On the homology of locally compact spaces with ends

Diestel, Reinhard; Sprüssel, Philipp

doi:10.1016/j.topol.2011.05.031

Cited by 7 publications

(12 citation statements)

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“…I suspect that this simplicial attempt to marry our topological cycle space to the usual homological setup is in fact inadequate, and that the way to do this lies not in changing the boundary operators but in restricting the chains. A singular approach that does capture the topological cycle space as its first homology group, just as in finite graphs, and where some infinite chains are allowed but not every subset (or sum) of edges is a legal 1-chain, has been constructed in [43]. It allows for some infinite chains, but not all; see Section 5 for more.…”

Section: Orthogonal Decompositionmentioning

confidence: 99%

“…The basic open sets of this space are the components of G − X left by deleting a finite set X of inner points of edges (and an edge-end belongs to the component in which all its rays have a tail). 43 If G is connected, as we assume, then this space is compact [81]. On the set of vertices and edge-ends, being separated by such a finite set X (i.e., by a finite set of edges) is an equivalence relation.…”

Section: Graphs With Infinite Degreesmentioning

confidence: 99%

“…As a step towards this goal, it will help to recast the topological cycle space theory of Sections 1 and 2, even for locally finite graphs, as a homology theory in the usual terms of algebraic topology [ 43]. We shall look at this problem next.…”

Section: Compactification Versus Metric Completionmentioning

confidence: 99%

“…A more subtle approach, which has been pursued in [41,43], is to see to what extent C(G) can be captured by the singular homology of |G|. After all, C(G) was defined via (the edge sets of) circles in |G|, which are just injective singular loops.…”

Section: Homology Of Locally Compact Spaces With Endsmentioning

confidence: 99%

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Locally finite graphs with ends: A topological approach, II. Applications

Diestel¹

2010

Discrete Mathematics

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This paper is intended as an introductory survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. This approach has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. The shift of paradigm it proposes is thus as much an answer to old questions as a source of new ones; many concrete problems of both types are suggested in the paper.This paper attempts to provide an entry point to this field for readers that have not followed the literature that has emerged in the last 10 years or so. It takes them on a quick route through what appear to be the most important lasting results, introduces them to key proof techniques, identifies the most promising open problems, and offers pointers to the literature for more detail.* This paper has appeared in two parts [33,34]. It is complemented by [44], the third part, which studies the algebraic-topological aspects of the theory.Given a standard subspace X and an end ω ∈ X, the maximum number of arcs in X that end in ω but are otherwise disjoint is the (vertex-) degree of ω in X; the maximum number of edge-disjoint arcs in X ending in ω is its edgedegree in X. Both maxima are indeed attained, but it is non-trivial to prove this [23]. End degrees behave largely as expected; for example, the connected standard subspaces in which every vertex and every end has (vertex-) degree 2 are precisely the circles. (Use Lemma 1.2 to prove this.) In Section 4.1 we shall define a third type of end degrees, their relative degree, which is useful for the application of end degrees to extremal-type problems about infinite graphs.Standard subspaces have the important property that connectedness and arc-connectedness are equivalent for them. This will often be convenient: while connectedness is much easier to prove (see Lemma 1.5), it is usually arc-connectedness that we need.Lemma 1.2. [35,39,90] Connected standard subspaces of |G| are locally connected and arc-connected.The proof that a connected standard subspace X is locally connected is not hard: an open neighbourhoodĈ ∩ X of an end ω will be connected if we choose the set S in its definition so as to minimize the number of C-S edges in X. But local connectedness is not a property we shall often use directly. Its role here is that it offers a convenient stepping stone towards the proof of arcconnectedness. 4 Direct proofs that X is arc-connected can be found in [39,46], and we shall indicate one in Section 3. Connected subspaces of |G| that are neither open nor closed need not be arc-connected [46]. Corollary 1.3. The arc-components of a standard subspace are closed.Proof. The closure in X of an arc-component of a standard subspace X is connected and itself standard, and hence arc-connected by Lemma 1.2.The edge set E(C) of any circle C will be called a circuit . Given any ...

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Section: Orthogonal Decompositionmentioning

confidence: 99%

Section: Graphs With Infinite Degreesmentioning

confidence: 99%

Section: Compactification Versus Metric Completionmentioning

confidence: 99%

Section: Homology Of Locally Compact Spaces With Endsmentioning

confidence: 99%

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Locally finite graphs with ends: A topological approach, II. Applications

Diestel¹

2010

Discrete Mathematics

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“…Conversely, the algebraic topology of non-triangulable spaces has produced some psychological unexpected results, and recently have appeared some papers [5,6,7], [8,Section 1.3] that may be interpreted as a search for a homology-theory that responds in a more natural way to non-tameness. On the other hand, many of these unexpected results seem to have its origin in the fact, that for defining all classical invariants of algebraic topology only algebraic structures have been used that are based on just considering finite sums and products, while for some spaces like Hawaiian Earrings the topology naturally also allows infinite concatenations.…”

Section: Introductionmentioning

confidence: 99%