  This paper is the last part of a comprehensive survey of a newly emerging
field: a topological approach to the study of locally finite graphs that
crucially incorporates their ends. Topological arcs and circles, which may pass
through ends, assume the role played in finite graphs by paths and cycles. The
first two parts of the survey together provide a suitable entry point to this
field for new readers; they are available in combined form in arXiv:0912.4213.
The topological approach indicated above has made it possible to extend to
locally finite graphs many classical theorems of finite graph theory that do
not extend verbatim. While the first part of this survey introduces the theory
as such and the second part is devoted to those applications, this third part
looks at the theory from an algebraic-topological point of view. The results
surveyed here include both a combinatorial description of the fundamental group
of a locally finite graph with ends and the homology aspects of this space.
