤ (cid:96). Note that, for each sequence x = ( x , . . . , x k ), there exists a shortest pathwhich passes through vertices x , . . . , x k in this order. We call such a shortestpath a path of x . If G is a tree, a path of x is unique for each sequence x .Let G be a tree. For a path x = ( a, x , . . . , x k − , b ) ∈ V ( G ) k +1 , we denoteby M C k,(cid:96) ( x ) the submodule of M C k,(cid:96) ( a, b ) generated by sequences whosepaths coincide with x . Clearly, M C ∗ ,(cid:96) ( x ) is a subcomplex of M C ∗ ,(cid:96) ( a, b )since each ∂ i ( x ) is 0 or has x as its path. Proposition 3.2.
Let G be a tree. We have the following direct sum de-composition M C ∗ ,(cid:96) ( a, b ) = (cid:77) x ∈ P ≤ (cid:96) ( a,b ) M C ∗ ,(cid:96) ( x ) , for each a, b ∈ V ( G ) and (cid:96) ≥ .Proof. Since we have seen that each sequence belongs to the unique compo-nent of the decomposition, it is sufficient to see that ∂ y ∈ M C k − ,(cid:96) ( x ) for y ∈ M C k,(cid:96) ( x ). Let y = ( y , . . . , y k ) ∈ M C k,(cid:96) ( x ). Then we have ∂ y = (cid:88) L( y ,..., ˆ y i ,...,y k )= (cid:96) ≤ i ≤ k − ( − i ( y , . . . , ˆ y i , . . . , y k ) . Since the path of each sequence ( y , . . . , ˆ y i , . . . , y k ) is unique, it must coincidewith x if the length L( y , . . . , ˆ y i , . . . , y k ) is preserved. Therefore we obtainthat ∂ y ∈ M C k − ,(cid:96) ( x ).In the following, we will construct a pair of simplicial complexes whoseassociated chain complex is isomorphic to the magnitude chain complex M C ∗ ,(cid:96) ( x ) for each (cid:96) ≥ x in G . For a path x = ( x , . . . x (cid:96) ),we consider a subsequence ϕ ( x ) = ( x , x i , . . . , x i m , x (cid:96) ) ≺ x satisfying6 < i s < (cid:96),d ( x i s − , x i s +1 ) < d ( x i s − , x i s ) + d ( x i s , x i s +1 ) , for every 1 ≤ s ≤ m . If G is a tree, it turns out that ϕ ( x ) consists of all“turning points” of x and end points by the following lemma. Lemma 3.3.
Let G be a tree, and let x = ( x , . . . x (cid:96) ) be a path in G . Forevery ≤ i ≤ (cid:96) − , we have ( x , x i , x (cid:96) ) ≺ ϕ ( x ) if and only if x i − = x i +1 .Proof. Every consecutive three points of a path in a tree must have eitherof the configuration of Figure 1.Figure 1: Consecutive three points in a tree.Thus we have x i − = x i +1 if and only if the triangle inequality is not anequality.Let ∆ (cid:96) − be the standard ( (cid:96) − P ( { , . . . , (cid:96) − } ) \ ∅ . For apath x = ( x , . . . x (cid:96) ) and its subsequence ϕ ( x ) = ( x , x i , . . . , x i m , x (cid:96) ), wedefine a subset ∆ x ⊂ ∆ (cid:96) − by∆ x = { σ | { i , . . . , i m } (cid:54)⊂ σ ∈ ∆ (cid:96) − } . For every σ ∈ ∆ x , any subset σ (cid:48) ⊂ σ is a simplex of ∆ x , which implies that∆ x is a subcomplex of ∆ (cid:96) − . Proposition 3.4.
There exists a chain isomorphism ( M C ∗ +2 ,(cid:96) ( x ) , ∂ ) ∼ = ( C ∗ (∆ (cid:96) − , ∆ x ) , − ∂ ) . Proof.
Note that every sequence y belonging to M C ∗ +2 ,(cid:96) ( x ) is a subsequenceof x , that is y = ( x , x j , . . . , x j k , x (cid:96) ) ≺ x .
