METRIC PROPERTIES OF THE TROPICAL ABEL-JACOBI MAP 1. Introduction In

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METRIC PROPERTIES OF THE TROPICAL ABEL-JACOBI MAP
MATTHEWBAKERANDXANDERFABER
Abstract. LetΓbeatropicalcurve(ormetricgraph), andfixabasepoint
p ∈ Γ. We define the Jacobian group J(G) of a finite weighted graph G,
andshowthattheJacobianJ(Γ)iscanonicallyisomorphictothedirectlimit
of J(G) over all weighted graph models G for Γ. This result is useful for
reducingcertainquestionsabouttheAbel-JacobimapΦp:Γ→J(Γ),defined
byMikhalkinandZharkov,topurelycombinatorialquestionsaboutweighted
graphs. WeprovethatJ(G)isfiniteifandonlyiftheedgesineach2-connected
component of G are commensurable over Q. As an application of our direct
limittheorem,wederivesomelocalcomparisonformulasbetweenρandΦ∗(ρ)
p
forthreedifferentnatural“metrics”ρonJ(Γ). Oneoftheseformulasimplies
that Φp is a tropical isometry when Γ is 2-edge-connected. Another shows
that the canonical measure µ on a metric graph Γ, defined by S. Zhang,
Zh
measureslengthsonΦp(Γ)withrespecttothe“sup-norm”onJ(Γ).
1. Introduction
In [1, 3, 5, 13, 14] and other works, combinatorial analogues of classical facts
about Riemann surfaces and their Jacobians are explored in the context of graphs
and tropical curves. (For the purposes of this article, a tropical curve is the same
thing as a compact metric graph of finite total length.) The Jacobian of a finite
(unweighted) graph G, as defined in [1], is a certain finite Abelian group whose
cardinality is the number of spanning trees in G. On the other hand, the Jacobian
of a tropical curve Γ of genus g is a real torus of dimension g. In both cases, given
a base point p, there is a natural map from G (resp. Γ) to its Jacobian, called the
Abel-Jacobi map, which is injective if and only if the graph (resp. tropical curve)
contains no bridges. (A bridge is an edge that is not contained in any non-trivial
cycle.) These are combinatorial counterparts of the classical fact that the Abel-
JacobimapfromaRiemannsurfaceX toitsJacobianisinjectiveifandonlyifthe
genus of X is at least 1. In the present paper, we explore some additional parallels
betweenthesealgebro-geometricandcombinatorialtheories,andpresentsomenew
combinatorial results with no obvious analogues in algebraic geometry.
The first part of the paper is devoted to constructing a rigorous framework for
understanding tropical curves and their Jacobians via models (cf. [2]). A model for
a tropical curve Γ is just a weighted graph G whose geometric realization is Γ. In
order to work with tropical curves and their Jacobians from this point of view, one
needs to be able to pass freely between different models for Γ, so it is desirable to
set up a general theory of Jacobians for weighted graphs which corresponds to the
usual notion of J(G) from [1, 3] when all edge lengths are equal to 1. Essentially
2000 Mathematics Subject Classification. 14H40(primary);05C50,05C38(secondary).
Keywordsandphrases. TropicalCurve,TropicalJacobian,PicardGroup,Abel-Jacobi,Metric
Graph,Foster’sTheorem.
1
2 MATTHEWBAKERANDXANDERFABER
every desired consequence works out exactly as one would hope: for each weighted
graphGthereisacanonicalisomorphismJ(G)∼=Pic0(G)(generalizingthegraph-
theoretic version of Abel’s theorem from [1]), and J(Γ) is canonically the direct
limit of J(G) over all models G for Γ. This allows us to give a very simple proof of
thetropicalversionofAbel’stheorem,firstprovedin[14]. Ourdirectlimitpointof
view — thinking of tropical curves as limits of models — reduces various questions
abouttropicalcurvesandtheirJacobianstoquestionsaboutfiniteweightedgraphs.
This discussion will occupy sections 2–4.
The Jacobian of a weighted graph G is sometimes finite and sometimes infinite;
it is natural to wonder when each case occurs. We provide a complete answer
to this question in §2.4: J(G) is finite if and only if the edges in each 2-connected
componentofGarecommensurableoverQ. Theproofisanapplicationofpotential
theory on metric graphs (cf. [2, 4]). At the other extreme, if the lengths of the
edges in each maximal 2-connected component are Q-linearly independent, then
the Jacobian group is free Abelian of rank #V(G)−#Br(G)−1, where V(G)
and Br(G) are the sets of vertices and bridges of G, respectively. If one fixes the
combinatorial type of G (i.e., the underlying unweighted graph), then it seems like
a difficult question to describe the possible group structures for J(G) as one varies
the edge lengths.
