Classiication o eective GKM graphs with combinatorial type K 4 Shintarô Kuroki Department o Applied Mathematics, Faculty o Science, Okayama Uniervsity o Science, 1-1 Ridai-cho Kita-ku, Okayama 700-0005, Japan Abstract. In this article, we classiy GKM graphs (Γ, α, ) with eective axial unction, where the graph Γ is combinatorially equivalent to the complete graph K 4. As a result, they are equivalent to one o the ollowing distinct GKM graphs: G st ; G 0 (1, 1); G 0 ( 1, 1); G 0 (k, ±1); G 1 (m); G 2 ; G 3, where k and m are integers which satisy k 2 and m 1. Keywords: GKM graph; torus action; equivariant cohomology. 1. Introduction The toric geometry may be regarded as the geometry o the spaces with complexity zero torus actions, i.e., the real 2n-dimensional (complex n-dimensional) with nice T n -actions. As is well-known, the theory o toric geometry is successully developed and it is widely studied in several dierent areas o mathematics: algebraic geometry, symplectic geometry or topology, etc (see e.g. [BP, Od]). Recently, the theory about maniolds with complexity one torus actions (i.e., (2n+2)-dimensional maniolds with n-dimensional torus actions) are also studied, in particular, in algebraic geometry and symplectic geometry (see e.g. [KT1, KT2, Ti]). However, unlike toric geometry, rom topological point o view, the spaces with complexity one torus actions is still developing (also see [Ku1]). In order to develop such theory, it may be useul to have good examples. In this article, we introduce some examples rom combinatorial point o view by classiying the GKM graphs (introduced in [GZ]) with combinatorial type K 4, i.e., the complete graph with our vertices. Recall that the GKM graphs are the combinatorial counterpart o the GKM maniolds which contains wide classes o maniolds with torus actions. For example, by classiication o toric maniolds, a toric maniold whose GKM graph is K 4 is nothing but the 3-dimensional complex projective space CP 3 with the standard torus action (also see Figure 1). Due to the deinition o GKM graph, other GKM graphs whose combinatorial type is K 4 must be induced rom the complexity one GKM maniolds (i there are geometric counterparts). Thereore, to classiy such GKM graphs should be useul to study the complexity one GKM maniolds in the uture. The goal o this article is to classiy GKM graphs with combinatorial type K 4 up to equivalence, and the main theorem can be stated as ollows (see propositions in Section 3): Theorem 1.1. The GKM graph (K 4, α, ) is equivalent to one o the ollowing GKM graphs: G st, G 0 (1, 1), G 0 ( 1, 1), G 0 (k, ±1), G 1 (m), G 2, G 3, where k and m are integers such that k 2 and m 1. The organization o this paper is as ollows. In Section 2, we recall the basics o GKM graphs. In Section 3, we prove Theorem 1.1. In the inal section (Section 4), we propose some questions derived rom our classiication. 2. GKM graph 2.1. Deinition. We irst recall the deinition o GKM graph. Let Γ = (V (Γ), E(Γ)) be an abstract graph with vertices V (Γ) and oriented edges E(Γ). For the given orientation on e E(Γ), we denote its 1
2 S. KUROKI initial vertex by i(e) and its terminal vertex by t(e). The symbol e E(Γ) represents the edge e with its orientation reversed. We assume no loops in E(Γ) and Γ is connected in this article. Set E p(γ) = {e E(Γ) i(e) = p}. A inite connected graph Γ is called an m-valent graph i E p(γ) = m or all p V (Γ). Let Γ be an m-valent graph. We next deine a label α : E(Γ) H 2 (BT ) on Γ. Recall that BT n (oten denoted by BT ) is a classiying space o an n-dimensional torus T, and its cohomology ring (over Z-coeicient) is isomorphic to the polynomial ring H (BT ) Z[a 1,..., a n ], where a i is a variable with deg a i = 2 or i = 1,..., n. So its degree 2 part H 2 (BT ) is isomorphic to Z n. Put a label by a unction α : E(Γ) H 2 (BT ) on edges o Γ. Set α (p) = {α(e) e E p(γ)} H 2 (BT ). An axial unction on Γ is the unction α : E(Γ) H 2 (BT n ) or n m which satisies the ollowing three conditions: (1): α(e) = α(e); (2): or each p V (Γ), the set α (p) is pairwise linearly independent, i.e., each pair o elements in α (p) is linearly