Infinite graphs, graph-like spaces and B-matroids

Abstract

The central theme of this thesis is to prove results about infinite mathematical objects by studying the behaviour of their finite substructures. In particular, we study B-matroids, which are an infinite generalization of matroids introduced by Higgs \cite{higgs}, and graph-like spaces, which are topological spaces resembling graphs, introduced by Thomassen and Vella \cite{thomassenvella}.

Recall that the circuit matroid of a finite graph is a matroid defined on the edges of the graph, with a set of edges being independent if it contains no circuit. It turns out that graph-like continua and infinite graphs both have circuit B-matroids. The first main result of this thesis is a generalization of Whitney's Theorem that a graph has an abstract dual if and only if it is planar. We show that an infinite graph has an abstract dual (which is a graph-like continuum) if and only if it is planar, and also that a graph-like continuum has an abstract dual (which is an infinite graph) if and only if it is planar. This generalizes theorems of Thomassen (\cite{thomassendual}) and Bruhn and Diestel (\cite{bruhndiestel}). The difficult part of the proof is extending Tutte's characterization of graphic matroids (\cite{tutte2}) to finitary or co-finitary B-matroids. In order to prove this characterization, we introduce a technique for obtaining these B-matroids as the limit of a sequence of finite minors.

In \cite{tutte}, Tutte proved important theorems about the peripheral (induced and non-separating) circuits of a $3$-connected graph. He showed that for any two edges of a $3$-connected graph there is a peripheral circuit containing one but not the other, and that the peripheral circuits of a $3$-connected graph generate its cycle space. These theorems were generalized to $3$-connected binary matroids by Bixby and Cunningham (\cite{bixbycunningham}). We generalize both of these theorems to $3$-connected binary co-finitary B-matroids.

Richter, Rooney and Thomassen \cite{richterrooneythomassen} showed that a locally connected, compact metric space has an embedding in the sphere unless it contains a subspace homeomorphic to $K_5$ or $K_{3,3}$, or one of a small number of other obstructions. We are able to extend this result to an arbitrary surface $\Sigma$; a locally connected, compact metric space embeds in $\Sigma$ unless it contains a subspace homeomorphic to a finite graph which does not embed in $\Sigma$, or one of a small number of other obstructions.