7e define a homomorphism t : M C ∗ +2 ,(cid:96) ( x ) −→ C ∗ (∆ (cid:96) − , ∆ x )by sending each sequence to its indices (Definition 2.11)( x , x j , . . . , x j k , x (cid:96) ) (cid:55)−→ { j , . . . , j k } and extending it linearly. We can easily see that this is well-defined by thedefinitions. We show that homomorphism t is bijective and is a chain map.First we show the injectivity. Suppose that we have t (cid:32) N (cid:88) α =1 c α ( x , x j α , . . . , x j αk , x (cid:96) ) (cid:33) = 0 . In the case that { j α , . . . , j αk } = { j α (cid:48) , . . . , j α (cid:48) k } for every 1 ≤ α, α (cid:48) ≤ N , it is clear that N (cid:88) α =1 c α ( x , x j α , . . . , x j αk , x (cid:96) ) = 0 . In general, the index set { , . . . , N } can be decomposed into pairwise disjointsubsets A , . . . , A M such that α, α (cid:48) ∈ A m implies that { j α , . . . , j αk } = { j α (cid:48) , . . . , j α (cid:48) k } , for every 0 ≤ m ≤ M . Hence we obtain N (cid:88) α =1 c α ( x , x j α , . . . , x j αk , x (cid:96) ) = M (cid:88) m =0 (cid:88) α ∈ A m c α ( x , x j α , . . . , x j αk , x (cid:96) ) = 0 , which implies that t is injective. Next we show the surjectivity. By defini-tion, [ { j , . . . , j k } ] (cid:54) = 0 ∈ C ∗ (∆ (cid:96) − , ∆ x ) implies that ϕ ( x ) ≺ ( x , x j , . . . , x j k , x (cid:96) ) . By the definition of ∆ x , the paths of ϕ ( x ) and ( x , x j , . . . , x j k , x (cid:96) ) coincide,which implies that ( x , x j , . . . , x j k , x (cid:96) ) ∈ M C k,(cid:96) ( x ) . t is surjective. Finally, the following calculation showsthat t is a chain map : ∂t (cid:32)(cid:88) α c α ( x , x αj , . . . , x αj k , x (cid:96) ) (cid:33) = ∂ (cid:32)(cid:88) α c α { j α , . . . , j αk } (cid:33) = (cid:88) α c α (cid:88) x . When m = 0, we have∆ x = ∅ . When m = (cid:96) −
1, we can see that ∆ x is homotopy equivalent tothe sphere S ( (cid:96) − . The following proposition shows that it is contractible inthe other cases. Proposition 3.5.
For every path x , the complex ∆ x is contractible when < m < (cid:96) − where ϕ ( x ) = ( x , x i , . . . , x i m , x (cid:96) ) .Proof. The complex ∆ x can be obtained from maximal faces { , . . . , ˆ i j , . . . , (cid:96) − } for 1 ≤ j ≤ m by glueing them at the common face { m +1 , m +2 , . . . , (cid:96) − } (bold parts in case m = 2 , l = 5. Theorem 3.6.
Let G be a tree, and let k, (cid:96) ≥ be integers. Then we have M H k,(cid:96) ( G ) = (cid:40) Z E ( G ) , k = (cid:96), , k (cid:54) = (cid:96), where E ( G ) denotes the cardinality of the edge set E ( G ) .Proof. By Proposition 2.9 and 3.2, we have
M H k,(cid:96) ( G ) = (cid:77) a,b ∈ V ( G ) (cid:77) x ∈ P ≤ (cid:96) ( a,b ) M H k,(cid:96) ( x ) . If x is a path such that ϕ ( x ) has less than ( (cid:96) + 1) points, then the x -component M H k,(cid:96) ( x ) is trivial by Proposition 3.4 and 3.5. In case ϕ ( x ) has( (cid:96) + 1) points, we have that x = x = x = . . . and x = x = x = . . . by Lemma 3.3. We can easily see that the amount of such paths is two timesthe cardinality of the edge set E ( G ). Then by Proposition 3.4, we have M H (cid:96),(cid:96) ( G ) = (cid:77) { x ,x }∈ E ( G ) M H (cid:96),(cid:96) ( x , x , x , . . . ) ⊕ M H (cid:96),(cid:96) ( x , x , x , . . . ) ∼ = (cid:77) { x ,x }∈ E ( G ) H (cid:96) − ( S ( (cid:96) − ) ⊕ H (cid:96) − ( S ( (cid:96) − ) ∼ = Z E ( G ) . For general graphs, we cannot decompose magnitude homology indexed bypaths as in the case of trees, since sequences may have more than one shortest10aths. Hence we develop a method to compute each ( a, b )-component ina similar way as in the tree case. Let G be a connected graph and let a, b ∈ V ( G ). We fix an integer (cid:96) ≥ Definition 4.1.
