Orthogonality and Minimality in the Homology of Locally Finite Graphs
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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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2013 ◽
Vol 22
(6)
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pp. 885-909
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2000 ◽
Vol 10
(05)
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pp. 591-602
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1971 ◽
Vol 69
(3)
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pp. 401-407
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1977 ◽
Vol 29
(1)
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pp. 165-168
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2006 ◽
Vol 96
(2)
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pp. 302-312
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