TThe tropical Poincaré-Hopf theorem
Johannes Rau
We express the beta invariant of a loopless matroid as tropical self-intersection number of the diagonal of its matroid fan (a “local”Poincaré-Hopf theorem). This provides another example of uncov-ering the “geometry” of matroids by expressing their invariants interms of tropicalised geometric constructions. We also prove a globalPoincaré-Hopf theorem and initiate the study of a more general tropi-cal Lefschetz-Hopf trace formula by proving the two special cases oftropical curves and tropical tori.
The Euler characteristic χ ( X ) of a compact manifold X is equal to the self-intersection number of the diagonal ∆ X ⊂ X × X . In short, deg ∆ X = χ ( X ). Thisis a reformulation of the Poincaré-Hopf theorem [Poi85; Hop27] (in view ofdeg ∆ X = deg T X ). We prove the following tropical (or matroid-theoretic) version: Theorem 1.1 (Local tropical Poincaré-Hopf)
Let M be a loopfree matroid of rank n + 1 . Denote by Σ M = Σ (cid:48) M / R1 its (projective) matroid fan and by ∆ the diagonal of Σ M in Σ M × Σ M . Then the self-intersection of ∆ in Σ M × Σ M is given by deg ∆ = ( − n β ( M ) . (1)Here, β ( M ) denotes the beta invariant of a matroid. It is the canonical re-placement for the Euler characteristic since e.g. ( − n β ( M ) = χ ( U ) if U is thecomplement of a hyperplane arrangement realizing M (also Lemma 4.5). Theproduct ∆ refers to the intersection product for tropical subcycles of matroid This work is part of the FAPA project “Matroids in tropical geometry” of the author atUniversidad de los Andes, Colombia (pending) M is realizable).The Poincaré-Hopf theorem can be regarded as a special case of the Lefschetz-Hopf trace formula (or fixed-point theorem) [Lef26; Dol95] applied to the endo-morphism ψ = id X . Thus, Theorem 1.1 naturally poses the question whether amore general tropical trace formula holds. Again, it is already interesting thatsuch a statement can be formulated in the tropical setup without any compact-ness requirements, as follows (for details, see section 4). Let X be a smoothtropical variety of dimension n without points of higher sedentarity (i.e. X islocally isomorphic to (an open subset of) a matroid fan) and let ψ : X → X be atropical endomorphism. We denote by Γ ψ and ∆ the graph and diagonal, respec-tively, of X in X × X . The intersection product Γ ψ · ∆ (again, in the sense of [Sha13;FR13]) can be regarded as the cycle of stable fixed points of ψ . On the trace side,we use the tropical Hodge type homology groups H p,q ( X ) [Ite+19; MZ14] andthe Borel-Moore versions H BM p,q ( X ) (e.g. [JRS18]), both with real coe ffi cients. If ψ is proper, we have induced pushforward maps ψ ∗ : H BM p,q ( X ) → H BM p,q ( X ). Wedenote the traces by Tr( ψ ∗ , H BM p,q ( X )). Conjecture 1.2 (Tropical Lefschetz-Hopf trace formula)
Let ψ : X → X be aproper tropical endomorphism of a smooth tropical variety X . Then we have deg( Γ ψ · ∆ ) = (cid:88) p,q ( − p + q Tr( ψ ∗ , H BM p,q ( X )) . (2)Theorem 1.1 is a special case of this formula for X = Σ M and ψ = id (cf.Lemma 4.5). In this paper, we also prove the following special cases. Theorem 1.3 (Global tropical Poincaré-Hopf)
The tropical Euler characteristic χ ( X ) := (cid:80) p,q ( − p + q dim H p,q ( X ) of a smooth tropical variety without points of highersedentarity is given by χ ( X ) = deg ∆ . Theorem 1.4 (Tropical Weil trace formula) If X is a smooth tropical curve, Con-jecture 1.2 holds. Theorem 1.5 If X is a tropical torus, Conjecture 1.2 holds. In a work in progress we hope to prove Conjecture 1.2 in the case of matroidal automorphisms (automorphisms which are induced by matroid automorphisms).2 emark 1.6
Currently, Conjecture 1.2 should be restricted to varieties X withoutpoints of higher sedentarity since we are lacking a definition for the intersectiontheoretic side in the presence of such points. We hope that the intersectionproduct (at least, its degree) can be defined such that the statement holds in thegreater generality. In fact, in the case of curves a definition for points of highersedentarity exists and we will prove Theorem 1.4 allowing such points. Thetropical tori of Theorem 1.5 do not contain such points by definition. Remark 1.7
As mentioned before, it might come as a surprise that the presentedresults hold in the non-compact (even local) setup of e.g. Theorem 1.1. Alongthese lines, we can make the following observation. There is a canonicallydefined cycle class map cyc : Z p ( X ) → H BM p,p ( X ) which associates to a tropical p -dimensional subcycle its fundamental cycle class. Moreover, the tropical ho-mology groups for smooth varieties satisfy Poincaré duality and carry variousintersection products [JSS19; JRS18; MZ14]. It is therefore tempting to hopethat all the aforementioned statements can be proven using these constructionssimilar to proofs of the classical statement (e.g. by writing down the Künnethdecomposition for cyc( ∆ ) and showing that the intersection pairings are com-patible). This, however, does not work since the tropical homology groups tendto be too small in the non-compact setting. In particular, for matroid fans wehave H BM p,p ( Σ M ) = 0 for p (cid:44) n and thus cyc( ∆ ) does not carry any information atall. Hence, Theorem 1.1 cannot be stated/proven using intersection products ontropical homology alone. In fact, our proof strategy has no classical analogue.Having said this, it would obviously be interesting to find classical coun-terparts of e.g. Theorem 1.1 and to establish connections to the intersectiontheory/ K -theory of (wonderful compactifications of) hyperplanes arrangementsand to similar expressions for the characteristic polynomial of a matroid in e.g.[FS12; Alu13; AHK18; MRS20; ADH20]. Remark 1.8
A priori, Conjecture 1.2 could be formulated using any of the trop-ical (co)homology versions H p,q ( X ), H BM p,q ( X ), H p,q ( X ) and H p,qc ( X ) (for H BM p,q ( X ),we allow locally finite chains; for H p,qc ( X ), we restrict to compactly supportedcochains). We consider all these groups with real coe ffi cients and hence drop R from the notation. By ordinary (not Poincaré) duality, we have H p,q ( X ) (cid:27) ( H p,q ( X )) ∗ as well as H BM p,q ( X ) (cid:27) ( H p,qc ( X )) ∗ and moreover ψ ∗ = ( ψ ∗ ) (cid:62) (we assume ψ proper in the second version). It follows that (cid:88) p,q ( − p + q Tr( ψ ∗ , H p,q ( X )) = (cid:88) p,q ( − p + q Tr( ψ ∗ , H p,q ( X )) , (3) (cid:88) p,q ( − p + q Tr( ψ ∗ , H BM p,q ( X )) = (cid:88) p,q ( − p + q Tr( ψ ∗ , H p,qc ( X )) . (4)3o we are left with two possibilities for the trace side. If ψ = id (Theorem 1.3),it follows from Poincaré duality [JSS19; JRS18] that all four versions do agree.Obviously, they also agree if X is compact as in Theorem 1.5. For generalendomorphisms, however, Equation 3 and Equation 4 may be di ff erent and the H p,q ( X ) version may not give the correct answer. For example, consider thestandard tropical line L ⊂ R and the map ψ : x (cid:55)→ dx , d ∈ N . Then the sum for H ∗ , ∗ ( X ) is 1 − d , whereas for H BM ∗ , ∗ ( X ) we get d −
2. The latter number agrees withthe intersection-theoretic side (see Lemma 4.11). This is why we decided to use H BM p,q in Conjecture 1.2. On the other hand, in view Lemma 4.9 and Remark 4.15one might argue that the given example is artificial and we could restrict toautomorphisms and use H p,q without much loss. So, the question which versionis more general/useful is probably still open for debate. Acknowledgements
This project started during my visit to Oslo University in March 2019. My specialthanks go to Kristin Shaw for the invitation, for bringing up this problem, andfor many useful discussions later on. I would also like to thank the members ofthe tropical and matroidal seminar during semester 2020-1 at Universidad delos Andes for helpful discussions on the topic.