Let K (cid:96) ( a, b ) be the set whose elements are subsets { ( x i , i ) , . . . , ( x i k , i k ) } ⊂ V ( G ) × { , , . . . , (cid:96) − } such that there exists a path( a, x , . . . , x (cid:96) (cid:48) − , b ) ∈ P ≤ (cid:96) ( a, b )with ( a, x i , . . . , x i k , b ) ≺ ( a, x , . . . , x (cid:96) (cid:48) − , b ).For simplicity, we abbreviate { ( x i , i ) , . . . , ( x i k , i k ) } to { x i , . . . , x i k } ifthere is no confusion. The set K (cid:96) ( a, b ) is a simplicial complex since { x j , . . . , x j k (cid:48) } ⊂ { x i , . . . , x i k } ∈ K l ( a, b )implies that there exists a path ( a, x , . . . , x (cid:96) (cid:48) − , b ) ∈ P ≤ (cid:96) ( a, b ) with( a, x j , . . . , x j k (cid:48) , b ) ≺ ( a, x i , . . . , x i k , b ) ≺ ( a, x , . . . , x (cid:96) (cid:48) − , b ) , that is { x j , . . . , x j k (cid:48) } ∈ K (cid:96) ( a, b ) . Clearly, the complex K (cid:96) − ( a, b ) is a subcomplex of K (cid:96) ( a, b ). We also definea subcomplex K (cid:48) (cid:96) ( a, b ) ⊂ K (cid:96) ( a, b ) by K (cid:48) (cid:96) ( a, b ) := {{ x i , . . . , x i k } ∈ K (cid:96) ( a, b ) | L( a, x i , . . . , x i k , b ) ≤ (cid:96) − } . Our goal is to construct an isomorphism between
M C ∗ ,(cid:96) ( a, b ) and the quo-tient chain complex C ∗− ( K (cid:96) ( a, b ) , K (cid:48) (cid:96) ( a, b )). We assume that d ( a, b ) ≤ (cid:96) ,since we have K (cid:96) ( a, b ) = ∅ for d ( a, b ) > (cid:96) . Lemma 4.2.
Let { x i , . . . , x i k } and { y j , . . . , y j k } be simplices of K (cid:96) ( a, b ) .If we have L( a, x i , . . . , x i k , b ) = L( a, y j , . . . , y j k , b ) = (cid:96) and x i s = y j s for ≤ s ≤ k , then we have i s = j s , for ≤ s ≤ k .Proof. By definition, there is a path( a, x , . . . , x (cid:96) − , b ) ∈ P ≤ (cid:96) ( a.b )with ( a, x i , . . . , x i k , b ) ≺ ( a, x , . . . , x (cid:96) − , b ) . ≤ s ≤ k , we have i s = i s − + d ( x i s − , x i s ) = s − (cid:88) n =0 d ( x i n , x i n +1 ) . Similarly we also have j s = s − (cid:88) n =0 d ( y j n , y j n +1 ) , for 1 ≤ s ≤ k . Since we have x i s = y j s for 1 ≤ s ≤ k , we obtain i s = j s .Next we define a homomorphism t : ( C ∗ ( K (cid:96) ( a, b ) , K (cid:48) (cid:96) ( a, b )) , − ∂ ) −→ ( M C ∗ +2 ,(cid:96) ( a, b ) , ∂ )by [ { x i , . . . , x i k } ] (cid:55)−→ ( a, x i , . . . , x i k , b ) . It is well-defined since ∂ i ( a, x i , . . . , x i k , b ) vanishes exactly when ∂ i shortensthe length of the sequence ( a, x i , . . . , x i k , b ), which is equivalent to sayingthat ∂ i { x i , . . . , x i k } ∈ C ∗ ( K (cid:48) (cid:96) ( a, b )). Theorem 4.3.
