The homology of a locally finite graph with ends

Abstract

We show that the topological cycle space of a locally finite graph is a canonical quotient of the first singular homology group of its Freudenthal compactification, and we characterize the graphs for which the two coincide. We construct a new singular-type homology for non-compact spaces with ends, which in dimension 1 captures precisely the topological cycle space of graphs but works in any dimension.

The fundamental group of a locally finite graph with ends–-a hyperfinite approach

“…(2) Observe that in the above argument, we did not use that Θ was injective. This fact is compatible with the standard proof that α is not nullhomologous appearing in [DS10] as the injectivity of the corresponding map π 1 (|Γ|) → F ∞ was not used either.…”

“…The original motivation for the work in [DS11a] was to show that certain loops in |Γ| are non-nullhomologous. The motivation for this is that these loops are trivial in the so-called topological cycle space, another homology theory that has proven to be more suitable than the usual one to study Fredudenthal compactifications (of spaces even more general than graphs) [DS10,DS11b]. The existence of such loops implies that these theories are actually different.…”

On the coincidence of zeroth Milnor–Thurston and singular homology

“…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.…”

Orthogonality and Minimality in the Homology of Locally Finite Graphs

“…The six subspaces of E mentioned above are the the ones most relevant to the homology of locally finite infinite graphs. See [5], Diestel and Sprüssel [6], and Georgakopoulos [7,8]. Our aim in this note is to facilitate orthogonality proofs for these spaces by showing that, whenever F is one of them, a set D of edges is orthogonal to F as soon as it is orthogonal to the minimal nonzero elements of F. This is easy when F is C fin or B fin or B skew :…”

“…The vertex-degree of ω is the maximum number of vertex-disjoint rays in ω. The mathematical background required for this paper is covered in [5,6]. For earlier results on the cycle and cut space see Bruhn and Stein [1,2].…”