In this section, we mainly present a variant of a construction from [FR13]. Itprovides a description of the diagonal of a matroid fan in terms of n tropicalrational functions. We will use this description later to compute the intersectionproduct ∆ . We start by fixing our basic notation for matroid fans. Throughout the paper, M will denote a loopless matroid of rank n + 1 on the ground set E = { , . . . , N } .We denote its rank function by rk, its lattice of flats by L ( M ) and the Möbiusfunction thereon by µ . Definition 2.1
The beta invariant of M is β ( M ) := ( − n +1 (cid:88) F ∈ L ( M ) µ ( ∅ , F ) rk( F ) . We refer to [Whi87, Section 7.3] for more background on the beta invariant. Itcan be computed asymmetrically as follows.4 emma 2.2
Fixing ∈ E , the beta invariant of M is equal to β ( M ) = ( − n (cid:88) F ∈ L ( M )0 (cid:60) F µ ( ∅ , F ) . Proof
E.g. [Whi87, Proposition 7.3.1 (d)] (cid:4)
Using β ( M ) as “Euler characteristic” is motivated by the following well-knownfact. Proposition 2.3 If U is the complement of a complex hyperplane arrangement whoseassociated matroid is M , then χ ( U ) = ( − n β ( M ) .Proof By the inclusion/exclusion properties of the Euler characteristic, we get χ ( U ) = (cid:88) F ∈ L ( M ) µ ( ∅ , F ) χ ( P n − rk( F ) C ) = (cid:88) F ∈ L ( M ) µ ( ∅ , F )( n + 1 − rk( F )) . Since (cid:80) F ∈ L ( M ) µ ( ∅ , F ) = 0, the statement follows. (cid:4) To a loopless matroid M , we can associate an a ffi ne matroid fan Σ (cid:48) M ⊂ R N +1 whose lineality space contains the line R1 . Here, denotes the all one vector(1 , . . . , projective matroid fan Σ M ⊂ R N +1 / R1 (cid:27) R N , see [Spe08]. We identify R N with R N +1 / R1 by fixing the section x = 0. We will mostly use the fan structure on Σ M associated to the lattice offlats L ( M ), called the fine subdivision of Σ M [AK06]. This subdivision contains acone σ F for any chain of flats F as follows: We will use the convention to write achain of flats as a decreasing sequence E = F (cid:41) F (cid:41) · · · (cid:41) F l (cid:41) F l +1 = ∅ . We call l ( F ) := l the length of F . To such a chain, we associate the cone σ F := R ≥ (cid:104) v F , . . . , v F l (cid:105) + R1 ⊂ R N +1 . Here, for any subset S ⊂ E , we denote by v S ∈ R N +1 the indicator vector for S whose i -th entry is − i ∈ S (max-convention!) and 0 if i (cid:60) S . If no confusionis likely, we use the same notations v S and σ F for the projections to R N . Then Σ (cid:48) M and Σ M can be described as to the union of all such cones in R N +1 and R N ,respectively. Note that Σ M is a unimodular fan of pure dimension n . We will usethe same notation Σ M for the collection of cones as well as the underlying set.Using chains C of arbitrary subsets C i ⊂ E , the cones σ C form a subdivision of R N (and R N +1 ) called the braid arrangement fan . It can be equivalently described5s the intersection of the hyperplane subdivisions x i = x j for all i (cid:44) j or as thenormal fan of the permutahedron. The fine subdivisions of matroid fans aresubfans of the braid arrangement fan.Following [Spe08], we recall that to any point x ∈ R N we can associate amatroid M x whose bases are the x -maximal bases of M . Here, the x -weight of abasis B is defined to be (cid:104) x, v B (cid:105) . A point x is contained in Σ M if and only if M x isloopfree. Using this description, it is easy to show that Σ (cid:48) M ⊕ M = Σ (cid:48) M × Σ (cid:48) M . Weare interested in the variant of this statement for projective fans.Let M be a parallel connection of two loopfree matroids M and M . Let E , E and E be the ground sets of M , M and M , respectively. By slight abuse ofnotation, we denote by 0 the element in all three ground sets along which theparallel connection is built. We have a canonical identification of ambient spaces R E \ = R E \ × R E \ which is compatible with our convention of setting x = 0. Lemma 2.4
Let M be a parallel connection of two loopfree matroids M and M .Then Σ M = Σ M × Σ M (as sets).Proof A basis of M can be uniquely written as a pair ( B , B ) where either 0 ∈ B ∩ B and B and B are bases of M and M , respectively, or 0 (cid:60) B ∪ B and B and B ∪ { } are bases of M and M , or the symmetric version of the second case.Clearly the ( x, y )-weight of ( B , B ) is equal to x -weight of B plus the y -weightof B (again, recall that x = 0 and y = 0 by convention). It follows easily that• the element 0 is contained in a basis of M ( x,y ) if and only if it is containedin a basis of ( M ) x and a basis of ( M ) y ,• in this case, M ( x,y ) is equal to the parallel connection of ( M ) x and ( M ) y along 0.In particular, M ( x,y ) is loopfree if and only if both ( M ) x and ( M ) y are loopfree,which proves the claim. (cid:4) Given two matroids
M, N on the ground set E , it is obvious that Σ N ⊂ Σ M (bothas sets and fans) if and only if L ( N ) ⊂ L ( M ) (or, in matroid terminology, N isa quotient of M ). In such a case, there exists a canonical sequence of matroids N = M , M , . . . , M s = M such that rk( M i ) = rk( N ) + i and Σ N = Σ M ⊂ Σ M ⊂ · · · ⊂ Σ M s = Σ M , (5)see [FR13, Corollary 3.6]. These matroids are given in terms of their rankfunctions by rk M i ( S ) = min { rk N ( S ) + i, rk M ( S ) } . (6)6f N and M correspond to hyperplane arrangements associated to the projectivesubspaces K ⊂ L ⊂ CP n , then the M i correspond to a chain of generic subspaces K = S ⊂ S ⊂ · · · ⊂ S s = L . Moreover, there exists an associated sequence ofrational functions g , . . . , g s : R N +1 → R such that Σ (cid:48) M s − i = g i · g i − · · · g · Σ (cid:48) M , (7)see [FR13, Proposition 3.10] (we refer to [AR10] for a definition of the intersec-tion with rational functions/the divisor construction). These functions are linearon each cone of the braid arrangement fan of R N +1 and hence determined bytheir values on the indicator vectors v S , S ⊂ E . These values are given by g i ( S ) = − M ( S ) ≥ rk N ( S ) + s + 1 − i, . (8)Here, we use the short-hand g i ( S ) instead of g i ( v S ). We are now going to use thisconstruction in the special case of diagonals. The construction from Equation 7 was used in [FR13] to construct functionsthat cut out the diagonal of a matroid fan. This description was then used todefine a general intersection product for tropical subcycles of matroid fans (andsmooth tropical varieties without points of higher sedentarity). We will use aslight variant of this construction here. Let us quickly explain the di ff erence.In [FR13], the construction of Equation 5 was applied to the a ffi ne matroid fans,i.e. to the diagonal ∆ (cid:48) of Σ (cid:48) M in Σ (cid:48) M × Σ (cid:48) M = Σ (cid:48) M ⊕ M . This is not quite what we want,since we