Theoriginalmotivationforthepresentworkwastobetterunderstandthecanon-
ical measure µ on a metric graph Γ of genus at least 1 defined by Zhang in [15].
Zh
The measure µ plays a role in Zhang’s theory analogous to the role played in
Zh
Arakelov theory by the canonical (1,1)-form ω on a Riemann surface X of genus
X
at least 1. (For the definition, see §6.3.) One of the most important descriptions
of the (1,1)-form ω is that it is obtained by pulling back the flat Riemannian
X
metric on the Jacobian J(X) under the Abel-Jacobi map Φ : X (cid:44)→ J(X), for
p
any choice of base point p ∈ X. It is natural to wonder whether Zhang’s measure
has a similar description in terms of the Abel-Jacobi map Φ : Γ → J(Γ) from a
p
tropicalcurveΓtoitsJacobian. Althoughthesituationdoesnotappeartobefully
analogous to the classical theory, we provide such a description in this paper: the
measure µ can be obtained by pulling back a canonical metric on J(Γ) which we
Zh
call the “sup-norm” or “Foster/Zhang” metric. More precisely, if e is an edge of
some model for Γ, then the length of Φ (e) with respect to the sup-norm metric is
p
µ (e). This gives a quantitative version of the fact that the edges of Γ contracted
Zh
to a point by Φ are precisely the bridges (which by Zhang’s explicit formula for
p
µ are exactly the segments of Γ to which µ assigns no mass). This analogy is
Zh Zh
described in greater detail in §6.3.
There are at least three natural “metrics” on J(Γ) for which one can explicitly
computethelengthofΦ (e)onJ(Γ)intermsofthelength(cid:96)(e)ofanedgeeinsome
p
model for Γ. In addition to the sup-norm metric, one can also define a “Euclidean
metric” on J(Γ), and we prove that the length of Φ (e) in the Euclidean metric is
p
(cid:112)
(cid:96)(e)µ (e). Althoughthisformulaisnotascleanastheformulaforthesup-norm,
Zh
it is striking that there is a simple answer in both cases. A third metric on J(Γ) is
the “tropical metric”: we prove that away from the bridge edges, Φ is a tropical
p
isometry from Γ onto its image. This result is the most natural of our three metric
comparisontheoremsfromthepointofviewoftropicalgeometry,whereasoursup-
norm theorem is arguably the most relevant one from the Arakelov-theoretic point
of view. We define and discuss these metric structures in §5.
METRIC PROPERTIES OF THE TROPICAL ABEL-JACOBI MAP 3
In the final section, we interpret the numbers F(e) = µ (e) (which we call
Zh
the Foster coefficients, after R. Foster [11]) in terms of electrical network theory,
orthogonal projections, weighted spanning trees, and random walks on graphs.
Unlike some authors, we have chosen in this paper to consider only tropical
curves with finite total length. This has no real effect on the generality of our
discussion of Jacobians; because infinite-length segments do not support harmonic
1-forms, they play no role in the construction of the Jacobian and are collapsed
under the Abel-Jacobi map.
Acknowledgments. Duringthewritingofthisarticle,thefirstauthorwassup-
portedinpartbyNSFgrantDMS-0600027,andthesecondauthorbytheCentrede
Recherches Math´ematiques and the Institut des Sciences Math´ematiques in Mon-
treal. TheauthorswouldliketothankFarbodShokriehforhelpfulfeedbackonthis
work. They also thank the anonymous referees for many thoughtful suggestions.
2. The Weighted Jacobian
The goal of this section is to define and investigate the Jacobian group of a
weighted graph. We prove the Jacobian is canonically isomorphic to the Picard
group, and that Jacobian groups behave well with respect to length-preserving
subdivision of edges. We also determine exactly when the Jacobian of a weighted
graph is a finite group.
