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#### The cycle space of an infinite graph

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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