On the homology of locally finite graphs

On the homology of locally finite graphs

Reinhard Diestel and Philipp Sprüssel
Abstract

We show that the topological cycle space of a locally finite graph is a canonical quotient of the first singular homology group of its Freudenthal compactification, and we characterize the graphs for which the two coincide. We construct a new singular-type homology for non-compact spaces with ends, which in dimension 1 captures precisely the topological cycle space of graphs but works in any dimension.

1 Introduction

Graph homology is traditionally, and conveniently, simplicial: a graph is viewed as a 1-complex, and one considers its first simplicial homology group. In graph theory, coefficients are typically taken from a field such as , or , which makes the group into a vector space called the cycle space of .

For reasons to become apparent later we denote this space as . For the moment it will suffice to take our coefficients from and interpret the elements of as sets of edges. For finite graphs , there are a number of classical theorems relating to other properties of , such as planarity. (Think of MacLane’s or Whitney’s theorem, or the Kelmans-Tutte planarity criterion.) The cycle space has thus become one of the standard aspects of finite graphs used in their structural analysis.

When is infinite, however, the space no longer adequately describes the homology of . Most of the theorems describing the interaction of with other properties of —including all those cited above—fail when is infinite. However, the traditional role of the cycle space in these cases can be restored by defining it slightly differently: when is locally finite, one takes as generators not the edge sets of the (finite) cycles in —as one would to generate —but the (possibly infinite) edge sets of all its topological circles, the homeomorphic images of the circle in the Freudenthal compactification of by its ends. (One also has to allow infinite sums in the generating process; for these to be well-defined, each edge may occur in only finitely many terms.) We denote this more general space, the topological cycle space of , by .

The space had not been considered in graph theory before [15] appeared, and it has been surprisingly successful at extending the classical cycle space theory of finite graphs to locally finite graphs; see e.g. [3, 4, 5, 7, 25, 36], or [13] for a survey. However, a question raised in [10] but still unanswered is how new, from a topological viewpoint, is the homology described implicitly by . It is the purpose of this paper to clarify this relationship.

Our first result says that there is indeed a classical homology theory whose first group is isomorphic to : the Čech homology of . However, the group of  as such does not carry all the information that makes it relevant to the study of the (combinatorial) structure of ; one also needs to know, for example, which group elements correspond to circuits and which do not. These details are lost in the transition between and the Čech homology, which is why we do not pursue this approach further.

Since topological circles are (images of simplices representing) singular 1-cycles in , it is also natural to ask how closely is related to the first singular homology group of . Indeed it is not clear whether the two coincide by some natural canonical isomorphism, so that would be just another way of looking at .

Our first major aim in this paper is to answer this question. We begin by studying the homomorphism that should serve as the desired canonical isomorphism if indeed there is one. Surprisingly, this homomorphism is easily seen to be surjective. However, it turns out that it usually has a non-trivial kernel. Thus , despite looking ‘larger’ because we allow infinite sums in its generation from elementary cycles, turns out to be a (usually proper) quotient of .

For the proof that has a non-trivial kernel we have to go some way towards the solution of another problem (solved fully in [20]): to find a combinatorial description of the fundamental group of the space  for an arbitrary connected locally finite graph .111Covering space theory does not apply since, trivial exceptions aside,  is not semi-locally simply connected at ends. We describe , as for finite , in terms of reduced words in the oriented chords of a spanning tree. However, when is infinite this does not work for arbitrary spanning trees; we have to allow infinite words of any countable order type; and reduction by cancelling adjacent inverse sequences of letters does not suffice. However, the kind of reduction we need can be described in terms of word reductions in the free groups on all the finite subsets of chords, which enables us to embed the group of infinite reduced words in the inverse limit of those , and handle it in this form. On the other hand, mapping a loop in to the sequence of chords it traverses, and then reducing that sequence (or word), turns out to be well defined on homotopy classes and hence defines an embedding of as a subgroup in . This combinatorial description of then enables us to define an invariant on 1-chains in that can distinguish some elements of the kernel of from boundaries of singular 2-chains, completing the proof that need not be injective.