are interested in the self-intersection of projective fans ∆ ⊂ Σ M × Σ M (the self-intersection of ∆ (cid:48) ⊂ Σ (cid:48) M ⊕ M is always zero). We could consider thelift (cid:101) ∆ ⊂ Σ (cid:48) M × Σ (cid:48) M of ∆ and compute the self-intersection (cid:101) ∆ . But note that (cid:101) ∆ = ∆ (cid:48) + R ( , ) has extra lineality space. In particular, unlike ∆ (cid:48) and ∆ , it is nota matroid fan (for the given embedding) and it is not clear to the author howto express (cid:101) ∆ as complete intersection or how to compute the self-intersection (cid:101) ∆ otherwise. Instead, we will work directly with the projective matroid fans(and justify why the results agree). This requires to break symmetry by choosingan element in E . However, we will later see that this corresponds nicely to theasymmetric formula for β ( M ) in Lemma 2.2. Let us give the details here.We denote by M ⊕ M the parallel connection M with it itself along 0. Theground set E (cid:48) of M ⊕ M is a disjoint union of E with itself, but with the two 0’sidentified. 7 onvention 2.5 We denote a subset in E (cid:48) as a pair ( F, G ), F, G ⊂ E with the condi-tion that either 0 ∈ F ∩ G or 0 (cid:60) F ∪ G .Using this convention, flats of M ⊕ M correspond to pairs of flats ( F, G ) of M .The rank function of M ⊕ M on flats is given byrk M ⊕ M ( F, G ) = rk F + rk G − δ ∈ F . (9)Here, by definition, δ ∈ F is 1 if 0 ∈ F and 0 otherwise. (The rank function rkwithout indices always refers to the rank function of M .) By Lemma 2.4, Σ M × Σ M = Σ M ⊕ M . We can describe ∆ ⊂ Σ M × Σ M as the matroid fan given by the rank functionrk ∆ ( F, G ) = rk( F ∪ G ) . (10)Of course, the union F ∪ G is understood to be the (non-disjoint) union in E here.It is straightforward to check that this indeed defines a matroid M ∆ and that theflats of M ∆ are of the form ( F, F ), F ∈ L ( M ), which shows that ∆ = Σ M ∆ (even onthe level of fan structures).We now apply the construction of Equation 5 to M ∆ and M ⊕ M (of rank n + 1and 2 n + 1, respectively). From Equation 8 we get functions g (cid:48) , . . . , g (cid:48) n : R N +1 → R given by g (cid:48) i ( F, G ) = − F ) + rk( G ) − δ ∈ F ≥ rk( F ∪ G ) + n + 1 − i, . (11)Note that these functions are “homogeneous” functions in the sense that theylive on R N +1 (not R N ) and g (cid:48) i ( ) = − g (cid:48) i ( E, E ) = 1. In order to obtain functionson R N , we break the symmetry again and dehomogenise these functions bysubtracting the coordinate function x . Then the functions g i = g (cid:48) i − x : R N → R are well-defined and determined by the values g i ( F, G ) = − (cid:60) F, rk( F ) + rk( G ) ≥ rk( F ∪ G ) + n + 1 − i, +1 0 ∈ F, rk( F ) + rk( G ) ≤ rk( F ∪ G ) + n + 1 − i, . (12)Our discussion so far can be summarized in the following statement, whichfollows directly from Equation 7. Proposition 2.6
The diagonal ∆ of Σ M in Σ M × Σ M can be described as the completeintersection ∆ = g n · · · g · ( Σ M × Σ M ) using the rational functions g i : R N → R from Equation 12. g , . . . , g n are not tropically polynomial norotherwise convex/regular.In order to compute ∆ , we will now just restrict the functions to ∆ , or rather,consider the pullbacks along d : x (cid:55)→ ( x, x ). Set f i := d ∗ ( g i ) : R N → R . Note that d is compatible with the braid arrangement fans of R N and R N (i.e. it maps acone σ C to the cone σ C (cid:48) where C (cid:48) is the chain of subsets obtained from replacingeach C in C by ( C, C )). Hence the functions f i are linear on the cones of the braidarrangement fan on R N and completely determined by the values f i ( F ) = − (cid:60) F and rk( F ) ≥ n + 1 − i, +1 0 ∈ F and rk( F ) ≤ n + 1 − i, . (13)Summarizing again, we can describe the left hand side of Theorem 1.1 as follows. Proposition 2.7
Let ∆ ⊂ Σ M × Σ M by the (projective) diagonal of a matroid fan ofloopless matroid M . Then deg ∆ = deg( f n · · · f · Σ M ) . (14) Proof
By Proposition 2.6 we can express ∆ as g n · · · g · ( Σ M × Σ M ). Hence deg ∆ =deg( g n · · · g · ∆ ) by [FR13, Theorem 4.5 (6)]. Finally, by the projection for-mula [AR10, Proposition 7.7] it follows that the latter expression is equal todeg( f n · · · f · Σ M ). (cid:4) We will now give a description of the intermediate intersection products f k · · · f · Σ M and prove the description by induction. Theorem 1.1 will then just followby inspection of the case k = n . The intermediate intersection products can bedescribed explicitly, and it would be interesting to see if they appear in othersituations or if they can be related to other canonical tropical subcycles of Σ M such as the CSM classes from [MRS20] or the intersection products appearing in[AHK18; ADH20].We set X k := f k · · · f · Σ M . In order to describe the X k , let us introduce someterminology first. We denote the rank function of M by rk. Let F be a chain offlats E = F (cid:41) F (cid:41) · · · (cid:41) F l (cid:41) F l +1 = ∅ . Definition 3.1
The (rank) gap sequence of F , denoted by gap( F ) = ( r , . . . , r l ), isthe sequence of numbers r i := rk F i − rk F i +1 − rk n + 1 n − k − · · · r · · · F F · · · s · · · n − kn − rE · · · k · · · F F F · · · F · · · ... type ( r, s )type ( k ) (cid:51) (cid:61) Figure 1
Chains of type ( r, s ) and type ( k ) To describe X k , we will only need chains F whose gap sequences have one ofthe following two shapes:gap( F ) = ( r, s, , . . . , , r + s = k, r, s ≥ , gap( F ) = ( k, , . . . , . (15)More specifically, we are only interested in the following two cases (c.f. Figure 1). Definition 3.2
A chain F is of type ( r, s ) if its gap sequence is ( r, s, , . . . ,
0) andadditionally 0 (cid:60) F (i.e. the only term of F containing 0 is E ).A chain F is of type ( k ) if its gap sequence is ( k, , . . . ,
0) and additionally 0 ∈ F .Note that the types ( k,
0) and ( k ) have the same gap sequence, but di ff er as towhether 0 is contained in F or not. By extension, the type of a cone in Σ M is thetype of the corresponding chain (if of any type at all). We can now describe thecycles X k (c.f. Figure 2). Proposition 3.3
The tropical cycle X k consists of the cones of type ( r, s ) , r + s = k , r, s ≥ , and the cones of type ( k ) . The weight of a cone σ of type ( r, s ) in X k is givenby ω ( σ ) = ( − rk( M/F ) − β ( M/F ) . (16) The weight of a cone σ of type ( k ) in X k is always ω ( σ ) = 1 . Before proving the proposition, let us check consistency by showing that thecase k = n implies Theorem 1.1. Proof (Theorem 1.1)
In the case k = n , the only chain of correct dimension is thetrivial chain F = ( E ⊃ ∅ ). We have gap( F ) = ( n ) and 0 (cid:60) F = ∅ . Hence, as a quitespecial case of our definitions, this is a chain of type (0 , n ). By Proposition 3.3,its weight is ( − n β ( M ). (cid:4) igure 2 The cycle X in the casethe standard hyperplane in R (i.e.the uniform matroid U , of rank3 on 4 elements). The rays of type(1), (1 ,