For (cid:96) ≥ , the above homomorphism t : ( C ∗ ( K (cid:96) ( a, b ) , K (cid:48) (cid:96) ( a, b )) , − ∂ ) −→ ( M C ∗ +2 ,(cid:96) ( a, b ) , ∂ ) is a chain map. Furthermore, it is an isomorphism for ∗ ≥ .Proof. We can show that the homomorphism t is a chain map by the samecomputation as in the proof of Proposition 3.4. Next we show that t isinjective. Suppose that t (cid:32) N (cid:88) α =1 c α [ { x αj , . . . , x αj k } ] (cid:33) = 0 . We can assume that[( x , x αj , . . . , x αj k , x l )] = [( x , x α (cid:48) j . . . , x α (cid:48) j k , x l )] (cid:54) = 0for any 1 ≤ α, α (cid:48) ≤ N as in the proof of Proposition 3.4. By Lemma 4.2, itturns out that their indices coincide. Hence we have N (cid:88) α =1 c α { x αj , . . . , x αj k } = 0 , which implies that t is injective. To prove surjectivity, let ( a, x i , . . . , x i k +1 , b )be a base element of M C k +2 ,l ( a, b ). Since we haveL( a, x i , . . . , x i k +1 , b ) = (cid:96), a, x , . . . , x (cid:96) − , b ) ∈ P ≤ (cid:96) ( a, b )such that ( a, x i , . . . , x i k +1 , b ) ≺ ( a, x , . . . , x (cid:96) − , b ) . Hence we obtain { x i , . . . , x i k +1 } ∈ K (cid:96) ( a, b ) , and we also have { x i , . . . , x i k +1 } (cid:54)∈ K (cid:48) (cid:96) ( a, b ) . Therefore t is an isomorphism on the module of each dimension.By Theorem 4.3, we can compute most part of magnitude homology forarbitrary graphs as follows. Corollary 4.4.
We have
M H k,(cid:96) ( a, b ) ∼ = H k − ( K (cid:96) ( a, b ) , K (cid:48) (cid:96) ( a, b )) for k, (cid:96) ≥ . Moreover, for k = 2 , we also have M H ,(cid:96) ( a, b ) ∼ = (cid:40) H ( K (cid:96) ( a, b ) , K (cid:48) (cid:96) ( a, b )) if d ( a, b ) < (cid:96), ˜ H ( K (cid:96) ( a, b )) if d ( a, b ) = (cid:96), where ˜ H ∗ denotes the reduced homology group.Proof. Suppose that (cid:96), k ≥
3. Since the homology of C ∗ ( K (cid:96) ( a, b ) , K (cid:48) (cid:96) ( a, b ))does not depend on the sign of the boundary map, we have M H k,(cid:96) ( a, b ) ∼ = H k − ( K (cid:96) ( a, b ) , K (cid:48) (cid:96) ( a, b ))from Theorem 4.3. Hence the statement follows. In case k = 2, we considera sequence of submodulesIm ∂ ⊂ Ker ∂ ⊂ M C ,(cid:96) ( a, b ) , where ∂ ∗ denotes differentials of M C ∗ ,(cid:96) ( a, b ). Then we have a short exactsequence0 −→ Ker ∂ / Im ∂ −→ M C ,(cid:96) ( a, b ) / Im ∂ −→ M C ,(cid:96) ( a, b ) / Ker ∂ −→ , which is isomorphic to0 −→ M H ,(cid:96) ( a, b ) −→ H ( K (cid:96) ( a, b ) , K (cid:48) (cid:96) ( a, b )) −→ Im ∂ −→ . Then, by the definition of ∂ , we haveIm ∂ ∼ = (cid:40) d ( a, b ) < (cid:96), Z if d ( a, b ) = (cid:96). H ( K (cid:96) ( a, b ) , K (cid:48) (cid:96) ( a, b )) ∼ = (cid:40) M H ,(cid:96) ( a, b ) if d ( a, b ) < (cid:96),M H ,(cid:96) ( a, b ) ⊕ Z if d ( a, b ) = (cid:96). Then we have
M H ,(cid:96) ( a, b ) ∼ = H ( K (cid:96) ( a, b ) , K (cid:48) (cid:96) ( a, b ))for d ( a, b ) < (cid:96) . If d ( a, b ) = (cid:96) , we see H ( K (cid:96) ( a, b ) , K (cid:48) (cid:96) ( a, b )) = H ( K (cid:96) ( a, b ))since K (cid:48) (cid:96) ( a, b ) = ∅ . Therefore, it also holds that M H ,(cid:96) ( a, b ) ∼ = ˜ H ( K (cid:96) ( a, b )) . Finally, we give an example of computation using the above method.
