  The aim of this paper is to see at what extent homological properties of an
inverse monoid are determined from those of its maximum group image. We provide
several evidences that the maximum group image contains vital homological
information which can be used to study certain properties of the monoid itself.
For instance, we prove that an inverse monoid $S$ is of type $FP_{\infty}$, if
and only if it contains a minimal idempotent and its maximum group image is of
the same type. Regarding cohomological dimensions, we show that the
cohomological dimension of a free Clifford monoid and that of its maximum group
image agree and are equal to one. Also we define the index of a full submonoid
of an inverse monoid in terms of their maximum group images and show that if
the index is finite then, the monoid is of type $\text{FP}_{\infty}$ if and
only if its submonoid is of the same type.