0) and (0 ,
1) are displayed inblack, blue and red, respectively. type (1 , ω = − , ω = 1type (1), ω = 1 Remark 3.4
The appearance of the beta invariant of the “factors” of the chain F is reminiscent of the definition of tropical CSM cycles in [MRS20, Definition2.8]. However, CSM cycles take into account cones with arbitrary gap sequencesand the beta invariant of each factor in the chain. In contrast, in X k only specialgap sequences and the beta invariant of the first factor occur. For k = n thedi ff erences disappear and Theorem 1.1 can also be stated as ∆ = csm n ( Σ M ) . We now want to prove Proposition 3.3. As a first step, we will have a look atthe balancing condition for the weighted fan X k . Lemma 3.5
Let G by a chain of flats corresponding to a codimension one face τ of X k and let G (cid:41) H denote the flats in G corresponding to the last non-zero entry of gap( G ) .Then the following holds:(a) Exactly one of the following four statements holds true.(A) rk( G ∪ { } ) ≤ n − k (B) rk( G ) ≥ n − k, (cid:60) G, (C) G = E, ∈ H , (D) G = E, (cid:60) H . (b) Assume that G belongs to one of the cases (A), (B) or (C). Then the facets of X k containing τ correspond bijectively to fillings G (cid:41) F (cid:41) H with flats F of rank rk F = rk H + 1 . Moreover, all such facets have identical type and weight ω and X k is balanced at τ . Explicitly, the balancing condition at τ is given by (cid:88) F ωv F = ωv G + ω (val − v H , (17) where F runs through all such fillings and val denotes the number of fillings. roof Let F be a chain corresponding to a facet of X k containing τ . By definitions, G is obtained from F by removing one of its flats, say F i . If i > F is oftype ( k ) and i >
1, it follows that G satisfies (A). If F is of type ( r, s ) and i = 2, weobtain case (B). Finally, if i = 1, we end up with (C) or (D), depending on whether0 ∈ F or not. It is clear that the cases are mutually exclusive. Hence (a) follows.For (b), we have a closer look at the previous argument. Note that the cases(A), (B), (C) correspond exactly to the case where either F is of type ( r, s ) and i ≥ F is of type ( k ) and i ≥
1. In both cases, rk( F i ) = rk( F i +1 ) + 1 and hence F corresponds to a filling as described in the statement. Moreover, it is obviousthat each filling occurs in this way. The type and weight of F is completelydetermined by the principal part E (cid:41) F of F which is still present in G andhence is fixed for given G . It remains to check the balancing condition in theform of Equation 17. After dividing by ω , this follows from the well-known factthat the sets F \ H , running through flats F with G (cid:41) F (cid:41) H and rk F = rk H + 1,form a disjoint partition of G \ H . (cid:4) We are now ready to prove Proposition 3.3.
Proof (Proposition 3.3)
The induction start k = 0 is trivial (note that β ( M ) = 1for any loopless matroid of rank 1). Let us prove the step k → k + 1. For eachcodimension one cone τ of X k , we need to compute its weight in f k +1 · X k . Forconvenience, we recall from Equation 13 that f k +1 is given by f k +1 ( F ) = − (cid:60) F and rk( F ) ≥ n − k, +1 0 ∈ F and rk( F ) ≤ n − k, . (18)Copying the notation from Lemma 3.5, we denote by G the chain associated to τ and by G (cid:41) H the last non-trivial step in G . We go through the cases (A), (B), (C),(D) according to Lemma 3.5 (a).Let r, s denotes integers such that r + s = k and r, s ≥
0. Note that G is of type( k + 1 ,
0) in case (D), of type ( k + 1) in case (C), of type ( r, s + 1) in case (B), and ofdi ff erent shape (not present in the description of X k +1 for case (A).If G is of type (A), then all the flats F occurring Equation 17 (including G and H ) satisfy rk( F ∪ { } ) ≤ n − k . Hence, their values under f k +1 are +1 or 0depending on whether 0 ∈ F or not. Plotting these values, we get the followingthree possible shapes. 0 ∈ H ∈ G \ H (cid:60) Gf k +1 ( G ) 1 1 0 f k +1 ( F ) 1 . . . . . . . . . f k +1 ( H ) 1 0 012Note that the single 1 in the middle of the f k +1 ( F ) line corresponds to H ∪ τ in f k +1 · X k andobtain zero in all three cases.Let us now assume G satisfies (B). Then the ranks of the flats involved inEquation 17 are rk( G ) ≥ n − k , rk( F ) = n − k − H ) = n − k −
2, and none ofthese flats contains 0. Hence the pattern of values under f k +1 is: f k +1 ( G ) 1 f k +1 ( F ) 0 . . . f k +1 ( H ) 0Hence the weight assigned to τ is ω = ( − r β ( M/F ), as required.We continue with case (C), so now E = G and 0 ∈ H = F . The ranks are nowgiven by rk( G ) = n + 1, rk( F ) = n − k and rk( H ) = n − k −
1. Since all flats contain0, we get the pattern of values f k +1 ( G ) 0 f k +1 ( F ) 1 . . . f k +1 ( H ) 1which gives weight 1 (note that ω = 1 in this case).Finally, we are left with case (D). So now E = G , 0 (cid:60) H = F and gap( G ) = ( k +1 , , . . . , X k that the facets of X k containing τ correspond to F = H ∪ F (cid:41) H with 0 (cid:60) F . Note that the balancingcondition around τ written in terms of the vectors primitive generators v F mayonly involve the additional vectors v E = and v H (with certain coe ffi cients). But f k +1 ( H ) = f k +1 ( E ) = 0 (since rk H = n − k − τ without knowing the coe ffi cients (in fact, it can be checked that theyare both equal to 1). To do so, note that rk( H ∪
0) = n − k and rk( F ) ≥ n − k forall the flats F (cid:41) H with 0 (cid:60) F , so they all evaluate to 1 under f k +1 . Moreover,the weight of the facet associated to H ∪ F it is( − rk( M/F ) − β ( M/F ). Hence the weight of τ in f k +1 · X k is equal to ω ( τ ) = 1 − (cid:88) F (cid:41) H (cid:60) F ( − rk( M/F ) − β ( M/F ) . To finish the calculation, we that µ can be defined as the inverse of the zetafunction of L ( M ) and hence for any interval H ⊂ G satisfies (cid:88) F ∈ L ( M ) H ⊂ F ⊂ G µ ( F, G ) = δ ( H , G ) , δ ( H , G ) = 0 unless H = G , in which case δ ( H , G ) = 1.Using Lemma 2.2 twice, we can now compute ω ( τ ) as follows. ω ( τ ) = 1 − (cid:88) F (cid:41) H (cid:60) F ( − rk( M/F ) − β ( M/F ) = 1 − (cid:88) F (cid:41) H (cid:60) F (cid:88) (cid:60) G µ ( F, G )= 1 − (cid:88) (cid:60) G (cid:88) H (cid:40) F ⊂ G µ ( F, G ) = 1 − (cid:88) (cid:60) G ( δ ( H , G ) − µ ( H , G ))= 1 − (cid:88) (cid:60) G µ ( H , G ) = ( − rk( M/H ) − β ( M/H ) . This agrees with Equation 16, so we are done. (cid:4)
This section deals with the tropical Lefschetz-Hopf trace formula (or fixed-pointtheorem) as formulated in Conjecture 1.2. We start by giving a few more detailsconcerning the notions used in the introduction.