Our second aim, then, is to begin to reconcile these different treatments of the homology of non-compact spaces between topology and graph theory. In Section 7 we present a first solution to this problem: we define a natural singular-type homology which, applied to graphs, captures precisely their topological cycle space. Essentially, we shall allow infinite sums of cycles and boundaries when building their respective groups, but start from finite chains with zero boundary as generators. Thus, topological circles are 1-cycles, as desired. But if is a 2-way infinite path, then its edges form an infinite 1-chain with zero boundary that is not a 1-cycle, because it is not a (possibly infinite) sum of finite 1-cycles. Our homology thus lies between the usual singular homology and the ‘open homology’ that is built from arbitrary locally finite chains without any further restriction (see, eg, [24, Ch. 8.8]).

Formally, we define our homology in Section 7 in a very general setting: all we require is a topological space in which some points are distinguished as ‘ends’. A drawback of this combination of simplicity with generality is that although we can define the groups as desired, our definitions do not lead to a homology theory in the full axiomatic sense. However, it is possible to do that too: to construct a singular homology theory that does satisfy the axioms and which, for graphs, is equivalent to the homology of Section 7 and hence to the topological cycle space. We may thus view our intuitive homology of Section 7 as a stepping stone towards this more general theory, to be developed in [21], which will work for any locally compact space with ends. In both settings, ends play a role that differs crucially from that of ordinary points, which enables this homology to capture the properties of the space itself in a way similar to how the topological cycle space describes a locally finite graph.

Our hope with this paper is to stimulate further work in two directions. One is that its new topological guise makes the cycle space accessible to topological methods that might generate some windfall for the study of graphs. And conversely, that as the approach that gave rise to  is made accessible to more general spaces and higher dimensions, its proven usefulness for graphs might find some more general topological analogues—perhaps based on the homology theory developed in [21] from the ideas presented in this paper.

2 Terminology and basic facts

In this section we briefly run through any non-standard terminology we use. We also list without proof a few easy lemmas that we shall need, and use freely, later on.

For graphs we use the terminology of [11], for topology that of Hatcher [29]. We reserve the word ‘boundary’ for homologousal contexts and use ‘frontier’ for the closure of a set minus its interior. Our use of the words ‘path’, ‘cycle’ and ‘loop’, where these terminologies conflict, is as follows. The word path is used in both senses, according to context (such as ‘path in ’, where was previously introduced as a graph or as a topological space). Note that while topological paths need not be injective, graph-theoretical paths are not allowed to repeat vertices or edges. The term cycle will be used in the topological sense only, for a (usually 1-dimensional) singular chain with zero boundary. When we do need to speak about graph-theoretic cycles (i.e., about finite connected graphs in which every vertex has exactly two incident edges) we shall instead refer to the edge sets of those graphs, which we shall call circuits. Our graphs may have multiple edges but no loops. This said, we shall from now on use the term loop topologically: for a topological path with . This loop is based at the point . Given any path , we write for the inverse path. An arc in a topological space is a subspace homeomorphic to .

Lemma 1 ([28, p. 208]).

The image of a topological path with distinct endpoints in a Hausdorff space contains an arc in between and .

All homotopies between paths that we consider are relative to the first and last point of their domain, usually . We shall often construct homotopies between paths segment by segment. The following lemma enables us to combine certain homotopies defined separately on infinitely many segments.

Lemma 2.

Let be paths in a topological space . Assume that there is a sequence of disjoint subintervals of such that and conincide on , while each segment is homotopic in to . Then and are homotopic.

Proof.

Write . For every let be a homotopy in between and . We define the desired homotopy between and as

Clearly, and . It remains to prove that is continuous.

Let and a neighbourhood of in be given. We find an so that ; the case is analogous. Suppose first that there is an such that . As the intervals are disjoint, this means that for some . Then , and hence . As is continuous, there is an with .

Now suppose that for every the interval meets . Then also , and hence . Pick with small enough that both and map into . Then . Indeed, for every and every we have . On the other hand, for every and we have for some . As and lie in , we have and hence . ∎

All the CW-complexes we consider will be locally finite: every point has an open neighbourhood meeting only finitely many cells. Note that a compact subset of such a complex can meet the closures of only finitely many cells, and that locally finite CW-complexes are metrizable [33, Ch. II, Prop. 3.8] and thus first-countable.

Locally finite CW-complexes can be compactified by adding their ends. This compactification can be defined, without reference to the complex, for any connected, locally connected, locally compact topological space with a countable basis. Very briefly, an end of is an equivalence class of sequences of connected non-empty open sets with compact frontiers and an empty overall intersection of closures, , where two such sequences and are equivalent if every contains all sufficiently late and vice versa. This end is said to live in each of the sets , and every together with all the ends that live in it is open in the space whose point set is the union of with the set of its ends and whose topology is generated by these open sets and those of . This is a compact space, the Freudenthal compactification of  [22, 23]. More topological background on this can be found in [2, Ch. I.9]; for applications to groups see e.g. [34, 35, 38, 39].