2.1. Weighted graphs. A weighted graph G in this paper will be an edge-
weighted, connected multigraph, possibly with loop edges, endowed with a fixed
orientation of the edges. (Most of our constructions, including the Jacobian and
Picard group of G, are independent of the choice of orientation.) More precisely,
G is given by specifying a nonempty vertex set V(G), an edge set E(G), a length
map1(cid:96):E(G)→R ,andanedgeassignmentmapι:E(G)→V(G)×V(G)such
>0
thatforanypairofdistinctverticesxandx(cid:48),thereexistsasequenceofverticesx=
x ,x ,...,x = x(cid:48) and a sequence of edges e ,...,e such that ι(e ) = (x ,x )
0 1 n 1 n i i−1 i
or ι(e ) = (x ,x ). For an edge e, the tail vertex e− and the head vertex e+
i i i−1
are defined by ι(e)=(e−,e+). Two vertices x and y are adjacent, written x∼y,
if there is an edge e such that ι(e)=(x,y) or ι(e)=(y,x). In either case, we write
x∈e to indicate that x is one of the vertices of e.
LetAbeeithertheringofintegersZorthefieldR. ThefreeA-modulegenerated
by the vertex set V(G) (resp. the edge set E(G)) is called the module of 0-
chains (resp. 1-chains) with coefficients in A, and is denoted by C (G,A)
0
(resp. C (G,A)). Note that C (G,Z) ⊂ C (G,R) for j = 0,1. Since C (G,A)
1 j j 0
is canonically isomorphic to its dual Hom(C (G,A),A), we may identify a 0-chain
0
(cid:80)
f = n .x with the function f :V(G)→A given by f(x)=n . A similar
x∈V(G) x x
remark applies to 1-chains.
Some authors follow a more canonical approach, replacing each edge e with two
edgeseande¯correspondingtoanedgewithtwopossibleorientations. A1-chainis
then a function on the edge space such that f(e¯)=−f(e). We have chosen instead
tofixanorientation, intheinterestofmakingintegrationalong1-chainslookmore
like the standard treatment in Riemannian geometry.
1Inourgeometricstudyofgraphswehavechosentouselengthsratherthanthemorestandard
notionofweightsw:E(G)→R>0 definedbyw(e)=(cid:96)(e)−1.
4 MATTHEWBAKERANDXANDERFABER
We may define inner products on C (G,R) and C (G,R) by
0 1
(cid:88)
(cid:104)f ,f (cid:105) = f (x)f (x), f ,f ∈C (G,R)
1 2 1 2 1 2 0
x∈V(G)
(cid:88)
(cid:104)α ,α (cid:105) = α (e)α (e)(cid:96)(e), α ,α ∈C (G,R).
1 2 1 2 1 2 1
e∈E(G)
Thedifferentialoperatord:C (G,R)→C (G,R)appliedtoa0-chainf ∈C (G,R)
0 1 0
gives the “slope of f along the edge e”:
f(e+)−f(e−)
(df)(e)= .
(cid:96)(e)
The adjoint operator d∗ :C (G,R)→C (G,R) acts on a 1-chain α by
1 0
(cid:88) (cid:88)
(d∗α)(x)= α(e)− α(e).
e∈E(G) e∈E(G)
e+=x e−=x
Withthesedefinitions, oneimmediatelychecksthatthematrixof∆=d∗drelative
to the basis of C (G,R) given by the vertices of G is equal to the usual weighted
0
Laplacian matrix of G. (See [2, §5] for the definition.)
We define H (G,R) = ker(d∗) and H (G,Z) = ker(d∗)∩C (G,Z). These will
1 1 1
be called the real (resp. integral) 1-cycles. By general linear algebra, we get a
canonical orthogonal “Hodge decomposition”
C (G,R)=ker(d∗)⊕im(d)=H (G,R)⊕im(d). (2.1)
1 1
A 1-form on G is an element of the real vector space with formal basis {de :
(cid:80)
e∈E(G)}. A 1-form ω = ω .de is harmonic if
e
(cid:88) (cid:88)
ω = ω for all x∈V(G). (2.2)
e e
e∈E(G) e∈E(G)
e+=x e−=x
Write Ω(G) for the space of harmonic 1-forms. Define integration of the basic
1-form de along an edge e(cid:48) by
(cid:40)
(cid:90) (cid:96)(e) if e=e(cid:48)
de=
0 if e(cid:54)=e(cid:48).
e(cid:48)
By linearity, we can extend this definition to obtain an integration pairing:
Ω(G)×C (G,R) −→ R
1
(cid:90)
(ω,α) (cid:55)→ ω.