For the purposes of this section, a smooth tropical variety without points of highersedentarity is a Hausdor ff topological space X together with finite atlas of smoothcharts ( U i , φ i ) (c.f. [MR19; FR13]). This means• the U i form a (finite) open cover of X ,• the maps φ i : U i → R N i are homeomorphisms onto their images,• the images are open subsets of a matroid fans Σ M i (for some loopfreematroids M i ).A tropical subcycle of X is a weighted closed subset whose restriction to a chart isan open subset of a tropical subcycle in R N i (balanced polyhedral set). Let Z k ( X )denote the group of k -dimensional tropical subcycles. The intersection-theoreticside of the trace formula is based on the intersection product Z k ( X ) × Z l ( X ) → Z n − k − l ( X )defined in [FR13, section 6]. Let ψ : X → X be a proper tropical endomorphism.The tropical subcycles ∆ and Γ ψ of X × X can be formally defined as the pushfor-wards of X along x (cid:55)→ ( x, x ) and x (cid:55)→ ( x, ψ ( x )), respectively.14 efinition 4.1 Let ψ : X → X be a proper tropical endomorphism of a smoothtropical variety X without points of higher sedentarity. The cycle of stable fixedpoints of ψ is the zero-dimensional cycle Γ ψ · ∆ (or rather, its projection to X ).Given a fan Σ ⊂ R N , we denote by F p ( Σ ) ⊂ (cid:86) p R N the framing group from[Ite+19] generated by wedges v ∧ · · · ∧ v p with the condition that all v i belong toa single cone of Σ . It is easy to see that this definition does only depend on thesupport set of Σ and not on a particular cone structure of Σ . In the following, wewill think of fans mostly as sets and pick a cone structure only when necessary.To each point x ∈ X we can associate a local fan Star X ( x ) ⊂ R N x (for x ∈ U i ,take the local fan of Σ M i at φ i ( x )). We may assume that Star X ( x ) spans R N x .Then it is well-defined (as a set) up to the action of GL( N x , Z ). We set F p ( x ) := F p (Star X ( x )). Tropical homology groups H p,q ( X ) were defined in [MZ14; Ite+19]as homology with local coe ffi cients groups F p ( x ) (analogously for cohomology H p,q ( X )). The Borel-Moore and compact support variants H BM p,q ( X ) and H p,qc ( X )appear in [JRS18]. We refer to these papers for more details. We only considerthe groups with real coe ffi cients and hence drop R from the notation.If ψ : X → X is a tropical endomorphism, we have induced pushforwardmaps ψ ∗ : H p,q ( X ) → H p,q ( X ). If ψ is proper, we also have maps ψ ∗ : H BM p,q ( X ) → H BM p,q ( X ). They are defined in the usual way with the help of the local multi-di ff erentials dψ x : F p ( x ) → F p ( ψ ( x )) on the level of coe ffi cients. Analogously,there are pullback maps for H p,q ( X ) and H p,qc ( X ) ( ψ proper). The trace side ofthe trace formula consists of the graded trace of the map ψ ∗ : H BM ∗ , ∗ ( X ) → H BM ∗ , ∗ ( X ).Here, graded means that the trace of the piece of degree ( p, q ) is counted withsign ( − p + q . As mentioned in Remark 1.8, we could equally well use the gradedtrace of ψ ∗ on H ∗ , ∗ c ( X ). However, the graded trace on H ∗ , ∗ ( X ) (or, equivalently, on H ∗ , ∗ ( X )) is di ff erent in general.In the following, we want to discuss three special cases of Conjecture 1.2.A global Poincaré-Hopf theorem corresponding to ψ = id, the case of tropicalcurves (even with points of higher sedentarity) and the case of tropical tori. Remark 4.2
We are currently working on a proof of Conjecture 1.2 in the caseof matroidal automorphisms (automorphisms which are induced by matroidautomorphisms). We decided to publish this separately since the extensioninvolves more complicated combinatorics and we hope that the specific case ofthe beta invariant is of particular appeal.