For graphs, ends and the Freudenthal compactification are more usually defined combinatorially, as follows [11, Ch. 8.5][27, 31]. Let be a connected locally finite graph. A 1-way infinite path in is a ray. Two rays are equivalent if no finite set of vertices separates them in , and the resulting equivalence classes are the ends of . It is not hard to see that this combinatorial definition of an end coincides with the topological one given earlier for locally finite complexes.222For graphs that are not locally finite, the two concepts differ [14]. The Freudenthal compactification of  is now denoted by ; its topology is generated by the open sets of itself (as a 1-complex) and the sets defined for every end and every finite set of vertices, as follows. is the unique component of in which lives (i.e., in which every ray of  has a tail, or subray), and is the union of with the set of all the ends of that live in and the (finitely many) open edges between and .333The definition given in [11] is formally more general, but equivalent to the simpler definition given here when is locally finite. Generalizations are studied in [32, 37]. Note that the frontier of in is a subset of , and that every ray converges to the end containing it. See [13] for (much) more on .

The end structure of is best reflected by a normal spanning tree; such trees exist in every connected countable graph [11, 30]. A spanning tree of  with root  is normal if the vertices of every edge of are comparable in the order which induces on . (Recall that if lies on the unique path of between and .) A key property of normal spanning trees is that the intersection of the down-closures of two vertices separates them in . This implies that every end of is represented by a unique ray in starting at , and hence that adding all the ends of to does not create any circles. More generally, it is not hard to prove the following:

Lemma 3.

Let be a normal spanning tree of , and let denote its closure in . Then for every closed connected set and there is a deformation retraction of onto .

Proof.

Let be a closed connected subset of , and let . Then is also closed in and hence arc-connected [17, Theorem 2.6]. For every there is a unique  arc in  [11, Theorem 8.5.7] which hence lies in . The space is metrizable so that every edge between levels and has length and hence every end has distance  from the root [12].  inherits this metric , note that for all . Further, if for some we have . We construct a homotopy in from the identity on to the map ; then we have for every and , and hence will be the desired homotopy for . For every and let be the unique point on at distance from .

For the proof that is continuous, we show that for every and ; then for every and every with and with we have

As and are closed, there is a last point on that is also in . As contains a unique arc between any two points in , we have and hence . If and , then . Otherwise both and are contained in or in . In particular, one of lies on the arc between the other and . Then . ∎

Lemma 3 implies that contains no topological circle. Equivalently: for any two points there is a unique arc in between and . We denote this arc by . The uniqueness of implies that none of its inner points can be an end. (Every arc containing an end also contains a vertex, and any two vertices of can also be joined by an arc in itself.)

When is a normal spanning tree of , every end in has a neighbourhood basis consisting of open sets such that is closed downwards, i.e. where implies . We call these sets the basic open neighbourhoods of the ends444The basic open neighbourhoods of a point are the connected open neighbourhoods of containing no vertex other than possibly . of  (given ). An important property of these sets is that for any two points we also have .

Now let , the (finite) set of vertices in and their neighbours. We call the subset of the inside of  around . Note that the neighbours of vertices , as well as the edges , also lie in .

More background on normal spanning trees, including an existence proof, can be found in [11, Ch. 8][18, 19].

Let us now introduce the topological cycle space of . This is usually defined over  (which suffices for its role in graph theory), but we wish to prove our main results more generally with integer coefficients. (The case will follow, but it should be clear right away that the non-injectivity of our homomorphism is not just a consequence of a wrong choice of coefficients.) We therefore need to speak about orientations of edges.

An edge of has two directions, and . A triple consisting of an edge together with one of its two directions is an oriented edge. The two oriented edges corresponding to are its two orientations, denoted by and . Thus, , but we cannot generally say which is which. However, from the definition of as a CW-complex we have a fixed homeomorphism . We call the natural direction of , and its natural orientation.

Given a set of edges in , we write for the set of their orientations, two for every edge in . Given a partition of the vertex set of , we write for the set of all its oriented edges with and , and call this set an oriented cut of .

Let be a path in . Given an edge of , if is a subinterval of such that and , we say that traverses  on . It does so in the direction of , or traverses . We then call its restriction to a pass of through , or , from to .