independent in H 2 (BT ); (3): or all e E(Γ), there exists a bijective map e : E i(e) (Γ) E t(e) (Γ) such that (1) e = 1 e, (2) e (e) = e, and (3) or each e E i(e) (Γ), the ollowing relation (called a congruence relation) holds: (2.1) α( e(e )) α(e ) 0 mod α(e) H 2 (BT ). The collection = { e e E(Γ)} is called a connection on the labelled graph (Γ, α); we denote the labelled graph with connection as (Γ, α, ). The conditions as above are called an axiom o axial unction. In addition, we also assume the ollowing condition: (4): or each p V (Γ), the set α (p) spans H 2 (BT ). The axial unction which satisies (4) is called an eective axial unction. Deinition 2.1 (GKM graph [GZ]). I an m-valent graph Γ is labeled by an axial unction α : E(Γ) H 2 (BT n ) or some n m, then such labeled graph is said to be an (abstract) GKM graph, and denoted as (Γ, α, ). I such α is eective, (Γ, α, ) is said to be an (eective) (m, n)-type GKM graph. Figure 1. A (3,3)-type GKM graph G st. We omit the label on the dierent direction because it is determined automatically by the axiom (1). In this article, we only consider a complete graph with our vertices, denoted as K 4, i.e., the 3- valent graph with our vertices. An eective GKM graph whose combinatorial structure is K 4 must be a (3, 3)-type or (3, 2)-type GKM graph (see Figure 1).
GKM GRAPH OF K 4 3 2.2. Equivalence relation. We next deine the equivalence relation among GKM graphs. We call two abstract graphs Γ and Γ are combinatorially equivalent i there is a bijective map : Γ = (V (Γ), E(Γ)) (V (Γ ), E(Γ )) = Γ such that (e) = (e) or all e E(Γ) and the ollowing diagram commutes: E(Γ) E(Γ ) i V (Γ) i V (Γ ) where i is the projection onto the initial vertex o an edge. Then, the equivalence relation on GKM graphs can be deined as ollows: Deinition 2.2. Let G = (Γ, α, ) and G = (Γ, α, ) be GKM graphs. We call G and G are equivalent (or ρ-equivalent), denoted as G G, i there is a combinatorial equivalent map : Γ Γ and an isomorphism ρ : H 2 (BT n ) H 2 (BT n ) such that the ollowing diagrams are commutative: (2.2) and E(Γ) E(Γ ) α H 2 (BT n ) ρ α H 2 (BT n ) (2.3) E i(e) (Γ) e E t(e) (Γ) or all e E(Γ). E i((e)) (Γ ) (e) E t((e)) (Γ ) 3. The classiication In this section, we classiy GKM graphs with combinatorial type K 4. 3.1. Classiication o abstract connections. We irst ignore the axial unctions and classiy the possible connections on K 4 up to equivalence. Namely we classiy the ollowing abstract connection on K 4: = { e e E(K 4), e = 1 e, e(e) = e}. We denote K 4 with an abstract connection as (K 4, ). We deine two (K 4, ) and (K 4, ) are equivalent i there is a combinatorial equivalent map : K 4 K 4 such that satisies the commutative diagram (2.3). The ollowing result is the classiication o such (K 4, ). Lemma 3.1. There are exactly seven abstract connections, say st and (k) (k = 1,..., 6), on K 4 up to equivalence. Proo. It is easy to check that there is the unique connection on G st in Figure 1. We call this connection the standard connection st. In order to get the other abstract connections, we may change the bijections st e (and st e ) or e E(K 4 ). By deinition e = 1 e, we may only consider the one direction o e E(K 4 ), i.e., we may only think bijections on 6 edges. Moreover, by using deinition e (e) = e and the act that K 4 is a 3-valent graph, there are two possible bijections on each edge e E(K 4 ). We irst consider the case when only one bijection st e on an edge e E(K 4 ) is changed, denote such connection as 1,e. Because K 4 is the complete graph, or any other edge e E(K 4 ) two (K 4, 1,e ) and (K 4, 1,e ) are equivalent. Thereore, the abstract connection which satisies that the only one bijection is dierent rom st is unique up to equivalence; thereore, we can denote it as (1). Because K 4 is the complete graph, we can apply the similar method or the other cases when bijections on k edges in the connection are dierent rom the standard connection. Namely, there is the unique connection (k) (up to equivalence) whose exactly k bijections (k) e are dierent rom those in the standard connection st, where k = 1,..., 6. This establishes the statement.
