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

The Eilenberg-Mac Lane cohomology of an inverse monoid and the maximum group image