A Survey of Graph-like Continua and Infinite Matroids
|Affiliation:||University of Waterloo|
|Room:||Mathematics 3 (M3) 3103|
In the last 20 years or so, there has been a lot of work done on extensions of results from finite graphs to compactifications of infinite graphs. These have found a natural home in the concept of a graph-like continuum: a compact, connected topological space X with a closed, totally disconnected subset V (the vertices) so that X\V consists of disjoint homeomorphs of the real line (the edges). The notion of a cycle space extends nicely to this context, as do spanning trees. The fundamental cycles relative to a spanning tree make a basis of the cycle space. The planarity characterizations of MacLane and Whitney also hold for graph-like continua.
With the algebra behaving so well, it was natural to consider matroids. Higgs' notion of B-matroid turns out to be the right definition of an infinite matroid; many of the equivalent definitions of matroid generalize to definitions of B-matroid that are all equivalent. There are interesting theorems that generalize results for finite matroids. For example, a 3-connected binary matroid has the property that its peripheral cycles span its cycle space and a matroid has no U(2,4)-minor if and only if it is representable using binary vectors -- with certain infinite sums allowed.
There is a conjectured version of the Matroid Intersection Theorem that generalizes the difficult Aharoni-Berger version of Menger's Theorem for infinite graphs.
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