Using that is compact and is Hausdorff, one easily shows that a path in contains at most finitely many passes through any given edge:

Lemma 4.

A path in traverses each edge only finitely often.

Proof.

Let be a path in , and let be an edge such that contains infinitely many passes through  (). As is compact, the sequence has an accumulation point , which is also an accumulation point of . But now fails to be continuous at , because for each  but each of and has a neighbourhood not containing the other. ∎

A loop that is injective on  is a circle in . (In most of our references, the term circle is used for the image of such a loop.) The set of all edges traversed by a circle is a circuit. It is easy to show that the image of a circle is uniquely determined by its circuit , being the closure of in .

Let denote the set of all integer-valued functions on the set of all oriented edges of that satisfy for all . This is an abelian group under pointwise addition. A family of elements of is thin if for every we have for only finitely many . Then is a well-defined element of : it maps each to the (finite) sum of those that are non-zero. We shall call a function obtained in this way a thin sum of those .

We can now define our oriented version of the topological cycle space of . When is a circle in , we call the function defined by

an oriented circuit in , and write for the subgroup of formed by all thin sums of oriented circuits.

We remark that is closed also under infinite thin sums [15, Cor. 5.2], but this is neither obvious nor generally true for thin spans of subsets of  [6, Sec. 3]. We remark further that composing the functions in with the canonical homomorphism yields the usual topological cycle space of  as studied in [3, 4, 5, 7, 8, 15, 16, 17, 25, 26, 36], the vector space of subsets of obtained as thin sums of (unoriented) circuits.

The topological cycle space can be characterized as the set of those subsets of that meet every finite cut of in an even number of edges [15, Thm. 7.1][11, Thm. 8.5.8]. The characterization has an oriented analogue:

Theorem 5.

An element of lies in if and only if for every finite oriented cut of .

The proof of Theorem 5 is not completely trivial. But it adapts readily from the unoriented proof given e.g. in [11], which we leave to the reader to check if desired.

When is fixed, we write for the group of singular -chains in  (with coefficients in unless otherwise mentioned), and for the corresponding groups of cycles and boundaries, and . We view all singular 1-simplices as maps from the real interval to . The homology class of a cycle  is denoted by .

A cycle that can be written as a sum of 1-simplices no two of which share their first point is an elementary cycle. Every 1-cycle is easily seen to be a sum of elementary 1-cycles, a decomposition which is not normally unique.

The following lemma enables us to subdivide or concatenate the simplices in a 1-cycle while keeping it in its homology class.

Lemma 6.

Let be a singular 1-simplex in , and let . Write and for the 1-simplices obtained from the restrictions of to and to by reparametrizing linearly. Then .       

When is a summand in a cycle , we shall say that the equivalent cycle obtained by replacing with in the sum arises by subdividing  (at  or at ). A frequent application of Lemma 6 is the following:

Corollary 7.

Every non-zero element of is represented by a sum of loops each based at a vertex.

Proof.

Pick a cycle representing a given homology class, and decompose it into elementary cycles. Use Lemma 6 to concatenate their simplices into a single loop. If such a loop  passes through a vertex, we can subdivide it there and suppress its original boundary point, obtaining a homologous loop based at that vertex. If does not pass through a vertex, then for some edge  (since non-trivial sets of ends are never connected), so is null-homotopic and . ∎

3 and the Čech homology

In this section we briefly describe the relationship between the topological cycle space of a graph with ends and its Čech homology. We shall see that their groups are canonically isomorphic, but also that this isomorphism is not enough to capture the relevance of to the structure of —the reason why graph theorists study cycle spaces in the first place. The material from this section will not be needed in the rest of the paper.

The Čech homology of a space is an alternative to singular homology for spaces that do not have a simplicial homology, and we begin by recalling its definition. Consider a space and an open cover of . Then defines a simplicial complex , the nerve of : The -simplices of are the elements of , and any elements of form an -simplex if and only if they have a nonempty intersection. For two open covers let if is a refinement of . In this case, it is easy to define a continuous map from to : For each -simplex of (i.e. ) there is a -simplex of (an element of ) that contains it. Map each to and extend this map linearly to the higher-dimensional simplices in so as to obtain a map . As can be contained in more than one element of , the choice of is not unique and neither is . But it is easy to see that all possible choices of induce homotopic maps and hence the induce a unique homomorphism on homology. Now the homology groups for all open covers together with the homomorphisms form an inverse family. Define the th Čech homology group to be the inverse limit of the .

