\documentclass[12pt]{article} \usepackage{e-jc} \specs{}{}{2014} \usepackage{ amsmath, amsthm, amssymb, enumitem} %\usepackage[arrow, matrix, curve]{xy} %\usepackage[notref,notcite]{showkeys} %\usepackage[applemac]{inputenc} %\usepackage{txfonts} \theoremstyle{plain} \newtheorem{theo}{Theorem} \newtheorem{theorem}[theo]{Theorem} \newtheorem{problem}[theo]{Problem} \newtheorem{lem}[theo]{Lemma} \newtheorem{kor}[theo]{Korollar} \newtheorem{cor}[theo]{Corollary} \newtheorem{Cor}[theo]{Corollary} \newtheorem{prop}[theo]{Proposition} \newtheorem{rem}[theo]{Remark} \newtheorem{Lem}[theo]{Lemma} \newtheorem{tm}[theo]{Theorem} \newtheorem{quest}[theo]{Question} \newtheorem{Prob}{Problem} \newtheorem{exer}[theo]{Exercise} \newtheorem{Conj}{Conjecture} \newtheorem{Claim}[theo]{Claim} \theoremstyle{definition} \newtheorem{defi}[theo]{Definition} \theoremstyle{definition} \newtheorem{Def}[theo]{Definition} \newtheorem{Exam}[theo]{Example} \newtheorem{Rem}[theo]{Remark} \newtheorem{fact}[theo]{Fact} \newcommand{\K}{\ensuremath{\mathbb{K}}} \newcommand{\R}{\ensuremath{\mathbb{R}}} \newcommand{\N}{\ensuremath{\mathbb{N}}} \newcommand{\F}{\ensuremath{\mathbb{F}}} \newcommand{\Z}{\ensuremath{\mathbb{Z}}} \newcommand{\Q}{\ensuremath{\mathbb{Q}}} \newcommand{\C}{\ensuremath{\mathbb{C}}} \newcommand{\T}{\ensuremath{\mathcal{T}}} \newcommand{\D}{\ensuremath{\mathcal{D}}} \newcommand{\V}{\ensuremath{\mathcal{V}}} \newcommand{\W}{\ensuremath{\mathcal{W}}} \newcommand{\lra}{\ensuremath{\Leftrightarrow}} \newcommand{\sm}{\ensuremath{\smallsetminus}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mle}{\preccurlyeq} \newcommand{\mge}{\succcurlyeq} \renewcommand{\P}{\mc{P}} \newcommand{\sub}{\subseteq} \newcommand{\subeq}{\subseteq} \newcommand{\supeq}{\supseteq} \newcommand{\Gtilde}{{\widetilde G}} \newcommand{\GPtilde}{{\widetilde G_\Psi}} %{{\GPtilde}} \newcommand{\GOtilde}{{\widetilde G_\Omega}} %{\widetilde {G_\Omega}} \newcommand{\AF}{\ensuremath{\mathcal A}} \newcommand{\BF}{\ensuremath{\mathcal B}} \newcommand{\CFtop}{\ensuremath{\mathcal C}_{\rm top}} \newcommand{\CF}{\ensuremath{\mathcal C}} \newcommand{\DF}{\ensuremath{\mathcal D}} \newcommand{\EF}{\ensuremath{\mathcal E}} \newcommand{\FF}{\ensuremath{\mathcal F}} \newcommand{\GF}{\ensuremath{\mathcal G}} \newcommand{\HF}{\ensuremath{\mathcal H}} \newcommand{\IF}{\ensuremath{\mathcal I}} \newcommand{\JF}{\ensuremath{\mathcal J}} \newcommand{\KF}{\ensuremath{\mathcal K}} \newcommand{\LF}{\ensuremath{\mathcal L}} \newcommand{\MF}{\ensuremath{\mathcal M}} \newcommand{\NF}{\ensuremath{\mathcal N}} \newcommand{\OF}{\ensuremath{\mathcal O}} \newcommand{\PF}{\ensuremath{\mathcal P}} \newcommand{\QF}{\ensuremath{\mathcal Q}} \newcommand{\RF}{\ensuremath{\mathcal R}} \newcommand{\SF}{\ensuremath{\mathcal S}} \newcommand{\TF}{\ensuremath{\mathcal T}} \newcommand{\UF}{\ensuremath{\mathcal U}} \newcommand{\VF}{\ensuremath{\mathcal V}} \newcommand{\WF}{\ensuremath{\mathcal W}} \newcommand{\XF}{\ensuremath{\mathcal X}} \newcommand{\YF}{\ensuremath{\mathcal Y}} \newcommand{\ZF}{\ensuremath{\mathcal Z}} \usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=blue]{hyperref} \newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\texttt{arXiv:#1}}} \newenvironment{txteq} { \begin{equation} \begin{minipage}[c]{0.85\textwidth} % set width to 0.9 x textwidth \em % switch on emph } {\end{minipage}\end{equation}\ignorespacesafterend} \DeclareMathOperator{\fin}{fin} \DeclareMathOperator{\sk}{skew} \DeclareMathOperator{\alg}{alg} \DeclareMathOperator{\coloneqq}{\mathrel{\mathop:}=} \title{\bf Orthogonality and minimality in the\\ homology of locally finite graphs} \author{Reinhard Diestel\\ \small Department of Mathematics\\[-0.8ex] \small University Hamburg\\[-0.8ex] \small Hamburg, Germany\\ \and Julian Pott\\ \small Department of Mathematics\\[-0.8ex] \small University Hamburg\\[-0.8ex] \small Hamburg, Germany\\ } \date{\dateline{Oct 25, 2013}{Aug 29, 2014}{}\\ \small Mathematics Subject Classification: 05C} \begin{document} \maketitle \begin{abstract} Given a finite set $E$, a subset $D\sub E$ (viewed as a function $E\to \F_2$) is orthogonal to a given subspace $\FF$ of the $\F_2$-vector space of functions $E\to \F_2$ as soon as $D$ is orthogonal to every $\sub$-minimal element of~$\FF$. This fails in general when $E$ is infinite. However, we prove the above statement for the six subspaces $\FF$ of the edge space of any $3$-connected locally finite graph that are relevant to its homology: the topological, algebraic, and finite cycle and cut spaces. This solves a problem of~Diestel (2010, \arxiv{0912.4213}). \bigskip\noindent \textbf{Keywords:} graph theory; locally finite graph; end; homology; topological; orthogonal; cycle space; bond space; cut space \end{abstract} \section{Introduction} Let $G$ be a $2$-connected locally finite graph, and let $\EF=\EF(G)$ be its edge space over $\F_2$. We think of the elements of $\EF$ as sets of edges, possibly infinite. Two sets of edges are \emph{orthogonal} if their intersection has (finite and) even cardinality. A set $D\in\EF$ is \emph{orthogonal} to a subspace $\FF\sub \EF$ if it is orthogonal to every $F\in\FF$. See \cite{DiestelBook10noEE, RDsBanffSurvey} for any definitions not given below. The topological \emph{cycle space} $\CFtop(G)$ of~$G$ is the subspace of $\EF(G)$ generated (via thin sums, possibly infinite) by the \emph{circuits} of~$G$, the edge sets of the topological circles in the Freudenthal compactification $|G|$ of~$G$. This space $\CFtop(G)$ contains precisely the elements of~$\EF$ that are orthogonal to $\BF_{\fin}(G)$, the finite-cut space of~$G$~\cite{DiestelBook10noEE}. The \emph{algebraic cycle space} $\CF_{\alg}(G)$ of~$G$ is the subspace of $\EF$ consisting of the edge sets inducing even degrees at all the vertices. It contains precisely the elements of~$\EF$ that are orthogonal to the \emph{skew cut space} $\BF_{\sk}(G)$~\cite{CR08}, the subspace of $\EF$ consisting of all the cuts of $G$ with one side finite. The \emph{finite-cycle space} $\CF_{\fin}(G)$ is the subspace of $\EF$ generated (via finite sums) by the finite circuits of~$G$. This space $\CF_{\fin}(G)$ contains precisely the elements of~$\EF$ that are orthogonal to $\BF(G)$, the cut space of~$G$~\cite{DiestelBook10noEE, RDsBanffSurvey}. Thus, \[ \CFtop = \BF_{\fin}^{\perp},\quad \CF_{\alg} = \BF_{\sk}^\perp,\quad \CF_{\fin} = \BF^{\perp}\text{.} \] \vskip-6pt \noindent Conversely, \[ \CFtop^{\perp}= \BF_{\fin},\quad \CF_{\alg}^\perp = \BF_{\sk},\quad \CF_{\fin}^{\perp}=\BF\text{.