Torsion in (Eulerian) Magnitude Homology of Graphs (TAGMaC)

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1. Torsion in (Eulerian) magnitude homology of graphs Patrick Martin II NC State University TAGMaC at NC State Homology and Knot Theory 2 November 2024 Martin (NC State) Torsion in (Eulerian) magnitude TAGMaC
2. magnitude of metric spaces Definition (Leinster ’13) The similarity matrix of a finite metric space (X , d) is  −d(x ,x )  −d(x ,x ) n 1 2 1 e ··· e   .. .. .. ζX (e −1 ) :=   . . . e −d(xn ,x1 ) · · · e −d(xn ,xn ) where X = {x1 , ..., xn }. If ζX (e −1 ) is invertible, then X has Möbius inversion with Möbius matrix µX = ζX−1 . The magnitude |X | of X is the sum of the entries in µX . Martin (NC State) Torsion in (Eulerian) magnitude TAGMaC
3. magnitude of a graph Definition (Leinster ’14) For a finite graph G = (V (G ), E (G )), the magnitude #G of G is the sum of the entries of µG (q) where µG (q)(v , w ) = q d(v ,w ) and d(v , w ) is the shortest-path distance. a3 a4 a2 a5 a1 a0 d(a1 , a5 ) = 2 d(a0 , a3 ) = 3 b3 b2 b0 b1 d(bi , bj ) = δij Observation (Leinster): ζG (0) = I , so det(ζG (q)) is invertible in Z[[q]]. Hence #G is a power series in q with integral coefficients. Martin (NC State) Torsion in (Eulerian) magnitude TAGMaC
4. categorification of the magnitude of a graph Definition (Hepworth-Willerton ’15) A k-path is a (k + 1)-tuple of vertices of G , v = (v0 , ..., vk ), such that vi ̸= vi+1 . Pk−1 The length of a k-path is len(v ) = i=0 d(vi , vi+1 ). The set of all k-paths in G of length ℓ is Tk,ℓ (G ). The (k, ℓ)-magnitude chain group is MCk,ℓ (G ) := Z⟨Tk,ℓ (G )⟩. The (k, ℓ)-magnitude homology group MHk,ℓ (G ) is the homology of the magnitude chain complex. Martin (NC State) Torsion in (Eulerian) magnitude TAGMaC
5. magnitude homology is a stronger invariant In 2018 Gu found examples of graphs with the same magnitude but differing magnitude homologies: Dodecahedral graph F020 A MH2,4 (F020 A) = Z60 Deasargues graph F020 B MH2,4 (F020 B) = 0 Hepworth and Willerton asked if there existed a graph with torsion in its magnitude homology. Martin (NC State) Torsion in (Eulerian) magnitude TAGMaC
6. the kaneta-yoshinaga construction Given a simplicial complex K the Kaneta-Yoshinaga graph associated to K is the graph G (K ) obtained as the underlying graph of the Hasse diagram b ). of P(K 1̂ K 2 [0, 1] K [0, 2] [1, 2] G (K ) 0 1 [0] [1] [2] 0̂ Martin (NC State) Torsion in (Eulerian) magnitude TAGMaC
7. The Kaneta-Yoshinaga Sequence Theorem (Kaneta-Yoshinaga ’18) Given a triangulable manifold K , there exists a chain isomorphism between Cn (∆(P(K ))) and MCn+2,rk(P(K b )) (0̂, 1̂): Hn (K ) ∼ = Hn (Bd(K )) ∼ = Hn (∆(P(K ))) ∼ = MHn+2,rk(P(K b )) (0̂, 1̂) Corollary The following graph contains Z2 as a subgroup in its (3, 4)-magnitude homology group. Martin (NC State) Torsion in (Eulerian) magnitude TAGMaC
8. torsion in magnitude homology Using the connected sum of lens spaces for the torsion part and the conected sum of solid torii for the free part we get the following: Theorem (Sazdanovic–Summers ’19) For any finitely generated abelian group A = Zr ⊕ Zpr1 ⊕ · · · ⊕ Zpmrm 1 there exists a graph G and integers k, ℓ ≥ 0 such that A ⊴ MHk,ℓ (G ). Martin (NC State) Torsion in (Eulerian) magnitude TAGMaC
9. the direct sum decomposition Definition (Asao-Izumihara ’20) Given vertices a, b ∈ V (G ) let MCk,ℓ (a, b) := Z⟨Tk,ℓ (a, b)⟩ where Tk,ℓ (a, b) are all k-paths of length ℓ in G which start at a and end at b. Observation (Asao-Izumihara ’20) MCk,ℓ (G ) = M MCk,ℓ (a, b). a,b∈V (G ) Martin (NC State) Torsion in (Eulerian) magnitude TAGMaC