4 S. KUROKI 3.2. Classiication o all axial unctions. Now we may prove the main theorem. In the beginning, we introduce the ollowing well-known act rom the toric geometry: Proposition 3.2. The (3, 3)-type GKM graph (K 4, α, ) is equivalent to G st (Figure 1). Thereore, we may only consider the case when (K 4, α, ) is a (3, 2)-type GKM graph. We irst show a rough classiication. Lemma 3.3. The (3, 2)-type GKM graph (K 4, α, st ) is equivalent to G 0 (m, n) (Figure 2), where m, n are non-zero integers which satisy one o the ollowing conditions: m = ±1, n = ±1, n = m or n = 2 m. Figure 2. A (3,2)-type GKM graph G 0 (m, n) with the standard connection st, where m and n are non-zero integers. Proo. To show the statement, we use the igures, see Figure 3 and Figure 4. Figure 3. The axial unctions around e. Figure 4. The axial unctions around e, e. Fix the axial unction around the middle vertex like in the Figure 3, i.e., α, β and mα+nβ or the ixed basis α, β H 2 (BT 2 ). Because they are pairwise linearly independent, we may choose m, n Z \ {0}. Recall that the connection is the standard connection st. So by using the congruence relation along e E(K 4), we also have integers k, h Z in Figure 3. Next, by using the congruence relation along e, e E(K 4 ) (see Figure 4), we have h = 1, k = m 1. By routine work, we also have j = n 1. In our article, we also assume that the eectiveness condition, i.e., the condition (4) in Section 2.1. Thereore, to satisy the condition (4) around the vertex s (see Figure 4), we have one o the ollowing equations: This establish the statement. m = ±1, n = ±1, or, m + n 1 = ±1. Proposition 3.4. The GKM graphs which appeared in Lemma 3.3 are equivalent to one o the ollowing distinct GKM graphs: where k 2. G 0 (1, 1), G 0 ( 1, 1), G 0 (k, 1), G 0 (k, 1)
GKM GRAPH OF K 4 5 Proo. It is easy to check that G 0 (m, n) G 0 (n, m) or all m, n Z \ {0}. Thereore, we may write G 0 (m, n) as G 0 (k, 1)( G 0 (1, k)), G 0 (k, 1)( G 0 ( 1, k)), G 0 (h, h)( G 0 ( h, h)) or G 0 (j, 2 j)( G 0(2 j, j)) or some integers k, h, j such that k 0, h 2, j 4. We irst consider G 0(k, 1) and G 0( k, k) as Figure 5. Then, the identity map on two graphs with Figure 5. The let is G 0 (k, 1) and the right is G 0 ( k, k) or k 2. the isomorphism ρ : H 2 (BT 2 ) H 2 (BT 2 ) deined by α α β and β kα + (k 1)β induces the equivalence G 0 (k, 1) G 0 ( k, k). We next consider G 0 (k, 1) and G 0 (k, 2 k) as Figure 6. Then, the identity map on two graphs with Figure 6. The let is G 0 (k, 1) and the right is G 0 (k, 2 k) or k 4. the isomorphism deined by α α β and β β induces the equivalence G 0(k, 1) G 0(k, 2 k). We next claim that G 0(k, 1) G 0(k, 1) or any integers k, k such that k, k 2 (also see the let graphs in Figure 5 and Figure 6). Note that i k 2 (resp. k 2), there are exactly two (2, 2)-type GKM subgraphs in G 0 (k, 1) (resp. G 0 (k, 1)), say K 1 and K 2 (resp. K 1 and K 2). Moreover, both o e = K 1 K 2 and e = K 1 K 2 are the edge whose axial unction is labelled by α (see Figure 5 and Figure 6). Assume that there is an equivalent map G 0 (k, 1) G 0 (k, 1). Then, this equivalent map preserves (2, 2)-type GKM subgraphs; thereore, e e and also (they are edges in the twisted position with e and e, see Figure 5 and Figure 6). Hence, this shows that every equivalent map satisies α ±α and kα ±(k α 2β). However, this gives a contradiction because i α ±α then kα ±kα. Consequently we have that G 0(k, 1) G 0(k, 1). We inally claim that G 0(1, 1) G 0(1, 1) G 0( 1, 1) G 0( 1, 1). Because G 0(m, n) G 0(n, m), we have that G 0(1, 1) G 0( 1, 1). See Figure 7. In Figure 7, by taking the isomorphism α α + β and β β, we have that G 0(1, 1) G 0(1, 1). Note that G 0(1, 1) has two pair o two edges with the same axial unctions, i.e., α and β (see the let o Figure 7). On the other hand, G 0 ( 1, 1) does not have such pair. I they are equivalent then such pair must be preserved. Hence, we have This establishes the statement o this proposition. G 0 (1, 1) G 0 ( 1, 1).