} \]\vskip6pt\noindent Thus, for any of the six spaces $\FF$ just mentioned, we have $\FF^{\perp\perp}=\FF$. \goodbreak Proofs of most of the above six identities were first published by Casteels and Richter \cite{CR08}, in a more general setting. Any remaining proofs can be found in~\cite{RDsBanffSurvey}, except for the inclusion~$\CF_{\alg}^\perp\supseteq \BF_{\sk}$, which is easy. The six subspaces of~$\EF$ mentioned above are the ones most relevant to the homology of locally finite infinite graphs. See~\cite{RDsBanffSurvey}, Diestel and Spr\"ussel~\cite{Hom1}, and Georgakopoulos~\cite{ltop,lhom}. Our aim in this note is to facilitate orthogonality proofs for these spaces by showing that, whenever $\FF$ is one of them, a set $D$ of edges is orthogonal to $\FF$ as soon as it is orthogonal to the minimal nonzero elements of~$\FF$. This is easy when $\FF$ is $\CF_{\fin}$ or $\BF_{\fin}$ or $\BF_{\sk}$: \begin{prop} Let $\FF$ be a subspace of $\EF$ all of whose elements are finite sets of edges. Then $\FF$ is generated (via finite sums) by its $\subseteq$-minimal nonzero elements. \end{prop} \begin{proof} For a contradiction suppose that some $F\in\FF$ is not a finite sum of finitely many minimal nonzero elements of $\FF$. Choose $F$ with $|F|$ minimal. As $F$ is not minimal itself, by assumption, it properly contains a minimal nonzero element $F'$ of $\FF$. As $F$ is finite, $F + F'=F\sm F'\in\FF$ has fewer elements than $F$, so there is a finite family $(M_i)_{i\le n}$ of minimal nonzero elements of $\FF$ with $\sum_{i\le n} M_i=F+F'$. This contradicts our assumption, as $F'+\sum_{i\le n} M_i=F$. \end{proof} \begin{cor}\label{cor:finite} If $\FF\in\{\CF_{\fin}, \BF_{\fin}, \BF_{\sk}\}$, a set $D$ of edges is orthogonal to $\FF$ as soon as $D$ is orthogonal to all the minimal nonzero elements of~$\FF$.\qed \end{cor} When $\FF\in\{\CFtop, \CF_{\alg},\BF\}$, the statement of Corollary~\ref{cor:finite} is generally false for graphs that are not $3$-connected. Here are some examples. For $\FF=\BF$, let $G$ be the graph obtained from the $\N\times\Z$ grid by doubling every edge between two vertices of degree $3$ and subdividing all the new edges. The set $D$ of the edges that lie in a $K^3$ of~$G$ is orthogonal to every bond $F$ of~$G$: their intersection $D\cap F$ is finite and even. But $D$ is not orthogonal to every element of $\FF=\BF$, since it meets some cuts that are not bonds infinitely. For $\FF=\CFtop$, let $B$ be an infinite bond of the infinite ladder $H$, and let $G$ be the graph obtained from $H$ by subdividing every edge in~$B$. Then the set $D$ of edges that are incident with subdivision vertices has a finite and even intersection with every topological circuit $C$, finite or infinite, but it is not orthogonal to every element of~$\CFtop$, since it meets some of them infinitely. For $\FF=\CF_{\alg}$ we can re-use the example just given for~$\CFtop$, since for 1-ended graphs like the ladder the two spaces coincide. However, if $G$ is $3$-connected, an edge set is orthogonal to every element of $\CFtop, \CF_{\alg}$ or $\BF$ as soon as it is orthogonal to every minimal nonzero element: \begin{theo}\label{maindual} Let $G=(V,E)$ be a locally finite $3$-connected graph, and $F,D\sub E$. \begin{enumerate}[label=\emph{(\roman*)}] \item $F\in \CFtop^\perp$ as soon as $F$ is orthogonal to all the minimal nonzero elements of $\CFtop$, the topological circuits of~$G$. \item $F\in \CF_{\alg}^\perp$ as soon as $F$ is orthogonal to all the minimal nonzero elements of $\CF_{\alg}$, the finite circuits and the edge sets of double rays in~$G$. \item $D\in\BF^\perp $ as soon as $D$ is orthogonal to all the minimal nonzero elements of $\BF$, the bonds of~$G$. \end{enumerate} \end{theo} Although Theorem~\ref{maindual} fails if we replace the assumption of $3$-connectedness with $2$-connectedness, it turns out that we need a little less than $3$-connectedness. Recall that an end $\omega$ of~$G$ has (combinatorial) \emph{vertex-degree $k$} if $k$ is the maximum number of vertex-disjoint rays in~$\omega$. Halin \cite{halin74} showed that every end in a $k$-connected locally finite graph has vertex-degree at least $k$. Let us call an end $\omega$ of~$G$ \emph{$k$-padded} if for every ray $R\in\omega$ there is a neighbourhood $U$ of~$\omega$ such that for every vertex $u\in U$ there is a \emph{$k$-fan} from $u$ to $R$ in~$G$, a subdivided $k$-star with centre $u$ and leaves on~$R$.\footnote{For example, if $G$ is the union of complete graphs $K_1,K_2,\dots$ with $|K_i|=i$, each meeting the next in exactly one vertex (and these are all distinct), then the unique end of~$G$ is $k$-padded for every $k\in\N$.} If every end of~$G$ is $k$-padded, we say that $G$ is \emph{$k$-padded at infinity}. Note that $k$-connected graphs are $k$-padded at infinity. Our proof of Theorem~\ref{maindual}(i) and~(ii) will use only that every end has vertex-degree at least $3$ and that $G$ is $2$-connected. Similarly, and in a sense dually, our proof of Theorem~\ref{maindual}(iii) uses only that every end has vertex-degree at least $2$ and $G$ is $3$-padded at infinity. \begin{theo}\label{mainbulky} Let $G=(V,E)$ be a locally finite $2$-connected graph. \begin{enumerate}[label=\emph{(\roman*)}] \item If every end of~$G$ has vertex-degree at least $3$, then $F\in \CFtop^\perp$ as soon as $F$ is orthogonal to all the minimal nonzero elements of $\CFtop$, the topological circuits of~$G$. \item If every end of~$G$ has vertex-degree at least $3$, then $F\in \CF_{\alg}^\perp$ as soon as $F$ is orthogonal to all the minimal nonzero elements of $\CF_{\alg}$, the finite circuits and the edge sets of double rays in~$G$. \item If $G$ is $3$-padded at infinity, then $D\in\BF^\perp $ as soon as $D$ is orthogonal to all the minimal nonzero elements of $\BF$, the bonds of~$G$. \end{enumerate} \end{theo} In general, our notation follows~\cite{DiestelBook10noEE}. In particular, given an end $\omega$ in a graph~$G$ and a finite set $S\sub V(G)$ of vertices, we write $C(S,\omega)$ for the unique component of $G-S$ that contains a ray $R\in\omega$. The \emph{vertex-degree} of $\omega$ is the maximum number of vertex-disjoint rays in~$\omega$. The mathematical background required for this paper is covered in~\cite{RDsBanffSurvey,Hom1}. For earlier results on the cycle and cut space see Bruhn and Stein~\cite{Degree,BruhnSteinEndDuality}. \section{Finding disjoint paths and fans} Menger's theorem that the smallest cardinality of an $A$--$B$ separator in a finite graph is equal to the largest cardinality of a set of disjoint $A$--$B$ paths trivially extends to infinite graphs. Thus in a locally finite $k$-connected graph, there are $k$ internally disjoint paths between any two vertices. In Lemmas~\ref{lem:disjoint_fans} and~\ref{cor:connected_witness} we show that, for two such vertices that are close to an end $\omega$, these connecting paths need not use vertices too far away from~$\omega$. In a graph $G$ with vertex sets $X,Y\sub V(G)$ and vertices $x,y\in V(G)$, a \emph{$k$-fan} from $X$ (or $x$) to~$Y$ is a subdivided $k$-star whose center lies in~$X$ (or is $x$) and whose leaves lie in~$Y$. A \emph{$k$-linkage} from $x$ to $y$ is a union of~$k$ internally disjoint $x$--$y$ paths. We may refer to a sequence $(v_i)_{i\in\N}$ simply by $(v_i)$, and use $\bigcup(v_i):=\bigcup_{i\in\N}\{v_i\}$ for brevity. \begin{lem}\label{lem:disjoint_fans} Let $G$ be a locally finite graph with an end~$\omega$, and let $(v_i)_{i\in\N}$ and $(w_i)_{i\in\N}$ be two sequences of vertices converging to~$\omega$. Let $k$ be a positive integer. \begin{enumerate}[label=\emph{(\roman*)}] \item If for infinitely many $n\in\N$ there is a $k$-fan from $v_n$ to $\bigcup(w_i)$, then there are infinitely many disjoint such $k$-fans. \item If for infinitely many $n\in\N$ there is a $k$-linkage from $v_n$ to $w_n$, then there are infinitely many disjoint such $k$-linkages. \end{enumerate} \end{lem} \begin{proof} For a contradiction, suppose $k\in\N$ is minimal such that there is a locally finite graph $G=(V,E)$ with sequences $(v_i)_{i\in\N}$ and $(w_i)_{i\in\N}$ in which either (i) or (ii) fails. Then $k>1$, since for every finite set $S\sub V(G)$ the unique component $C(S,\omega)$ of $G-S$ that contains rays from $\omega$ is connected and contains all but finitely many vertices from $\bigcup(v_i)$ and $\bigcup(w_i)$. For a proof of (i) it suffices to show that for every finite set $S\sub V(G)$ there is an integer $n\in \N$ and a $k$-fan from $v_n$ to~$\bigcup (w_i)$ avoiding~$S$. Suppose there is a finite set $S\sub V(G)$ that meets all $k$-fans from $\bigcup(v_i)$ to~$\bigcup(w_i)$. By the minimality of~$k$, there are infinitely many disjoint $(k-1)$-fans from $\bigcup(v_i)$ to~$\bigcup(w_i)$ in $C \coloneqq C(S,\omega)$. Thus, there is a subsequence $(v'_i)_{i\in\N}$ of $(v_i)_{i\in\N}$ in~$C$ and pairwise disjoint $(k-1)$-fans $F_i\sub C$ from $v'_i$ to~$\bigcup(w_i)$ for all $i\in\N$. For every $i\in\N$ there is by Menger's theorem a $(k-1)$-separator $S_i$ separating $v'_i$ from $\bigcup(w_i)$ in~$C$, as by assumption there is no $k$-fan from $v'_i$ to~$\bigcup(w_i)$ in~$C$. Let $C_i$ be the component of $G-(S\cup S_i)$ containing $v_i'$. Since $F_i$ is a subdivided $|S_i|$-star, $S_i\sub V(F_i)$. Hence for all $i\ne j$, our assumption of $F_i\cap F_j=\emptyset$ implies that $F_i\cap S_j=\emptyset$, and hence that $F_i\cap C_j=\emptyset$. But then also $C_i\cap C_j=\emptyset$, since any vertex in $C_i\cap C_j$ could be joined to $v_j'$ by a path $P$ in $C_j$ and to $v_i'$ by a path $Q$ in $C_i$, giving rise to a $v_j'$--$\bigcup (w_i)$ path in $P\cup Q\cup F_i$ avoiding $S_j$, a contradiction. As $S\cup S_i$ separates $v'_i$ from $\bigcup(w_i)$ in~$G$ and there is, by assumption, a $k$-fan from $v'_i$ to~$\bigcup(w_i)$ in~$G$, there are at least $k$ distinct neighbours of~$C_i$ in $S\cup S_i$. Since $|S_i|=k-1$, one of these lies in~$S$. This holds for all $i\in\N$. As $C_i\cap C_j=\emptyset$ for distinct $i$ and $j$, this contradicts our assumption that~$G$ is locally finite and $S$ is finite. This completes the proof of (i). For (ii) it suffices to show that for every finite set $S\sub V(G)$ there is an integer $n\in\N$ such that there is a $k$-linkage from $v_n$ to $w_n$ avoiding $S$. Suppose there is a finite set~$S\sub V(G)$ that meets all $k$-linkages from $v_i$ to $w_i$ for