α
Lemma 2.1. The kernel on the left of the integration pairing is trivial, while the
kernelontherightisim(d). Inparticular, integrationrestrictedtoΩ(G)×H (G,R)
1
gives a perfect pairing — i.e., an isomorphism H (G,R)→∼ Ω(G)∗.
1
Proof. Foreache∈E(G),lets ∈Rbearealnumber. Itfollowsimmediatelyfrom
e
(cid:80)
the definitions that the 1-form ω = s .de is harmonic if and only if the 1-chain
e
α = (cid:80)s .e is a 1-cycle. As (cid:82) ω = (cid:80)s2(cid:96)(e) = 0 if and only if s = 0 for all e, we
e α e e
find that the integration pairing restricted to Ω(G)×H (G,R) is perfect.
1
METRIC PROPERTIES OF THE TROPICAL ABEL-JACOBI MAP 5
(cid:82)
To finish the proof, it suffices by (2.1) to show that ω = 0 for all 0-chains
df
f ∈C (G,R) and all ω ∈Ω(G). If we write ω =(cid:80)ω .de, then
0 e
(cid:90) ω = (cid:88) (cid:88) f(e+)−f(e−) ω (cid:90) de(cid:48) = (cid:88) (cid:2)f(e+)−f(e−)(cid:3)ω
(cid:96)(e) e(cid:48) e
df e∈E(G)e(cid:48)∈E(G) e e∈E(G)
 
(cid:88)  (cid:88) (cid:88) 
= f(x) ωe− ωe.
 
x∈V(G) e∈E(G) e∈E(G)
e+=x e−=x
The final expression vanishes by the definition of harmonic form. (cid:3)
For each 1-chain α ∈ C (G,R), we can define an integration functional (cid:82) :
1 α
Ω(G) → R as above. Lemma 2.1 shows every element of Ω(G)∗ arises in this way.
Consider the following subgroup given by integration on integral 1-chains:
(cid:26)(cid:90) (cid:27)
Ω(G)(cid:93) = ∈Ω(G)∗ :α∈C (G,Z) .
1
α
AgainreferringtoLemma2.1,wemayidentifythegroupofintegralcyclesH (G,Z)
1
with a subgroup of Ω(G)(cid:93) via α(cid:55)→(cid:82) .
α
Definition 2.2. The Jacobian of a weighted graph G is given by
J(G)=Ω(G)(cid:93)/H (G,Z).
1
From the construction of J(G) one sees that it is a finitely generated Abelian
(cid:82)
group. A canonical set of generators is given by functionals of the form for
e
e∈E(G).
Remark 2.3. There are several other definitions of the Jacobian group in the liter-
ature that deserve comment:
(1) AdefinitionofJacobiangroupforunweightedgraphswasgivenin[1],which
establishes several basic properties of J(G), including the Abel-Jacobi iso-
morphism. Inthefinalsection,theauthorssuggest“Therewouldbenodif-
ficultytoextendallpreviousconsiderationstographswithpositiveweights
on vertices and edges.” However, their definition of Jacobian group does
not immediately generalize well to the weighted case because H (G,Z) is
1
not an integral lattice. Our definition not only agrees with theirs in the
unweighted case, but it also behaves well with respect to length-preserving
edge subdivisions and passage to the limit over all such subdivisions. Nev-
ertheless, we would like to acknowledge the inspiration gained from [1].
(2) For a weighted graph G, define J(G)R := Ω(G)∗/H1(G,Z) equipped with
the inner product structure arising by duality from Lemma 2.1. The “ja-
cobienne” in [5] and the “Albanese torus” with its “flat structure” in [13]
are both defined to be the real torus H (G,R)/H (G,Z) with the inner
1 1
product defined in §2.1. They are canonically isomorphic to J(G)R, which
contains our Jacobian group J(G) as a subgroup. The “Jacobian torus” in
[13] is dual to the Albanese torus.
Example 2.4. Let G be the weighted graph with two vertices x,y, two edges e ,e
1 2
such that ι(e ) = ι(e ) = (x,y), and (cid:96)(e ) = 1−(cid:96)(e ) = r < 1. Then H (G,Z) =
1 2 1 2 1
6 MATTHEWBAKERANDXANDERFABER
Z.(e − e ). If we identify H (G,R) with Ω∗(G) as in Lemma 2.1, then (cid:82) =
1 2 (cid:82) 1 e1
r.(e −e ) and =(r−1).(e −e ). It follows that
1 2 e2 1 2
(cid:16) (cid:17)(cid:46)
J(G)= Z.r(e −e )+Z.(1−r)(e −e ) Z.(e −e )
1 2 1 2 1 2
(cid:40)
Z if r (cid:54)∈Q,
∼
=
Z/nZ if r =m/n and gcd(m,n)=1.