For locally finite graphs the first Čech homology group and the topological cycle space coincide:

Theorem 8.

For a locally finite graph we have a canonical isomorphism .

Proof.

To compute the inverse limit of the groups it suffices to to consider a family of open covers of that contains a refinement for every open cover of , and to compute the inverse limit of the inverse family . We will now construct a suitable .

Let be a normal spanning tree of and denote the subtree induced by the first levels by . Now for each and each let contain an open cover consisting of the following sets: An open star of radius around each vertex , finitely many open subintervals of length of each edge , and the sets for each end of omega. Note that is a finite family as has only finitely many components.

Using that is compact, it is not hard to see that for each open cover of some is a refinement of . Clearly, every retracts to the graph obtained from by contracting all components of , and hence the homology group is a direct product of ’s, one for each chord of with at least one endvertex in . Thus also is the direct product of copies of , one for each chord of . As the same is true for , we have that and are canonically isomorphic. ∎

Although the first Čech homology group is isomorphic to the group of the topological cycle space, it does not sufficiently reflect the combinatorial properties of . For example, a number of classical results about the cycle space say which circuits generate it—as do the non-separating chordless circuits in a -connected graph, say. In the Čech homology, however, it is not possible to decide whether a homology class in corresponds to a circuit in . One might think that since a homology class corresponds to a family of homology classes in the groups , the class should correspond to a circuit if every with sufficiently large corresponds to a circuit in . But this is not the case: the limit of a sequence of cycle space elements in the can be a circuit even if the elements of the sequence are not circuits in the .

In order to have a homology that reflects the properties of , we thus need to take a singular approach.

4 Comparing with

Let be a connected locally finite graph. Our aim in this section is to compare the first singular homology group of  (with integer coefficients) with the oriented topological cycle space of , the group of thin sums of oriented circuits in . When is finite then , and all circuits and their thin sums are finite. Hence in this case is just the first simplicial homology group of , so the two groups are indeed the same.

When is infinite, however, both circuits and thin sums can be infinite too. So they are not just the simplicial 1-cycles in . But there is an obvious singular 1-cycle in associated with an oriented circuit : the circle , viewed as a singleton 1-chain. Our aim is to extend this correspondence to one between and .

Our approach will be to define a homomorphism that counts for a given homology class how often the 1-simplices of a cycle representing , when properly concatenated, traverse a given edge ; we then let map to this number.555The precise definition of will be given shortly. We shall prove that always lies in  and, perhaps surprisingly, that maps onto . However, we find that is not normally injective. Our first main result characterizes the graphs for which it is:

Theorem 9.

The map is a group homomorphism onto , which has a non-trivial kernel if and only if contains infinitely many (finite) circuits.

Thus,  turns out to be a canonical—but usually non-trivial—quotient of . Taking this result mod 2 answers our original question: the topological cycle space of  is a canonical—but usually non-trivial—quotient of the singular homology group of  with coefficients.

We remark that the last condition in Theorem 9 can be rephrased in various natural ways: that has a spanning tree with infinitely many chords; that every spanning tree of has infinitely many chords; or that contains infinitely many disjoint (finite) circuits [11, Ex. 37, Ch. 8]. The remainder of this section and the next two sections will be devoted to the proof of Theorem 9.

Let us define formally. Let denote the unit circle in the complex plane. The elements of are represented by the loops , , . Write for the group isomorphism . For every edge of , let wrap round in its natural direction, defining as and putting . Note that is continuous.

The following lemma is easy to prove using homotopies in , combined by Lemma 2:

Lemma 10.

Let be a loop based at a vertex. If traverses exactly times in its natural direction and exactly times in the opposite direction, then .

Proof.

Composing a pass of through  (in its natural direction) with yields a map from a subinterval of to which, after reparametrization, is homotopic to .

The domains of distinct passes of through are closed subintervals of meeting at most in their boundary points. The rest of is a finite disjoint union of open intervals (or or ). Each of these is in turn a disjoint union, possibly infinite, of open intervals which maps to  and closed intervals which maps to . Since , by definition, contains no pass through ,  always maps and to the same endvertex of . Then is homotopic to the constant map to that vertex, and is homotopic to the constant map to 1. These homotopies combine to a homotopy of to the constant map with value 1.

