Finite graph homology may seem 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 circles, the homeomorphic images of $S^1$ 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 cycle–cocycle duality and Whitney's theorem, Tutte's generating 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.