6 S. KUROKI Figure 7. The let is G 0 (1, 1) and the right is G 0 (1, 1). With the method similar to that demonstrated in the proo o Proposition 3.4, we have the ollowing series o propositions: Proposition 3.5. The (3, 2)-type GKM graph (K 4, α, (1) ) is equivalent to G 1 (m) (Figure 8) or some m N; namely, G 1 (m) G 1 (m ) i and only i m = m ( 0). Figure 8. A (3,2)-type GKM graph G 1 (m) (m N) with the connection (1), i.e., (1) e st e. This axial unction is nothing but the axial unction o G 0 (m, 1). Proposition 3.6. The (3, 2)-type GKM graph (K 4, α, (2) ) is equivalent to G 2 (Figure 9). Figure 9. A (3,2)-type GKM graph G 2 with the connection (2), i.e., (2) e and (2) e st e st e. This axial unction is nothing but the axial unction o G 0(1, 1). Proposition 3.7. The (3, 2)-type GKM graph (K 4, α, (3) ) is equivalent to G 3 (Figure 10). We also have the ollowing proposition. Proposition 3.8. For k = 4, 5, 6, there are no axial unctions α such that (K 4, α, (k) ) is a GKM graph. Consequently, we establish Theorem 1.1.
GKM GRAPH OF K 4 7 Figure 10. A (3,2)-type GKM graph G 3 with the connection (3), i.e., (3) e st e, (3) e st e and (3) e or any m, n Z \ {0}. st e. This axial unction is not induced rom G 0(m, n) 4. Final remarks We inally remark some acts and ask some questions or uture works. 4.1. The group o axial unctions. We have introduced the group o axial unctions A(Γ, α, ) or GKM graph (Γ, α, ) in [Ku2]. The rank o this ree abelian group evaluates the maximal axial unction which is the extension o α. It is not so diicult to compute them or our GKM graphs in Theorem 1.1. The ollowing is the results o computations. Theorem 4.1. The group o axial unctions or each GKM graph in Theorem 1.1 is as ollows: A(G 0(1, 1)) A(G 0( 1, 1)) A(G 0(k, ±1)) Z 3 ; A(G 1 (m)) A(G 2 ) A(G 3 ) Z 2. Thereore, we see that G 0(1, 1), G 0( 1, 1) and G 0(k, ±1) can be induced rom G st. 4.2. GKM cohomology. The most important invariant o GKM graphs is the equivariant cohomology o GKM graph (introduced in [GZ]). We call them a GKM cohomology in this paper and denote it as H T (Γ, α, ). It is well-known that H T (G st ) Z[τ 1, τ 2, τ 3, τ 4 ]/ τ 1 τ 2 τ 3 τ 4, where deg τ i = 2 (i = 1, 2, 3), and its H (BT 3 )-algebraic structure is obtained by α τ 1 τ 4; β τ 2 τ 4; γ τ 3 τ 4. Because G 0 (m, n) is obtained by changing γ to mα + nβ (also see Section 4.1), its GKM cohomology is as ollows: where H T (G 0(m, n)) H T (G st)/i, I = (τ 3 τ 4 ) m(τ 1 τ 4 ) n(τ 2 τ 4 ). Note that we do not need to use the connection to deine the GKM cohomology, though the connection is essential to deine the group o axial unctions (see [Ku2]). Thereore, This establishes that the ollowing theorem: H T (G 1 (m)) H T (G 0 (m, 1)), H T (G 2 ) H T (G 0 (1, 1)).