Example 2.5. Extending the previous example, let G be the weighted graph with
n+1 vertices and n+1 edges arranged in a single directed cycle. Let (cid:96) ,...,(cid:96) be
0 n
(cid:80)
the lengths of the edges, and suppose that (cid:96) =1. Then one can verify that
j
J(G)∼=(Z.(cid:96) +···+Z.(cid:96) )/ Z ⊂ R/Z.
0 n
If (cid:96) ,...,(cid:96) are rational, then J(G) is a (finite) torsion group. At the other ex-
0 n
treme,if(cid:96) ,...,(cid:96) areQ-linearlyindependent,thenJ(G)isfreeAbelianofrankn.
0 n
Compare with Theorems 2.15 and 2.16.
2.2. The Picard Group and the Jacobian. Let G be a weighted graph. Re-
call that we have a homomorphism d : C (G,R) → C (G,R) and its adjoint
0 1
d∗ : C1(G,R) → C0(G,R). Define im(d)Z = im(d)∩C1(G,Z). Elements of im(d)Z
correspond to functions f : V(G) → R with integer slopes, modulo constant func-
tions.
The divisor group of G, denoted Div(G), is defined to be C (G,Z). The
0
(cid:80) (cid:80)
degreeofadivisorD = D(x).xisdeg(D)= D(x). Definethefollowing
x∈V(G)
subgroups of Div(G):
Div0(G) = {D ∈Div(G):deg(D)=0}
Prin(G) = d∗(im(d)Z).
A simple calculation shows that if α∈C (G,R) is any 1-chain, then deg(d∗α)=0.
1
Consequently, we find Prin(G)⊂Div0(G).
Definition 2.6. The (degree zero) Picard Group of a weighted graph G is
given by
Pic0(G)=Div0(G)/Prin(G).
We want to show that the homomorphism d∗ : C (G,Z) → C (G,Z) = Div(G)
1 0
descends to an isomorphism h : J(G) → Pic0(G). To that end, consider the fol-
lowing diagram of homomorphisms (the map h will be discussed in the theorem
METRIC PROPERTIES OF THE TROPICAL ABEL-JACOBI MAP 7
below):
0 0 (2.3)
(cid:15)(cid:15) (cid:15)(cid:15)
0 (cid:47)(cid:47)H1(G,Z) (cid:47)(cid:47)H1(G,Z)⊕im(d)Z d∗ (cid:47)(cid:47)Prin(G) (cid:47)(cid:47)0
(cid:15)(cid:15) (cid:15)(cid:15)
0 (cid:47)(cid:47)H (G,Z) (cid:47)(cid:47)C (G,Z) d∗ (cid:47)(cid:47)Div0(G) (cid:47)(cid:47)0
1 1
(cid:15)(cid:15) (cid:15)(cid:15)
J(G)(cid:95) (cid:95) (cid:95) h(cid:95) (cid:95) (cid:95)(cid:47)(cid:47)Pic0(G)
(cid:15)(cid:15) (cid:15)(cid:15)
0 0
Lemma 2.7. Let G be a weighted graph. Then the diagram (2.3) (without the map
h) is commutative and exact.
Proof. Thisisanexerciseinunwindingthedefinitions. NotethatthemapC (G,Z)→
1
J(G) is defined by α(cid:55)→(cid:82) . (cid:3)
α
Theorem 2.8. Let G be a weighted graph. There exists a unique homomorphism
h : J(G) → Pic0(G) that makes the diagram (2.3) (with the map h) commutative.
Moreover, h:J(G)→Pic0(G) is an isomorphism.
Proof. Apply the Snake Lemma to (2.3). (cid:3)
For unweighted graphs, a different proof of this theorem is given in [1, Prop. 7].