Example 4.5.
A graph Sq ([7, pp 14-15]) is described in Figure 3. Wecompute the magnitude homology M H ∗ , ( Sq ).Figure 3: Graph Sq .The magnitude chain complex can be decomposed into 36 components whichare classified into 8 types( a, a ) , ( a, b ) , ( a, c ) , ( a, d ) , ( b, b ) , ( b, c ) , ( b, f ) , ( b, e ) . First, consider ( a, a )-type components. We have two ( a, a )-type compo-nents, ( a, a )- and ( d, d )-component. These components have the same chaincomplexes by symmetry. The paths belonging to ( a, a )-component whoselength are 4 are( a, b, a, b, a ) , ( a, b, a, f, a ) , ( a, b, c, b, a ) , ( a, b, f, b, a ) , ( a, f, a, f, a ) , ( a, f, a, b, a ) , ( a, f, e, f, a ) , ( a, f, b, f, a ) . The paths whose length are less than 4 are not related to the homologysince they vanish by the quotient operation. Assigning (4 −
2) = 2-simplicesfor this 8 paths, we can construct the simplicial complex K ( a, a ) and thesubcomplex K (cid:48) ( a, a ) as shown in Figure 4.14igure 4: The 2-dimensional simplicial complex K ( a, a ) and the subcomplex K (cid:48) ( a, a ) (bold part).By Corollary 4.4, we have M H k, ( a, a ) ∼ = ˜ H k − ( | K ( a, a ) | / | K (cid:48) ( a, a ) | ) for k ≥ . Since | K ( a, a ) | / | K (cid:48) ( a, a ) | is homotopy equivalent to S ∨ S ∨ S ∨ S ∨ S ∨ S , we have M H k, ( a, a ) ∼ = (cid:40) Z ( k = 4) , k (cid:54) = 4 , k ≥ . In addition, we can also identify the generators of the magnitude homologyfrom Figure 4.( a, d )-type is one of types which give non-trivial elements to the 3rd mag-nitude homology. There are two ( a, d )-type components, or ( a, d )-componentand ( d, a )-component. The paths belong to ( a, d )-component are( a, b, c, e, d ) , ( a, b, f, e, d ) , ( a, f, b, c, d ) , ( a, f, e, c, d ) . The resulting simplicial complexes K ( a, d ) and K (cid:48) ( a, d ) are shown in Figure5.Figure 5: The simplicial complex K ( a, d ) and the subcomplex K (cid:48) ( a, d )(bold part). 15 K ( a, d ) | / | K (cid:48) ( a, d ) | is homotopy equivalent to S ∨ S , so we have M H k, ( a, d ) ∼ = (cid:40) Z ( k = 3) , k (cid:54) = 3 , k ≥ . We can also compute the other components, and the result is following.rank (a, a) (a, b) (a, c) (a, d) (b, b) (b, c) (b, f) (b, e) k = 2 0 0 0 0 0 0 0 0 k = 3 0 0 8 4 0 0 0 0 k = 4 12 40 0 0 32 0 20 8The rank of the 3rd magnitude homology is 12 and that of the 4th magnitudehomology is 112, so that these coincide with the result [7, TABLE 3]. Acknowledgements
The authors would like to express gratitude to Professor Osamu Saeki forbringing two of us together and supporting our research. Especially, thesecond author is grateful to Professor Saeki for supervising him as a mastercourse student. The second author is also grateful to Naoki Kitazawa, Hi-roaki Kurihara and Dominik Wrazidlo for helpful correction and comments.The first author is supported by the Program for Leading Graduate Schools,MEXT, Japan. He is also supported by JSPS KAKENHI Grant Number17J01956.