We start with a definition. 15 efinition 4.3
The tropical Euler characteristic of a smooth tropical variety X without points of higher sedentarity is χ ( X ) := (cid:88) p,q ( − p + q dim H p,q ( X ) . The local tropical Euler characteristic at a point x is χ ( x ) := (cid:80) np =0 ( − p F p ( x ). Remark 4.4
Of course, the definition also makes sense for more general tropicalspaces (non-smooth or with points of higher sedentarity). As mentioned in Re-mark 1.8, by ordinary and Poincaré duality [JSS19; JRS18] we could equivalentlyuse the variants H BM p,q ( X ), H p,q ( X ) or H p,qc ( X ) in the definition.Summarizing some well-known facts, the following lemma asserts that Theo-rem 1.1 agrees with the special case of Conjecture 1.2 for ψ = id and X = Σ M . Lemma 4.5
Let M be a loopless matroid of rank n + 1 . Then ( − n β ( M ) = χ ( Σ M ) = χ (0) . Proof
For any fan Σ , it is obvious (e.g. using the cellular description) that theonly non-zero homology groups are H p, ( Σ ) = F p ( Σ ) . (19)Hence the right hand side equality is clear. Moreover, it was shown in [Zha13]that (cid:77) p F p ( Σ M ) (cid:27) (cid:77) p OS p ( M ) , (20)where the right hand side denotes the Orlik-Solomon algebra of M . Finally, bye.g. [OT92] ( − n β ( M ) = (cid:88) p ( − p dim OS p ( M ) , (21)and hence the claim follows. (cid:4) We call x ∈ X a vertex of X if the lineality space of Star X ( x ) is { } . Since eachchart contains at most one vertex, the number of vertices of X is finite. The setof vertices is denoted by Vert( X ). We can now restate Theorem 1.3 (equivalently,Conjecture 1.2 for ψ = id) in the following refined form. Theorem 4.6 (Global tropical Poincaré-Hopf theorem)
The tropical Euler char-acteristic of a smooth tropical variety without points of higher sedentarity is equalto χ ( X ) = (cid:88) x ∈ Vert( X ) χ ( x ) = deg ∆ . roof It follows from Theorem 1.1, Lemma 4.5 and the locality of the tropicalintersection product [FR13, Section 6] that ∆ = (cid:88) x ∈ Vert( X ) χ ( x ) · x (under the natural isomorphism ∆ (cid:27) X ). This proves the right hand side equality.To prove the left hand side, we will use the cosheaf versions F p of the framinggroups F p to compute H p,q ( X ) [MZ14, Section 2.4]. We call an open subset W of a fan Σ ⊂ R n convex if it the intersection of a convex set in R n with Σ . Iffurthermore 0 ∈ W , then then H ( W , F p ) = F p ( Σ ) and H q ( W , F p ) = 0 for q > X , we may assume that the U i and all theconnected components of their intersections U I = (cid:84) i ∈ I U i are convex after beingmapped by one of the φ i . Since ( U , F p ) is acyclic, we can compute H ∗ ,q ( X ) as the Č ech homology of ( U , F p ). Let C p,q ( U ) = (cid:77) | I | = q F p ( U I )a piece of the associated Č ech complex. Since these pieces are finite-dimensional,by the Hopf trace lemma [GD03, §9, Theorem 2.1] we get χ ( X ) = (cid:88) p,q ( − p + q dim C p,q ( U ) . Consider a convex set W which does not contain a vertex. It means that theunderlying fan Σ has non-trivial lineality space, which implies that n (cid:88) p =0 ( − p F p ( W ) = n (cid:88) p =0 ( − p F p ( Σ ) = 0 . Indeed, this can be checked either by a simple direct argument or by invokingTheorem 1.1 again (note that Σ M has lineality space if and only if M is discon-nected if and only if β ( M ) = 0, e.g. [FR13, Lemma 2.3]). Finally, without lossof generality we may assume that each vertex x is contained in exactly one U i ,denoted U x . It then follows (cid:88) p,q ( − p + q dim C p,q ( U ) = (cid:88) x ∈ Vert( X ) (cid:88) p ( − p dim F p ( U x ) = (cid:88) x ∈ Vert( X ) χ ( x ) . This proves the left hand side equality. (cid:4) emark 4.7 In the proof, we used that χ ( x ) (cid:44) x is a vertex. Wecould hence write χ ( X ) = (cid:82) X χ ( x ) dx . For general tropical spaces (not necessarilysmooth) at least the only if direction holds. Note also that csm n ( X ) := (cid:80) x χ ( x ) x is the natural extension of CSM classes [MRS20] from matroid fans to smoothtropical varieties.It is interesting to note that the tropical Poincaré-Hopf theorem can be lo-calised at the vertices of X (the stable fixed points of ψ = id). On a technicallevel, this is of course related to the fact that the tropical intersection product isdefined on the cycle level without the need to pass to rational equivalence. In this subsection, we prove Theorem 1.4. In honour of Weil’s formula foralgebraic curves, we call this special case the tropical Weil trace formula.Throughout this section, C denotes a connected smooth tropical curve. Ad-ditionally to the charts described previously, we also allow points of highersedentarity with local model −∞ ∈ T = R ∪ {−∞} . We denote by Vert( C ) the setof vertices including the subset Vert ∞ ( C ) of points of higher sedentarity. We call C m := C \ Vert ∞ ( C ) the mobile part of C . Note that C is irreducible in the sensethat the group of 1-dimensional tropical subcycles is Z ( C ) = Z C (cid:27) Z . Definition 4.8
Let f : C → D be a proper tropical morphism of connectedsmooth tropical curves C and D . The degree of ψ is the integer deg( ψ ) ∈ N ∪ { } such that ψ ∗ ( C ) = deg( ψ ) · D .An open edge of C is a connected component e of C \ Vert( C ). Its local degree deg e ( ψ ) is the absolute value of the local stretching factor dψ x ∈ Z , x ∈ e .Here, we refer to the pushforward of tropical cycles defined for example in[AR10]. Since we are only interested in endomorphisms, the following lemmafocuses on this case. For l ∈ R > , we denote by S l = R /l Z the tropical curveconsisting of a circle of length l . Lemma 4.9
Let f : C → C be a proper tropical endomorphism of a connected smoothtropical curve C . Then the following holds.(a) If deg( ψ ) = 0 , ψ is constant.(b) If deg( ψ ) > , ψ is surjective and e (cid:55)→ ψ ( e ) is a bijection on the set of open edges.(c) If X (cid:29) S l , then deg e ( ψ ) = deg( ψ ) for any open edge e .(d) If deg( ψ ) = 1 , ψ is an automorphism.(e) If deg( ψ ) ≥ , then either C (cid:27) S l , or X m (cid:27) ( −∞ , or X m (cid:27) Σ M , where M is aloopfree matroid of rank . roof By definition of the pushforward ψ ∗ ( C ) [AR10, Construction 7.3], thedegree can computed at generic points of C by counting preimages x with(positive) weights | dψ x | . Since ψ ( C ) ⊂ C is a connected subgraph, this proves (a)and (since ψ is proper) the first part of (b). The set of open edges is finite andthe image ψ ( e ) of an open edge is either a point or contained in an open edge(balancing condition). Hence, surjectivity implies that in fact ψ ( e ) is equal to anopen edge and that this assignment is bijective.An open edge is isometric to one the following three models: (0 , l ) , l ∈ R > ,(0 , + ∞ ), R or S l . The restriction ψ | e : e → ψ ( e ) is an a ffi ne, surjective map in thefirst two cases and therefore bijective. By the previous remarks, statement (c)follows.If deg( ψ ) = 1, ψ is invertible (over Z ) on C \ Vert( C ) by (c) and it is clearthat this can be extended to the vertices (note that it follows from the previousstatements that ψ also induces a bijection on Vert ∞ ( C )). This proves (d).For (e), note that an open edges of isometry type (0 , l ) must be mapped to anedge of type (0 , deg e ( ψ ) l ). Hence, using (c), such edges cannot exist if deg( ψ ) ≥ (cid:4) Remark 4.10
