We propose a homology theory for locally compact spaces with ends in which
the ends play a special role. The approach is motivated by results for graphs
with ends, where it has been highly successful. But it was unclear how the
original graph-theoretical definition could be captured in the usual language
for homology theories, so as to make it applicable to more general spaces.

We show that the topological cycle space of a locally finite graph is a
canonical quotient of the first singular homology group of its Freudenthal
compactification, and we characterize the graphs for which the two coincide. We
construct a new singular-type homology for non-compact spaces with ends, which
in dimension~1 captures precisely the topological cycle space of graphs but
works in any dimension.