8 S. KUROKI Theorem 4.2. The ollowing isomorphism holds: H T (G 0 (1, 1)) Z[τ 1, τ 2, τ 4 ]/ τ 1 τ 2 τ 4 (τ 1 + τ 2 τ 4 ) ; H T (G 0( 1, 1)) Z[τ 1, τ 2, τ 4]/ τ 1τ 2τ 4( τ 1 τ 2 + 3τ 4) ; H T (G 0(k, 1)) Z[τ 1, τ 2, τ 4]/ τ 1τ 2τ 4(kτ 1 + τ 2 kτ 4) ; H T (G 0(k, 1)) Z[τ 1, τ 2, τ 4]/ τ 1τ 2τ 4(kτ 1 + τ 2 + (2 k)τ 4) ; H T (G 1(m)) Z[τ 1, τ 2, τ 4]/ τ 1τ 2τ 4(mτ 1 + τ 2 mτ 4) ; H T (G 2) Z[τ 1, τ 2, τ 4]/ τ 1τ 2τ 4(τ 1 + τ 2 τ 4). I we divide them by τ 1 τ 4 and τ 2 τ 4, i.e., H >0 (BT 2 ), we obtain the ordinary cohomology counterparts, say H (Γ, α, ). Corollary 4.3. The ollowing isomorphism holds: H (G 0 (m, n)) H (G 1 (m)) H (G 2 ) Z[τ 4 ]/ τ 4 4 H (CP 3 ). So the ollowing case is still remaining: Problem 4.4. Compute H T (G 3) without using geometry (see [FIM]). Remark 4.5. Note that the complex quadric Q 3 = SO(5)/SO(3) SO(2) with T 2 -action is a GKM maniold whose combinatorial type o GKM graph is K 4. Thereore, by Corollary 4.3 (and H (CP 3 ) H (Q 3 ) over integer coeicients), we have that the GKM graph G 3 must be obtained rom Q 3 with T 2 -action. So by [GKM] HT (G 3 ) HT (Q 3 ). In addition, because every G 0 (m, n) is deined by the restricted T 2 -action o the standard T 3 -action on CP 3 (see Section 4.1). So it is natural to ask whether there are some nice geometric objects or the other GKM graphs G 1(m), G 2. Problem 4.6. Are there any equivariantly ormal, simply connected GKM maniolds which deine GKM graphs G 1(m) and G 2? Acknowledgment This work was supported by JSPS KAKENHI Grant Number 15K17531. Reerences [BP] V. M. Buchstaber, T. E. Panov, Toric topology, Mathematical Surveys and Monographs, 204, AMS, Prov., RI, 2015. [FIM] Y. Fukukawa, H. Ishida, M. Masuda, The cohomology ring o the GKM graph o a lag maniold o classical type, Kyoto J. Math. 54 (2014), 653 677. [GKM] M. Goresky, R. Kottwitz and R. MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), 25 83. [GZ] V. Guillemin, C. Zara One-skeleta, Betti numbers, and equivariant cohomology, Duke Math. J. 107 (2001), 283 349. [KT1] Y. Karshon and S. Tolman, Centered complexity one Hamiltonian torus actions, Trans. Amer. Math. Soc. 353 (2001), 4831 4861. [KT2] Y. Karshon and S. Tolman, Classiication o Hamiltonian torus actions with two-dimensional quotients, Geom. Topol. 18 (2014), 669 716. [Ku1] S. Kuroki, Classiications o homogeneous complexity one GKM maniolds and GKM graphs with symmetric group actions, RIMS Kôkyûroku 1922 (2014), 135 146. [Ku2] S. Kuroki, Upper bounds or the dimension o tori acting on GKM maniolds, arxiv:1510.07216 (preprint 2016). [Ti] D. Timashev, Torus actions o complexity one, Toric topology, 349 364, Contemp. Math., 460, Amer. Math. Soc., Providence, RI, 2008. [Od] T. Oda, Convex Bodies and Algebraic Geometry. An Introduction to the Theory o Toric Varieties, Ergeb. Math. Grenzgeb. (3), 15, Springer-Verlag, Berlin, 1988. E-mail address: kuroki@xmath.osu.ac.jp