Finite graph homology may be trivial, but for infinite graphs things become interesting. We present a new `singular' approach that builds the cycle space of a graph not on its finite cycles but on its topological \emph{circles}, the homeomorphic images of the unit circle in the space formed by the graph together with its ends.
Our approach permits the extension to infinite graphs of standard results about finite graph homology -- such as Tutte's generating theorem, cycle-cocycle duality and Whitney's theorem, MacLane's planarity criterion, the Tutte/Nash-Williams tree packing theorem -- whose infinite versions would otherwise fail. A~notion of end degrees motivated by these results opens up new possibilities for an `extremal' branch of infinite graph theory.
Numerous open problems are suggested.
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