10. torsion without triangulable manifolds Theorem (Martin-Sazdanovic ’24) Given an even positive integer k ≥ 4 and a derangement σ ∈ Sk−1 , a poset Pkσ can be constructed such that Z2 ⊴ H1 (∆(Pkσ )). (1 2 3) P4 Notice that these cannot be triangulations of a manifold since the top elements cover more than three other elements. Corollary (Martin-Sazdanovic ’24) Let Gkσ represent the graph of the Hasse diagram of Pbkσ . Then Z2 ⊴ MH3,4 (Gkσ ). Martin (NC State) Torsion in (Eulerian) magnitude TAGMaC
11. eulerian magnitude homology Definition (Giusti–Menara ’24) A k-path v = (v0 , ..., vk ) in a graph G is Eulerian if vi ̸= vj for all i, j ∈ {0, ..., k}, i ̸= j. ETk,ℓ (G ) is the set of all eulerian k-paths of length ℓ. The (k, ℓ)-Eulerian chain groups are EMCk,ℓ (G ) := Z⟨ETk,ℓ (G )⟩ with the same boundary maps as Magnitude homology. The Eulerian magnitude homology EMHk,ℓ (G ) of a graph G is the homology of the Eulerian chain complex. Lemma (Menara ’24) EMCk,ℓ (G ) = M EMCk,ℓ (a, b). a,b∈V (G ) a̸=b Martin (NC State) Torsion in (Eulerian) magnitude TAGMaC
12. torsion in eulerian magnitude homology Conjecture (Menara ’24) EMHk,ℓ (G ) is torsion free for every graph G and every integer k, ℓ ≥ 0. Theorem (Martin-Sazdanovic ’24) Given a finite ranked poset P, ∼ MHk,rk(P) b (0̂, 1̂) = EMHk,rk(P) b (0̂, 1̂). Corollary Given a finitely generated abelian group A, there exists a graph G and integers k, ℓ ≥ 0 such that A ⊴ EMHk,ℓ (G ). Corollary For all k ≥ 4 and derangements σ ∈ Sk−1 , Z2 ⊴ EMH3,4 (Gkσ ). Martin (NC State) Torsion in (Eulerian) magnitude TAGMaC
13. proof idea 1 2 Use the Kaneta-Yoshinaga embedding and the fact that “vertices" cannot repeat in the chain of a poset, OR Use the Asao-Izumihara complex which yields a geometric realization associated to a given graph. Martin (NC State) Torsion in (Eulerian) magnitude TAGMaC
14. asao and izumihara’s geometric approach Definition (Asao-Izumihara ’20) Given a graph G , for vertices a, b of G define the simplicial complexes whose vertices are ordered pairs: Kℓ (a, b) = {{(xi1 , i1 ), ..., (xik , ik )}|k ≤ ℓ − 1, len(a, xi1 , ..., xik , b) ≤ ℓ} Kℓ′ (a, b) = {{(xi1 , i1 ), ..., (xik , ik )}|k ≤ ℓ − 1, len(a, xi1 , ..., xik , b) < ℓ}. Observation Note that Kℓ′ (a, b) is a subcomplex of Kℓ (a, b). Definition (Menara ’24) ETℓ (a, b) is the subcomplex of Kℓ (a, b) consisting of Eulerian paths. Martin (NC State) Torsion in (Eulerian) magnitude TAGMaC
15. asao-izumihara isomorphism Theorem (Asao-Izumihara ’20) For the following relative chain groups, C∗ (Kℓ (a, b), Kℓ′ (a, b)) ∼ = MC∗+2,ℓ (a, b). Corollary (Asao-Izumihara ’20 ) Hk (Kℓ (a, b), Kℓ′ (a, b)) ∼ = MHk+2,ℓ (a, b). Theorem (Menara ’24) For the following relative chain groups, C∗ (ETℓ (a, b), ETℓ−1 (a, b)) ∼ = EMC∗+2,ℓ (a, b) Corollary (Menara ’24) Hk (ETℓ (a, b), ETℓ−1 (a, b)) ∼ = EMHk+2,ℓ (a, b). Martin (NC State) Torsion in (Eulerian) magnitude TAGMaC
16. torsion via Asao-Izumihara Lemma (Martin–Sazdanovic ’24) For the graph G representing Pb of any ranked poset P, it follows that K ′ b (0̂, 1̂) = ∅ = ETrk(P)−1 (0̂, 1̂). b rk(P) Corollary (Martin–Sazdanvoic ’24) b = H∗ (Kℓ (0̂, 1̂)) ∼ H∗ (∆(P)) = MH∗+2,rk(P) b (0̂, 1̂) b = H∗ (ETℓ (0̂, 1̂)) ∼ H∗ (∆(P)) = EMH∗+2,rk(P) b (0̂, 1̂) Martin (NC State) Torsion in (Eulerian) magnitude TAGMaC
17. open questions What properties does magnitude detect in a graph? What properties does torsion in (Eulerian) magnitude homology detect in a graph? Are there graphs that have torsion in magnitude homology but not Eulerian magnitude homology? (Menara ’24) DMCk,ℓ (G ) = MCk,ℓ (G )/EMCk,ℓ (G ). What can we learn about the graph G from the long exact sequence in homology? Martin (NC State) Torsion in (Eulerian) magnitude TAGMaC