TAILIEUCHUNG - Báo cáo toán học: "Relating different cycle spaces of the same infinite graph"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Relating different cycle spaces of the same infinite graph. | Relating different cycle spaces of the same infinite graph R. Bruce Richter and Brendan Rooney Submitted Jul 5 2009 Accepted Jun 11 2001 Published Jun 21 2011 Mathematics Subject Classification 05C10 Abstract Casteels and Richter have shown that if X and Y are distinct compactifications of a locally finite graph G and f X Y is a continuous surjection such that f restricts to a homeomorphism on G then the cycle space Zx of X is contained in the cycle space Zy of Y. In this work we show how to extend a basis for Zx to a basis of Zy . 1 Introduction Bonnington and Richter 1 introduced the cycle space of a locally finite graph as the edge sets of subgraphs in which every vertex has even degree. Diestel and Kuhn 4 introduced a different cycle space for a locally finite graph G as the space generated by embeddings of circles into the Freudenthal compactification F G of G. The space F G is obtained from G by adding one point at infinity for each end of G. Vella and Richter 7 unified these notions by introducing edge spaces and showing that a nice cycle space theory holds for compact weakly Hausdorff edge spaces for definitions see Section 2 . In particular the cycle space of Bonnington and Richter can be viewed as the Diestel-Ktihn cycle space but in Alexandroff s 1-point compactification A G of the locally finite graph G rather than in F G . Our first principal result clarifies the relation between the cycle spaces of a given edge space and its image under a continuous edge-preserving map. In particular we show how to use spanning trees to extend a basis of the smaller space to a basis of the larger space. This recalls the fact that a basis of the cycle space of a given finite graph G can be extended to a basis of the cycle space of any graph G resulting from a sequence of vertex identifications of G. In order to better comprehend the statement loosely speaking an edge space X E consists of a topological space X with a specified set E of disjoint open arcs each arc also

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