We deduce that is homotopic to a concatenation of loops in of which (after reparametrization) are equal to  and are equal to the inverse loop , and the rest are constant with value 1. The result follows. ∎

Given , we now let assign to the natural orientation of :

This completes the definition of , which is clearly a group homomorphism.

Lemma 11.

.

Proof.

By Theorem 5 it suffices to show that for every finite oriented cut of and every we have . Let and be given, let , and assume for simplicity that the orientations of these edges are their natural orientations. Since is a homomorphism, we may assume that is represented by an elementary 1-cycle, which we may choose by Corollary 7 to consist of a loop  based at a vertex. We shall prove that traverses the edges in as often from to as it does from to . Then

by Lemma 10.

Let be the domains of the passes of through edges of . In order to prove that as many of these passes are from a vertex in to one in as vice versa, it suffices to show that each of the segments has either all its vertices in or all its vertices in , assuming for simplicity that is based at . If the starting vertex of lies in , say, put

We wish to show that . If not, then is an end, and this end lies both in the closure of and in the closure of . But these closures are disjoint: the set of vertices incident with an edge in  is finite, and since separates from , the neighbourhood of any end avoids either or . ∎

Next, we prove that is surjective. At first glance, this may seem surprising: after all, we have to capture arbitrary thin sums of oriented circuits, which may well be disjoint, by finite 1-cycles.

Lemma 12.

.

Proof.

Let be an arbitrary thin sum of oriented circuits, where each is a circle in . Ignoring any that are constant with value 0, we may assume that each is based at a vertex . (Recall that if the image of  contains no vertex it must lie inside an edge, because non-trivial sets of ends cannot be connected.) We shall construct a loop in such that .

Let be a spanning tree of  and pick a root . Write for the set of vertices at distance  in  from , and let be the subtree of induced by . Our first aim will be to construct a loop in that traverses every edge of once in each direction and avoids all other edges of . We shall obtain as a limit of similar loops in . We shall then incorporate our loops into , to obtain . When we describe these maps informally, we shall think of as measuring time, and of a loop as a journey through .

Let be the unique (constant) map . Assume inductively that is a loop traversing every edge of exactly once in each direction. Assume further that pauses every time it visits a vertex, remaining stationary at that vertex for some time. More precisely, we assume for every vertex that is a disjoint union of as many non-trivial closed intervals as has incident edges in , and of one more such interval in the case of . Let us call the restriction of to such an interval a pass of through . We are thus assuming that is the union of its passes through the vertices and edges of .

Let be obtained from by replacing, for each leaf of , the unique pass of through by a topological path that starts out remaining stationary at for some time, then visits all the neighbours of in in turn, pausing at each and shuttling back and forth between and those neighbours, and finally returns to to pause there. Outside the passes of through leaves of , let agree with . Note that satisfies our inductive assumptions for : it traverses every edge of exactly once each way, pauses every time it visits a vertex, and is the union of its passes through the vertices and edges of .

Let us now define . Let be given. If the values coincide for all large enough , let for these . If not, then for every , and is a ray in ; let map to the end of containing that ray.

Clearly every is continuous, and is continuous at points not mapped to ends. To show that is continuous at every point mapped to an end , let a neighbourhood of in be given. Put . Choose large enough that the tree spanned in by the vertices above —those vertices for which the path in contains —avoids the finite set . We claim that maps the interval to . Since agrees with on the boundary points of but not on , we know that is a neighbourhood of  in , so this will complete the proof that is continuous. Let be given. Induction on shows that for every . Hence if is not an end, then for some . But if is an end, then this is the end of a ray that starts at and lies in . Hence so does .

Let be obtained from by replacing for every vertex one of the passes of through with a concatenation of all the circles with and . Note that these are finitely many for each , because has only finitely many edges at  and is a thin sum.

Let us prove that is continuous. As before, this is clear at points which does not map to an end: for such the map agrees on suitable intervals and with or some , which we know to be continuous. The proof that is continuous at points which maps to ends is similar to our earlier continuity proof for . The only difference now is that we have to choose large enough also to ensure that none of the with passes through a vertex of . Such a choice of is possible, because only finitly many edges are incident with vertices in  and the form a thin family of functions. Then contains not only but also the images of all with , because is connected but does not meet the frontier  of .

