Orthogonality and Minimality in the Homology of Locally Finite Graphs

  • Reinhard Diestel
  • Julian Pott
Keywords: graph theory, locally finite graph, end, homology, topological, orthogonal, cycle space, bond space, cut space


Given a finite set $E$, a subset $D\subseteq E$ (viewed as a function $E\to \mathbb F_2$) is orthogonal to a given subspace $\mathcal F$ of the $\mathbb F_2$-vector space of functions $E\to \mathbb F_2$ as soon as $D$ is orthogonal to every $\subseteq$-minimal element of $\mathcal F$. This fails in general when $E$ is infinite.

However, we prove the above statement for the six subspaces $\mathcal F$ of the edge space of any $3$-connected locally finite graph that are relevant to its homology: the topological, algebraic, and finite cycle and cut spaces. This solves a problem of Diestel (2010, arXiv:0912.4213).

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