2.3. Compatibility under refinement. Given two weighted graphs G and G ,
1 2
we say that G refines G if there exist an injection a : V(G ) (cid:44)→ V(G ) and a
2 1 1 2
surjection b : E(G ) (cid:16) E(G ) such that for any edge e of G there exist vertices
2 1 1
a(e−) = x ,x ,...,x = a(e+) ∈ V(G ) and edges e ,...,e ∈ E(G ) satisfying
0 1 n 2 1 n 2
(1)b−1(e)={e ,...,e },(2)(cid:80)n (cid:96)(e )=(cid:96)(e),and(3)ι (e )=(x ,x )foreach
1 n i=1 i 2 i i−1 i
i = 1,...n. (Here ι is the associated edge map for G .) See Figure 2.1 below for
2 2
an example. Roughly, one refines a weighted graph by subdividing its edges in a
lengthpreservingfashionandendowingthenewedgeswiththeinducedorientation.
In the interest of readable exposition, we will identify the vertices of G with their
1
images in V(G ) by a, and we will say that the edge e of G is subdivided into
2 1
e ,...,e if b−1(e)={e , ...,e }. Henceforth, we will suppress mention of a and
1 n 1 n
b when we speak of refinements.
If G is a refinement of G , we write G ≤G . This is a partial ordering on the
2 1 1 2
collection of all weighted graphs. There is a canonical refinement homomorphism
r : C (G ,R) → C (G ,R) defined by r (e) = (cid:80)n e if e is an edge of G
21 1 1 1 2 21 i=1 i 1
that is subdivided into e ,...,e . Identifying a 1-chain α ∈ C (G ,R) with the
1 n 1 1
corresponding function on the edges of G , we have r (α)(e ) = α(e) for all i =
1 21 i
1,...,n. IfG ≤G ≤G ,thenitisclearthatwehaver =r r . Ifnoconfusion
1 2 3 31 32 21
will arise, we avoid writing the subscripts on the refinement homomorphisms.
8 MATTHEWBAKERANDXANDERFABER
The refinement homomorphism r : C (G ,R) → C (G ,R) induces a push-
1 1 1 2
forwardhomomorphismr on1-formsdefinedbyr (de)=(cid:80)n de ,inthenotation
∗ ∗ i=1 i
of the last paragraph.
Lemma 2.9. Let G and G be two weighted graphs such that G refines G .
1 2 2 1
(1) The refinement homomorphism r induces an isomorphism of real vector
spacesH (G ,R)→∼ H (G ,R)suchthatr(H (G ,Z))=H (G ,Z). More-
1 1 1 2 1 1 1 2
over, if d :C (G ,R)→C (G ,R) denotes the differential operator on G ,
i 0 i 1 i i
then r(im(d1)Z)⊂im(d2)Z.
∼
(2) The push-forward homomorphism r induces an isomorphism Ω(G ) →
∗ 1
Ω(G ), and it is compatible with the integration pairing in the sense that
2
for each α∈C (G,R) and ω ∈Ω(G ), we have (cid:82) r (ω)=(cid:82) ω.
1 1 r(α) ∗ α
(3) The refinement homomorphism induces an isomorphism Ω(G )∗ →∼ Ω(G )∗
1 2
given by (cid:82) (cid:55)→ (cid:82) for α ∈ H (G ,R). In particular, it induces an iso-
α r(α) 1 1
morphism
Ω(G )∗/H (G ,Z)→∼ Ω(G )∗/H (G ,Z).
1 1 1 2 1 2
Proof. By induction on the number of vertices in the refinement, we may assume
thatG isobtainedfromG bysubdividinganedgee intoe ande . Letι (e )=
2 1 0 1 2 1 0
(x ,x ). Let the new vertex x on G be defined by ι (e )=(x ,x ) and ι (e )=
0 2 1 2 2 1 0 1 2 2
(x ,x ). (See Figure 2.1.)
1 2
e e e
0 1 2
e− e+ x x x
0 0 0 1 2
Figure 2.1. An illustration of the edge e subdivided into edges
0
e ,e , with arrows indicating the agreement of the orientations.
1 2
We now begin the proof of (1). Define d∗ and d∗ to be the adjoints of the
1 2
correspondingdifferentialoperatorsonG andG ,respectively. Letα∈C (G ,R).
1 2 1 1
If x (cid:54)= x ,x ,x , then the definition of (d∗α)(x) and (d∗r(α))(x) doesn’t involve
0 1 2 1 2
e ,e or e . Hence (d∗r(α))(x)=(d∗α)(x). For the remaining vertices, we see that
0 1 2 2 1
(cid:88) (cid:88)
(d∗r(α))(x )= r(α)(e)− r(α)(e) −r(α)(e )
2 0 1
e∈E(G2) e(cid:54)=e1∈E(G2)
e+=x0 e−=x0
(cid:88) (cid:88)
= α(e)− α(e) −α(e )
0 (2.4)
e∈E(G1) e(cid:54)=e0∈E(G1)
e+=x0 e−=x0
=(d∗α)(x ),
1 0
(d∗r(α))(x )=r(α)(e )−r(α)(e )=α(e )−α(e )=0.