Finally, recall that  traverses every edge of once in each direction, and that it does not traverse any other edges. Therefore , and hence as desired. ∎

To complete the proof of Theorem 9 it remains to show that has a non-trivial kernel if and only if contains infinitely many circuits. The forward implication of this is easy. Indeed, suppose that contains only finitely many circuits, and let be a normal spanning tree of . Then  has only finitely many chords, so  is homotopy equivalent to a finite graph (Lemma 3). Hence, as is well known,  equals the first simplicial homology group of viewed as a 1-complex, which in turn is clearly isomorphic to . Therefore must be injective.

The converse implication, surprisingly, is quite a bit harder. Assuming that contains infinitely many circuits, we shall define a loop in that traverses every edge equally often in both directions (so that ), and which is easily seen not to be null-homotopic. To prove that , however, i.e. that is not a boundary, will be harder: it turns out that we first have to understand the fundamental group of  a little better. With this knowledge we shall then be able to define an invariant of 1-chains that can distinguish from boundaries.

The next section, therefore, contains an interlude in which we describe the fundamental group of an arbitrary locally finite connected graph  combinatorially. We shall then complete the proof of Theorem 9 in Section 6.

5 A combinatorial characterization of 

Our aim in this section is to prove some aspects of a combinatorial description of  that we need for our proof of Theorem 9. In [20], we give a more comprehensive such description; see Theorem 18 below.

When is finite,  is the free group on the set of chords (arbitrarily oriented) of any fixed spanning tree, the edges of that are not edges of the tree. The standard description of is given in terms of reduced words of those oriented chords. The map assigning to a path in the sequence of chords it traverses defines the canonical group isomorphism between and .

Our description of for infinite  will be similar in spirit, but more complex. We shall start not with an arbitrary spanning tree but with a normal spanning tree. (The trees that work are precisely the topological spanning trees defined in [17] or [11, Ch. 8.5], which include the normal spanning trees.) Then every path in defines as its ‘trace’ an infinite word in the oriented chords of that tree, as before. However, these words can have any countable order type, and it is no longer clear how to define reductions of words in a way that captures homotopy of paths.

Consider the following example. Let be the infinite ladder, with a spanning tree  consisting of one side of the ladder and all its rungs (drawn bold in Figure 1). The path running along the bottom side of the ladder and back is a null-homotopic loop. Since it traces the chords all the way to  and then returns the same way, the infinite word should be reducible. But it contains no cancelling pair of letters, such as or .

Figure 1: The null-homotopic loop

This simple example suggests that some transfinite equivalent of cancelling pairs of letters, such as cancelling inverse pairs of infinite sequences of letters, might lead to a suitable notion of reduction. However, one can construct graphs which, for any suitable spanning tree, contain null-homotopic loops whose trace of chords contains no such cancelling subsequences (of any order type).666For example, consider for the binary tree  the loop constructed in the proof of Lemma 12, which traverses every edge of exactly once in each direction. This loop is null-homotopic in  (Lemma 3), but no sequence of edges, of any order type, is followed immediately by the inverse of that sequence. The edges of aren’t chords of a spanning tree, but this can be achieved by changing the graph: just double every edge and subdivide the new edges once. The new edges then form a normal spanning tree in the resulting graph , whose chords are the original edges of our , and is still a (null-homotopic) loop in .

We shall therefore define the reduction of infinite words differently, in a non-recursive way: just as a homotopy can shrink a loop simultaneously (rather than recursively) in many places at once, our reduction ‘steps’ will be ordered linearly but not be well-ordered. This definition is less straightforward, but it has an important property: as for finite , our notion of reduction will be purely combinatorial and make no reference to the topology of .

The main step then will be to show that embeds as a subgroup in the group of reduced words, and how. We shall see that, as in the finite case, the map assigning to a path in its trace of chords and reducing that trace is well defined on homotopy classes. In [20] we shall prove that the map it induces on these classes is injective. Then  can be viewed as a subgroup of the group of reduced infinite words, which in turn can be viewed as a subgroup of the inverse limit of the free groups with generators the finite sets of oriented chords of any fixed normal spanning tree (Theorem 18).

Let us make all this precise. Let be a locally finite connected graph, fixed throughout this section. Let be a fixed normal spanning tree of , and write for its closure in . Unless otherwise mentioned, the endpoints of all paths considered from now on will be vertices or ends, and any homotopies between paths will be relative to .

When we speak of ‘the passes’ of a given path , without referring to any particular edges, we shall mean the passes of through chords of . If has only finitely many chords, then is homotopy equivalent to a finite graph, by Lemma 3. Let us therefore assume that