We say C is of finite type if every chart ( U , φ ) can be extended toa chart ( U (cid:48) , φ (cid:48) ), U ⊂ U (cid:48) , such that φ ( U ) ⊂ φ (cid:48) ( U ) (the closure is taken in R N )[MR19, Definition 6.1.14]. For curves, this equivalent to the requirement thatfor any open edge e isometric to (0 , l ) or (0 , + ∞ ) the limit in C for x → x → l in the first case). For such curves, one can showthat any tropical endomorphism ψ : C → C is either constant or proper (andhence surjective). Indeed, if ψ is non-constant, one can show that ψ ( C ) is open(using local irreducibility) and that C m ⊂ ψ ( C ) (using finite type). Hence, again, e (cid:55)→ ψ ( e ) defines a bijection of open edges which can be restricted to those edgeswhose closure contains a point of higher sedentarity. This implies surjectivityand properness.Next, we discuss the local cases of computing Γ ψ · ∆ . Let M be a loopfreematroid of rank 2 and let ψ : Σ M → Σ M be a proper (i.e. non-constant) tropicalendomorphism. Without loss of generality, we may restrict the ambient space R N to the span of Σ M and, equivalently, assume that all rank 1 flats are single-tons. Under this assumption, the permutation of rays of Σ M under ψ inducesa bijection ψ (cid:48) : E → E . We denote by fix( ψ (cid:48) ) = ψ (cid:48) ) the number of elementsfixed by ψ (cid:48) . Lemma 4.11
Let M be a loopfree matroid of rank and let ψ : Σ M → Σ M be a propertropical endomorphism such that ψ (0) = 0 . Then deg( Γ ψ · ∆ ) = deg( ψ ) + 1 − fix( ψ (cid:48) ) . (22)19 roof We set d := deg( ψ ). According to Proposition 2.6, we can compute Γ ψ · ∆ as γ ∗ ( g ) · Σ M , where g is the function from Equation 12 and γ : x (cid:55)→ ( x, ψ ( x )). Theimage of the primitive generator v { i } under γ in terms of primitive generatorsfor Σ M ⊕ M is γ ( v { i } ) = v ( { i } , { ψ (cid:48) ( i ) } ) + ( d − v ( ∅ , { ψ (cid:48) ( i ) } ) i (cid:44) (cid:44) ψ (cid:48) ( i ) ,v ( { } , { } ) + ( d − v ( E, { } ) i = 0 = ψ (cid:48) ( i ) ,v ( { } ,E ) + dv ( ∅ , { ψ (cid:48) ( i ) } ) i = 0 (cid:44) ψ (cid:48) ( i ) ,v ( { i } , ∅ ) + dv ( E, { } ) i (cid:44) ψ (cid:48) ( i ) . It follows that γ ∗ ( g )( v { i } ) = − i = ψ (cid:48) ( i ) (cid:44) , i (cid:44) (cid:44) ψ (cid:48) ( i ) (cid:44) i, + d i = 0 = ψ (cid:48) ( i ) , +1 i = 0 (cid:44) ψ (cid:48) ( i ) , + d i (cid:44) ψ (cid:48) ( i ) . Since deg( Γ ψ · ∆ ) = (cid:80) i ∈ E γ ∗ ( g )( v { i } ), the claim follows. (cid:4) In the presence of points of higher sedentarity, we also have to compute thecontribution of such points to Γ ψ · ∆ . To do so, we use the extension of theintersection product to points of higher sedentarity for smooth tropical surfaces [Sha15] (see also [MR19]). The only extra ingredient here is the assignment of anintersection multiplicity for two lines in T meeting in ( −∞ , −∞ ). Given the prim-itive generators ( a , a ), ( b , b ) ∈ N , this multiplicity is set to be min { a b , a b } [MR19, Definition 3.5.1]. Lemma 4.12
Let ψ : [ −∞ , → [ −∞ , be a proper tropical endomorphism. Then deg( Γ ψ · ∆ ) = 1 .Proof Any such ψ is of the form x (cid:55)→ dx with d = deg( ψ ). The two cycles ∆ and Γ ψ are two rays in [ −∞ , with primitive direction vectors (1 ,
1) and (1 , d ). Theirintersection is equal to the point ( −∞ , −∞ ) with multiplicity min { · d, · } = 1. (cid:4) Remark 4.13
Note that in the situation of the lemma the only non-zero Borel-Moore homology group is H BM1 , ( X ) = R and Tr( ψ ∗ , H BM1 , ( X )) = d . The discrepancyon the trace side gets corrected if we extend ψ to a map T → T , since now thereis an extra fixed point 0 with intersection multiplicity d − C m (cid:27) ( −∞ ,
0) and deg( ψ ) > heorem 4.14 (Tropical Weil trace formula) Let ψ : C → C be a tropical endo-morphism of a connected smooth tropical curve C such that C m (cid:29) ( −∞ , or deg( ψ ) =1 . Then we have deg( Γ ψ · ∆ ) = (cid:88) p,q ( − p + q Tr( ψ ∗ , H BM p,q ( C )) . (23) Proof
Let us first deal with a few special cases. If deg( ψ ) = 0, then by part (a) ofLemma 4.9, ψ ≡ c is constant and deg( Γ ψ · ∆ ) = deg( C · { c } ) = 1. So, Equation 23holds true after replacing H BM p,q with H p,q . Since ψ is proper, C is compact whichimplies H BM p,q = H p,q .The case X (cid:27) S l is covered by Theorem 4.18, so we exclude this case here.Let us now assume deg( ψ ) ≥
2. By Lemma 4.9 (e) and the exclusions madeso far, this implies C m = Σ M where M is a loopfree matroid of rank 2. Notethat we can assume ψ (0) = 0, since even in the case C m = R there exists afixed point in C m , given that deg( ψ ) ≥
2. We use the same assumptions andnotation as in Lemma 4.11. Additionally, we denote by fix( ψ ∞ ) the numberof points in Vert ∞ ( C ) fixed by ψ . Lemma 4.11 and Lemma 4.12 imply thatdeg( Γ ψ · ∆ ) = deg( ψ ) + 1 − fix( ψ (cid:48) ) + fix( ψ ∞ ).To compute the trace side, first note that H BM1 , ( C ) = 0, H BM1 , ( C ) = Z C andTr( ψ ∗ , H BM1 , ( C )) = deg( ψ ). For p = 0, we again use the Hopf trace lemma [GD03,§9, Theorem 2.1] in order to compute the trace on the level of simplicialchain complexes (with respect to the obvious decomposition into rays and ver-tices). We get C BM0 , ( C ) = R E → C BM0 , ( C ) = R Vert( C ) , Tr( ψ ∗ , C BM0 , ( C )) = fix( ψ (cid:48) ) andTr( ψ ∗ , C BM0 , ( C )) = 1 + fix( ψ ∞ ). In total, we see that this agrees with deg( Γ ψ · ∆ ).We are left with the case deg( ψ ) = 1, so ψ is an automorphism by Lemma 4.9(d). By subdividing the open edges isometric to (0 , l ) or R which get flipped by ψ , we obtain a simplicial structure for C such that ψ maps cells to cells and theorientation of fixed cells is preserved. In particular, the locus of fixed points of ψ can be written as a union of such cells. By abuse of terminology, for the rest ofthis argument we call the cells of this decomposition the vertices and edges of C . Let us denote by ψ and ψ the induced bijections on the vertices and edges,respectively. Moreover, for any x ∈ Fix( ψ ) we denote by ψ (cid:48) x the permutation ofthe edges of C emanating from x . Using Lemma 4.11 and Lemma 4.12 again, weget deg( Γ ψ · ∆ ) = fix( ψ ∞ ) + (cid:88) x ∈ Fix( ψ ) x (cid:60) Vert ∞ ( C ) − fix( ψ (cid:48) x ) = (cid:88) x ∈ Fix( ψ ) − fix( ψ (cid:48) x ) . As before, we compute the trace side on the level of the simplicial chain complex.21etting Tr p,q := Tr( ψ ∗ , C BM p,q ( C )), we getTr , = fix( ψ ) , Tr , = (cid:88) x ∈ Fix( ψ ) fix( ψ (cid:48) x ) − , Tr , = fix( ψ ) , Tr , = fix( ψ ) . Indeed, the cases p = 0 are obvious. In the cases p = 1, we use deg( ψ ) = 1.Moreover, considering Tr , , the contribution of each x ∈ Fix( ψ ) \ Vert ∞ ( C ) isequal to Tr( dψ x , F ( x )). With the assumption from Lemma 4.11, we can resolve F ( x ) by 0 → R1 (cid:44) → R E → F ( x ) → . It follows that Tr( dψ x , F ( x )) = fix( ψ (cid:48) x ) −