all $i\in\N$. By the minimality of $k$ there is an infinite family $(L_i)_{i\in I}$ of disjoint $(k-1)$-linkages $L_i$ in $C\coloneqq C(S,\omega)$ from $v_i$ to $w_i$. As earlier, there are pairwise disjoint $(k-1)$-sets $S_i\sub V(L_i)$ separating $v_i$ from $w_i$ in~$C$, for all $i\in I$. Let $C_i,D_i$ be the components of $C-S_i$ containing $v_i$ and $w_i$, respectively. For no $i\in I$ can both $C_i$ and $D_i$ have $\omega$ in their closure, as they are separated by the finite set $S\cup S_i$. Thus for every $i\in I$ one of $C_i$ or $D_i$ contains at most finitely many vertices from $\bigcup_{i\in I} L_i$. By symmetry, and replacing $I$ with an infinite subset of itself if necessary, we may assume the following: % \begin{txteq}\label{finitely_many} The components $C_i$ with $i\in I$ each contain only finitely many\\ vertices from $\bigcup_{i\in I} L_i$. \end{txteq} If infinitely many of the components $C_i$ are pairwise disjoint, then $S$ has infinitely many neighbours as earlier, a contradiction. By Ramsey's theorem, we may thus assume that % \begin{equation}\label{not_empty} C_i\cap C_j\neq\emptyset \text{ \em for all }i,j\in I. \end{equation} Note that if $C_i$ meets $L_j$ for some $j\ne i$, then $C_i\supseteq L_j$, since $L_j$ is disjoint from $L_i\supseteq S_i$. By~(\ref{finitely_many}), this happens for only finitely many $j>i$. We can therefore choose an infinite subset of~$I$ such that $C_i\cap L_j=\emptyset$ for all $ii$. Deleting those $j$ from $I$, and repeating that argument for increasing $i$ in turn, we may thus assume that $P_i\sub S\cup C_{i+1}$ for all $i\in I$. By (3) and (4) we deduce that $P_i\sm S$ are now disjoint for different values of $i\in I$. Hence $S$ contains a vertex of infinite degree, a contradiction. \end{proof} Recall that $G$ is \emph{$k$-padded} at an end $\omega$ if for every ray $R\in\omega$ there is a neighbourhood $U$ such that for all vertices $u\in U$ there is a $k$-fan from $u$ to~$R$ in~$G$. Our next lemma shows that, if we are willing to make $U$ smaller, we can find the fans locally around $\omega$: \begin{lem}\label{cor:connected_witness}\label{lem:connected_witness} Let $G$ be a locally finite graph with a $k$-padded end $\omega$. For every ray $R\in\omega$ and every finite set $S\sub V(G)$ there is a neighbourhood $U\subseteq C(S,\omega)$ of $\omega$ such that from every vertex $u\in U$ there is a $k$-fan in $C(S,\omega)$ to $R$. \end{lem} \begin{proof} Suppose that, for some $R\in\omega$ and finite $S\sub V(G)$, every neighbourhood $U\sub C(S,\omega)$ of~$\omega$ contains a vertex $u$ such that $C(S,\omega)$ contains no $k$-fan from $u$ to $R$. Then there is a sequence $u_1,u_2,\dots$ of such vertices converging to $\omega$. As $\omega$ is $k$-padded there are $k$-fans from infinitely many $u_i$ to $R$ in~$G$. By Lemma~\ref{lem:disjoint_fans}(i) we may assume that these fans are disjoint. By the choice of $u_1,u_2,\dots$, all these disjoint fans meet the finite set $S$, a contradiction. \end{proof} \section{The proof of Theorems~\ref{maindual} and \ref{mainbulky}}\label{proofofmaindual} As pointed out in the introduction, Theorem~\ref{mainbulky} implies Theorem~\ref{maindual}. It thus suffices to prove Theorem~\ref{mainbulky}, of which we prove (i) first. Consider a set $F\neq\emptyset$ of edges that meets every circuit of~$G$ evenly. We have to show that $F\in \CFtop^{\perp}$, i.e., that $F$ is a finite cut. (Recall that $\CFtop^\perp$ is known to equal $\BF_{\fin}$, the finite-cut space~\cite{RDsBanffSurvey}.) As $F$ meets every finite cycle evenly it is a cut, with bipartition $(A,B)$ say. Suppose $F$ is infinite. Let $\RF$ be a set of three disjoint rays that belong to an end $\omega$ in the closure of~$F$. Every $R$--$R'$ path $P$ for two distinct $R,R'\in\RF$ lies on the unique topological circle $C(R,R',P)$ that is contained in $R\cup R'\cup P\cup\{\omega\}$. As every circuit meets $F$ finitely, we deduce that no ray in $\RF$ meets $F$ again and again. Replacing the rays in~$\RF$ with tails of themselves as necessary, we may thus assume that $F$ contains no edge from any of the rays in $\RF$. Suppose $F$ separates $\RF$, with the vertices of $R\in\RF$ in~$A$ and the vertices of $R',R''\in\RF$ in $B$ say. Then there are infinitely many disjoint $R$--$(R'\cup R'')$ paths each meeting $F$ at least once. Infinitely many of these disjoint paths avoid one of the rays in~$B$, say $R''$. The union of these paths together with $R$ and $R'$ contains a ray $W\in\omega$ that meets $F$ infinitely often. For every $R''$--$W$ path $P$, the circle $C(W,R'',P)$ meets $F$ in infinitely many edges, a contradiction. Thus we may assume that $F$ does not separate $\RF$, and that $G[A]$ contains $\bigcup\RF$. As $\omega$ lies in the closure of $F$, there is a sequence $(v_i)_{i\in\N}$ of vertices in~$B$ converging to~$\omega$. As~$G$ is $2$-connected there is a $2$-fan from each $v_i$ to $\bigcup \RF$ in~$G$. By Lemma~\ref{lem:disjoint_fans} there are infinitely many disjoint $2$-fans from $\bigcup(v_i)$ to~$\bigcup\RF$. We may assume that every such fan has at most two vertices in $\bigcup\RF$. Then infinitely many of these fans avoid some fixed ray in $\RF$, say $R$. The two other rays plus the infinitely many $2$-fans meeting only these together contain a ray $W\in\omega$ that meets $F$ infinitely often and is disjoint from $R$. Then for every $R$--$W$ path $P$ we get a contradiction, as $C(R,W,P)$ is a circle meeting $F$ in infinitely many edges. For a proof of~(ii), note first that the minimal elements of $\CF_{\alg}$ are indeed the finite circuits and the edge sets of double rays in~$G$. Indeed, these are clearly in~$\CF_{\alg}$ and minimal. Conversely, given any element of~$\CF_{\alg}$, a set $D$ of edges inducing even degrees at all the vertices, we can greedily find for any given edge $e\in D$ a finite circuit or double ray with all its edges in~$D$ that contains~$e$. We may thus decompose $D$ inductively into disjoint finite circuits and edge sets of double rays, since deleting finitely many such sets from~$D$ clearly produces another element of~$\CF_{\alg}$, and including in each circuit or double ray chosen the smallest undeleted edge in some fixed enumeration of~$D$ ensures that the entire set $D$ is decomposed. If $D$ is minimal in~$\CF_{\alg}$, it must therefore itself be a finite circuit or the edge set of a double ray. Consider a set $F$ of edges that fails to meet some set $D\in\CF_{\alg}$ evenly; we have to show that $F$ also fails to meet some finite circuit or double ray evenly. If $|F\cap D|$ is odd, then this follows from our decomposition of $D$ into disjoint finite circuits and edges sets of double rays. We thus assume that $F\cap D$ is infinite. Since $|G|$ is compact, we can find a sequence $e_1,e_2,\dots$ of edges in $F\cap D$ that converges to some end~$\omega$. Let $R_1,R_2,R_3$ be disjoint rays in~$\omega$, which exist by our assumption that $\omega$ has vertex-degree at least~3. Subdividing each edge~$e_i$ by a new vertex~$v_i$, and using that $G$ is 2-connected, we can find for every~$i$ a 2-fan from $v_i$ to $W = V(R_1\cup R_2\cup R_3)$ that has only its last vertices and possibly~$v_i$ in~$W$. By Lemma~\ref{lem:disjoint_fans}, with $w_1,w_2,\dots$ an enumeration of~$W$, some infinitely many of these fans are disjoint. Renaming the rays~$R_i$ and replacing $e_1,e_2,\dots$ with a subsequence as necessary, we may assume that either all these fans have both endvertices on~$R_1$, or that they all have one endvertex on~$R_1$ and the other on~$R_2$. In both cases all these fans avoid~$R_3$, so we can find a ray~$R$ in the union of $R_1$, $R_2$ and these fans (suppressing the subdividing vertices~$v_i$ again) that contains infinitely many~$e_i$ and avoids~$R_3$. Linking $R$ to a tail of~$R_3$ we thus obtain a double ray in $G$ that contains infinitely many~$e_i$, as desired. To prove (iii), let $D\sub E$ be a set of edges that meets every bond evenly. We have to show that $D\in\BF^\perp$, i.e., that $D$ has an (only finite and) even number of edges also in every cut that is not a bond. As~$D$ meets every finite bond evenly, and hence every finite cut, it lies in $\BF_{\fin}^\perp=\CFtop$. We claim that % \begin{equation}\tag{$\star$}\label{disjoint_union} \begin{minipage}[c]{0.85\textwidth} \emph{$D$ is a disjoint union of finite circuits.} \end{minipage} \end{equation} To prove~(\ref{disjoint_union}), let us show first that every edge $e\in D$ lies in some finite circuit $C\sub D$. If not, the endvertices $u,v$ of $e$ lie in different components of $(V, D\sm\{e\})$, and we can partition $V$ into two sets $A,B$ so that $e$ is the only $A$--$B$ edge in $D$. The cut of~$G$ of all its $A$--$B$ edges is a disjoint union of bonds~\cite{DiestelBook10noEE}, one of which meets $D$ in precisely $e$. This contradicts our assumption that $D$ meets every bond of~$G$ evenly. For our proof of~(\ref{disjoint_union}), we start by enumerating $D$, say as $D=:\{e_1,e_2,\dots\}=:D_0$. Let $C_0\sub D_0$ be a finite circuit containing $e_0$, let $D_1:= D_0\sm C_0$, and notice that $D_1$, like $D_0$, meets every bond of~$G$ evenly (because $C_0$ does). As before, $D_1$ contains a finite circuit $C_1$ containing the edge $e_i$ with $i=\min \{j\mid e_j\in D_1\}$. Continuing in this way we find the desired decomposition $D=C_1\cup C_2\cup\dots$ of $D$ into finite circuits. This completes the proof of~(\ref{disjoint_union}). As every finite circuit lies in $\BF^\perp$, it suffices by~(\ref{disjoint_union}) to show that $D$ is finite. Suppose $D$ is infinite, and let $\omega$ be an end of~$G$ in its closure. Let us say that two rays $R$ and $R'$ \emph{hug} $D$ if every neighbourhood $U$ of~$\omega$ contains a finite circuit $C\sub D$ that is neither separated from $R$ by~$R'$ nor from $R'$ by~$R$ in~$U$. %some neighbourhood $U'\sub U$ of $\omega$ containing $C$. We shall construct two rays $R$ and $R'$ that hug~$D$, inductively as follows. Let $S_0=\emptyset$, and let $R_0,R'_0$ be disjoint rays in~$\omega$. (These exist as~$G$ is $2$-connected~\cite{halin74}.) For step $j\ge 1$, assume that $S_i, R_i$, and $R'_i$ have been defined for all $i