2 1 1 2 0 0
The computation for the vertex x , similar to that for x , shows (d∗r(α))(x ) =
2 0 2 2
(d∗α)(x ). As H (G,R) = ker(d∗) for any weighted graph G, these computations
1 2 1
imply that r(H (G ,R))⊂H (G ,R).
1 1 1 2
Fortheoppositeinclusion,takeβ ∈H (G ,R). Sinced∗β(x )=β(e )−β(e )=
1 2 2 1 1 2
0,wemaydefineacycleαonG bysettingα(e )=β(e )=β(e )andα(e)=β(e)
1 0 1 2
METRIC PROPERTIES OF THE TROPICAL ABEL-JACOBI MAP 9
for all other edges e. By construction r(α) = β, hence r(H (G ,R)) = H (G ,R).
1 1 1 2
As r is injective, it induces an isomorphism between cycle spaces as desired.
Fromthefactthatr(α)isintegervaluedifandonlyifαisintegervalued,wede-
ducer(H (G ,Z))⊂H (G ,Z). Theoppositeinclusionfollowsbythecomputation
1 1 1 2
given above for R-valued cycles.
Now suppose f ∈C (G ,R), and define g ∈C (G ,R) by
0 1 0 2

f(x) if x(cid:54)=x
 1
g(x)=
f(x2)(cid:96)(cid:96)((ee1))++(cid:96)f((ex0))(cid:96)(e2) if x=x1.
1 2
Then a direct computation shows d g =r(d f). Hence r(im(d ))⊂im(d ). But r
2 1 1 2
preserves integrality of 1-chains, so we have proved the final claim of (1).
The computation (2.4) shows r induces an isomorphism as desired in asser-
∗
tion (2). Indeed, the defining relation for cycles (d∗α = 0) is identical to that
for harmonic 1-forms as in (2.2). To check compatibility of the integration pair-
ing, it suffices by linearity to check (cid:82) r (de(cid:48)) = (cid:82) de(cid:48) for any pair of edges
r(e) ∗ e
e,e(cid:48) ∈ E(G ). If e (cid:54)= e(cid:48), then both integrals vanish. If e = e(cid:48) (cid:54)= e , then r(e) = e
1 0
andr (de(cid:48))=de(cid:48),sotheequalityofintegralsisobviousinthiscaseaswell. Finally,
∗
suppose e=e(cid:48) =e . Then
0
(cid:90) (cid:90) (cid:90)
r (de )= (de +de )=(cid:96)(e )+(cid:96)(e )=(cid:96)(e )= de .
∗ 0 1 2 1 2 0 0
r(e0) e1+e2 e0
Finally, the proof of (3) is given by composing the duality isomorphism from
Lemma 2.1 with the isomorphism in part (1):
Ω(G )∗ →∼ H (G ,R)→∼ H (G ,R)→∼ Ω(G )∗.
1 1 1 1 2 2
The middle isomorphism preserves integral cycles, which finishes assertion (3). (cid:3)
Theorem 2.10. If G and G are weighted graphs such that G refines G , then
1 2 2 1
the refinement homomorphism r descends to a canonical injective homomorphism
ρ : J(G ) → J(G ). If G is a weighted graph that refines G , so that G ≤ G ≤
1 2 3 2 1 2
G , then the injections ρ :J(G )→J(G ) satisfy ρ =ρ ρ .
3 ij j i 31 32 21
Proof. By Lemmas 2.7 and 2.9(1), the refinement map r induces a homomorphism
J(G1)∼=C1(G1,Z)/(H1(G1,Z)⊕im(d1)Z)
−→C1(G2,Z)/(H1(G2,Z)⊕im(d2)Z)∼=J(G2).
The remaining assertions of the theorem are left to the reader. (cid:3)
Fix a weighted graph G and define R(G ) to be the set of all weighted graphs
0 0
that admit a common refinement with G . That is, G ∈ R(G ) if there exists a
0 0
weighted graph G(cid:48) such that G ≤ G(cid:48) and G ≤ G(cid:48). Then R(G ) is a directed set
0 0
under refinement, and the previous theorem shows that {J(G) : G ∈ R(G )} is a
0
directedsystemofgroups. Wenowdescribethestructureofthe“limitJacobian:”2
2A special case of Theorem 2.11 has been proved independently by Haase, Musiker, and Yu
[12].
