  We show how topological methods developed in a previous article can be
applied to prove new results about topological and homological finiteness
properties of monoids. A monoid presentation is called special if the
right-hand side of each relation is equal to $1$. We prove results which relate
the finiteness properties of a monoid defined by a special presentation with
those of its group of units. Specifically we show that the monoid inherits the
finiteness properties $F_n$ and $FP_n$ from its group of units. We also obtain
results which relate the geometric and cohomological dimensions of such a
monoid to those of its group of units. We apply these results to prove a
Lyndon's Identity Theorem for one-relator monoids of the form $\langle A \mid
r=1 \rangle$. In particular we show that all such monoids are of type
$F_{\infty}$ (and $FP_{\infty}$), and that when $r$ is not a proper power, then
the monoid has geometric and cohomological dimension at most $2$. The first of
these results resolves an important case of a question of Kobayashi from 2000
on homological finiteness properties of one-relator monoids. We also show how
our topological approach can be used to prove results about the closure
properties of various homological and topological finiteness properties for
amalgamated free products and HNN-extensions of monoids. To prove these results
we introduce new methods for constructing equivariant classifying spaces for
monoids, as well as developing a Bass-Serre theory for free constructions of
monoids.