1. Finally, a point x ∈ Fix( ψ ) ∩ Vert ∞ ( C )contributes zero to Tr , since F ( x ) = 0 in this case. We can now conclude that (cid:88) p,q ( − p + q Tr( ψ ∗ , H BM p,q ( C )) = fix( ψ ) − (cid:88) x ∈ Fix( ψ ) (fix( ψ (cid:48) x ) −
1) = (cid:88) x ∈ Fix( ψ ) − fix( ψ (cid:48) x ) . This proves the claim. (cid:4)
Remark 4.15
In fact, in the case deg( ψ ) ≤ ψ constant or automor-phism, even without the properness assumption), our proof works equally wellwith usual homology H p,q ( C ). Indeed, the only change required is that nowTr , = Tr , = fix( ψ b ) only counts compact fixed edges. Example 4.16
Let X be a tropical curve whose underlying graph is the Θ -graph G and with vertices v and v (see Figure 3). Its homology groups can be easilycalculated as H , ( X ) = Z , H , ( X ) = F ( v ) (cid:27) Z , H , ( X ) = H ( G ) (cid:27) Z and H , ( X ) = Z · X = Z .Let ψ : X → X be the automorphisms which exchanges v and v and flipsevery edge. Then Γ ψ · ∆ consists of the midpoints of the three edges, and each ofthem occurs with intersection multiplicity 2 by Lemma 4.11. On the trace side, ψ induces id for ( p, q ) = (0 ,
0) and (1 ,
1) and − id for (1 ,
0) and (0 , − ( − − ( −
2) + 1 = 6 as well.Assume now that two of the edges of X have the same length. Then thereexists an automorphism ψ : X → X which exchanges the two edges (but keepsthe vertices and the third edge fixed). The set-theoretic fixed point locus of ψ consists of the third edge, but only the vertices are stable fixed points. Eachvertex has intersection multiplicity 1 in Γ ψ · ∆ . The pushforward is still idfor (0 ,
0) and (1 , ,
0) and (0 , ψ ) ∗ permutes the basis elements. Hence these traces are zero and we get1 + 0 + 0 + 1 = 2. 22 2 2 11 Figure 3
The two endomorphisms ψ and ψ and their stable fixed points withmultiplicities Let Λ ⊂ R n be a lattice in R n (i.e. a discrete free abelian subgroup). The quotient X = R n / Λ is called a tropical torus . Note that the “tropical” (here, integral-a ffi ne)structure on X is induced by the lattice Z n ⊂ R n . For more information ontropical tori, we refer to [MZ08]. We denote by Mat( Λ ) the n × n matrices withreal entries such that A Λ ⊂ Λ . Lemma 4.17
Let X = R n / Λ be a tropical torus and ψ : X → X a tropical endomor-phism. Then ψ is of the form x (cid:55)→ Ax + v mod Λ for some matrix A ∈ Mat( Z n ) ∩ Mat( Λ ) and v ∈ R n . Moreover, A is uniquely deter-mined by ψ .Proof For any x ∈ X , we can canonically identify T x X = R n with lattice of integertangent vectors T Z x X = Z n . By definition of tropical morphisms (see e.g. [MR19]),the di ff erential map x (cid:55)→ dψ x : T x X → T x X is locally constant with values inMat( Z n ). Hence, it is globally constant, and setting A = dψ x and v such that[ v ] = ψ (0), we see that ψ the required form. The compatibility with Λ implies A ∈ Mat( Λ ). (cid:4) Theorem 4.18
Let X = R n / Λ be a tropical torus and ψ : X → X a tropical endomor-phism with di ff erential dψ = A ∈ Mat( Z n ) ∩ Mat( Λ ) . Then deg( Γ ψ · ∆ ) = det(id − A ) = (cid:88) p,q ( − p + q Tr( ψ ∗ , H p,q ( X )) . (24) Proof
Let us first do the linear algebra behind the statement. Let χ ( t ) = det(id − tA )23e the characteristic polynomial of A (for s = 1 /t ). By standard expansion ofdeterminants, the coe ffi cient of ( − t ) k in χ is equal to the sum of minors det( A I )of size k . Here, A I denotes the diagonal submatrix of A with rows and columnsgiven by I ⊂ { , . . . , n } . On the other hand, consider the map A ∧ k : (cid:86) k R n → (cid:86) k R n induced by A . Clearly, its trace is also equal to the sum of minors det( A I ) ofsize k . (A fancier way of saying the same thing is that χ ( t ) is equal to the graded trace (i.e. graded pieces are counted with alternating signs) of the map tA ∧∗ : (cid:86) ∗ R n → (cid:86) ∗ R n induced by tA on the exterior algebra.)Now, since the framing groups F p ( x ) are constant on tropical tori, the ( p, q )groups are equal to H p,q ( X ) = F p ([0]) ⊗ H q ( X, R ) = p (cid:94) R n ⊗ q (cid:94) R n . Moreover, under this identification, ψ ∗ is equal to A ∧∗ ⊗ A ∧∗ . By, the computationfrom above the graded trace of ψ ∗ is hence equal toTr( ψ ∗ ) = Tr( A ∧∗ ) = det(id − tA ) , which proves the right hand side of Equation 24.It remains to check the left hand side. Consider the subspaces D = { ( x, x ) } and G = { x, Ax } of R n . Clearly, ∆ and Γ ψ are translations of the projections of D and G , respectively, modulo Λ × Λ . Hence, for any fixed point of ψ we can computeits contribution to Γ ψ · ∆ as the tropical intersection multiplicity of G and D . Thismultiplicity can be computed by combining lattice bases of G and D in a matrixand taking the absolute value of its determinant (cf. [Rau16; MR19]). In our case,we get the absolute value ofdet (cid:32) id id A id (cid:33) = det (cid:32) id 0 A id − A (cid:33) = det(id − A ) . If det(id − A ) = 0, this implies Γ ψ · ∆ = 0 since all intersection multiplicities arezero. Now assume det(id − A ) (cid:44)
0. In this case, the equation (id − A ) x = v hasexactly | det(id − A ) | many solutions modulo Λ (since | det(id − A ) | also computesthe degree of the map id − A : R n / Λ → R n / Λ ). Hence ψ has exactly | det(id − A ) | fixed points, and each contributes | det(id − A ) | to deg( Γ ψ · ∆ ), which proves theleft half of Equation 24. (cid:4) Remark 4.19
Note that the tropical formula is essentially a product of the classical trace formula for ψ (considered as map between manifolds) with the tropicalformula for the map A : R n → R n . Indeed, from the classical point of view, eachof the | det(id − A ) | fixed points intersect transversally, and the homology groups24o consider are H ∗ ( X, R ) = (cid:86) ∗ R n . The first part of the proof is then a proof of theclassical version (when using the correct signs). On the other hand, the tropicalhomology groups for R n are H ∗ , ∗ ( R n ) = F ∗ ( R n ) = (cid:86) ∗ R n as well. Again, the correctpieces of the previous proof also prove the trace formula for A : R n → R n . Finally,combining the two parts gives the “squared” version of Theorem 4.18. References [AHK18] Karim Adiprasito, June Huh, and Eric Katz.
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