10 MATTHEWBAKERANDXANDERFABER
Theorem 2.11. Let G be a weighted graph. There is a canonical isomorphism
0
lim J(G)→∼ Ω(G )∗/H (G ,Z).
−→ 0 1 0
G∈R(G0)
Before proving the theorem, let us recall a few definitions and facts regarding
cycles from algebraic graph theory. (See [6].) A path in the weighted graph G is a
sequence x ,e ,x ,e ,...,x of alternate vertices and edges such that {e−,e+} =
0 1 1 2 n i i
{x ,x }.3 That is, the vertices of the edge e are precisely x and x , but
i−1 i i i−1 i
perhaps not in the order dictated by the orientation. A path is closed if x =x .
0 n
GivenapathP andanedgee,wecanassociateaninteger(cid:15)(P,e)thatisthenumber
of times the sequence e−,e,e+ occurs in the path P less the number of times the
sequence e+,e,e− occurs in P. Intuitively, it is the number of times the path P
traverses the edge e counted with signs depending on the orientation of e. Define a
(cid:80)
1-chain associated to the path P by α = (cid:15)(P,e).e. If P is a closed path, then
P e
α is a 1-cycle.
P
Let T be a spanning tree in G. Given any edge e in the complement of T, we
can define a 1-cycle associated to it as follows. There is a unique minimal path in
T beginning at e+ and ending at e−. If we preface this path with e−,e, we get a
closedpathP. Notethat|(cid:15)(P,e)|≤1foralledgese. Wecallα (asdefinedabove)
P
the fundamental cycle associated to T and e, and denote it by α . For a fixed
T,e
spanning tree T, the cycles {α :e∈E(G)(cid:114)E(T)} form a basis of H (G,Z).
T,e 1
Proof of Theorem 2.11. First note that when G varies over R(G ), a natural di-
0
rected system is formed by the groups Ω(G)∗/H (G,Z) and the isomorphisms of
1
Lemma 2.9(3). Since Ω(G)(cid:93) ⊂ Ω(G)∗ for any weighted graph G, it follows that
thereisalsoacanonicalinjectivehomomorphismJ(G)(cid:44)→Ω(G)∗/H (G,Z). Direct
1
limit is an exact functor, so the maps J(G)(cid:44)→Ω(G)∗/H (G,Z) induce a canonical
1
injective homomorphism
limJ(G)(cid:44)→limΩ(G)∗/H (G,Z), (2.5)
1
−→ −→
where the limits are over all weighted graphs G∈R(G ).
0
Choose a spanning tree T for G , and enumerate the edges in the complement
0
of T by e ,...,e . Write α = α for the jth fundamental cycle associated to
1 g j T,ej
T. The fundamental cycles α ,...,α form a basis for H (G ,Z), and for any
1 g 1 0
refinement G of G , the cycles r(α ),...,r(α ) form a basis for H (G,Z), where
0 1 g 1
r :C (G ,Z)→C (G,Z)istherefinementhomomorphism. Toprovesurjectivityof
1 0 1
themapin(2.5),itsufficestoshowthatforanyrealnumberst ,...,t ∈[0,1),there
1 g
is a refinement G of G and a 1-chain β ∈C (G,Z) such that (cid:82) =(cid:82) .
0 1 (cid:80) β P(cid:80)jtjr(αT,ej)
Define a basis for Ω(G ) as follows. If α = α (e).e, set ω = α (e).de.
0 j e j j e j
Nowforeachj =1,...,g,chooseanintegern andapositiverealnumberu <(cid:96)(e )
j j j
so that
g (cid:90)
(cid:88)
n u = t ω .
j j i j
i=1 αi
Subdivide the edge e into two edges e(cid:48) and e(cid:48)(cid:48) such that
j j j
• p is the new vertex with ι(e(cid:48))=(e−,p ) and ι(e(cid:48)(cid:48))=(p ,e+), and
j j j j j j j
• (cid:96)(e(cid:48))=u and (cid:96)(e(cid:48)(cid:48))=(cid:96)(e )−u .
j j j j j
3In[6],thisiscalledawalk. Thereitiscalledapath ifej−1(cid:54)=ej forj=2,...,n.