Cycle spaces in topological spaces

Abstract

Abstract:We develop a general model of edge spaces in order to generalize, unify, and simplify previous work on cycle spaces of infinite graphs. We give simple topological criteria to show that the fundamental cycles of a (generalization of a) spanning tree generate the cycle space in a connected, compact, weakly Hausdorff edge space. Furthermore, in such a space, the orthogonal complement of the bond space is the cycle space. This work unifies the two different notions of cycle space as introduced by Diestel

The bond and cycle spaces of an infinite graph

“…Let T be a spanning tree in a connected, compact, weakly Hausdorff edge space (X, E) and let e ∈ E \ T . It is a consequence of [7,Thm. 3] that T + e contains an edge cycle containing e (use e together with the edge set of a minimal, closed, connected set in T containing the ends of e) -the corresponding cycle is the fundamental cycle C T (e).…”

“…In this short section, we review a few results from [7] that are central to the theory of cycle and bond spaces. An edge space (X, E) is an edge cycle if: (i) X is connected; (ii) for each e ∈ E, X − e is connected; and (iii) for distinct e, f ∈ E, X − {e, f } is not connected.…”

“…Both A \ E and B \ E are closed, so δ(A) is a bond -this is the fundamental bond B T (e). The following is [7,Lemma 16]. …”

“…They prove that the fundamental cycles of an end-faithful spanning tree generate this cycle space. In the later work [4], this is extended to a certain more general topological spaceG that can be obtained from some slightly more general infinite graph G. Vella and Richter [7] show that these results are special cases of a more general theory that includes, for locally finite graphs, the cycle space notion of Bonnington and Richter [1], in which a set of edges of a locally finite graph is in this new cycle space if every vertex is incident with an even number of edges in the set. This cycle space is larger than the Diestel-Kühn space for a locally finite graph having more than one end.…”

“…In [1], the closure of an embedding of a locally finite graph into the sphere is considered. See [7] for details.…”